(Universitext) Emmanuele DiBenedetto - Degenerate Parabolic Equations-Springer (1993)

Emmanuele DiBenedetto
Degenerate
Parabolic
Equations
,(
•
i Springer-Verlag
Degenerate
Parabolic
Equations
With 12 Figures
Springer-Verlag
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Northwestern University
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and
University of Rome II
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Editorial Board
(North America):
I.H. Ewing F. W. Gebring
Department of Mathematics Department of Mathematics
Indiana University University of Michigan
Bloomington, IN 47405 Ann Arbor, MI 48109
USA USA
P.R. Halmos
Department of Mathematics
Santa Clara University
Santa Clara, CA 95053
USA
AMS Subject Classifications (1991): 35K65
Ubrary of Consress Catalo&ing-in-Publication Data

DiBenedetto, Emmanue1e.
Degenerate parabolic equationslEmmanue1e DiBenedetto.
p. em. - (Universitcxt)
Includes bibliographical references.
ISBN 0-387-9402C).() (New York: acid-free). - ISBN 3-540-9402C).()
(Berlin: acid-free)
1. Differential equations, Parabolic. I. Tide.
QA377.062 1993
5W.353-dc20 93-285
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Preface
1. Elliptic equations: Harnack estimates and HOlder

continuity
Considerable progress was made in the early 1950s and mid-l960s in the theory
of elliptic equations, due to the discoveries of DeGiorgi [33] and Moser [81,82].
Consider local weak solutions of
u E Wl~;(n), n a domain in RN
(l.l) {
(aijUz;)x., = 0 in n,
where the coefficients x --+ aij(x), i,j = 1,2, ... ,N are assumed to be only
bounded and measurable and satisfying the ellipticity condition
(1.2) ajieiej ~ 'xleI 2 , a.e. n, Ve E R N , for some ,X > o.
DeGiorgi established that local solutions are HOlder continuous and Moser proved
that non-negative solutions satisfy the Harnack inequality. Such inequality can be
used, in turn, to prove the HOlder continuity of solutions. Both authors worked with
linear p.d.e. 's. However the linearity has no bearing in the proofs. This pennits an
extension of these results to quasilinear equations of the type
(1.3) {
U E wl!:(n), p> 1
div a(x, u, Du) + b(x, u, Du) = 0, in n,
with structure conditions
a(x,u,Du)· Du ~ 'xIDuI P - tp(x), a.e. nT, p> 1
(1.4) { la(x, u, Du)1 :5 AIDul p - 1 + tp(x),
Ib(x, u, Du)1 :5 AIDul p - 1 + tp(x) .
vi Preface
Here 0 < A ~ A are two given constants and cp E Ll::(n) is non-negative. As a

prototype we may take
(1.5) div IDuIP-2 Du = 0, in n, p> 1.
The modulus of ellipticity of(1.5) is IDuI P- 2. Therefore at points where IDul =0,
the p.d.e. is degenerate if p > 2 and it is singular if 1 < p < 2.
By using the methods of DeGiorgi, Ladyzbenskaja and Ural'tzeva [66] es-
tablished that weak solutions of (1.4) are finder continuous, whereas Serrin [92]
and Trudinger [96], following the methods of Moser, proved that non-negative
solutions satisfy a Harnack principle. The generalisation is twofold, i.e., the prin-
cipal part a(x, u, Du) is pennitted to have a non-linear dependence with respect
to UZi , i = 1,2, ... , N, and a non-linear growth with respect to IDul. The latter
is of particular interest since the equation in (1.5) might be either degenerate or
·singular.
2. Parabolic equations: Harnack estimates and Holder

continuity
The first parabolic version of the Harnack inequality is due to Hadamard [50] and
Pini [86] and applies to non-negative solutions of the heat equation. It takes the
following fonn. Let u be a non-negative solution of the heat equation in the cylin-
drical domain n T == n x (0, T), 0 < T < 00, and for (xo, to) E nT consider the
cylinder
(2.1) Qp == Bp(xo) x (to - p2, tol, Bp(xo) == {Ix - xol < p} .

There exists a constant "y depending only upon N, such that if Q2p c nT, then
(2.2) u(xo, to) ~ "y sup u(x, to _ p2) .

8 p (zo)
The proof is based on local representations by means of heat potentials. A striking

result of Moser [83] is that (2.2) continues to hold for non-negative weak solutions
of
u E V 1 ,2(nT) == Loo (O,T;L2(n»)nL2 (o,T;Wl,2(n» ,

(2.3) {
Ut-(ai;(x,t)uZi)zoJ =0, in nT
where ai; E L 00 (nT ) satisfy the analog of the ellipticity condition (1.2). As before,
it can be used to prove that weak solutions are locally II)lder continuous in T • n
Since the linearity of (2.3) is immaterial to the proof, one might expect, as in the
elliptic case, an extension of these results to quasilinear equations of the type
u E V~,p(nT) == Loo (0, T; L2(n»nV (O,~; Wl,p(n»),

(2.4) {
Ut - dlva(x,t,u,Du) = b(x, t,u, Du), m nT,
where the structure condition is as in (1.4). Surprisingly however, Moser's proof
could be extended only for the case p = 2, i.e., for equations whose principal
3. Parabolic equations and systems vii
part has a linear growth with respect to IDul. This appears in the work of Aron-
son and Serrin [7] and Trudinger [97]. The methods of DeGiorgi also could not be
extended. Ladyzenskaja et al. [67] proved that solutions of (2.4) are R>Ider contin-
uous, provided the principal part has exactly a linear growth with respect to IDul.
Analogous results were established by Kruzkov [60,61,62] and by Nash [84] by
entirely different methods.
Thus it appears that unlike the elliptic case, the degeneracy or singularity of
the principal part plays a peculiar role, and for example, for the non-linear equation
(2.5) Ut - div IDu IP-2 Du = 0,

one could not establish whether non-negative weak solutions satisfy the Harnack
estimate or whether a solution is locally HOlder continuous.
3. Parabolic equations and systems

These issues have remained open since the mid-1960s. They were revived however
with the contributions ofN.N. Ural'tzeva [100] in 1968 and K. Uhlenbeck [99] in
1977. Consider the system
Ui E W,~:(n), p> 1, i=I,2, ... n,
(3.1)
in n.
When p > 2, Ural'tzeva and Uhlenbeck prove that local solutions of (3.1) are of
class c,t~:(n), for some aE (0, 1). The parabolic version of (3.1) is
{
u == (Ulo U2, ••. , un), Ui E V1,p(nT), i=l, 2, ... n,
(3.2)
Ut - div IDul p - 2 DUi = 0, in aT.
Besides their intrinsic mathematical interest, this kind of system arises from geom-
etry [99], quasiregularmappings [2,17,55,89] and fluid dynamics [5,8,56,57,74,75].
In particular Ladyzenskaja [65] suggests systems of the type of (3.2) as a model
of motion of non-newtonian fluids. In such a case u is the velocity vector. Non-
newtonian here means that the stress tensor at each point of the fluid is not linearly
proportional to the matrix of the space-gradient of the velocity.
The function w = IDul 2 is formally a subsolution of
(3.3)
where
at,k
_ {fJt,k + (P -
=
2)Ui,Zt Ui ,zlo }
IDul 2 •
This is a parabolic version of a similar finding observed in [99,I00J for elliptic

systems. Therefore a parabolic version of the Ural'tzeva and Uhlenbeck result re-
quires some understanding of the local behaviour of solutions of the porous media
equation
(3.4) U~O, m>O,
viii Preface
and its quasilinear versions. Such an equation is degenerate at those points of OT

where u=O ifm> 1 and singular ifO<m< 1.
The porous medium equation has a life of its own. We only mention that
questions of regularity were first studied by Caffarelli and Friedman. It was shown
in [21] that non-negative solutions of the Cauchy problem associated with (3.4) are
HOlder continuous. The result is not local.
A more local point of view was adopted in [20,35,90]. However these con-
tributions could only establish that the solution is continuous with a logarithmic
modulus of continuity.
In the mid-1980s, some progress was made in the theory of degenerate p.d.e. 's
of the type of (2.5), for p > 2. It was shown that the solutions are locally HOlder
continuous (see [39]). Surprisingly, the same techniques can be suitably modified
to establish the local HOlder continuity of any local solution of quasilinear porous
medium-type equations. These modified methods, in tum, are crucial in proving
that weak solutions of the systems (3.2) are of class cl~; (OT).
Therefore understanding the local structure of the solutions of (2.5) has im-
plications to the theory of systems and the theory of equations with degeneracies
quite different than (2.5).
4. Main results
In these notes we will discuss these issues and present results obtained during
the past five years or so. These results follow, one way or another, from a sin-
gle unifying idea which we call intrinsic rescaling. The diffusion process in (2.5)
evolves in a time scale determined instant by iQstant by the solution. itself, so that,
loosely speaking, it can be regarded as the heat equation in its own intrinsic time-
configuration. A precise description of this fact as well as its effectiveness is linked
to its technical implementations.
We collect in Chap. I notation and standard material to be used as we proceed.
Degenerate or singular p.d.e. of the type of (2.4) are introduced in Chap. II. We
make precise their functional setting and the meaning of solutions and we derive
truncated energy estimates for them. In Chaps. III and VI, we state and prove
theorems regarding the local and global HOlder continuity of weak solutions of
(2.4) both for p > 2 and 1 < p < 2 and discuss some open problems. In the singular
case 1 < p < 2, we introduce in Chap. IV a novel iteration technique quite different
than that of DeGiorgi [33J or Moser [83].
These theorems assume the solutions to be locally or globally bounded. A
theory of boundedness of solutions is developed in Chap. V and it includes equa-
tions with lower order terms exhibiting the Hadamard natural growth condition.
The sup-estimates we prove appear to be dramatically different than those in the
linear theory. Solutions are locally bounded only if they belong to L ,oc ({}T) for
some r ~l satisfying
(4.1) Ar == N(p - 2) + rp > 0

and such a condition is sharp. In Chap. XII we give a counterexample that shows
that if (4.1) is violated, then (2.5) has unbounded solutions.
The HOlder estimates and the Loo-bounds are the basis for an organic the-
ory of local and global behaviour of solutions of such degenerate and/or singular
equations.
4. Main results ix
In Chaps. VI and VII we present an intrinsic version of the Harnack estimate

and attempt to trace their connection with HOlder continuity. The natural parabolic
cylinders associated with (2.5) are
(4.2)
We show by counterexamples that the Harnack estimate (2.2) cannot hold for non-
negative solutions of (2.5), in the geometry of (4.2). It does hold however in a
time-scale intrinsic to the solution itself. These Harnack inequalities reduce to (2.2)
when p = 2. In the degenerate case p > 2 we establish a global Harnack type
estimate for non-negative solutions of (1.5) in the whole strip ET == RN X (0, T).
We show that such an estimate is equivalent to a growth condition on the solution
as Ixl - 00. If max{l; J~l} < p < 2, a surprising result is that the Harnack
estimate holds in an elliptic form, i.e., holds over a ball Bp at a given time level.
This is in contrast to the behaviour of non-negative solutions of the heat equation
as pointed out by Moser [83] by a counterexample. These Harnack estimates in
either the degenerate or singular case have been established only for non-negative
solutions of the homogeneous equation (2.5). The proofs rely on some sort of non-
linear versions of 'fundamental solutions'. It is natural to ask whether they hold
for quasilinear equations. This is a challenging open problem and parallels the
Hadamard [50] and Pini [86] approach viafundamental solutions, versus the 'non-
linear' approach of Moser [83].
The number p is required to be larger than 2Nj (N + 1) and such a condition
is sharp for a Harnack estimate to hold. The case 1 < P ~ 2Nj (N + 1) is not
fully understood and it seems to suggest questions similar to those of the limiting
Sobolev exponent for elliptic equations (see Brezis [19]) and questions in differen-
tial geometry. Here we only mention that as
of the type of motion by mean curvature.
p'"1, (2.5) tendsformally to a p.d.e.
HOlder and Harnack estimates as well as precise sup-bounds coalesce in the
theory of the Cauchy problem associated with (2.4). This is presented in Chap. XI
for the degenerate case p > 2 and in Chap. XII for the singular case 1 2, we identify the optimal growth of the initial datum as Ix I- 00 for a solution,
local or global in time, to exist. This is the analog of the theory of Tychonov [98],
Tacklind [94] and Widder [105] for the heat equation. When 1 < p < 2 it turns out
that any non-negative initial datum U o E Lfoc(RN) yields a unique solution global
in time. In general
2N
I<P$N+l'
Therefore the main difficulty of the theory is to make precise the meaning of solu-
tion. We introduce in Chap. XII a new notion of non-negative weak solutions and
establish the existence and uniqueness of such solutions. We show by a counterex-
ample that these might be discontinuous. Thus, in view of the possible singulari-
ties, the notion of solution is dramatically different than the notion of 'viscosity'
solution. Issues of solutions of variable sign as well as their local and global be-
haviour are open.
In Chaps. VIII-X, we tum to systems of the type (3.2) and prove that
(4.3) u~? E C/:'c(ilT), i = 1, 2, ... ,n, j = 1, 2, ... , N,

provided p > 2Nj(N + 1). Analogous estimates are derived for all p > 1 for
solutions in L[oc(ilT), where r~1 satisfies (4.1). Again such a condition is sharp
x Preface
for (4.3) to hold. Near the lateral boundary of ilT we establish C a estimates/or
all a E (0, 1), provided p > max {I; ~~2}' Estimates in the class c1,a near the
boundary are still lacking even in the elliptic case.
A similar spectrum of results could be developed for equations of the type

(3.4). We have avoided doing this to keep the theory as organic and unified as
possible.
We have chosen not to present existence theorems for boundary value prob-
lems associated with (2.4) or (3.2). Theorems of this kind are mostly based on
Galerkin approximations and appear in the literature in a variety of forms. We re-
fer, for example, to [67] or [73]. Given the a priori estimates presented here these
can be obtained alternatively by a limiting process in a family of approximating
problems and an application of Minty's Lemma. These notes can be ideally divided
in four parts:
1. HOlder continuity and boundedness of solutions (Chapters I-V)

2. Harnack type estimates (Chapters VI-VII)
3. Systems (Chapters VIII-X)
4. Non-negative solutions in a strip ET (Chapters XI-XII).
These parts are technically linked but they are conceptually independent, in
the sense that they deal with issues that have developed in independent directions.
We have attempted to present them in such a way that they can be approached
independently.
The motivation in writing these notes, beyond the specific degenerate and
singular p.d.e., is to present a body of ideas and techniques that are surprisingly
flexible and adaptable to a variety of parabolic equations bearing, in one way or
another, a degeneracy or singularity.
Acknowledgments
The book is an outgrowth of my notes for the Lipschitz Vorlesungen that I delivered
in the summer of 1990 at the Institut fUr Angewandte Math. of the University of
Bonn, Germany. I would like to thank the Reinische Friedrich Wilhelm Universilit
and the grantees of the Sonderforschungsbereich 256 for their kind hospitality and
support.
I have used preliminary drafts and portions of the manuscript as a basis for lec-
ture series delivered in the Spring of 1989 at 1st. Naz. Alta Matematica, Rome Italy,
in July 1992 at the Summer course of the Universidad Complutense de Madrid
Spain and in the Winter 1992 at the Korean National Univ. Seoul Korea. My thanks
to all the participants for their critical input and to these institutions for their sup-
port.
I like to thank Y.C. Kwong for a critical reading of a good portion of the
manuscript and for valuable suggestions. I have also benefited from the input of
M. Porzio who read carefully the first draft of the first four Chapters, V. Vespri and
Chen Ya-Zhe who have read various portions of the script and my students J. Park
and M. O'Leary for their input.
Contents
Preface
§1. Elliptic equations: Harnack estimates and HiUder continuity ....... v
§2. Parabolic equations: Harnack estimates and Hi>lder continuity . . . . .. vi
§3. Parabolic equations and systems. . . . . . . . . . . . . . . . . . . . . . . . .. vii
§4. Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. viii
I. Notation and function spaces

§1. Some notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I
§2. Basic facts about W1,P(n) and w:,p(n). . . . . . . . . . . . . . . . . . . . 3
§3. Parabolic spaces and embeddings . . . . . . . . . . . . . . . . . . . . . . . . . 7
§4. Auxiliary lemmas .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 12
§5. Bibliographical notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 15
II. Weak solutions and local energy estimates

§1. Quasilinear degenerate or singular equations ................. 16
§2. Boundary value problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 20
§3. Local integral inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 22
§4. Energy estimates near the boundary ....................... 31
§5. Restricted structures: the levels k and the constant 'Y •••••••••••• 38
§6. Bibliographical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 40
xii Contents
III. HOlder continuity of solutions of degenerate

parabolic equations
§1. The regularity theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 41
§2. Preliminaries ...................................... 43
§3. The main proposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 44
§4. The first alternative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 49
§5. The fllSt alternative continued . . . . . . . . . . . . . . . . . . . . . . . . . .. 52
§6. The first alternative concluded .......................'. . .. 55
§7. The second alternative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 58
§8. The second alternative continued . . . . . . . . . . . . . . . . . . . . . . . .. 62
§9. The second alternative concluded. . . . . . . . . . . . . . . . . . . . . . . .. 64
§lO. Proof of Proposition 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 68
§II. Regularity up to t = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 69
§12. Regularity up to ST. Dirichlet data ....................... 72
§13. Regularity at ST. Variational data . . . . . . . . . . . . . . . . . . . . . . .. 74
§14. Remarks on stability ................................. 74
IV. HOlder continuity of solutions of singular

parabolic equations
§1. Singular equations and the regularity theorems . . . . . . . . . . . . . . .. 77
§2. The main proposition .............................:.. 79
§3. Preliminaries ...................................... 81
§4. Rescaled iterations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 84
§5. The first alternative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 88
§6. Proof of Lemma 5.1. Integral inequalities. . . . . . . . . . . . . . . . . . .. 92
§7. An auxiliary proposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 95
§8. Proof of Proposition 7.1 when (7.6) holds ................... 97
§9. Removing the assumption (6.1) .......................... 101
§10. The second alternative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 102
§11. The second alternative concluded . . . . . . . . . . . . . . . . . . . . . . . .. 106
§12. Proof of the main proposition ........................... 109
§13. Boundary regularity .................................. no
§14. Miscellaneous remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 114
v. Boundedness of weak solutions

§1. Introduction....................................... 117
§2. Quasilinear parabolic equations. . . . . . . . . . . . . . . . . . . . . . . . .. 118
§3. Sup-bounds ....................................... 120
§4. Homogeneous structures. The degenerate case p > 2 ........... 122
§5. Homogeneous structures. The singular case 1 < p < 2 .......... 125
§6. Energy estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 128
Contents xiii
§7. Local iterative inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

§8. Local iterative inequalities (P> .............
max { 1; J~2}) 134
§9. Global iterative inequalities. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 135
§1O. Homogeneous structures and 1 <p$max {I; J~2} ............ 137
§11. Proof of Theorems 3.1 and 3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
§12. Proof of Theorem 4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
§13. Proof of Theorem 4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
§14. Proof of Theorem 4.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
§15. Proof of Theorem 4.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
§16. Proof of Theorems 5.1 and 5.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
§17. Natural growth conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
§18. Bibliographical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
VI. Harnack estimates: the case p>2

§1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
§2. The intrinsic Harnack inequality . . . . . . . . . . . . . . . . . . . . . . . . . . 157
§3. Local comparison functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
§4. Proof of Theorem 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 163
§5. Proof of Theorem 2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 167
§6. Global versus local estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
§7. Global Harnack estimates ....... '. . . . . . . . . . . . . . . . . . . . . .. 171
§8. Compactly supported initial data . . . . . . . . . . . . . . . . . . . . . . . . . 172
§9. Proof of Proposition 8.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 174
§ 10. Proof of Proposition 8.1 continued . . . . . . . . . . . . . . . . . . . . . . .. 177
§ 11. Proof of Proposition 8.1 concluded . . . . . . . . . . . . . . . . . . . . . . .. 179
§ 12. The Cauchy problem with compactly supported initial data . . . . . . .. 180
VII. Harnack estimates and extinction profile for

singular equations
§ 1. The Harnack inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
§2. Extinction in finite time (bounded domains) . . . . . . . . . . . . . . . . . . 188
§3. Extinction in finite time (in RN) ......................... 191
§4. An integral Harnack inequality for all 1 < p < 2 . . . . . . . . . . . . . . . 193
§5. Sup-estimates for J~l <p<2 .......................... 198
§6. Local subsolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 199
§7. Time expansion of positivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
§8. Space-time configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
§9. Proof of the Harnack inequality . . . . . . . . . . . . . . . . . . . . . . . . . . 206
§ 10. Proof of Theorem 1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 211
xiv Contents
VIII. Degenerate and singular parabolic systems

§1. Introduction....................................... 215
§2. Boundedness of weak solutions . . . . . . . . . . . . . . . . . . . . . . . . .. 218
§3. Weak differentiability of IDul2j1 Du and energy estimates for IDul 223
§4. Boundedness of IDul. Qualitative estimates .. , . . . . . . . . . . . . . . 231
§5. Quantitative sup-bounds of IDul . . . . . . . . . . . . . . . . . . . . . . . . . 238
§6. General structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
IX. Parabolic p-systems: HOlder continuity of Du

§1. The main theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
§2. Estimating the oscillation of Du . . . . . . . . . . . . . . . . . . . . . . . . . 248
§3. HOlder continuity of Du (the case p > 2) . . . . . . . . . . . . . . . . . . 251
§4. HOlder continuity of Du (the case 1 < p < 2 ) . . . . . . . . . . . . . . . . 256
§S. Some algebraic Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258
§6. Linear parabolic systems with constant coefficients . . . . . . . . . . . . . 263
§7. The perturbation lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 268
§S. Proof of Proposition l.l-(i) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275
§9. Proof of Proposition l.l-(ii) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278
itO. Proof of Proposition l.l-(iii) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282
§ 11. Proof of Proposition 1.1 concluded . . . . . . . . . . . . . . . . . . . . . . . . 284
§12. Proof of Proposition 1..2-(i) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286
§13. Proof of Proposition 1.2 concluded. . . . . . . . . . . . . . . . . . . . . . . . 288
§14. General structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291
X. Parabolic p-systems: boundary regularity

§1. Introduction....................................... 292
§2. Flattening the boundary .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294
§3. An iteration lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297
§4. Comparing w and v (the case p > 2) . . . . . . . . . . . . . . . . . . . . 299
§5. Estimating the local average of IDwl (the case p > 2) .......... 304
§6. Estimating the local averages of w (the case p > 2) . . . . . . . . . . . 305
§7. comparingwandv(thecasemax{1;;~2}<p<2) ......... 309
§8. Estimating the local average of IDwl . . . . . . . . . . . . . . . . . . . . . 313
XI. Non-negative solutions in ET • The case p> 2

§ 1. Introduction....................................... 316
§2. Behaviour of non-negative solutions as Ixl -+ 00 and as t "" 0 .... 317
§3. Proof of (2.4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319
§4. Initial traces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322
Contents xv
§5. Estimating IDulp - 1 in ET ............................ 323

§6. Uniqueness for data in LtoARN) ........................ 326
§7. Solving the Cauchy problem . . . . . . . . . . . . . . . . . . . . . . . . . . . , 330
XII. Non-negative solutions in E T . The case 1 <p<2

§l. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334
§2. Weak solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 337
§3. Estimating IDul ................................... 340
§4. The weak Harnack inequality and initial traces . . . . . . . . . . . . . . . . 344
§5. The uniqueness theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346
§6. An auxiliary proposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 350
§7. Proof of the uniqueness theorem. . . . . . . . . . . . . . . . . . . . . . . . .. 362
§8. Solving the Cauchy problem . . . . . . . . . . . . . . . . . . . . . . . . . . . , 362
§9. Compactness in the space variables. . . . . . . . . . . . . . . . . . . . . . . . 363
§10. Compactness in the t variable . . . . . . . . . . . . . . . . . . . . . . . . . .. 366
§1l. More on the time-compactness . . . . . . . . . . . . . . . . . . . . . . . . . . 370
§12. The limiting process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371
§13. Bounded solutions. A counterexample . . . . . . . . . . . . . . . . . . . . . . 376
Bibliography ...................................... 381

I
Notation and function spaces
1. Some notation
Let fl be a bounded domain in RN of boundary afl and for 0 < T < 00 let flT
denote the cylindrical domain flx (0, TJ. Also let,
8T == afl x [0, TJ, r == 8 T u(flx {O})
denote the lateral boundary and the parabolic boundary of flT respectively.
If fl is a sphere of radius P > 0 centered at some Xo E R N , we denote it by
Bp(xo) == {Ix - xol < p}, and if Xo coincides with the origin, we let Bp(O) == Bp.
The boundary afl will be assumed to satisfy the property of positive geomet-
ric density, i.e.,
there exists 0:* E (0,1) and Po> 0 such that 'r/xo E afl,
(1.1) { for every ball Bp(x o) centered at Xo and radius p::; Po
ISliBp(xo)I ::; (1 - o:*)IBp(xo)l,

where 1171 denotes the Lebesgue measure of a measurable set E.
At times it will be necessary to assume that afl is of class C1,A for some
A E (0, 1). That is, there exist a positive number Po such that for all Xo E afl, the
portion of afl within the ball BpJx o) can be represented, in a local system of
coordinates, as the graph of a C1,A function ¢(x o ) such that ¢(xo)(x o ) =0. We set
(1.2)
2 I. Notation and function spaces
Here for a smooth function fjJ defined on a compact subset IC of Rft. for some
positive integer n
(1.3) [fjJh IC == sup IfjJ(x)1 + sup IfjJ(x) - fjJ~Y)I.

'zEIC (Z,II)EIC Ix - YI
The boundary all is piecewise smooth if it satisfies (1.1) and is the union of finitely
many portions of (N-1) -dimensional hypersurfaces of class Cl,.x.
a
If II is piecewise smooth, we say that a certain quantity, say C or 'Y, depends
upon the structure ofall if it can be calculated apriori only in terms of the numbers
o· ,Po in the definition (l.1), the number of the components making up all and
the III . 11I1H norm of each of the components making up all.
If 1 E Lq(ll), 1 ~ q ~ 00, denote by II/lIq,n the Lq(ll)-norm of I. The
function 1 is in Lfoc(ll) if II/lIq,lC is finite for all compact subsets IC of ll.
Let q, r ~ 1. A function 1 defined and measurable in llT belongs to
if
11/11 ••.,,,, '" (l ([ IfI'dz) dT) f 1 < 00.
Also 1 E Lf::C(llT), iffor every compact subset IC of II and every subinterval

[t1' t2J C (0, TJ
J(/ I/lqdx)
tl IC
i dr < 00.
Whenever q = r we set Lq,r(llT) == Lq(llT), Lf::C(llT) == Lfoc(llT) and

II/lIq,q;nT == II/lIq,nT' These definitions are extended in the obvious way when
either q or r are infinity.
If 1 E CI(llT), we denote by DI == (f"'11 1"'2"" ,I"'N) the gradient of 1
with respect to the space variables only.
The spaces WI,p(n) and W!,P(ll). p~ 1, are defined by
W l,p (ll) is the completion of Coo (ll) under the norm

IIvllwl,p(n) == livllp,n + IiDvllp,n, 11 E COO(ll)nLP(ll).
W~'P(ll) is the completion ofC~(ll) under the norm
IIvllw~,p(n) == IIDvlip,n. 11 E C~(ll).
Equivalently WI,P(ll) is the Banach space of functions v E LP(ll) whose gener-

alised derivatives v"" belong to LP(n) for all i= 1, 2, ... , N.
A function v E Lfoc (ll) is in W,!;: (ll) if for every compact subset IC ell, v E
Wl,P(K).
2. Basic facts about WI.pCo) and W~·P(O) 3
We let WI,oo(n) denote the space of functions v E LOO(n) whose disttibu-

tional derivatives vz • are in LOO(n), for i = 1,2, ... , N. The space W,~:o(n) is
defined analogously.
2. Basic facts about W 1,P(il) and W~'P(il)

We collect here a few facts that will be of frequent use in what follows. The first
is about the Gagliardo-Nirenberg multiplicative embedding inequality.
THEOREM 2.1. Let v E w;,"(n), p ~ 1. For every fixed number s ~ I there
exists a constant C depending only upon N, p and s such that
(2.1)
where Q E [0, 1], p, q ~ 1, are linked by
(2.2)
and their admissible range is:
(2.2-i) q N=I,
qE[S,OO], QE [O'P+S~_I)];
(2.2-ii) if 1 ~ p < N, a E [0, I] and
qE [s, :!p] if s~ :!p'

qE[:!p'S] if S~:!p;
(2.2-iii) if p~N > 1, q E [s, 00) and
oE [0, NP+~:-N»).
COROLLARY 2.1. LetvE W;,"(n), and assume pE [I,N). There exists a con-
stant 'Y depending only upon N, and p, such that
Np
(2.1)' where q=--.
N-p
PROOF: We may take Q = 1 and S = 1 in (2.2-ii).

H 8n is piecewise smooth, functions v in WI,"(n) are defined up to 8n via
their traces. We will denote by vlan the trace on 8n of a function v E Wl,"(n) .
THEOREM 2.2. Assume that an is piecewise smooth. There exists a constant C

depending only upon N,P and the structure of an such that
(2.3)
where
1 (N - I)P] if < p < N,

qE [ ' N -P
(2.3-i) 1
'
(2.3-ii) q E [1,00), if p= N.
If an is piecewise smooth. the space W;,"(n) can be defined equivalently

as the set of functions v E WI,,, (n) whose trace on an is zero.
Remark 2.1. The embedding inequalities of Theorem 2.1 and Corollary 2.1 con-
tinue to hold for functions v in WI,"(n). not necessarily vanishing on an in the
sense of the traces. provided we assume further that an is piecewise smooth and
that
(2.4) f
n
v(x)dx = o.
In such a case the constant C depends upon s, p, q, a, N and the structure of an.
However it does not depend on the size of n. i.e., it does not change under dilations
ofn.
Let k be any real number and for a function v E WI,"(n) consider the trun-
cations of v given by
(2.5) (v - k)+ == max{(v - k) ; O}, (v - k)_ == max{ -(v - k) ; O}.
LEMMA 2.1. LetvEW1,"(n). Thenforall kER, (v=Fk)± E Wl'''(n). Assume

in addition that the trace of v on an is essentially bounded and
Ilvlloo,an :5 ko, for some ko > o.

Then for all k~ko, (v - k)± EW;,"(n).
COROLLARY 2.2. Let Vi EW1,"(n),i=I,2, ... ,nEN. Then
w == min {Vl,V2, ... ,vn } E W1,"(n).
PROOF: Assume first n=2. Then
'. { .} VI - (V2 - VI)+ V2 - (VI - V2)+

mm Vb V2 = 2 + 2 .
The general case is proved by induction.

If v is a continuous function defined in n and k < l is a pair of real numbers.
we set
2. Basic facts about W1;P({}) and W.,';p({}) 5
a Ivex) > l},

r>~
- {x E
(2.6) [v < k] - {x E alv(x) < k},
[k < v < lj - {x E n Ik < vex) < l} .
LEMMA 2.2. Let v E WI,I(Bp(X o)) nC(Bp(xo)) for some p > 0 and some
Xo E RN and let k and I be any pair of real numbers such that k < I.
There exists a constant "1 depending only upon N, p and independent of
k, l, v, x o , p. such that
(2.7) (l- k)1 [v > 1]1 ~ "11 [~< kjl

N+1 I IDvldx.
[k<v<IJ
Remark 2.2. The conclusion of the lemma continues to hold for functions v E
WI,I (n)nC( n) provided n is convex. We will use it in the case a is a hemisphere
or a cube.
Remark 2.3. The continuity is not necessary to the conclusion of Lemma 2.2. The
function v has been assumed to be continuous to give an unambiguous meaning
to the definitions (2.6). If v is only in Wl,l (a). one could fix an arbitrary repre-
sentative out of the equivalence class v say v and define (2.6) accordingly. The
conclusion of the Lemma is independent of the choice of v.
2-(i). Poincare-type inequalities

Inequality (2.7) is due to De Giorgi [33] and it is a particular case of a more general
Poincare-type inequality. The embedding (2.1)' of Corollary 2.1 gives a majorisa-
tion of the Lq (n) -norm of u solely in terms of the V (n) -norm of its gradient.
This is possible because one knows that u vanishes on aa in the sense of the traces.
A Poincare-type inequality bounds some integral norm of a function u E Wl,p (n)
in terms only of some integral norm of its gradient, provided some information is
available on the set where u vanishes.
PROPOSITION 2.1. Let n be a bounded convex set in RN and let I.{) E C (1i)
satisfy
{
0 ~ I.{) ~ 1, "Ix E n,
(2.8)
the sets [I.{) > k] are convex, Vk E (0,1).
Let vE WI,p(a), p~ 1, and assume that the set
e == [v = Ojn[l.{) = 1]
has positive measure. There exists a constant C depending only upon Nand p and
independent of v and I.{), such that
(2.9)
6 I. Notation and fwiction spaces
PROOF: We fust prove (2.9) for p= 1. For every zEe and XE 0,
Iz-zl
Iv(x)1 = Iv(x) - v(z)1 = / J :p v(z+j:::jP) dp/
o
Iz-zl
$ J /VV(Z+RP)/ dp.
o
Multiply this inequality by cp(x) and integrate in dx over 0 and in dz over e. This
gives
Iz-zi
(2.10) lei J cplvldx $ J dz J dx J CP(x)/DV(z+j:::jp)/ dp.
n E n 0
By virtue of the assumption (2.8)
cp(x)/Vv(z+j::=j"p)/ $ cpJDvl(z+j:::jP).
We put this inequality in (2.10) and compute the integral in dx on the right hand
side, in polar coordinates with pole at z and radial variable r = Ix - zl. We let w
be the angular variables and denote by 'R.(w) the polar representation of ao with
pole at z.
diam n ) 1l(... )
lei Jcplvldx $ JdZ J rN- 1dr J J IxcpIDul(x) N1
- z1N-1 r - dr
( dw
n E 0 1... 1=1 0
$ -y(diamO)N Je dz nJI:~~11~1 dx
$ -y(diamOt suPJ
zEn
IX - d~N_1
Z
JcplDvldx.
e n
Next, for all x E O. and for all 6> 0,
dz
Ix - zlN-l +
J dz
Ix - z1N-1
En{lz-zl~6}
Minimising with respect to the parameter 6 gives
(2.11) Jcplvl dx
n
$ -y (~;:N JcplVvl dx,
n
3. Parabolic spaces and embeddings 7
for a constant 1'=1'(N,p). By replacing v with Ivl P in (2.11), we obtain
J cplvl P dx <
- P'Y
(diamfl)N
1t:11--k
J cplvlp-1lDvl dx
n n
<
- 2
~jcplvlPdx + 1'(P) [(diam~N1P
1t:1 1-
jCPIDvIPdx.
n n
Remark 2.4. Inequality (2.7) follows by applying (2.9) with cp == 1 and p = 1 to

the function
w= {:in{V,l} -
k if v > k
if v ~ k.
By Corollary 2.2 such a function is in W1,1(fl).
3. Parabolic spaces and embeddings
We introduce spaces of functions, depending on (x, t) E flT, that exhibit differ-

ent regularity in the space and time variables. These are spaces where typically
solutions of parabolic equations in divergence fonn are found.
Let m, p ~ 1 and consider the Banach spaces
and
Vom,P(flT) == L oo (0, Tj Lm(fl))nLP (0, Tj W~,p(n)),
both equipped with the nonn. v E Vm,P( flT ).
When m=p. we set V!,P(flT ) == V!(flT) and VP,P(flT ) ==VP(flT). Both

spaces are embedded in Lq (flT ) for some q > p. In a precise way we have
PROPOSITION 3.1. There exists a constant l' depending only upon N, p, m such
that for every v E Vom,p( fl T )
(3.1) jj1v(x,t)lqdxdt
nT
",' ([/IDV(X,t)IPdxdt ) ( esssupj Iv(x, t)lmdx) ." ,

O<t<T
n
where
N+m
q=p--.
N
Moreover
(3.2)
The multiplicative inequality (3.l ) and the embedding (3.2) continue to hold for
n
functions v E V m ,,,( T ) such that
J
n
v(x, t)dx = 0, fora.e. tE(O,T),
provided an is piecewise smooth. In such a case the constant "f depends also on
the structure of an.
PROPOSITION 3.2. Assume that an is piecewise smooth. There exists a constant
"f depending only upon N,p, m and the structure of an, such that for every v E
Vm,"(ilT ) ,
N+m
(3.3) q=p--.
N
PROOF OF PROPOSITION 3.1: Assume first that N(p-m)+mp > O. Write the
embedding inequality (2.1) for the function x -+ v(x, t) for a.e. t E (0, T) and for
the choice of the parameters s =m and
N+m
a=?!.; q=p--- ; N(P - m) + mp>O.
q N
Taking the qthpower in the resulting inequality and then integrating over (0, T)
proves (3.1). If N (p - m) + mp ::5 0, we must have p < N. Therefore applying
Corollary 2.1,
T
JJ
nT
Ivl q dx dt = JJ
0 n
Ivl"lvl mN dx dt
~
$ ! (j Ivl~dz (j
T
) IV1mdz)
N
~
$ ([f IDV1Pdzdt) (~~'f [IV(X,tl1mdz) ~
To prove (3.2), we rewrite (3.1) as
3. Parabolic spaces and embeddings 9
and apply Young's inequality.

PROOF OF PROPOSITION 3.2: IfvEVm,p(nT ), consider the function
w(·, t) = v(·, t) - I~I J

n
v(x, t)dx, a.e. t E (0, T),
which has zero integral average over n for a.e. t E (0, T). By Remark 2.1, x-+
w(x, t) satisfies the embedding inequality (2.1) for a.e. t E (0, T] and with constant
C depending also upon the structure of an.
Proceeding as before, we arrive at (3.2) for w. For a.e. t E (0, T) ,
IIDwllp,nT = IIDvllp,nT' IIw(" t)lIm,n :::; 21Iv(·, t)lIm,n.
Moreover
(I U
1
IIwll •.a. " II-II •. ",. - 1111 hI. I_1 m dx) ;1; <it) •
Therefore
. The last term is majorised by

1
( Inl N(p~ml+mp)
Nm
9 esssup IIv("
O<t<T
t)lIm,n,
and the proposition follows.
We will use the following Corollaries obtained from the previous Propositions
by taking m = p and by applying the HOlder inequality.
COROLLARY 3.1. Let p > 1. There exists a constant 'Y depending only upon N
and p, such that for every v E Vo"( T ). n
(3.4) IIvll:,nT :::; 'Yllvl > olmllvll~p(nT)'
COROLLARY 3.2. Letp> 1. There exists a constant'Y depending only upon N,p
and the structure of an. such that for every v E V P ( nT ).
(3.5) IIvll:,nT :5 'Y (1 + InlT)'lIIh
N Ilvl > 01 mpIIvll~p(nT)'
The next two Propositions hold in the case m = p.
PROPOSITION 3.3. There exists a constant 'Y depending only upon N and p such
that for every v E VJ'( (h ),
(3.6)
where the numbers q, r ~ 1 are linked by

1 N N
(3.7) -+-=-
r pq p'
and their admissible range is
q E (p,ool, r E [P2,00) ; if N = 1,
(3.8)
{ qE [p, :!p], r E [P,ool ; if 1Sp < N,
q E [P, 00), r E (~, 00] ; if 1 < N S p.
PROOF: Let v E V!(nT ) and let r ~ 1 to be chosen. From (2.1) with s = pit
follows that
Choose ar=p. Then conditions (2.2)-(2.2-iii) imply (3.7)-(3.8), and the Proposi-
tion follows.
The next Proposition holds for functions v E VP( nT ) not necessarily vanish-
ing on the lateral boundary of nT.
PROPOSITION 3.4. There exists a constant 'Y depending only upon N, p, m and
the structure of 8 n, such that for every v E VP ( nT ),
(3.9) Iivliq,r;iJT S 'Y (1 + 1;11 )t IivlivP(iJT)
where q and r satisfy (3.7) and (3.8).

PROOF: Apply (2.1) to the function
w(·, t) = v(·, t) - I~I J

n
v(x, t)dx, a.e. t E (0, TI,
n
which has zero average in for a.e. t E (0, T). Proceeding as in the proof of Propo-
sition 3.3 we arrive at (3.6) for w where 'Y now also depends upon the structure of
8n. From this
3. Parabolic spaces and embeddings II
IIvll •.•,,,. ~ ~lIvllv.(".) + ~Inlt-l (1 U T)ldx) ·dT)

Iv(x. 1
:$1'lI v llvP(I1T ) +1' C!1~/N); ~:~~lIv(·,'T)lIp,I1'

We conclude this section by stating a parabolic version of Lemma 2.1 and
Corollary 2.2 concerning the truncated functions (v - k)±.
LEMMA 3.1. Let v E V m ,P(IlT)' Then/or all k E R, (v - k)± E V m ,P(!1T)'
Assume in addition that the trace 0/ x - v( x, t) on all is essentially bounded and
esssupllv(·,t)lIoo,al1:$ k o, /orsome ko > 0.
O<t<T
Then/or all k?, ko • (v =t= k)± E Vom ,P(!1T ).
COROLLARY 3.3. Let Vi ELP (0, Tj W 1,P(Il)), i=1,2, ... ,nEN. Then
w == min{v1,v2, ... ,vn } E LP (0,TjW 1,P(Il)).
3-(i). Steklovaverages
°
Let v be a function in L1 (!1T) and for < h < T introduce the Steklov averages
Vh(-,t) defined for all 0< t < T by
Vh
_ *J v(·,'T)d'T,
= {
t+h
t
t E (O,T - h},
0, t>T-hj
vii.
_{* j
= t-h
v(','T)d'T, t E (h,T],
0, t < h.
LEMMA 3.2. LetvE Lq,r(IlT ). Then, as h-O, Vh converges to v in Lq,r(IlT _t:)
for every e E (0, T).lfv E C (0, Tj Lq(f1)). then as h - 0, Vh(-, t) converges to
v(·, t) in Lq(Il)/or every tE (0, T - e), 'VeE (0, T).
A similar statement holds for vii.. The proof of the lemma is straightforward
from the theory of LP spaces.
4. Auxiliary lemmas
4-(;). Fast geometric convergence
We state and prove two lemmas concerning the geometric convergence of se-
quences of numbers.
LEMMA 4.1. Let {Yn } , n=O, 1,2, ... , be a sequence of positive numbers, sat-
isfying the recursive inequalities
(4.1)
where C, b,> 1 and a > 0 are given numbers. If
(4.2)
then {Yn } converges to zero as n-oo.

The proof is by induction.
LEMMA 4.2. Let {Yn } and {Zn} , n=O, 1,2, ... , be sequences ofpositive num-
bers, satisfying the recursive inequalities
Yn+l :5 Cbn (Y,t+a + Z~+"Yna) ,
(4.3) {
Zn+1 :5 Cbn (Yn + Z~+,,)
where C, b> 1 and It, a > 0 are given numbers. If
1+ ..
~
(4.4) Y.o + Z 1+" < (2C)- " b--;;Z-
0 _ , where u = min{lt; a},
then {Yn } and {Zn} tend to zero as n - 00.
PROOF: Set Mn =Yn + Z~+" and rewrite the second of (4.3) as
(4.5)
Consider the tetm in braces in the first of (4.3). If Z!+" :5 Yn• such a term is
majorised by 2M~+a. If Z!+" ~ Yn• then the same term can be majorised by
y~+a + (Z~+,,)l+a :5 M~+a.

Combining this with (4.5) we deduce that in either case
M n+l <
_
2C 1+K.b(1+K.)nMl+
n '
min {K.,a}
The proof is concluded by induction as in Lemma 4.1.

4. Auxiliary lemmas 13
4-(ii). An interpolation lemma
LEMMA 4.3. Let {Yn } , n = 0,1,2 ... , be a sequence of equibounded positive

numbers satisfying the recursive inequalities
(4.6)
where C, b> 1 and a E (0, 1) are given constants. Then
(4.7) 2C
Yo ~ ( bl-~
)±
Remark 4.1. The Lemma turns the qualitative infonnation of equiboundedness
of the sequence {Yn } into a quantitative apriori estimate for Yo.
PROOF OF LEMMA 4.3: From (4.6), by Young's inequality
VeE (0,1), n=O,1,2, ....
By iteration
C )~ n-l .
Yo ~ enYn + ( e1- a Q ~ (b±e)'.
Choose b± !
e = so that the sum on the right hand side can be majorised with a
series convergent to 2. Letting n -+ 00 proves the Lemma.
4-(iii). An algebraic lemma
We conclude this section by recording two algebraic inequalities needed in what

follows.
LEMMA 4.4. Letp~2.ThenVa,bERm,mEN
(4.8)
where 'Yo depends only upon p, m.

Let 1<p<2. Then Va, bERm
(4.9)
where 'Yl depends only upon p, m.

PROOF:
J(p) = (laI P - 2 a - Iblp - 2b, a - b)

1
~ (/ ! I" + (1 - 8)bl'-'(80 + (1 - 8)b)da, 0 - b)

!
1
= Isa + (1 - s)bI P- 2 Ia - bl 2 ds
o
!
1
+ (p - 2) Isa + (1 - s)bI P- 4 1(sa + (1 - s)b, a - b}1 2 ds.

o
1
Ifp~2. J(p) ~ la - bl 2 J Isa + (1 - s)bI P- 2ds. If lal ~ Ib - al. we have
o
Isa + (1 - s)bl ~ lial- (1 - s)la - bll ~ sla - bl
and (4.8) follows. If lal < Ib - al.
!
1
la - bl 2 Isa + (1 - s)bI P- 2dx

o
>
-
la _ b12 ! 1
(Isa + (1- s)bl )P
(2 - s)la - bl 2
2 /2
ds
o
1 )P/2
~~ (
/ Isa + (1- s)bl 2 ds
= ~ ~/2 (la1 2 + Ibl2 + (a, b) )P/2

~ 'iola - bjP.
Remark 4.2. The reverse inequality to (4.8) is. in general. false. Indeed. if a E R + •
b=a+l
J(p) = (p - 1)~P-2; for some ~ E (a, a + 1).
Letting a -+ 00 shows that the inequality J(p) :5 "rIa - bl P cannot hold with "r
independent of a and ~.
Next, if 1 < p < 2,
!
1
J(p) :5 (p - 1)la - bl 2 Isa + (1 - s)bI P- 2 ds.

o
If lal ~ la - bl. since p < 2,
5. Bibliographical notes 15
Isa + (1- s)bI P- 2 ~ Ilal- (1- s)la - bIlP-2~ sp- 2 la - bl p - 2

and (4.9) follows since sp-2 is integrable. If lal < Ib - ai, let s. be defined by
(1 - s.)la - bl = lal.
Then estimate
1
/(P) ~ (p -I)la - bl /lIa l -2 (1- s)la - b!lP-2ds

o
8.
~ la - bl P / :s (lal - (1 - s)la - bDp-1ds

o
1
+ la - bl /
s.
! (Ial - (1 - s)la - bl)p-1ds
Remark 4.3. The reverse inequality is false, in general, with 'Y independent of
a,b.
5. Bibliographical notes
For the theory of Sobolev spaces we refer to the monographs of Adams [I] and
Mazja [76]. The embedding theorems 2.1 and 2.2 are special cases of more gen-
eral embedding theorems. No attempt has been made to state them in the most
general setting and under the best assumptions on the regularity of afJ. For a
proof of Lemma 2.1 we refer to Mazja [76] and Stampacchia [93]. Lemma 2.2
is due to DeGiorgi [33]. Also the statement in Remark 2.2 follows from the proof
in [33]. The parabolic spaces Vm,P(fJT ) and Vom,P(fJT ) are standard in the the-
ory of parabolic partial differential equations and we refer for example to [67,73].
The embedding theorems 3.2 and 3.3 are a modification of similar statements and
proofs in [67]. The lemmas on rapid geometric convergence are stated in [67]; we
have given a different proof. The interpolation inequality of Lemma 4.3 is taken
from Campanato [22,23]. Lemma 4.4 is taken from [27].
II
Weak solutions and local energy
estimates
1. Quasilinear degenerate or singular equations

We introduce a class of quasilinear parabolic equations with the principal part in
divergence fonn, that are either degenerate or singular due to the vanishing of the
gradient IDul of their solutions.
(1.1) Ut - diva(x,t,u,Du) = b(x,t,u,Du) in'D'(nT)'

n n
The functions a : T x R N + 1 -+ RN and b : T x RN +1 -+ R are only assumed
to be measurable and satisfying the structure conditions
a(x, t, u, Du)· Du ~ ColDul P - f{)o(x, t),

la(x, t, u, Du)1 S C11Dul p - 1 + f{)l (x, t),
s
Ib(x, t, u, Du)1 C21Dui P + f{)2(X, t)
for p > 1 and a.e. (x, t) E nT • Here Ci , i = 0, 1,2, are given positive constants
and f{)i, i =0, 1, 2, are given non-negative functions, defined in (IT and subject to
the condition
-'!.- • •
f{)o, f{)1p-I ,f{)2 E Lq,r(rl
UT )
where q, r ~ 1 satisfy
1 N .
(As) -=r + -:::
pq
= 1- 1\:1.
and
1. Quasilinear degenerate or singular equations 17
(As-i)
(As-ii)
(A5 -iii)
A measurable function u is a local weak sub(super)-solution of (1.1) in nT if
(1.2) u E Gloc (0, T; L~oc(n»nLfoc (0, T; wl~:(n)) ,

and for every compact subset JC of n and for every subinterval [h, t2J of (0, TJ
t t2
(1.3) fUrpdXlt2 + f f{ -urpt + a(x, T, u, Du)·Drp} dxdT
/C 1 tl/C
t2
:5 (~) f fb(X, T, u, Du)rp dxdT,

tl/C
for all locally bounded testing functions
(1.4) rp ~ o.
The local boundedness of the testing functions rp is required to guarantee the con-
vergence of the integral on the right hand side of (1.3).
A function u that is both a local subsolution and a local supersolution of (1.1)
is a local solution.
Remark 1.1. If p = 2. then (1.1) is non-degenerate. In such a case it is known

that locally bounded weak: solutions are locally mlder continuous; moreover the
assumptions (AI )-( As) are optimal for a mlder modulus to hold.
It would be technically convenient to have a formulation of weak solution
that involves Ut. Unfortunately solutions of (1.1). whenever they exist. possess a
modest degree of regularity in the time variable and. in general, Ut has a meaning
only in the sense of distributions.
The following notion of local weak: sub(super)-solution involves the discrete
time derivative of u and is equivalent to (1.3).
Fix t E (0, T) and let h be a small positive number such that 0 < t < t + h < T.
In (1.3) take tl = t, t2 = t + h and choose a testing function rp independent of
18 ll. Weak solutions and local energy estimates
the variable T E (t, t + h). Dividing by h and recalling the definition of Steklov
averages we obtain
(1.5) J {Uh,tCP + la(x, T, u, Du)Jh ·Dcp -Ib(x, T, U, DU)]h cp} dx ::;;

K:x{t}
(~)O,
for all 0 < t < T - h and for all cp E WJ,p(,qnL~c(.fl), cp ~ O.
To recover (1.3), fix a subinterval O<tl < t2 <T,choose h so small thatt2+h ::;; T
and in (1.5) take a testing function as in (1.4). Such a choice is admissible, since
the testing functions in (1.5) are independent of the variable T E (t, t + h) but may
be dependent upon t. Integrating over It I ,t2J and letting h -+ 0 with the aid of
Lemma 3.2 of Chap. I gives (1.3).
1-(i). Subsolutions and parabolic equations

Tbe structure conditions (AI)-(As) are not sufficient to characterize parabolic
p.d.e. 's. For example the 'principal part'
a(x, t, u, Du} = Du - Du/IDul
satisfies (Ad-(As) with p= 2. However its 'modulus of ellipticity' changes type

at IDul = 1. In what follows we assume that (1.1) is weakly parabolic in the sense
that it satisfies (AI )-( As) and in addition, whenever u is a weak solution of ( 1.1),
(A6) for all k E R the truncated functions (u - k)±

are weak subsolutions of (1.1) in the sense of (1.3).
with a(x, t, u, Du) replaced by
±a(x, t, k ± (u - k)±, ±D(u - k)±)
and b(x, t, u, Du} replaced by
±b(x, t, k ± (u - kh, ±D(u - k)±).
To clarify the connection between subsolutions and parabolic structures, we

derive some sufficient conditions on a(x, t, u, Du) for (A6) to be verified. Let u
be a local weak solution of (1.1), and in (1.5) take the testing function
e>O and cp satisfying (1.4).
We integrate in dt over It I , t2J C (0, T) and let first h -+ 0 and then e -+ 0 to obtain
1. Quasilinear degenerate or singular equations 19
t2
ju+C{)(x,t)I:: + fj{-u+C{)t+a(x,t,u+,DU+).DCP} dxdr

n tIn
t2
(1.6) =f f b(x, t, u+, Du+) cpdxdr

tIn
LEMMA 1.1. Assume that
a(x,t,u,11)·11 ~ 0,
Then (1.1) is weakly parabolic.

PROOF: It suffices to verify (A6) for (u - k)+ and k = 0. This is the content of
(1.6).
A more general condition for (As) to hold can be given by the notion of
monotonicity. We say that 11-+a(X, t, u, 11) is monotone i£<l)
(1.7) (a(x,t,u,11d - a(x, t,u, 112),111 - "12) ~ 0, V11i ERN, i=I,2.
LEMMA 1.2. Assume that 11-+a(x, t, u, 11) is monotone and
(1.8)
Then (A6) holds.

PROOF: Write the last integral on the right hand side of (1.6) as
t2
[(a(X,t,u,DU+) -a(x,t,u,O)).Du+l d d
e fy ()2 cp X r
U++e
tl n
t2
-f}diV a(x, t, u, 0) ~
U++e
cp dxdr
tl n
t2
- f"fa(x,t,u,O).Dcp~dxdr.
J
tIn
U++e
°
We let e '\. and discard the non - negative contribution of the first integral. The
sum of the last two terms tends to zero since
(1) The monotonicity assumption (1.7) is natural in the existence theory. It permits one
to apply Minty's Lemma [78] to identify the weak limit of the principal part of the p.d.e.
when (1.1) is approximated by a sequence of regularised problems.
20 n. Weak solutions and local energy estimates
(1.9)
t2
=j j div a(x, t, u+, O)<pdxdT

tin
t,
=- jja(x,t,u+,O)'D<PdxdT.
tin
One checks that the assumptions of the lemma are verified for example by
equations with principal part
where 1/Jo is bounded, non-negative and .p;,x; E Ll(nT) and the matrix (ai;) is
only measurable and positive definite.
Remark 1.1. The 'regularity' assumption (1.8) is only needed to justify the limit
in (1.9). It can be dispensed with when working with a sequence of approximating
solutions.
2. Boundary value problems

We will give regularity results for weak solutions of (1.1) up to the lateral boundary
ST, provided u satisfies appropriate Dirichlet or Neumann boundary conditions.
°
We also prove that weak solutions are HBlder continuous up to t = if the initial
datum is HBlder continuous.
Since the arguments are local in nature, for these results to hold, the pre-
scribed boundary data have to be taken only locally. However, for simplicity of
presentation we will state them globally, in tenns of boundary value problems.
2-(i). The Dirichlet problem
Consider fonnally the Dirichlet problem
ut .- div a~, t,.u, Du) = b(x, t, u, Du), in nT,

(2.1) { u( ,t)18n - g( ,t), a.e. t E (0, T),
u(',O) = uoO,
where the structure conditions (Al)-(As) are retained. On the Dirichlet data 9
and U o we assume
2. Boundary value problems 21
(D) 9 is continuous onST with modulus of continuity, say wg (·),

(Uo ) Uo is continuous in n with modulus of continuity, say woO.
A weak: sub(super)-solution of the Dirichlet problem (2.1) is a measurable function
u, satisfying
(2.2)
and for all te (0, T]
(2.3)· I uY'(x, t) dx I I {-uY't + a(x, T, u, Du)·DY'} dxdT

n nc
~ (~) luoY'(X, 0) dx + Ilb(X, T,U, DU)Y'dxdT,
n nc
for all bounded testing functions
In addition the second of (2.1) holds in the sense that u ~ ~ 9 on an in the sense
of the traces of functions in W1,p(n) for a.e. t e (0, T).
A function u that is both a sub-solution and a super-solution of (2.1) is a
solution of the Dirichlet problem.
The formulation can be rephrased in terms of Steldov averages as in the pre-
vious section, namely
(2.5) l{uh,tY' + [a(x, T, u, Du)lh ·DY' - [b(x, T, u,Du)lh Y'} dx ~ (~) 0,

,Ux{t}
for all 0 < t < T - h and for all Y' e w!,p(n)nLOO(n), Y' ~ o.
Moreover the initial datum is taken in the sense of L 2 (n), i.e.,
(2.6)
2-(ii). Variational boundary data

Assume that an is piecewise smooth, so that the outward unit normal, which we
denote with n, is defined a.e. on an and consider formally the Neumann problem
Ut - div a(x, t, u, Du) = b(x, t, u, Du), in nT,

(2.7) { a(x, t, u, Du)·n = ",,(x, t, u), on ST,
u(·,O) = u o (-).
We retain the structure conditions (A1HAs) and the assumption (U o ) on the

initial datum. We assume that ",,(., t, u(·, t» admits, fora.e. tE (0, T),anextension
into n which we denote by 1fi(., t, u( ,t», such that
11fi1 :5 ""ou + ""10 a.e. nT,

(N) { l1fiul :5""0'
l1fiz.1 :5 ""1. i =1, 2, ... ,N,
where ""1, ""0 are given non-negative functions satisfying
(N-i) 1/Jr,1/Jrr E Ltl,r(nT), whereq and f satisfy (As).
To give a notion of weak sub(super)-solution we let /C be an arbitrary compact
subset of RN and consider testing functions
(2.8) cp ~ 0.
A function u,
(2.9)
is a weak sub(super)-solution of the Neumann problem (2.7) if for every compact

subset /C of RN and for every subinterval [t1' t2J of [0, T]
t t2
(2.10) j UCPdXlt2 +j j{-uCPt + a(x,T,U,Du)·Dcp}dxdT
ICnn 1 tllCnn
~ ~
:5 (~) j jb(x,T, U, Du)cpdxdT + j j 1/J(x, t, u)cpdudT,

tllCnn tl ICnan
where du denotes the surface measure on an. We remark that the testing functions
cp vanish, in the sense on the traces, on the boundary of /C and not on the boundary
of n. The variational datum is reflected in the boundary integral on the right hand
side of (2.10). The formulation in terms of Steklov averages is
(2.11) j {Uh,tCP + [a(x, T, u, Du)J h ·Dcp - Ibex, T, u, Du)J h cp} dx

(ICnn]x{t}
:5 (~) j [1/J(x, t, u)Jh cp du,

(ICnan]x{ t}
for all °< t < T - h and for all cp E W~·P(/C), cp ~ 0,
and the initial datum is taken in the sense of (2.6).
3. Local integral inequalities

We will derive some integral inequalities in the interior of nT. which will be the
main tools in establishing local HOlder estimates for the solutions. Analogous es-
3. Local integral inequalities 23
timates near the lateral boundary ST as well as near t = 0 will be derived in the
next section.
Let K p denote the N-dimensional cube centered at the origin and wedge 2p.
i.e.•
If Xo E RN. we let [xo + Kp] denote the cube of centre Xo and wedge 2p which is
congruent to Kp. i.e.•
[xo + Kp] == {x E RNll~~N IXi - xo.il < p}.
Let 6 be a given positive number and consider the cylinder
Q (6, p) == Kp x {-6, O} ,
and if (xo, to) ERN+!. we let [(xo, to) + Q (6, p)] denote the cylinder with 'ver-
tex' at (xo, to) congruent to Q (6, p). i.e .•
[(xo, to) + Q (6,p)] == {x E RNll~~N IXi - xo,il < p} x {to - 6, to}.
We will refer to these as cubes and cylinders of 'radius' p and height 6.

Fix (xo, to) E nT and let p and 6 be so small that [(xo, to) + Q (6, p)] E nT.
Let (denote a piecewise smooth cutoff function in [(xo, to) + Q (6, p)] such that
(3.1) (E [0,1], ID(I < 00, and ((x, t) = 0, for x outside [xo + Kp].
Assume that
(3.2)
and construct the truncated functions (u - k)±. We will choose levels k satisfying
(3.3) esssup I(u - k)±1 == Ht ~ 0,

[(z",t,,)+Q(II,p»)
where 0 is a positive parameter to be chosen later.

Remark 3.1. Suppose (3.3) is written for (u - k)+ and assume the number
o is small. Then the levels k are forced to be near the essential sup of u in
[(xo, to) + Q (6, p)]. Likewise if (3.3) is written for (u - k)_. then k has to be
close to the essential inf of u within [(xo, to) + Q (6, p)].
Roughly speaking. the function u is HOlder continuous if. within [(xo, to)+
Q(6, p)]. it is close in some integral norm to its integral average. Accordingly
the sets where the function is near its supremum or near its infimum. within
[(xo, to) + Q (6, p)]. have relatively small measure. Our energy inequalities re-
flect this through the sets
24 II. Weak solutions and local energy estimates
In estimating the contribution of the lower order terms !.pi, i = 0, 1, 2, it is conve-

nient to introduce the numbers q, r,,,, constructed starting from q, f''''1 as follows:
qp(1 + ",) fp(1 + ",)
(3.5) q= q- 1 ;
A r = r-
A 1 ;
It is seen from (As-i)-(As-iii) that they satisfy

1 N N
(3.6)
;+pq=p2'
and their admissible range is
qE(P'OO)' rE(p2,oo); if N = 1,
(3.7) { qE (p, :!p), r E (p,oo); if1 < p < N,
qE(P,OO), rE(~'OO); if! < N ~ p.
1be statement that a constant "'I depends only upon the data means that it can be
determined a priori only in terms of the numbers N, p, q, r, "', the constants Ci , i =
0,1,2, and the norms
3-(i). Local energy estimates
PROPOSITION 3.1. Let u be a locally bounded weak solution 0/(1.1) in nT .

There exist constants "'I and 00 that can be determined a priori only in terms 0/ the
data such that/or every cylinder [(x o, to) + Q (0, p)] c nT and/or every level k
satisfying (3.3) for 0 ~ 00
(3.8)
sup
to-8<t<to
I(u -k)~(P(x, t)dx + "'1- 1 frJ flD(u - k)±(IPdxdT
[xo+Kp] [(X o,to)+Q(8,p)]
$ I(U - k)~(P(x,to - O)dx + "'I II(U - k)~ID(IPdxdT

[xo+Kp] [(Xo,to)+Q(8,p)]
~
+"'1 II(u -k)~(P-1(tdxdT +

[(x o,to)+Q(8,p)]
"'I {J IAt,(T)l
0- 8
i dT} •
PROOF: After a translation we may assume that (x o, to) coincides with the origin
and it will suffice to prove (3.8) for the cube Q (0, p). In the weak formulation (1.5)
take the testing functions
"P = ±(Uh - k)::I::(P

and integrate over (-e, t), t E (-e, 0). Estimating the various terms separately
we have flfSt
Therefore integrating by parts and letting h - 0 with the aid of Lemma 3.2 of
Chap. I,
~ j(u - k)~(P(x,t)dx - ~ j(u - k)~(P(x, -e)dx

Kp Kp
t
- ~I fiu - k)~(p-l(t dxd-r.

-8Kp
In estimating the remaining parts we first let h _ 0 and then use the structure
conditions (AI)-(As). To simplify the notation, set
tE(-e,O).
Then
(3.9) ± j j[a(x, t, u, DU)]h·D ((Uh - k)::I::(P) dxd-r ~

Q'
II a(x, t, u, Du)· [±D(u - k)::I::(P ± p(u - kh(P-1 D(] dxd-r

Q'
~ Co IIID(U - khlP(PdxdT - II "Po(P X [(u - k)::I:: > 0] dxd-r

Q' Q'
- pCI IIID(U - k)::I::lp-l(U - k)::I::(P-IID(ldxd-r

Q'
- p I j "PI (u - k)::I::(P-IID(ldxdT,
Q'
where x( E) denotes the characteristic function of the set E. By Young's inequality

(i) pCI / /ID(U - k)±IP-I(U - k)±(P-IID(ldxdr

Q'
~ ~o //ID(U - k)±IP(Pdxdr + 'Y(Co) //(U - k)~ID(IPdxdr,

Q' Q'
(ii) p / / CPI(U - k)±(,,-IID(ldxdr:5: //(U - k)~ID(IPdxdr

Q' Q'
+1' //cprX[(U-k)± >Ojdxdr.

Q'
Combining this in (3.9) we arrive at
/ /a(x, r, u, Du)·D ((u - k)±(P) dxdr

Q'
~ ~o / /ID(U - k)±(IPdxdr - l' / /(U - k)~ID(IPdxdr

Q' Q'
-1' //(CPo+cp(=r)X[(U-k)± >Ojdxdr.

Q'
Finally
(3.10) / /lb(X, r, u, Du)(u - k)±(Pldxdr

Q'
:5: C2 //ID(U - k)±IP(u - k)±(Pdxdr + / / CP2(U - k)±(Pdxdr.

Q' Q'
Now if we impose on the levels k the restriction
Co
(3.11) esssup I(u - k)±1 :5: 6 ==
0 4C '
Q(9,p) 2
we deduce from (3.10)

(3.10') f flb(x, r, u, Du)(u - k)±(PI dxdr

Qt
:5 ~o fflD(u - k)±IP(Pdxdr + II C{)2(U - k)±(Pdxdr

Qt Qt
:5 ~o !!ID(U -
Qt
k)±(IPdxdr + Do !!
Qt
C{)2X [(u - k)± > 0] dxdr
+')' f I(u - k)~ID(IPdxdr.

Qt
Combining these estimates and recalling that t E ( -6, 0) is arbitrary, we obtain
sup f(u - k);(P(x, t) dx + C2° jr flD(u - k)±(IPdxdr

-9<t<O
Kp
J
Q(9,p)
:5 I (u - k);(P(x, -6)dx
Kp
+ ')' f I(u - k)~ID(IPdxdr + ')' If(u - k);(p-l(t dxdr

Q(9,p) Q(9,p)
+')' fl(C{)o + C{)r +C{)2)x[(u-k)± >Ojdxdr.

Q(9,p)
By HOlder's inequality
11(C{)o+C{)r +C{)2)x[(u-k)± >Ojdxdr

Q(9,p)
1'-1
:5 II". +"r + '1'211.,.,,,, {} A~p (T >I'" '" dT } . .

Therefore recalling the definition (3.5) of the numbers q, r, It, inequality (3.8) fol-
lows.
Remark 3.2. The proof shows that the number Do in (3.11) has to be chosen small
according to the constant C2 • If in (1.1) b(x, t, u, Du) =0, then Do can be taken to
be infinite and no restriction is imposed on the levels k.
Remark 3.3. If the lower order terms C{)i are all zero, then in (3.8) the last term can
be discarded.
3-(ii). Local logarithmic estimates

Introduce the logarithmic function
(3.12) 1/1 (Hr, (u - k)±,c) == In+ {± Hr

Hie - (u - k)± + c
}, 0 < c < Hr,
where Hr is defined in (3.3) via the levels k, and for s > 0
In+ s == max{lns; OJ.

In the cylinder [( x o, to) + Q «(J, p)I we take a cutoff function satisfying (3.1) and
(3.13) ( is independent of t E (to - (J, tol.
PROPOSITION 3.2. Let u be a locally bounded weak solution of (1.1) in th.
There exist constants "( and 60 that can be determined a priori only in terms of the
data. such that for every cylinder (Xo, to) + Q«(J,p) E {h andforevery level k
satisfying (3 J) for 6 S 60
(3.14) sup
to-9<t<to
!
[zo+K,,]
1/12 (Hr, (u - k)±, c) (x, tKP(x) dx
< f 1/12 (Hr,(u-k)±,c) (x,to-(JKP(x)dx

[zo+K,,]
+ 'Y !! 1/111/11£ (Hr, (u - k)±, c) 12 -

[(zo,to)+Q(9,p)]
P IV(I P dxdr
~
+; (1 +In ~t) ll.IAUT)li dT } ·

PROOF: As before, we may take (x o, to) == (0, 0) and will work within the cylin-
der Qt introduced earlier. Also, to simplify the symbolism let us set
(3.15)
In (1.5) take the testing function
I{) = ~h [1/I2(Uh)] (P = [1/12(Uh)]' (P.

By direct calculation
[1/I2(Uh)]" = 2(1 + 1/1)1/112 E Lioc(lh)

which implies that such a I{) is an admissible testing function in (1.5). Since 1/1 (Uh)
vanishes on the set where (Uh - k)±=O,
//!
Qt
1£h [1/J2]' (PdxdT = //!
Qt
1/J2(PdxdT
=/ 1/J2(1£hKPdx - /1/J2(1£hKP dX.

Kpx{t} Kpx{-9}
Therefore letting h -+ 0
//!
Qt
1£h [1/J2]' (PdxdT ----+ /
Kpx{t}
1/1 2 (Ht, (1£ - k)±,c) (Pdx
- / 1/1 2 (Ht, (1£ - k)±, c) (Pdx.

Kpx{ -9}
To estimate the remaining terms we let h -+ 0 first and then make use of the
structure conditions (At}-(As).
/ / a(x,T,1£, D1£)·DIPdxdT
Qt
//(1 +
~ 2Co
Qt
1/J) 1/J,2ID1£I P(PdxdT
-2//(1 +
Qt
1/J) 1/Jf2IPo (x, TKPdxdT
- 2pCl / /ID1£ IP- 11/J1/J'(P- 1ID(ldxdT

Qt
- 2pCl / / 1/J1/J'IPl(X, TKP-1ID(ldxdT.

Qt
From this, by repeated application of Young's inequality
/ /a(x,T,1£, D1£)·DIPdxdT
Qt
//(1 +
~ Co
Qt
1/J)1/J,2ID1£I P(PdxdT
-2//(1 +Qt
1/J) 1/J,2 IPo (x, TKPdxdT
- -y(P) //1/J (1/J,)2- p ID(IPdxdT

Qt
- -y(P) / / 1/J (1/J,)2 IPr (PdxdT.

Qt
For the lower order terms we have

/ /lb(X, T, U, DU)1/J'I/l(PldxdT :5 C2 //IDuIP (1 + 1/J) ,t/P1/J,-l(Pdxdr

Qt Qt
+ //'P21/J1/J'(Pdxdr.
Qt
Next we observe that by virtue of (3.3) and the definition (3.12) and (3.15) of 1/J
1/J'-1 = Ht - (u - k)± + c < 26

and
Therefore by virtue of the choice (3.11) of the levels k we have
/ /lb(x,T, U, Du)1/J1/J'(PldxdT :5 ~o //IDUIP (1 + 1/J) 1/J12(PdxdT

Qt Qt
+ ~ In ( ~t) //1'P2IX [(u - k)± > OJ dxdr.

Qt
Collecting these estimates we arrive at
/ !lI2 (Ht, (u - k)±, c) (Pdx

Kpx{t}
:5 /!lI 2 (Ht, (u - k)±,c) (Pdx + "Y //1/JI1/J'1 2 - PID(I PdxdT

Kpx{ -9} Q(9,p)
+; (1+1n~t) //('Po +'Pr +'P2)X[(u-k)±>OJdxdT

Q(9,p)
where we have used the fact that c < 1. Treating the last integral as before proves
the Proposition.
Remark 3.4. If the constant C2 in (A3) is zero, then we may take 6 =00 and there
is no restriction on the levels k. Also if 'Pi =0, i=O, 1,2, then the last term on the
right hand side of (3.14) can be discarded.
Remark 3.5. In any case, whence the constant 60 has been chosen according to
(3.11). the constant "Y on the right hand side of either (3.8) or (3.14), is independent
o/u.lt is only the levels k that might depend upon the solution u via (3.11).
4. Energy estimates near the boundary 31
4. Energy estimates near the boundary

We assume u is a weak solution of either the Dirichlet problem (2.1) or the Neu-
mann problem (2.7), satisfying in addition
(4.1) uEV>C'J({h) and UEL 2 (0,T;W 1 ,PUh»).
The assumptions (D), (Vo), (N), (N - (i» on the boundary data will be retained.
We will derive energy and logarithmic estimates, similar to those of Propositions
3.1 and 3.2, near the lateral boundary ST as well as at t = O.
Fix a point (x o, to) on ST, and construct the box [(x o, to) + Q (8, p)], where
8 is so small that to - 8>0. In [(xo, to) + Q (8, p)] introduce a piecewise smooth
cutoff function (x, t) - «x, t) satisfying (3.1). We observe that for all t E (t o -
8, to), x-«x, t) vanishes on the boundary of[xo + Kp] and not on the boundary
of [x o + Kp] n n.
Here the interior quantities introduced in the previous section are modified
as follows
(4.2) esssup I(u - k)±1 == D~ $ 6,
[(xo,t o)+Q(9,p)]nnT
where 6 $ 60 and 60 is a parameter chosen according to (3.11). Analogously we
define the logarithmic function
(4.3) !Ii(D~,(U-k)±,C)==ln+{ Dk± - (uD~)

- k ±+
},C<D~' C
and introduce the sets
(4.4) B~/T) == {x E [xo + Kp] n nl (U(X,T) - k)± > O}.

4-(i). Variational boundary data
Let u be a weak solution of (2.7) satisfying (4.1) and assume in addition that
(4.5) on is of class Cl+ A forsome AE (0,1).
PROPOSITION 4.1. There exist constants "( and 60 that can be determined a pri-
ori only in terms of the data and the quantities lIulloo,nT and IIlonllh+).,suchthat
for every (xo, to) E ST./or every cylinder [(xo, to) + Q (8, p)] such that to-O>O
and for every level k satisfying (4.2) for 6 $ min {60 ; I}
(4.6) sup j(u - k)~(P(x, t) dx + ,,(-1 jr f ID(u - k)±(IPdxdT

to-9<t<to
[xo+Kp]nn
J
[(x o,to)+Q(9,p)]nnT
$ f (u - k)'i(p(x, to - 8) dx + "(
[xo+Kp]nn
ff (u - k)~ID(IPdxdT
[(xo,t o)+Q(9,p)]nnT
1!.ll±.cl
+"( jj(U-k)~(P-l(tdxdT+"({lIB~p(T)lidT} r
[(x o ,to)+Q(9,p)]nnT 0- 9
Moreover if ( is independent oft e (to - fJ, to),
(4.7) sup
to-9<t<to
J!li2 (D~, (u - k)±, c) (x, t)(P(x) dx
[zo+Kp)nn
~ J!li 2 (D~, (u - k)±, c) (x, to - fJ)(P(x) dx

[zo+Kp)nn
+ ')' JJ !lil!li
u (D~, (u - k)±,c) 1
2 P
- ID(I P dxdT
[(Zo,to )+Q(9,p»)nnT
~
+; (I +In D!) Ll.IB~p(T>I:dT} ·

where the numbers q, r, It satisfy (3.6)-(3.7).
Remark 4.1. If the Neumann data are homogeneous, i.e., .,pi 0, i = 0, 1, and =
'Pi =0, i=O, 1,2, and b(x, t,u,Du) =0. then we may take 6=00 and the levels
k are not restricted.
Remark 4.2. The proof below is local in nature and it shows that (4.6)-(4.7) hold
true for weak solutions that satisfy the Neumann data on a portion of ST. Accord-
ingly, only such a portion is required to be of class CHA. Also. no reference to
initial data is necessary.
PROOF OF PROPOSITION 4.1: Fix (xo, to) e ST, assume that (xo, to) coincides
. with the origin and work with cubes Kp and cylinders Q(fJ,p). Since an is of
class CHA, for sufficently small p, the portion of an within the cube K p. can be
represented in a local system of coordinates as a portion of the hyperplane x N = 0
and KpnnC {XN >O}. Set
(4.8) K: =Kpn{XN > O} and Q+ (fJ, p) =Q (fJ, p)n{XN > O}.

Without loss of generality we may assume that (2.11) is written in such a coordinate
system. To derive (4.6), in (2.11) we take the testing functions
and let h -+ O. All the terms in (2.11) are treated as in the proof of Proposition 3.1,
except for the boundary integral. We arrive at
(4.9) sup
-8<t<O
I (u - k)~(P(x, t)dx + C2° jr f ID(u - k)±(IPdxdT
J
K: Q+(8,p)
~ I
Kp
(U - k)~(P(x, -9)dx
+1' II k)~ID(IPdxdT II
(u - + l' (u - k)~(p-l(tdXdT
Q+(8,p) Q+(8.p)
+1' II (<,00 + <,Or + <,02)

Q+(8.p)
X [(U - k)± > OJ dxdT
o
+1' II
-8 Kp
¢(X, T, U)(U - k)±(PdxdT.
where Kp =Kpnan is the (N-l)-dimensional cube
We estimate such a boundary integral by transforming it into an interior integral

as follows
IJI
-8K p
.,(x, T, u)(u - k)±C"dXdTI
~ 11(/a:N
p
(,i(x, T, u)(u - k)±(p) dx N) dXdT
~1' II (I~ZN
Q+(8,p)
I(u - k)±(P + 1~IID(u - k)±(I) dxdT
+1' II
Q+(9,p)
(1,z,I(u - k)±(P-1ID(1 + l~ullD(u - k)±I(u - k)±(p) dxdT.
By virtue of assumption (N),
II I~ZNI(u-
Q+(9,p)
k)±(PdxdT ~ Dt II
Q+(9.p)
¢lX[(u-k)± > OjdxdT,
where Dt is defined in (4.2). Also by Young's inequality

II I~I {ID(u - k)±(1 + (u - k)±(P-1ID(I} dxdr

Q+(9.p)
~ ~o II ID(u - k)±(IPdxdr + 'Y(P) II (u - k)'±ID(IPdxdr

Q+(9.p) Q+(9.p)
+ 'Y (p, lIulloo.UT) II (?/Io + ?/Il)~ X [(U - k)± > OJ dxdr.

Q+(9.p)
Finally
II l~uIlD(u - k)±I(u - k)±(Pdxdr

Q+(9.p)
~ 'Y I I I~ul {ID(u - k)±(I(u - k}±(p-l + (U - k)~(P-IID(I} dxdr

Q+(9.p)
~ ~o II ID(u-k)±(IPdxdr+'Y(p,60 ) II(u-k)'±ID(IPdxdr
Q+(9.p) Q+(9.p)
+ 'Y (p, lIulloo.uT) II ?/1ft X [(u - k)± > OJ dxdr.

Q+(9.p)
Combining these estimates implies that the boundary integral on the right hand
side of (4.9) can be estimated by
~o IIID(U - k)±(IPdxdr + 'Y(P, lIulloo.uT) II (u - k)'±ID<lPdxdr

Q+(9.p) Q+(9.p)
+'Y(p,60 ) II (1+?/Io+?/Id~x[(u-k)± >Ojdxdr.

Q+(9.p)
We put this in (4.9) and, to conclude the proof, estimate the integral involving the
functions !Pi, i=O, 1,2, and ?/Ii, i=O, I, as in the proof of Proposition 3.1.
The proof of the logarithmic estimate (4.7) near the lateral boundary ST is
similar to the proof of the interior logarithmic estimate (3.14), modulo the modi-
fications indicated above and we omit the details.
4-(;;). Dirichlet boundary data

Let u be a weak solution of the Dirichlet problem (2.1), which in addition satisfies
(4.1). The assumption (D) on the boundary datum 9 is retained.
Fix (xo,to) EST and consider the cylinder [(xo,t o) + Q(8,p)j, where 8 is
so small that to - 8 > O. Local energy estimates for u near (x o, to) are obtained by
taking, in the weak formulation (2.5), the testing functions
integrating over [(xo, to) + Q (8, p)) and letting h -+ O. Such a choice of testing
functions is admissible if for a.e. t E ( to - 8, to),
(4.10) (u(·, t) - k)± (P(x, t) E W~'P ([xo + Kp) n 0).

Since x-+((x, t) vanishes on the boundary of [x o + Kp) and not on the boundary
of [x o + Kp) n 0, condition (4.10) will be verified iffor a.e. tE (to -8, to)
(u - k)± = 0 in the sense of the traces on a[xo + Kp) n O.

In view of Lemma 3.1 of Chap. I, this can be realised for the function (u - k)+ if
k is chosen to satisfy
(4.11) k ~ sup g.
[(zo ,to )+Q(lI,p »)nST
Analogously the functions -(Uh - k)_(P can be taken as testing functions in (2.5)
if
(4.12)
With these choices of k we may repeat calculations in all analogous to those of

Proposition 3.1 and derive energy inequalities for u near ST. Analogous consider-
ations hold for a version of the logarithmic estimates along the lines of Proposition
3.2. We summarise
PROPOSITION 4.2. There exist constants "( and 60 that can be determined a pri-
ori only in terms ofthe data and such that for every (xo, to) E ST .for every cylinder
[(x o, to) + Q (8, p)) such that to-8 > 0 andfor every level k satisfying (4.2)for
6 ~ 60 and in addition (4.11)for the functions (u - k)+ and (4.12) for (u - k)_ ,
the following inequalities hold:
(4.13) sup
to-lI<t<to
I(u - k)~(P(x, t)dx + ,,(-1
J
Jr
f ID(u - k)±(IPdxdT
[zo+Kp)nn [(zo,to)+Q(II,p»)nnT
~ !(U-k)~(P(x,to-(})dX+"( !!(u-k)~ID(IPdxdT
[zo+Kp)nn [(zo,to)+Q(II,p»)nnT
~
+"( I I (u - k)~(P-l(tdxdT + "(

[(zo,to)+Q(II,p»)nnT
{J 0-6
IBt,p(T)lidT} r
Moreover if the cutofffunction ( is independent of t E (to - 8, to),

(4.14) sup 1!li2 (nt, (u - k)±, c) (x, t)(P(x)dx

to-9<t<to
[zo+Kp]nn
:5 ! !li 2 (nt,(u-k)±,c) (x,to-O)(P(x)dx

[zo+Kp]nn
+')' II !lil!liu (nt,(u-k)±,c) 12 - Pln(I Pdxdr

[(Zo,to)+Q(fJ,p)]nnT
~
+ ~ (1+ In Dn LlIBUT) It dT } ·
where the numbers q, r, It satisfy (3.6)-(3.7).
Local considerations as those in §3 apply to the present case. In particular the
Proposition continues to hold for weak solutions that satisfy the Dirichlet data on
a portion of ST.
4-(iii). Initial data
Consider a weak solution of (1.1) that takes the initial datum U o in the sense that
k!
h
(4.15) u(·, r)dr -+ Uo in L~oc(fl) as h-+O.

o
Thus u could be a solution of either the Dirichlet problem (2.1) or the Neumann
problem (2.1). In either case the assumption (Uo ) is in force.
Fix (x o, to) E flT and consider the cylinder [(xo, to) + Q (0, p)] where 0 is
such that to-O=O. Therefore [(x o, to) + Q (0, p)]lies on the bottom of the cylin-
drical domain flT • Consider a cutoff function ( satisfying (3.1) and in addition
( is independent of E (0, to).
°
Local energy estimates for u near t = are derived by taking in the weak formu-
lation (1.5) testing functions
"': = ±(Uh - k)±(P,

integrating over (0, t), t E (0, to)' and letting h-+O. The fllSt term in (1.5) gives
~ !(uh - k)l(x,t)(Pdx - ~ !(Uh - k)l(x,O)(Pdx.

~+K~ ~+K~
If k is chosen so that k~sUP[zo+Kp] u o , then in view of (4.15) we have

j (Uh(X, 0) - k)! <Pdx - + 0 as h - o.

[xo+Kp]
Also from the definition (3.12) of the function lli(·). it follows that
lli (Dt, (u - k)±, c) =0 whenever (u - k)± = o.
j lli 2 (Dt, (Uh - k)+, c) (x,O)<P(x)dx -+ 0 as h - O.

[xo+Kp]
Analogous considerations hold for (Uh - k) _ <p. We summarise

PROPOSITION 4.3. There exist constants 'Y and 60 that can be determined a pri-
ori only in terms of the data such that for every (xo, to) E {h,for every cylinder
[(xo, to) + Q (8, p)] such that t o-8 = 0 andfor every level k satisfying (4.2)for
6 $ 60 and in addition
(4.16) {k ~ sUP[xo+Kp] Uo for the function (u - k)+

for the function (u - k) _,
k$ inf[xo+K p] U o
the following inequalities hold:
(4.17) sup j(u - k)~(x, t)<P(x) dx + jr flD(u - k)±<IPdxdT

to-9<t<to
[xo+Kp]
J
[(xo,t o )+Q(9,p)]
~
$ 'Y j j (u - k)'±ID<IPdxdT + 'Y {JIB~/T)lidT} r
[(x o ,to)+Q(9,p)]nnT 0
Moreover
(4.18) sup jlli2 (Dt,(u- k)±,c) (x,t)<P(x)dx

to-9<t<to
[xo+Kp]nn
+ 'Y j j llillliu (Dt, (u - k)±, c) 12 - P ID<I P dxdT

[(X o ,to )+Q(9,p)]
li!±.!tl
~ J(1+ In D!) tl.IB~)T)lidT} ·

where the numbers q, r, K, satisfy (3.6)-(3.7).
Remark 4.3. Local considerations apply to this case along the lines of similar
remarks in the previous sections.
38 D. Weak solutions and local energy estimates
Remark 4.4. The constant 'Yon the right hand sides of either (4.13)-(4.14) or
(4.17)-(4.18) is independent ofu. It is only the levels k that might depend upon
the solution u via (3.11). Moreover if "Pi == 0, i = 0, 1, 2, and C 2 = 0, the levels k
are independent of u.
Remark 4.5. We conclude this section by observing that all the energy estimates
as well as logarithmic estimates for (u - k)+ hold true if merely u is a subsolution
of (1.1) and for (u - k)_ if u is a supersolution of (1.1).
5. Restricted structures: the levels k and the constant 'Y
We will make a few remarks on the dependence of the constant 'Y in the energy
and logarithmic estimates and on the restrictions to be placed on the levels k.
5-(i). About the constant 'Y

For the interior estimates of Propositions 3.1 and 3.2, the constant 'Y depends only
upon the data and it is independent of the apriori knowledge of lIulloo,nT. It can
be calculated apriori only in terms of the numbers N, p, r, It, the constants Ci, i =
0,1,2, and the norms
lI"Po, "Pr , "P2114,r; nT·
1be same dependence holds for estimates near the parabolic boundary of nT in
the case of Dirichlet data (see §4-(ii) and §4-(iii».
In the case of variational data, 'Y depends also upon the structure of an (see
§1, Chap. I), and the norms
5-(;;). Restricted structures

The choices (3.11) and (4.2) of 60 impose a restriction on the levels k. Such a re-
striction is needed to handle the lower order terms b( x, t, u, Du) in (1.1). It follows
from (3.1 0) and (3.10)' that the choice (3.11) of 60 permits the absorption of the
term
C2 jjlD(U - k):l:I"(u - k):l:C"dxdr :5 6 C2 j jID(U -

0 k):l:I"C"dxdr
~ q
into the tenns generated by the principal part of the operator in (1.1). Also. the
coefficient of the integral involving "P2 depends only upon the data (i.e., Co, C2 ),
if the levels k are chosen according to (3.11).
5. Restricted structures: the levels k and the constant 'Y 39
Such a choice of ~o impose~ on k to be close to either the supremum or the

infimum ofuin Q(6, p). Thus, in particular, the apriori knowledge of li u lioo,Q(8,p),
is required.
We will introduce structure conditions on (1.1) that yield energy and logarith-
mic estimates analogous to (3.8) and (3.14) for the truncated functions (u - k)±.
with no restriction on the levels k. We will limit ourselves to the interior estimates
of Propositions 3.1 and 3.2.
First, it is obvious from the remarks above that Propositions 3.1 and 3.2 con-
tinue to hold for all the levels k if b(x, t, u, Du) == O. A more general condition
is
(A 3)
where
(A~)
and q, r satisfy (As) and (As-i)-(As-iii). The structure condition (A3) implies
(A3)' The source term '{J2 is required to be more integrable than the corresponding
source tenn in (A3).
Let us consider local weak solutions u of (1.1) with the structure conditions
(Ad, (A2), (A3), (A4), (A~), (As) and (As-i)-(As-iii). We do not require
that u be locally bounded. To derive local energy and logarithmic estimates for
u we proceed as in the proof of Propositions 3.1 and 3.2. The lower order tenns in
(3.10) are now estimated by repeated use of the Young's inequality as follows.
! !lb(X, r, u, Du)(u - k)±("ldxdr

Q&
~ ~o ! !ID(U - k)±(I"dxdr + 'Y

Q&
!!
Q&
'{J2(U - k)±("dxdr
+ 'Y !!(u -
Qt
k)~ max {ID(I" ; ("} dxdr.
We conclude that, for such solutions, inequalities (3.8)-(3.14) hold true for every
level k, with a constant 'Y independent of u, provided the integral
! j(U - k)~ID(I"dxdr
[(x o ,t o )+Q(8.p)]
in (3.8) is replaced by
j j(U - k)~ max {ID(I"j ("} dxdr

[(xo ,t o )+Q(8,p)]
and the integral

JJ!lil!li (Ht, (u - k)±, c) 1-"ID(I"dxdr

u
2
[(xo,to )+Q(9,p))
in (3.14) is replaced by
JJ!lil!li (Ht, (u - k)±, c) 1

u
2 -" max {ID(I"; ("} dxdr.
[(x o ,to )+Q(9,p))
When p = 2, assumptions (A d-( As) are optimal to obtain a HOlder modulus of

continuity for the solutions (see [67]).
The weak fonnulation of local and global weak sub(super)-solutions is stan-
dard and we refer for example to [67,73].
When 1 < p < 2, it seems more suitable to work with cubes of the type of
K p rather than balls. For this reason we have introduced a unified geometry. The
notation [xo + Kp] to denote a cube about Xo is introduced in Krylov-Safonov
(64).
The idea of deriving energy inequalities for the truncated functions (u - k) ±
seems to appear fust in Bernstein (12), in a global way, i.e., with the integrals
extended to the whole nT • A local version of such estimates by use of local cutoff
functions was introduced in the celebrated paper of DeGiorgi (33). Since then they
have been widely used eSJ)C':ially in the russian literature (see for example (67) and
references therein).
Logarithmic estimates seem to be crucial in the study of the local behaviour of
solutions of elliptic and parabolic equations in divergence fonn. For this we refer to
Kruzkov [60,61,62), Moser [81,82,83] and Senin (92). The logarithmic function
in (3.12) has been introduced in (35) and is now a standard tool in studying the -
local behaviour of degenerate and singular p.d.e. 's.
III
Holder continuity of solutions of
degenerate parabolic equations
1. The regularity theorem
Consider solutions u of (1.1) or of Chap. II for the case P > 2. The equation is
degenerate since the modulus of ellipticity vanishes when IDul =O. We will prove
that if u E L~(nT), then it is HOlder continuous within its domain of defini-
tion. It will shown in Chap. V that local weak: solutions of such degenerate equa-
tions are indeed locally bounded. To simplify the presentation we will assume that
u E L 00 (nT ). If u is only locally bounded, it will suffice to work: within a fixed
compact subset of nT. In the theorems below, the statement that a constant 'Y de-
pends upon the data means that it can be determined a priori only in terms of the
norm II ull oo,nT , the constants Ci , i=O, 1,2, and the norms liepa, cpr ,cp2I1ti,T;nT
appearing in the structure conditions (A 1 )-( A3)' We let /C denote a compact sub-
set of nT and let p - dist (/C j r) be the intrinsic parabolic distance from /C to the
parabolic boundary of nT, i.e.,
(1.1) p - dist (/Cj rjp) == (.,.t)elC

inf
(1I •• )er
( ~n It - sll/p .
Ix - yl + lIull~' T
)
l-(i). Interior HOlder continuity
THEOREM 1.1. Let u be a bounded local weak solution of (1.1) of Chap. 1/ in

n T . Then (x, t) - u(x, t) is locally Holder continuous in nT. Moreover there
42 lll. HOlder continuity of solutions of degenerate parabolic equations
/or every pair o/points (x}, h), (X2' t2) EK-.I/thelowerordertermsb(x, t, u, Du)
satisfy (A~) 0/§5 o/Chap. II. then 'Y and a are independent o/liulloo.nT.
1-(ii). Boundary regularity (Dirichlet data)
THEOREM 1.2. Let u be a bounded weak solution 0/ the Dirichlet problem (2.1 )
o/Chap. II and let (D) and (Uo ) hold. The boundary an is assumed to satisfy
the propertyo/positive geometric density (1./) o/Chap.l. Then uEC caT)' and
there exists a continuous positive non-decreasing/unction s -. w( s) : R + -. R + •
such that
IU(XI' tt) - U(X2' t2)j :5 W (ixi - x21 + It I - t21 i ) ,
/orevery pair o/points (Xl, tt), (X2' t2) E nT.lnparticular if the boundary datum
9 is Holder continuous in ST with exponent say a g • and if the initial datum U o is
Holder continuous in a with exponent say a uo ' then u is Holder continuous in aT
aiad there exist constants 'Y> 1 and a E (0, 1) such that
lu(xt, tt} - U(X2' t2)1 :5 'Yllulloo.nT~XI - x21 + lIulI:f.nT It I - t211/Pr '
for every pairo/points (Xl, tt}, (X2' t2) E aT.

The constants 'Y and a depend only upon the data. Moreover the constant a
depends also upon the HOlder exponents a g , Quo 0/ 9 and U o respectively.
I/the lower order terms b(x, t, u, Du) satisfy (A~) 0/§5 o/Chap.lI. then 'Y
and a are independent 0/ lIulloo.nT.
Even though we have stated the Theorem in a global way the proof has a
local thrust. For example the boundary datum 9 could be continuous or HOlder
continuous only on a open portion of ST (open in the relative topology of ST), say
E. Then the solution u of the Dirichlet problem would be continuous (respectively
HOlder continuous) up to every compact subset of E.
Analogous remarks hold in the case U o is only locally continuous or locally
HOlder continuous. In particular to establish the continuity (HOlder continuity re-
spectively) of u up to nx {O}, no reference is needed to the Dirichlet problem or
any boundary value problem.
2. Preliminaries 43
1-0;;). Boundary regularity (Variational data)

To stress such a locality we state our next theorem as if no information were avail-
able on the initial datum uO •
THEOREM 1.3. Let u be a weak solution of the Neumann problem (2.7) of
Chap. II, satisfying
u E LaO (fi X [E, TJ) , EE (0, T).
Assume that an is of class Cl,~ and let (N) and (N - i) hold. Then u is Holder
continuous in n x [Eb T],for all E < El < T, and there exist constants "y and Q
such that
IU(Xb td - U(X2, t2) I

~ 'Yllulloo,iix[€,T] (IXl - x21 + lIull.:rox(€,T)ltl - t211/P) Q ,
for every pair of points (Xl, tl), (X2, t2) E n x [El, T]. The constants 'Y > 1 and
Q depend only upon E, lIulloo,iix(€,T] and the data, including the structure of an
and the norms IItPl, tPrr IIq,r;UT appearing in (N - i). In addition the constant
'Y depends upon the distance (El - E).
If the Neumann data are homogeneous, i.e., if tPo == tPl == 0, and if in addition
the lower order terms b(x, t, u, Du) satisfy (Aa) of§5 of Chap. II, then 'Y and Q
are independentofllulloo,iix(€,T]'
Remark 1.1. The continuity of u can be claimed up to t = 0 provided (Uo ) of
Chap. II holds. Also, if U o is HOlder continuous in n then u is HOlder continuous
innT.
2. Preliminaries
The HOlder continuity of u, either in the interior of nT or at the parabolic boundary,
will be, heuristically, a consequence of the following fact. The function (x, t) -+
u( x, t) can be modified in a set of measure zero to yield a continuous representative
out of the equivalence class u E Vj!:(nT ), if for every (xo, to) E nT there exist
a family of nested and shrinking cylinders [(xo, to) + Q (6n , Pn)] with the same
vertex such that the essential oscillation Wn of u in [(xo, to) + Q (6n, Pn)] tends
to zero as n -+ 00 in a way quantitatively determined by the structure conditions
(AD!-(A6).
The key idea of the proof is to work with cylinders whose dimensions are
suitably rescaled to reflect the degeneracy exhibited by the equation. To make this
precise, fix (xo, to) E nT and construct the cylinder
44 m. ltilder continuity of solutions of degenerate parabolic equations
where E is a small positive number to be determined later. After a translation we
=
may assume that (xo, to) (0, 0). Set
p.+ = esssup u, p.- = ess inf u, W= essosc =p. + -p. -
Q(RP-',2R) Q(RP-',2R) Q(RP-e,2R)
and construct the cylinder
(2.1) 2. = (~)"-2
ao A
where A is a constant to be determined later only in terms of the data. We will
assume that
(2.2) (W)"-2
A > [lE.
This implies the inclusion
(2.3)
and the inequality

essosc u < w.
Q(aoRP,R) -
By cylinders rescaled to rejlectthe degeneracy, we mean boxes of the type
(2.1) where the length has been suitably stretched to accommodate the degeneracy.
If p = 2, these are the standard parabolic boxes reflecting the natural homogeneity
of the space and time variables.
3. The main proposition
PROPOSITION 3.1. There exist constants Eo, " E (0, 1) and C, A > 1, that can
be determined a priori depending only upon the data, satisfying the following.
Construct the sequences
Ro=R, wo=w
andfor n= 1, 2, ... ,
R". = C-nR,
Construct also the family of cylinders
~ __ (Wn ),,-2, n=0,1,2, ....
an A
Thenfor all n=O, 1,2, ...
Q(n+1) c Q(n) and
A consequence of this Proposition is:

3. The main proposition 45
LEMMA 3.1. There exist constants 'Y > 1 and 0 E (0, 1) that can be determined a
priori only in terms of the data. such that for all the cylinders
o < p 5: R, Q (aopp, p) , ~ = (~)P-2,

ao A
essosc u 5: 'Y (w + ,neo)

Q(aop",p)
(-RP)Q .
PROOF: From the iterative construction of Wn it follows that Wn+1 5: 7JWn +
C R~o and by iteration
We may assume without loss of generality that eo is so small that" 5: C-Eo. Then
Let now 0 < P 5: R be fixed. There exists a non-negative integer n such that
c-(n+1) R 5: P 5: C- n R.
This implies the inequalities
Therefore
o=mm . { ;"2eo} .
0 1
To conclude the proof we observe that since Wn 5: w. the cylinder Q (aopp , p) is

included in Q(n) =Q (an~' Rn). so that
essosc u5:wn.
Q(aop",p)
Statements of HOlder continuity over a compact set now follow by a standard cov-
ering argument.
46 ill. HOlder continuity of solutions of degenerate parabolic equations
Remark 3.1. The proof of Proposition 3.1 will show that indeed it is sufficient to
work with the number w and the cylinder Q (aoRP, R) linked by
(3.1) essosc u < w.

Q(aoRp,R) -
This fact is in general not verifiable, for a given box, since its dimensions would
have to be intrinsically defined in terms of the essential oscillation of u within it.
Therefore the role of having introduced the cylinder Q (RP-E, 2R) and hav-
ing assumed (2.2) is that (3.1) holds true for the constructed box Q (aoRP, R). It
will be part of the proof of Pf<)position 3.1 to show that at each step the cylinders
Q(n) and the essential oscillation of u within them satisfy the intrinsic geometry
dictated by (3.1).
To begin the proof, inside Q (aoRP , R) consider subcylinders of smaller size
constructed as follows. The number w being fixed, let So be the smallest positive
integer such that
w
(3.2) -28 0 <6
_ 0,
where the number 60 is introduced in (3.11) of Chap. II in the derivation of the

local energy estimates. Then construct cylinders
(3.3) [(0, t) + Q (dRP, R)] ,
Figure 3.1
These are contained inside Q (aoRP, R) if the number A is chosen larger that
280 and if t ranges over
_ {AP-2 _ (280 }P-2} RP < t < o.

wp - 2
The structure of the proof is based on studying separately two cases. Either
we can find a cylinder of the type of [(0, l) + Q (dRP, R)] where u is mostly large,
or such a cylinder cannot be found. In either case the conclusion is that the essential
oscillation of u in a smaller cylinder about (xo, to) decreases in a way that can be
quantitatively measured. In the arguments to follow we assume (2.2) is in force
and determine later the numbers A, e and eo.
Remark 3.2. For later use we estimate the quantity
w ) -2 1!1!±.cl
G(w, R} == 'YRNIt ( 28 0 dr,
where 'Y is a constant depending only upon the data and K. is defined in (3.5) of
Chap. II. From the defu\ition of din (3.3) it follows that
(3.4) G(w , R} _ A I RNltw- b,

< where b = 2 + (p _ 2) p(1 + K.}
r
and
Al = A2+(P-2)P(1;,,) •
Along the proof we will encounter quantities of the type AiRNltw-b, i =
1,2, ... ,t, where Ai are constants that can be determined a priori only in terms of
the data and are independent of w and R. We may assume without loss of generality
that they satisfy
(3.5)
Indeed if not, we would have w:5 C Reo for the choices
C = max A~/b and

l~i9 t
and the first iterative step of the Proposition would be trivial.

Remark 3.3. The proof below and (2.2) show that the numbers e and eo can be
taken as
NK.
eo = b' e = (p- 2}eo.
In the estimates to follow we denote with 'Y a generic positive constant that
can be calculated a priori depending only upon the data and that may be different
in different contexts.
48 ill. HOlder continuity of solutions of degenerate parabolic equations
3-(i). About the dependence on lIulloo.nT

We will use the energy and logarithmic estimates of Propositions 3.1 and 3.2
of Chap. II for the truncated functions (u - k)± over cylinders contained in
Q (aoRP , R). When working with (u - k) _ we will use the levels
for some i ~ o.
These levels are admissible since
lI(u - k)-lIoo.Q(IJoRP.R) ::;; 60 •

When working with (u - k) + we will take levels
for some i ~ o.
These are also admissible since
II (u - k)+ lloo.Q(lJoRP.R) ::;; 60 •

Let us fix 60 as in (3.11) of Chap. II. Then, sincew::;;2I1ull oo .nT' (3.2) holds
true if we choose So so large that
280 = 8~2I1ulloo.nT.
Having chosen So this way. (3.2) is verified when working within any subdomain of
DT • The a priori knowledge of the norm lIulloo.nT is required through the number
So. If the lower order terms b(x, t, u, Du) in (1.1) satisfy (A~) of §5 of Chap. II,
then. as remarked there. the energy and logarithmic inequalities hold true for the
truncated functions (u - k)± with no restriction on the levels k. Thus in such a
case So can be taken to be one and no a priori knowledge of lIulloo.nT is needed.
The numbers A and Ai introduced in (3.5) will be chosen to be larger than
280 • In the proof below we will choose them of the type
and .-
A . - 2;0+h, , i = 0,1,2, ... ,
where hi ~ 0 will be independent of lIulloo.nT. We have just remarked that if the
lower order terms b(x, t, u, Du) satisfy (A~) of§5 of Chap. II, then So can be taken
to be one. We conclude that for equations with such a structure, the numbers Ai
can be determined a priori only in terms of the data and independent of the norm
lIulloo.nT·
4. The first alternative 49
4. The first alternative

LEMMA 4.1. There exists a number Vo E (0, 1) independent 0/ w, R, A such that
if/or some cylinder of the type [(0, f) + Q (dRP, R)]
I(X, t) E [(0, f) + Q (dRP, R)] lu(x, t} < IL- + 2~0 15 volQ (dRP, R) I,
then
(4.1) u(x,t) > IL- + 2S~+1 a.e. (x,t) E [(O,f) +Q(d(f)",f)]·
PROOF: Fix a cylinder for which the assumption of the lemma holds. Up to a
=
translation we may assume that (0, f) (0, O), and we may work within cylinders
Q(dPP,p) , O<p~R. Let
R R
Rn ="2 + 2n+1' n = 0,1,2, ... ,
construct the family of nested cylinders Q (dR~, Rn) and let (n be a piecewise
smooth cutoff function in Q (dR~, Rn) such that
(4.2)
We will use the energy inequalities of Proposition 3.1 of Chap II. written over the
cylinders Q (d~, Rn), for the functions (u - kn )-. where forn=O, 1,2, ... ,
k = -
n IL +~ W
280+1 + 280+1+n '
In this setting. (3.8) of Chap. II takes the form
(4.3) esssup J(u - k n ): (!(x, t)dx + jr flD (u - knL (nlPdxd1'

-dR~<t<O J ,
KRn Q(dR~,Rn)
57: {JJ(U-kn)~dXd1'+~
Q(dR~,Rn)
!!(U-kn):dXd1'}
Q(dR~,Rn)
,,(1+,,)
+7 { J
-dR~
IAkn,Rn (1')1 i d1'} r
We will show that as n -+ 00

50 DI. HOlder continuity of solutions of degenerate parabolic equations
Ilx[(U-knL >O]dxdr-+O.
Q(dR!:.Rn)
Since kn '\. koo = J,L- + 2.::'+1' this would imply that
thereby proving the lemma. We observe that
and estimate above the first two tenns on the right hand side of (4.3) by
7 ~: (2: )2 (2:
0 0 r- 2
II X[(u - knL > 0] dxdr
Q(d~.Rn)
+7: (2:J ~ II X [(u - knL > 0] dxdr

2
Q(dR~,R,,)
:57: (2:J" Ilx[(U-knL >O]dxdr,

Q(d~.R,,)
where we have used the defmition (3.3) of d. Combining these remarks in (4.3)
and dividing through by d, we arrive at
(4.4) esssup I(u - kn)~ (:(x, t) dx + -d1 I liD (u - knL (nl"dxdr

-dR~<t<O 11
KRn Q(d~.R,,)
:57~: (2:J" ~ Ilx[(U-knL >O]dxdr

Q(d~.Rn)
( W ),,-2 d,.
~ 1 I 0 }~
~dR~IAkn.R,,(r)l·dr
1:
{
+7 2 Bo
In (4.4) we introduce the change of time-variable z = tid which transfonns

Q (d~, Rn)into
Qn == Q(~,Rn) == KR" x{-~,O}.
Setting also
v(·,z) = u(·,zd) and (n(-,z) = (n(·,zd),
the inequality (4.4) can be written more concisely as
4. The first alternative 51
where we have set

o
and IAnl = J IAn(z)ldz.
-Rl:
Since (11 - knL <n vanishes on the lateral boundary of Qn, by Corollary 3.1 of
Chap. I we have
(4.5) II (11 - knL 1I:,Q,,+1 $11 (11 - knL <nll:,Qn

$ II (v - ~ P
knL 'nllp~,Qn IAn INT.;P
$11 (v - knL <nlltP(Qn)IAnlNT.; .
The left hand side of (4.5) is estimated below by
11(11 - knL 1I:,Qn+l ~ Ikn - kn+1 IPIAn+1I ~ 2P(~+2) (;.'J PIAn+1l·

Combining these estimates gives
IA 11+~
(4.6) IAn+d ~ -y4np nRP
+ 14-' (2~' r' dol':"' lA_I'm LIA,,( t Ii

z) dz }
Ell±!!.l.
Divide by IQn+11 and introduce the quantities
Using also the fact that. by virtue of Remark 3.2 and (3.5)
RNIC (.!!!...)
2 80
-2 dP(1:IC) <1
- ,
we obtain from (4.6) in dimensionless fonn

52 m. H6lder continuity of solutions of degenerate parabolic equations
y.n+ 1 <
_I"'4np {y'l+~
n + Y.~Zl+"}
n n , 'tin = 0,1,2, ....
Next by the .embedding of Proposition 3.3 of Chap. I
Therefore
Z n+l < np {y. + Zl+"}

_ ",4
I nn , 'tin = 0, 1,2, ....
From Lemma 4.2 of Chap. I it follows that Yn and Zn tend to zero as n -+ 00,
provided
where 80 =min { N;"p; It}.
5. The frrst alternative continued
Suppose the assumptions of Lemma 4.1 are verified for some box of the type
[(0, l) + Q (dRP, R)]. We will exploit the fact that at the time level
-8 =t - (2BoW)2-P (R)P
"2'
the functionx-+u(x, -8) is strictly above the level ",- + 2.::'+1' in the cube K R / 2 •
To simplify the symbolism let us set p = ~ and construct the cylinder
Q(8,p) == Kpx(-8,0), p= R/2.
The length 8 of such a cylinder satisfies
(5.1) (-2-W)2-P < -pP8 -< (W)2-P

0 -
-A 2 P•
The next lemma asserts that, owing to (4.1), the set where u(·, t) is close to I.e ,
within the smaller cube KR/4' can be made arbitrarily small for all time levels
-8~t~0.
S. The fllSt alternative continued S3
LEMMA 5.1. For every number Vi E (0,1), there exists a positive integer 81.
depending only upon the data and independent of w, R, such that
'lit E (-9,0).
PROOF: Consider the logarithmic estimates of Proposition 3.2 of Chap. II, written
over the cylinder Q (0, p), for
(u-k)_,
As a number c in the definition of W, we take
W
c-~"':"":'"":"
- 2s o+ i + n '
n> 1,
where n is to be chosen. Thus we take
where
Hi; = esssup
Q(fJ.p)
(u - (JL- + 2 W+ i ) )
So _
::; 2 w+ i
80.
.
For t= -0, by virtue of(4.1) we have (u - (JL- + 2.~+dL =0, and therefore
W(x, -0) = 0,
These remarks in (3.14) of Chap. II yield
(5.2) Iw
Kp
2(x, t)(P(x)dx ::; ; II W!Wu !2- Pdxdr
Q(fJ.p)
n(lIA',p(T) dT) · ,
.ell±!!l
+ ~(2":'+' r' ~ + In H, (2<.:,+0> Ii

where A;.i·) is defined in (3.4) of Chap. II and x -+ (x) is a piecewise smooth
cutoff function in Kp that equals one on Kp/2 and such that ID(I::; 4/ p. Next
W::;ln( ~ ) =nln2
~
and
!WU !2-P = !Hk_- (u - k)_
W n !P-2 ::; (W
+ 2so +l+ 2 80
)P-2
•
Therefore, in also view of (5.1), the first term on the right hand side of (5.2) is
estimated above by
54 m. Itilder continuity of solutions of degenerate parabolic equations
; ffl[lll[lul2-PdXdT S 'Yn; (2~JP-2IKp/21 S 'YnAP-2IKp/21,

Q(9,p)
where l' and A are constants depending only upon the data. and A has to be deter-
mined later. The second tenn is estimated by using the conditions (3.6). (3.7) of
Chap. II. linking the parameters r, q, K.. This gives
n
Ei!±cl
~(2'.:,+or'(l + In H; (2•.:'+0 (lIA'-"<T)I~ dT) .

w
S'Y n ( 2so +1+n
)-2 (W)-Ei!±cl(P-2)
A r R NIt IKp / 2 1.
The number n will be detennined shortly. depending only upon the data and inde-
pendent of w, p. Therefore by virtue of Remark 3.2 and (3.5) we may estimate
( w )-2 (~)_P(1:")(P_2) Nit <

n 2Bo +l+ n A R _ 1.
Combining these remarks into (5.2) yields
(5.3) f 1[12(x,t)dx S 'YnAP- 2 Kp/2 I I,

Kp/2
where we have used the fact that (== 1 on K p/2. The integral in (5.3) is estimated
below by extending the integration to the smaller set
{x E K p/ 2 1 u(x, t) < ,.,.- + 2Bo~l+n} , t E (-0,0).
On such a set
and since the right hand side of this inequality is a decreasing function of Hi; • we
have
1[12 ~ In2( 20:+1 ) = (n _ 1)21n2 2.
2°o+ n
Putting this into (5.3) gives that for all t E (-0,0)
(5.4) Ix E K p / 2 Iu (x, t) <,.,.- + 2Bo~l+n IS 'Y AP- (n ~ 1)2I Kp/21·

2
To prove the lemma we have only to choose n sufficiently large.

6. The fmt alternative concluded 55
6. The first alternative concluded
The infonnation in Lemma 5.1 will imply that u is strictly bounded away from p.-
in a smaller cylinder. To make this precise. consider the box
Q(O,~) =Kp/2X(-0,0), P = R/2,
where (J satisfies the bounds in (5.1).

LEMMA 6.1. The numbers V1 E (0,1) and 81 » 1 can be chosen a priori depen-
dent only upon the data and independent of w and R. so that
w
u(x, t) > p.- +2 81 +1' a.e. (x, t) E Q ((J,~) .
PROOF: We will use the local energy estimates of Proposition 3.1 of Chap. II in
the following setting. Let
n = 0,1,2, ... ,
construct the cylinders Q ((J, Pn) and let x -+ (n (x) be a piecewise smooth cutoff
function in KPn that equals one on KPn+1 and such that ID(n 1:$ 2n+3 / p. Write the
inequalities (3.8) of Chap. II over Q (0, Pn) for the functions (u - kn )_. where
w w
kn = p.- + 2 81 + 1 + 2s l+ Hn '
and observe that. owing to the conclusion of Lemma 4.1.
(u - knL (x, -0) = 0, IrIx E KPn, n = 0,1,2, ....

With these choices. the energy inequalities yield
(6.1) sup
-8<t<0
j(u - kn)~ '~dx + ,),-1
JJ
r riD (u - knL (niP dxdr
KPn Q(8,Pn)
£lli!!.!.
::; ')'~P jj(u - kn)~ dxdr +"( {JIAkn,pJr)lidr} r
Q(8,Pn) -8
The first tenn on the left hand side is estimated below. for all t E (-0,0). by
j(u - kn)~ '~dx ~ (2~1) 2-p j(U - kn)~ '~dx

KPn KPn
> 2-P (2 )P-2 ~ j(u - k )P I"Pdx > ~ j(u - k )P I"Pdx
81
- A pP n - '>n - n - '>n , pP
KPn KPn
56 DI. HOlder continuity of solutions of degenerate parabolic equations
if 81 is chosen so large as to satisfy the conclusion of Lemma 5.1 and the inequality
2- P (281 I A)p-2 ~ 1. We put this in (6.1), divide through by BI pP and introduce
in the cylinders Q (B, Pn), the change of variable z = tPP lB. This maps Q (B, Pn)
into the boxes
Qn == KPn x(-PP,O).
Let us also set v(x, z) = u(x, zBI PP) and
o
IAnl = jIAn(z)ldZ.
-pP
By the embedding of Corollary 3.1 of Chap. I,
2-(n+2)1' (2~1 )"IAn+l1 ~ jj(v - kn)~ dxdz

Qn+1n[v<kn+11
~ jj(v - kn)~ (~dxdz

Qn
~ 111 (v - knL (nll~p(Q,,)IAnlwT,;

< 1 2n1' (~)p IA Il+~
- pP 28 1 n
I!i!±!!l
+~(~t"¥" 1A.lm {}An(Z>li dz } •
I
Divide through by the coefficient of An+ 11 and set
Using also (5.1) and (3.5) to estimate

6. The first alternative concluded 57
we arrive at
Yn+1 :5 -y4np {y~+m:;; + Yn;vT,; Z~+/( } .
Proceeding as in the proof of Lemma 4.1, we have
Zn+1 :5 -y4 np {Yn + Z~+/(}.
By Lemma 4.2 of Chap. I it follows that Yn and Zn tend to zero as n -+ 00, provided
(6.2)
where 90 = min {NT,; K}.

j To prove the lemma, we fix III as in (6.2) and pick
81according to Lemma 5.2.
We summarise the results obtained so far.
PROPOSITION 6.1. There exists numbers 110 , TJo E (0, 1) and Al » 1 depending
only upon the data and indepeiuJent o/w, R, such that if/or some cylinder o/the
type [(0, l) + Q (dRP, R»),
(6.3) I(X, t) E [(0, l) + Q (dRP, R)llu(x, t) < jJ- + 2~o I

:5 1I0 IQ(dRP,R) I,
then either
(6.4)
or
(6.5) essosc u:5 TJo W
Q(d( I)".f)
where b is introduced in (3.4).
PROOF: Assume (6.4) is violated. By Lemma 6.1, we can determine a positive
number 81 such that
essm.f u > jJ- +-- W
Q( 8,-f) - 281 +1
where 9 satisfies (5.1) with p = R/2. Change the sign of this inequality and add
the quantity esssuPQ(8.i) u to the left hand side and jJ+ to the right hand side.
This gives
ess OBC U :5
Q(8.f)
(1 - 21+1) w.
81
Therefore the proposition follows with TJo = (1 - 2.:+1), since

58 m. IIUder continuity of solutions of degenerate parabolic equations
Remark 6.1. Let us trace the dependence of "10 and Al upon lIulloo,nT. The num-
bers 110 and III depend only upon the data and are independent of u. The number
81 is given by 81 = 8 0 + n where n is chosen from (5.4). Thus
depends upon lIulloo,nT via 8 0 through (3.2). Also A and Al are of the type Ai =
2so +h ,. where hi, i = 0, 1,2, ... , can be determined a priori only in terms of the
data and are independent of II u II OO,nT. We conclude that if the lower order terms
b(x, t, u, Du) satisfy the structure condition (Aa) of§5 of Chap. II. we have 8 0 = 1
and therefore "10' A, Ai can be determined a priori only in terms of the data and
are independent of lIulloo,nT.
7. The second alternative

We assume in this section that the assumptions of Lemma 4.1 are violated. i.e. for
every subcylinder [(0, t) + Q (dRP, R)] cQ (aom', R)
I(X,t) E [(O,t) +Q(dm',R)]lu(x,t) < p.- + 2~o I> 1I0lQ (dRP, R) I·

Since
we rewrite this as
(7.1) I(X, t) E [(0, t) + Q (dRP, R)lIu(x, t) > p.+ - 2~o I

~ (1- 110) IQ(dRP,R) I,
valid/or all cylinders
[(0, t) + Q (dRP, R)] c Q (aoRP, R) ~ = (~)P-2.

ao A
In view of (7.1) we will study the behaviour of u near its supremum p. + and will
be working with the truncated functions (u - k) + for the levels
i ~ 0.
LEMMA 7.1. Let [(0, t) + Q (dRP, R)] cQ (aom', R) befixedand let (7.1) hold.
There exists a time level
t· E [f - dRP , f - 110 dRP]
2 '
such that
7. Tho second alternative 59
PROOF: If not, for aU tE [i - dJlP, i - ~dJlP],
Ix E KR I u(x, t) > p.+ - 2~o I> (11_-v:i2) IKRI

and
I(X,t) E [(O,t) + Q(dJlP,R)] I u(x,t) > p.+ - 2~o I

t-!f-dRP
~ f
t-dRP
Ix E KR Iu(x, r) > p.+ - 2~o Idr
> (l-vo ) IQ(dR"',R) I,
contradicting (7.1).
The lemma asserts that at some time level t* the set where u is close to its
supremum occupies only a portion of the cube K R. The next lemma claims that this
indeed occurs for all time levels near the top of the cylinder [(0, t) + Q (dJlP, R)].
LEMMA 7.2. There exists a positive integer 82 > 80 such that
Ix E I
KR u(x,t) > p.+ - 2~21 :5 (1- (~f) IKRI,
fora/ltE [i - ~dJlP,fj.
PROOF: Consider the logarithmic inequalities (3.14) of Chap. II written over the
box K R X (t* , t) for the function (u - k) + for the levels k =p. + - 2":0 . As for the
number c in the definition of the function 1[1, we take
c- -2-
w
-
Bo+n'
n °
> to be chosen.
Thus we take
(7.2)
where
Ht == ess sup (u - ( p.+ - -W ) ) .
[(O,f)+Q(dRP ,R)] 280 +
The cutoff function x-+(x) is taken so that ( = 1 in the cube K(l-O')R, uE (0, I),
and ID(I:5 (UR)-l. With these choices, inequality (3.14) of Chap. II yields for all
tE(t*,t)
60 m. HiUder continuity of solutions of degenerate parabolic equations
t
(7.3) I
K(l-,,)R
p 2(X,t)dx:5 j llf2(x,to<)dx+ «(1~)P j j llf lllful2- PdxdT
KR tOKR
E1.!±.!!.l
+0 (2'~+') -2 [1 + I. Hi C.~"") -1] {}At(Tl11dT} ·

The various tenns in (7.3) are estimated as follows. First
llf:5nln2; Illfu 2P :5 2P (2"0W)P-2 ; [l+lnHt(2"0+n

l- W )-1] :5-ynln2.
Next, from (7.2) it follows that llf vanishes on the set [u<JL+ - 2":0]. Therefore,
using Lemma 7.1, the first integral on the right hand side of (7.3) is estimated
above by
The second integral is estimated by

t
«(1~)P ff
tOKR
llflllfuI2-PdxdT:5 ;pnIKRI,
since f - to< :5dRP, and d is given by (3.3). Finally for the last tenn, we have
E1.!±.!!.l
o (2'~+.r' [1+ I.Ht (2.:'+.r'] {jIAt(Tl11dr} ·

:5 -ynA2w- bRNItIKRI,
where A2 = 2(B o +n)b and b is defmed in (3.4). By Remark 3.2 and (3.5) we may
assume that nA2W-b RNIt $1. Combining these remarks in (7.3) we conclu(,ie that
for all tE (to<, f)
(7.4) j 22
llf2(x, t) dx :5 n 1n 2 (11_-II:i2) IKRI + ;P nIKRI·
K(1-,,)R
The left hand side of (7.4) is estimated below by integrating over the smaller set
{x E K(1-<T)Rlu(x, t) > 1'+ - 2B~+n } .
On such a set, since the function llf in (7.2) is a decreasing function of Ht, we
estimate
7. The second alternative 61
After carrying this in (7.4) and dividing through by (n - 1)21n2 2 we obtain
Ix E K(l-u)R I u(x, t) > J1.+ - 211~+n I
On the other hand

~ (n: lr (/--V:/2) IKRI + u;n IKR1 ·
Ix E KR I u(x,t) > J1.+ - 28~+n I

~ Ix E K(l-u)R Iu(x, t) > J1.+ - 2s~+n 1+ IKR\K(l-u)RI
~ Ix E K(l-u)R I u(x, t) > J1.+ - 2B~+n 1+ NuIKRI·
Therefore
Ix E KR Iu(x, t) > J1.+ - 2s~+n I

~ [(n: 1) (/--v:i2) + u;n + NU]IKRI,
2
for all t E (t* , f). Choose u so small that u N ~ ~ v~ and then n so large that
Then for such a choice of n the lemma follows with 82 =80 + n.

Remark 7.1. Since the number Vo is independent of w and R, also 82 is indepen-
dent of these parameters. The number A that determines the length of Q (aoRP, R)
is still to be chosen. We will determine it later independent of w and R and subject
to the condition A > 282 •
Since (7.1) holds for all cylinders of the type [(0, f) + Q (dRP, R)], the con-
clusion of Lemma 7.2 holds true for all time levels satisfying
where ao and d are defined in (2.1) and (3.3) respectively. If the number A is
chosen sufficiently large, we deduce
62 lli. HOlder continuity of solutions of degenerate panboIk equations
COROLLARY 7.1. ForalltE(-!fR",O),
From now on we will focus on the cylinder Q (If R", R) and to simplify the
symbolism we set
A.(t) = {x E KRlu(x,t) > J.I.+ - ;.},
As = {(x,t) E Q (~ RP,R) 11£(x,t) > J.I.+ - ;.}.
8. The second alternative continued

The information of Corollary 7.1 will be employed to deduce that the set where
u is close to its supremum J.I.+, within the cylinder Q (!fR", R), can be made
arbitrarily small. In this section we will also detennine the length of the cylinder
Q (aoR", R) by detennining the number A.
LEMMA 8.1. For every II. E (0, 1) there exists a number S. >S2 itulepetulento!
w and R, such that
Remark 8.1. Assume for the moment that the number B. has been chosen. Then
we detennine the length of the cylinder Q (aoR", R) by choosing
(8.1)
PROOF OF LEMMA 8.1: Consider the local eneqy estimates (3.8) of Chap. II
written over the box Q (aoRP, 2R), for the functions (1£ - k)+. The levels k are
given by
w
k=,,+--8
r- 2 '
where 82 :::; 8 :::; B. and B. is to be chosen. We take a cutoff function ( that equals
one on Q (!f RP, R), vanishes on the parabolic boundary of Q (ooR", 2R) and
such that
Neglecting the firsttenn on the left hand side of these energy estimates, and using
the indicated choices, we obtain
8. The second alternative continued 63
(8.2) IIID(U-k)+IPdxdT~;;' II(U-k)~dxdT

Q(.y.RP,R) Q(a RP,2R)
o
+ aO~ II(U-k)!dxdT+'Y{ JIAt.2R(T)lidT} r
Q(ao RP,2R) -a';RP

The various tenn on the right hand side of (8.2) are estimated as follows. First
(i) ;;, II(U-k)~dxdT~ ;P (;rIQ(~RP,R) I·

Q(a Rp,2R)
o
Next by virtue of the choice (8.1) of the parameter A, and the defmition (2.1) of
ao ,
(ii) ao~ If (u - k)! dxdT ~ ;" (;

Q(a RP,2R)
o
r IQ C; R", R) I·
Finally making use of Remark 3.2 and (3.5)
~
(iii) ~ {j~At.2R(T)I\dT} · .
~ ;" (;r IQ (~ R",R) I(A 3w- b RNK )
~;" (;rIQ(~RP,R) I,
where A3 = 2b8 • and b is defined in (3.4). These estimates in (8.2) give
(8.3) fflDulPdxdT ~ ;" (;r IQ (~ RP,R) I·

A.
Next we use Lemma 2.2 of Chap. I applied to the function u(·, t) for all times
-!f RP ~ t ~ 0, and for the levels
w
(I - k) = 2 +1 .
8
Notice that by virtue of Corollary 7.1 we have
Applying Lemma 2.2 of Chap. I in this setting, gives

64 III. RUder continuity of solutions of degenerate parabolic equations
w
2S+1IAs+1(t)1 :5 v~ IKRI
4-yR N H JIDul dx,
A.(t)\AO+l (t)
for all t E ( - ~ flP , 0). From this, integrating over such a time interval we get
2:1 lAsH I :5 ~ R JJ IDul dxdr

A.\A'+1
1
,; ~R ([fIDU1PdxdT) 'IA,\A.+d7'.
Take the ;tr power, estimate the integral on the right hand side by (8.3) and divide
through by (2;+1 ) ~ . This gives
IAs+1l~ :5 -y(vo)-t!r IQ (~ RP,R) 1;;!-rIAs\As+1l.

These inequalities are valid for all 82 :5 8:5 8 •• We add them for
8 = 82,82 + 1,82 + 2, ... 8. - 1.
The right hand side can be majorized by a convergent series bounded above by
IQ I.
(~RP, R) Therefore
(8. - 82) IAs.l~ :5 -y(vo)-t!r \Q (~ RP,R) \~.

To prove the lemma we divide by (8. - 82) and take 8. so large that
-y
----.:..--E.::.!--..,-I :5 v•.
v~ (8. - 82) p
Remark 8.2. If v. is independent of wand R, also 8. and hence A are independent

of these quantities.
Remark 8.3. The process desCribed in Lemma 8.1 has a double scope. Given v.,
it determines a level p. + - 2":' and a cylinder so that the measure of the set where
u is above such a level can be made smaller than v., on that particular cylinder.
9. The second alternative concluded

Next we show that indeed u is strictly below its supremum p.+ in a smaller box
coaxial with Q (~RP, R) and with the same vertex. To simplify the symbolism
~~~ 2
a. = !ao = ! (~)P-
2 2 w
,
and write accordingly Q (~RP, R) =Q (a.flP, R).
9. The second alternative concluded 65
LEMMA 9.1. The number II. (and hence s. and A) can be chosen so that
u(x, t) :5 JI.+ - 2S~+l '
PROOF: We will apply the local energy estimates of Proposition 3.1 of Chap. II
over the boxes Q (a. R~, Rn) to the function (u - kn ) +, where for all n =
0,1,2, ... ,
and
The cutoff functions (n are taken to satisfy
0< (n(X, t), Vex, t) E Q (a.R~, Rn) , and

(n == 1 in Q (a.R~+l,Rn+d j
(n = 0 on the parabolic boundary of Q (a.~, Rn),

IDr I<
.. n -
2"+1
R'
0 < 8 r < 2 P (n+1)
- 8T ..n - a. RP ,
.!.
a..
= 2 (!:!L)P-2
A '
A = 2S ••
With these choices. inequalities (3.8) of Chap. II take the form
(9.1) esssup I(u - k n )! (~(x, t)dx + f f ID (u - k n )+ (nlPdxdr

-a.R!:<t<O
- - KRn
11
Q(a.R:;,R,,)
:5 'Y Rp2np ff (u-kn)~dxdr+ :.RP

11 2np ff
11 (u-kn)!dxdr
Q(a.R:;,Rn) Q(a.R!:,R,,)
p(1+")
+~{j~At..''<T)lldT} ·
First by the definition of a.
I(U - kn )! (!:(x, t) dx ~ (2~.) 2-pI (u - kn)~ (!:(x, t) dx

KRn KRn
~ 2a.1I (u - kn )+ (nll:,KRn (t).
Next. using again the definition of a., the first two terms on the right hand side of
(9.1) are estimated above by
66 m. HOlder continuity of solutions of degenerate parabolic equations
Substituting this in (9.1) and dividing through by aoo gives
By (3.5) and Remark 3.2 we may estimate
a *:">-1 -< (~)P

• 2".
A w-b < (~)11 R-NIC
2". 4 - '
where A4 = Ab. Next, in the cylinders Q (a .. ~, Rn) we introduce the change of

variable z =tfa.. which maps Q (aoo~, Rn) into Qn =KR" X (-~, 0). Setting
v(·,z) = u(·,aooz),
and
o
IAnl =
-R~
J IAn(z)ldz,
inequality (9.2) can be rewritten more concisely as
This inequality and Corollary 3.1 of Chap. I give

9. The second alternative concluded 67
2-(n+2)p (~)P
28 •
IAn+l I
= (kn+l - knt I(x, z) E Qn+! I v(x, z) > kn+ll
:5 II (v - k n )+ lI~n+l
:5 II (v - kn )+ (nll~n
:5 'YIAnl ~ II (v - k n)+ (nllt"(Qn)
<
-
'"I (~)P
28 •
np
2Rp IA n 11+~
Ei!.±!!l
+~ (2~.r R-N<IA.I*' tlIA,.(Z>!:dZ} •

Thus setting
we have the recursive inequalities
Yn+! :5 'Y4np {Y~+~ + Yn~ Z!+I< } ,

Zn+l :5 'Y 4np {Yn + Z!+I<} .
It follows from these with the aid of Lemma 4.2 of Chap. I that Yn and Zn tend to
zero as n - 00 provided
1 + " 1 + ..
Yo + Z~+I< :5 'Y -er 4- p er == II.,
where 00 = min {~; "'}.
The following proposition summarises the results of the second alternative
and it is proved arguing as in the proof of Proposition 6.1
PROPOSITION 9.1. There exists numbers 110 ,111 E (0,1) and A2 » 1 depending
only upon the data and independent of w and R, such that if for all cylinders of
the type [(0, f) + Q (dRP, R)]
I(x, t) E [(0, f) + Q (dRP, R)] lu(x, t) > Jl.+ - 2~o I

:5 (1 - 110 ) IQ (dRP, R) I,
then either
(9.3)
68 m. UUder continuity of solutions of degenerate parabolic equations
or
(9.4) essosc u < 111 W

[Q(a.( ft.f)] -
where b is introduced in (3.4).
Remark 9.1. The constants 111 E (0,1) and A2 depend only upon the data and, in
general, also upon the nonn lIulloo.nT via the number So. If the lower order tenn
b(x, t, u, Du) satisfies the structure condition (A~) of §5 of Chap. II, we have
So = 1 and therefore 111, A, A2 can be detennined a priori only in tenns of the data
and are independent of lIulloo.nT'
10. Proof of Proposition 3.1
The two alternatives just discussed can be combined to prove the main Proposition
3.1. Let us recall that
~ = (~)P-2.
ao A
The concluding statement of the first alternative is that, starting from the cylinder
and going down to the smaller cylinder
Q(d(ft,f),
the essential oscillation w decreases by a factor 110 E (0,1), unless w :5 A1R~,
where A 1 is a large constant that can be computed a priori only in tenns of the
data and the number So is introduced in (3.2). Analogously, the conclusion of the
second alternative is that starting from the same cylinder and going down to the
smaller box
Q(!f(ft.f) ,
the number w decreases by a factor 111 E (0, 1), unless w:5 A2R~ , where A2 is a
constant that can be computed a priori in tenns of the data. We combine these two
facts into
LEMMA 10.1. There exist constants
that can be determined a priori only in terms of the data, such that either
w:5.AR~ or
11. Regularity up to t = 0 69
We comment further on the content of Remark 3.1. The arguments presented

do not require that the starting cylinder be Q (RP-E, 2R). It would have been suf-
ficient to have started from the box
if we had known a priori that
(l0.1) essosc U < w.

Q(aoRP,R} -
Next we will construct a box for which information of the type of (l0.1) can be
derived. Set
WI == max {Tlw;AR~} and ~ = (WI)P-2,

al A
and let us estimate from below the length of the cylinder Q ( d ( i)P ,i) for which
the conclusion of Lemma 10.1 holds. We have
where
and
It follows that, for the cylinder Q (aIRf, R I ), inequality (l0.1) is verified and the
process can now be repeated starting from such a box, thereby proving Proposition
3.1.
As indicated in §3 this implies the interior mlder continuity stated in Theo-
rem 1.1. The constant dependence indicated in the statement of the theorem follows
from the arguments of §3-( I) and Remarks 6.1 and 9.1.
11. Regularity up to t = 0
Let u be a weak solution of (1.1) of Chap. II that takes initial data U o in n. We

assume U o is continuous with modulus of continuity, say wo (')' The regularity of
u up to t = 0 will follow from a proposition analogous to Proposition 3.1. Fix
n n.
(xo, 0) E x {O}, and R> 0 so that [xo + K2RJ c After a translation we may
assume Xo = 0 and construct the cylinder
where c is a positive number to be chosen. As before. set
p.+ = esssup
Qo(RP-~ ,2R)
u, p.- = essinf
Qo(RP-< ,2R)
u, w = essosc
Qo(RP-~ ,2R)
u.
Let So be the smallest positive integer satisfying (3.2) and construct the box
For all R> 0, these boxes are lying on the bottom of ilT.
PROPOSITION 11.1. There exist constants co, ij E (0, 1) and C, A > 1 that can
be determined a priori depending only upon the data. satisfying the following.
Construct the sequences Ro = R, Wo =W and
Rn = C- nR, Wn+l = max {iiWn; CR~o}, n= 1, 2, ... ,

and the family of boxes
~ __ (~)P-2, n = 0,1,2, ....

an A
and essoscu :5 max

Q~")
{w n ; 2essoscuo }.
Kiln
The proof of the continuity (or the HOlder continuity) of u up to t = follows

from a simple variant of Lemma 3.1. Statement and proof of such a variant goes
°
along the lines of similar arguments in §3. Here we indicate how to prove Proposi-
tion 11.1. Assume without loss of generality that p.+ ~ Ip.-I and that dRP < RP- e,
.
I.e.. ( 2':0 )P-2 > R£. Indeed otherwise we would have
E
. C: o = --.
p-2
This implies that Qo (dRP, R) is all contained in the box Qo (RP-e, 2R). and we
may work within Qo (dRP, R). Also without loss of generality we may assume
that So ~ 2. Set
and consider the two inequalities

_ w _
+
(11.1) p. - 280
w
< P.o+ , p. + 2 > P.o •
80
If both hold, subtracting the second from the first we obtain

11. Regularity up to t =0 71
and there is nothing to prove. Let us assume. for example. that the second of (11.1)
is violated. Then for all 8:::: 8 0 • the levels
_ w
k = Jl. +2 8 '
satisfy the second of (4.16) of Chap. II. Therefore we may derive energy and log-
arithmic estimates for the truncated functions (u - k) _. These take the form
(11.2) sup
O<t<dRP
}(u - k):(x, t}("dx + JfJf ID(u - k)-(rdxdT
- - KR Qo(dRP,R)
.ei.!±.cl
$1' II(U-k)"-'D('''dxdT+1'{jiBk.R(T)'~dT} ,. ,
Qo(dRP,R) 0
(11.3) sup 1!li2(Dk",(u-k)-,c)("(x)dx

O<t<dRp
- - KR
$ l' II !lil!liu (Dk", (u - k)_,c) 1

2 -"ID(I"dxdT
Qo(dRP,R)
+; (Hln ~') {fBk.n(fll'df ('"' ,

where Dtand Bt,R are defined as in (4.2) and (4.4) of Chap. II. The proof can
now be completed as follows. First by using the logarithmic estimates (11.3) and
proceeding as in Lemma 5.1. given any eo E (0, 1) we can find positive numbers
to and Ao. depending only upon the data. such that either
(11.4) (b defined in (3.4»
or. for all t E (0, dJlP).
Ix E KR/21 u(x, t) < Jl.- + 2~0 I< eoIKR/21·

Second. using the energy inequalities (11.2) and the procedure of Lemma 6.1. we
conclude that if (11.4) does not hold. then
(11.5) esSl'nf u > Jl.- + --.

W
Qo(d(ft.f) 12 0+1
Changing the sign of (11.5) and adding ess sUPQo( d( ft. f) u to the left hand side
and Jl.+ to the right hand side we obtain
72 ill. ltilder continuity of solutions of degenerate parabolic equations
1
ij = 1 - 2to +1 .
If the frrst of (11.1) is violated, we write the energy and the logarithmic inequalities
for (u - k)+, k=JI.+ - ~ for s~so and proceed as before.
To summarise, going down from Qo (RP-E, 2R) to the smaller box
the essential oscillation decreases by a factor of ij, unless either
W <
-
2essoscuo
KR
or
LEMMA 11.1. There exist constants Ao > 1 and ijE (0,1), that can be computed
a priori only in terms of the data, such that either
or
To prove Proposition 11.1 we iterate this process over a sequence of boxes

all lying on the bottom of nT • This is done by arguments similar to those in the
previous sections.
12. Regularity up to ST. Dirichlet data

Let (x o, to) be fixed and consider the cylinder !(xo, to) + Q (RP-£, 2R)], where
e = to(P - 2),
where the number b is defined in (3.5) and It is introduced in (3.2) of Chap. II. We
let R> 0 be so small that to - RP-E ~ 0, and change variables so that (xo, to) ==
(0,0). The function u solves (1.1) of Chap. II and takes boundary data 9 on ST
in the sense of the traces of functions in V 2,p(nT). The Dirichlet datum (x, t)--+
g(x, t) is continuous in ST with modulus of continuity wg (·). Set
JI.+ = esssup u, JI. = essinf u, W = essosc u,

Q(RP-< ,2R)nI1T Q(RP-£ ,2R)nI1T Q(RP-£ ,2R)nI1T
and construct the box
Q(dRP,R) ,
where the number So is introduced in (3.2). Let also

JI.; = sup g, JI.- =
9
inf
Q(dRP,R)nST
g.
Q(dRP,R)nST
12. Regularity up to Sr. Dirichlet data 73
If the two inequalities
(12.1)
are both true, subtracting the second from the first gives
w ~ 2 osc g,
Q(dRP,R)nST
and the oscillation of u over Q (dJtP, R) n {h is comparable to the oscillation

of 9 over Q (dRP, R) n ST. Let us assume, for example, that the first of (12.1) is
violated. Then the levels
satisfy (4.11) of Chap. II. and we may derive energy estimates for (u - k) +. Since
(u - k)+ vanishes on Q (dJtP, R)nST. wemayextendittothe wholeQ (dJtP, R)
by setting it to be zero outside nT within the box Q (dRP, R). Also. in (4.13)
of Chap. II we take a cutoff function vanishing on the parabolic boundary of
Q (dRP, R). Taking into account these remarks. we obtain the energy estimates
(12.2) sup f(U-k)!(P(x,t)dx+jrrID(u-k)+(IPdxdr

-dRp<t<O
- - KR
J
Q(dRP,R)
~ "Y f f(u - k)~ ID(IPdxdr + "Y ff(u - k)! (p-l(t dxdr

Q(dRP,R) Q(dRP,R)
~
"'{1.IBt.R(T)11dT} . ,
where Bt,R(r) is defined in (4.4) of Chap. II.
We observe that the conclusion of Lemma 7.2 is automatically verified for
(u - k)+. Indeed the function x -+ (u(x, t) - k)+, vanishes outside nnKR. for
all t E (-dJtP, 0) and an satisfies the property of positive geometric density of
Chap. I. Therefore we may use Lemma 8.1 and its proof to deduce that for all
el E (0,1). there exist positive numbers Al and £1 that can be detennined a priori
only in tenns of the data such that either w < Al RlYf' or
I(X, t) E Q (dRP, R) 1 u(x, t) > p.+ - 2~1 I< ellQ (dRP, R) I·

An application of Lemma 9.1 now gives
LEMMA 12.1. There exist numbers Al > 1 and ijE (0,1) that can be computed
a priori only in terms of the data such that either
w < AIRl!f- or essosc

Q(d( ~)",~)
u:5 max {ijwi osc
Q(dR",R)nST
g}.
The proof of the theorem can now be completed by stating a proposition sim-
ilar to Proposition 11.1.
13. Regularity at ST. Variational data

First we remark that the proof of interior regularity is only based on the energy
and logarithmic estimates of §3 of Chap. II. In particular if such estimates were
available for some locally bounded function U E vt:':( nT), then the conclusion of
Theorem 1.1 would hold for u, irrespective of the differential equation u might
satisfy.
Keeping this in mind, one realises that the proof of Theorem 1.3 is the same as
that of interior HOlder continuity, owing to the energy and logarithmic inequalities
of Proposition 4.1 of Chap. II.
If (xo, to) E ST is fixed, after a translation to (xo, to) == (0,0) and a local
flattening of on, inequalities (4.6) and (4.7) of §4 of Chap. II, can be viewed as
written over cylinders of the type Q+(8, p), defined in (4.8) of Chap. II.
The cutoff function x -+ (x, t) vanishes on the boundary of Kp and not on
the boundary of K"t. This affects the proof only in the application of the embed-
ding Corollary 3.1 of Chap. I. Such an embedding was applied, after rescaling, to
functions VE V"(Qn), where Qn==KR" x {-~,O} (see Lemmas 4.1,6.1,8.1).
Now for these domains, the ratio T Ilnl,,/N is a constant depending only upon the
dimension N.
We also remark that the application of Lemma 2.2 of Chap. I, in the context
of half cubes K"t, is possible since such a lemma holds for convex domains (see
Remark 2.2 of Chap. I).
14. Remarks on stability

As p'\. 2 the equation becomes less degenerate. The proof presented in the previ-
ous sections shows that "Y and Q are stable in the sense that
lim "Y(p) = "Y(2) < 00 and lim Q(p) = Q(2) E (0,1).
",2 ",2
Thus the classical results of HOlder continuity of weak solutions of quasilinear
non-degenerate parabolic equations can be recovered from our results by letting
p'\. 2 in the structure conditions of § I of Chap. II.
14-(i). Continuous dependence on the operator

A similar stability holds for the local behaviour of solutions of a family of equa-
tions of the type of (1.1) of Chap. II. To be specific, let us consider as an example
the family of equations
8 .
8T UA - dlvaA (x, t, UA, DUA) = bA (x, t, UA, DUA)
UA E C'oe (0, Ti L~oc(n))nLfoe (0, Ti wl~:(n») ,

where A ranges over some subset I of the real numbers. Assume that for all AE I,
the functions
satisfy the structure conditions (At}-(A5) uniformly in A, i.e., for constants Ci

and functions "Pi, i =0, 1, 2, independent of A. Assume moreover that
uniformly in A. Then
LEMMA 14.1. {UA} is a family of uniformly Holder continuous functions over
compact subsets of nT.
Results of this kind are referred to in the literature as continuous dependence
of the solution on the operator. Stability results also hold for a family of equations
where also the parameter p ranges over a compact subset of [2, 00).

Questions regarding the local behaviour of solutions of equations of the type
of the p-Iaplacian were raised by Ladyzenkaja-Solonnikov-Ural'tzeva [67],
Aronson-Serrin [7] and Trudinger [97]. The first results for the elliptic case ap-
pear in Ural'tzeva [100] and Uhlenbeck [99] and, for the parabolic case, in [39].
These results hold also for systems and we will comment further on them in
Chap. VIII. The proof presented here is taken from [36,37]. The structure con-
ditions (Ad-(A5) are optimal for Theorems 1.1-1.3 to hold, as pointed out in
[67] in the non-degenerate case p = 2. The iteration technique of Lemma 4.1 is
a parabolic version of a similar elliptic technique due to DeGiorgi [33]. The new
input regards the space-time geometry intrinsically defined by the solution itself.
A first version of this technique appears in [38] in a simpler situation. It turns out
that the same idea can be used to establish the local HOlder continuity of solutions
of the porous medium equations and its generalisations. Here we mention the con-
tributions of [37] and [24]. It can also be used to prove the local HOlder continuity
of doubly degenerate equations. To be specific consider the p.d.e. (1.1) of Chap. II
satisfying the structure conditions
(AI) a(x, t, u, Du)·Du ~ Co~(Iul)IDuIP - lPo(x, t),

(A2) la(x,t,u,Du)1 $ C1~(luI)IDulp-l +~;(U)lPl(X,t),
(A3) Ib(x, t, u, Du)1 $ C2~(lu1)1Du1P + lP2(X, t).
The non-negative functions lPi, i=O, 1,2, satisfy (A4)-(A5) of§1 of Chap. II. The
function ~( .) is degenerate near the origin in the sense that there exists a number
°
lTo > such that
° °
for given positive constants < 1'1 $ 1'2 and $ f32 $ /31. This behaviour has to
hold only near the degeneracy, i.e., for s near zero. For s > lTo it will suffice that
~(s) be bounded above and below by given positive constants, i.e., for example,
We require that
u E Cloc (O,TjL~oc(.a»), ~;6(u)IDul E Lfoc(!1r).
Let F(.) denote a primitive of ~;6 (.). Then the p.d.e. can be interpreted weakly
by requiring that
F(u) E Lfoc (O,T;W,!::(il»).
If~(s) = 1, 'r:Is >0, then (1.1) is of the p-Iaplacian type. Ifp=2 and ~(s) =sm-1
for some m > 0, then 0.1) exhibits a degeneracy (m > 1), or singularity (0 < m < 1)
of the type of porous medium equation. In the latter case a weak. solution is required
to satisfy
lul m E L~oc (o,TjW,!:(ilT»).
The mlder continuity of solutions of such doubly degenerate equations can be
proved by methods similar to the ones presented here and has been established
independently by Porzio-Vespri [88] and Ivanov [52]. The technique is also flex-
ible enough to handle equations bearing a power-like degeneracy at two values of
the solutions. These arise in the flow of immiscible fluids in a porous medium and
have as a prototype
Ut = Llu(1 - u) = 0, O:5u:51.
Results on continuous dependence appear in [9] in a different context.
IV
Holder continuity of solutions of
singular parabolic equations
1. Singular equations and the regularity theorems

Evolution equations of the type of (1.1) of Chap. II for 1 < p < 2 are singular since
I
their modulus of ellipticity becomes unbounded when Dul =0. We will lay out a
theory of local and global HOlder continuity of solutions 11. of such singular p.d.e. 'so
We assume that 11. E L 00 (nT ). If 11. is only locally bounded it will suffice to work
within compact subsets K. of nT. The intrinsic p-distance dist (K.j rjp) from K.
to the parabolic boundary of nT is defined as in (1.1) of Chap. ID. In the theorems
below, the statement that a constant 'Y depends upon the data means that it can be
determined a priori only in terms of lIulloo,nT • the constants Gi , i=O, 1, 2, and the
norms lI'i'o, 'i'r ,'i'2I1q,r';nT appearing in the structure conditions (At}-(A5). For
pin the singular range 1 <p< 2,let ,p- dist (K.j r) denote the intrinsic parabolic
distance from K. to the parabolic boundary of nT, i.e,
P - dist (K;j r) inf ( lIuli oo

== (".'lEIC Cz
' T
1)
P n Ix - yl + It - sip .
(1I •• lEr
1-(i). HOlder continuity in the interior
THEOREM 1.1. Let 11. be a bounded local weak solution of (1.1) of Chap. 11 and
let (At}-(A5) hold. Then 11. is locally HOlder continuous in nT • and there exists
constants 'Y> 1 and a E (0,1) depending only upon the data. such that VK. c nT.
78 IV. HBlder continuity of solutions of singular parabolic equations
/oreverypairo/points (Xl, td, (X2, t2) E IC. I/the lower order terms b(x, t, u, Du)
satisfy (A~) 0/§5 o/Chap.lI, then 'Y and a are independento/llulloo.or .
1-(ii). Boundary regularity (Dirichlet data)
THEOREM 1.2. Let u be a bounded weak solution o/the Dirichlet problem (2.1)
o/Chap. II and let (D) and (Uo ) hold. Assume also that the boundary an has
the property 0/ positive geometric density (1.1) 0/ Chap. I. Then u E C (liT) and
there exists a continuous non-decreasingfunction s -+ w(s) : R+ -+ R+, such
that w(O);:O and
lu(XI. tl) - U(X2' t2)1 :5 w (ixi - x21 + It I - t21t;) ,
/orevery pairo/points (XI. tt), (X2, t2) E nT.lnparticu/ar, ifthe boundmydatum

9 is Holder continuous in ST with exponent say a g , and if the initial datum U o is
Holder continuous in Ii with exponent say auo' then (x, t) -+ u(x, t) is Holder
continuous in nT and there exist constants 'Y > 1 and a E (0, 1) such that
for every pair o/points (Xl, tl), (X2, t2) E liT,

The constants 'Y and a depend only upon the data and the number a* 0/
(1.1) o/Chap.l. Moreover the constant a also depends upon the Holder exponents
a g , auo 0/ 9 and U o respectively.
I/the lower order terms b(x, t, u, Du) satisfy (A~) 0/§5 o/Chap.lI, then 'Y
and a are independent o/llulloo.or .
1-(iii). Boundary regularity (variational data)
THEOREM 1.3. Let u be a bounded weak solution 0/ the Neumann problem (2.7)
o/Chap.1I and let (N) and (N - i) hold. Assume that the boundary an is 0/ class
CI.~. Then u is Holder continuous in nT and there exist constants 'Y and a such
that
for every pair o/points (XI. td, (X2, t2) E nT.

The constants 'Y > 1 antfo: only depend upon lIulloo.i'iT and the data, including
the structure ofafl and the norms IItP1, tPf=r IIq.r;,aT appearing in the assumptions
(N) - i.
If the Neumann data are homogeneous. i.e., iftPo=tP1 =0, and ifin addition
the lower order terms b(x, t, u, Du) satisfy (Aa) of§5 of Chap. II. then 'Y and 0:
are independent ofllulloo.i'iT.
l-(iv). Some comments
The last two Theorems have been stated in a global way. The proof however uses
only local arguments so that they could be stated within any compact portion, say
/(, of fl. Accordingly, the hypotheses on the boundary data need only to hold within
/(,. For example, in the case of Dirichlet data, the boundary datum 9 could be con-
tinuous or HOlder continuous only on a open portion of ST (open in the relative
topology of ST), say E. Then the solution u of the Dirichlet problem would be
continuous (respectively HOlder continuous) up to every compact subset of E.
Analogous considerations can be made for Neumann data satisfying (N)-(N-i) on
relatively open portions of ST.
Similar remarks hold if U o is only locally continuous or locally HOlder con-
tinuous. In particular, to establish the continuity (lli)lder continuity respectively)
of u up to fl x {O}, no reference is needed to any boundary data on ST.
Finally we comment on the assumption that u be locally bounded. It will be
shown in the next Chapter, that when p > 2, solutions of (1.1) are locally bounded.
This is no longer true, in general, if 1 <p< 2. A weak solutions of u of (1.1) is in
L~(flT)' only if
u E L1oc(flT ) for some r ~ 1 satisfying N(p - 2) + rp>O

and such a condition is sharp. Thus, unlike the degenerate case, when p is near one,
the local boundedness is not implicit into the notion of weak solution and must be
obtained by other information such as boundary data. We refer to Chap. V for a
systematic study of local and global boundedness.
2. The main proposition

The lli)lder continuity of u, either in the interior of flT or at the parabolic boundary,
will be, heuristically, a consequence of the following fact. The function (x, t) -+
u( x, t) can be modified in a set of measure zero to yield a continuous representative
out of the equivalence class uE Vi!;;(flT ), if for every (xo, to) E flT there exist a
family of nested and shrinking cylinders [(xo, to) + Q (On' Pn)] with same vertex
such that the essential oscillation Wn of u in [(xo, to) + Q (On, Pn)] tends to zero as
80 IV. HOlder continuity of solutions of singular parabolic equations
n - 00 in a way that can be quantitatively detennined by the structure conditions

(AlHA 5). \
To begin the proof of Theorem 1.1 we introduce a space-time configuration
n
that reflects the singularity exhibited by the p.d.e. Fix (xo, to) E T and construct
the cylinder
[(X o, to) + Q (RP, R l - e)] C nT,
where e is a small positive number to be detennined later. After a translation one
may assume that (xo, to) == (0, 0) and set
1'+ =Q(RP,RI-C)
esssup u, 1'- = ess inf u,
Q(RP,Rl-<)
W= essosc u==J.L+ - 1'-.
Q(RP,Rl-<)
Consider the box

E=!
(2.1) Q (RP, CoR) , where Co = (~) P
and where A is a constant to be detennined later only in tenns of the data. If we

assume that
(2.2)
then we have
and essosc u < w.

Q(RP,coR) -
Cylinders of the type of (2.1) have the space variables stretched by a factor
(w / A) ~ , which is intrisically detennined by the solution. If p 2 these are the
=
standard parabolic cylinders with the natural homogeneity of the space and time
variables.
PROPOSITION 2.1. There exist constants eo," E (0,1) and C,A,A > 1, that
can be determined a priori depending only upon the data, satisfying the following.
Construct the sequences Ro = R, WO =w
Rn = c-nR, Wn+1 = max{1JWn;.A~t}, n=I,2, ... ,
and the boxes
Q(n) == Q (R", enR,,), en = (:;) ~, n=O, 1,2, ....
Then for all n=O, 1,2, ...
A consequence of this proposition is

3. Preliminaries 81
LEMMA 2.1. There exist constants 'Y> 1 and aE (0, 1) that can be determined a
priori only in terms of the data, such that for all the cylinders
0< p ~ R,
essosc
Q(pP,cop)
U~'Y(w+JtEO)(RP)Q.
This is the analog of Lemma 3.1 of Chap. III. The proof is the same and it
implies the HOlder continuity of u over compact subsets of {}T via a covering
argument.
Remark 2.1. The proof of Proposition 2.1 will show that it would suffice to work
with the number w and the cylinder Q (IV', CoR) linked by
(2.3) essosc u < w.

Q(RP,coR) -
This fact. is in general not verifiable for a given box since its dimensions would
have to be intrinsically dermed in terms of the essential oscillation of u within it.
The reason for introducing the cylinder Q (RP, R1-E) and assuming (2.2) is
that (2.3) holds true for the constructed box Q (RP, coR). It will be part of the proof
of Proposition 2.1 to show that at each step the cylinders Q(n) and the essential
oscillation of u within them satisfy the intrinsic geometry dictated by (2.3).
Remark 2.2. Such a geometry is not the only possible. For example, one could in-
troduce a scaling with different parameters in the space and time variables. Exam-
ples of such mixed scalings will occur along the proof of Proposition 2.1. Here we
mention that the proof could be structured by introducing the boxes Q (RP-E , 2R)
and Q (aoIV', R) formally identical to those of §2 of Chap. III and rephrasing the
Proposition 2.1 in terms of such a geometry.
3. Preliminaries
Inside Q (RP, CoR) consider subcylinder: of smaller size constructed as follows.

The number w being fixed, let So be the smallest positive integer such that
w
(3.1) -280 <fJ
_ 0'
where fJo is introduced in (3.11) of Chap. II. Then construct cylinders
E.::l
(3.2) [(ii,O) + Q (RP, doR)1 , do = (~)
2
80
P
82 IV. ltiider continuity of solutions of singular parabolic equations
.-
Figure 3.1
These are contained inside Q (RP , coR) if the number A is larger that 260 and if x
ranges over the cube K'R.(w). where
(3.3)
!=J!
= Lo (doR) • where Lo == (:'0)" - 1.
One may view these as boxes moving inside Q (Rf', CoR) as the coordinates x of
dleir vertices range over the cube K'R.(w)' The cylinders [(x. 0) + Q (Rf' , doR) I
can also be viewed as the blocks of a partition of Q (Rf', coR). Indeed we may ar-
range that Lo be an integer and view the cube KeoR as the union, up to a set of mea-
sure zero, of L: disjoint cubes each congruent to KdoR. Analogously Q (Jl1', CoR)
is the disjoint union, up to a set of measure zero of L:
open boxes each congruent
to Q (Jl1', doR). The proof of Theorem 1.1. is based on studying the following two
cases. Let 110 be a small positive number. Then either
the first alternative
there exists a cylinder of the type of [(x, 0) + Q (Jl1', doR)]. making up the parti-
lion of Q (Jl1' , coR), such that
(3.4) meas{ (x,t) E [(x,O) +Q(RP,doR)11 u(x,t) < #r + 2~0}

< 1I0 IQ(RP,doR) I,
or
the second alternative
for all cylinders [(x,O) + Q (RP, doR)] making up the partition of Q (Jl1', CoR),
3. Preliminaries 83
(3.5) meas { (x, t) E [(x,O) + Q (R", doR)] 1u(x, t) < JJ- + 2~0 }
~ volQ (R", doR) I·
In either case the conclusion is that the oscillation of u in a smaller cylinder
with vertex at the origin, decreases in a way that can be quantitatively measured.
In the arguments to follow we assume (2.2) holds. Indeed if not,
pE
eo = (2 _ p)
and the fll'St iterative step of Proposition 2.1 would be trivial.
Remark 3.1. Along the proof we will encounter quantities of the type
i=I,2, ... ,IEN,
where Ai are constants that can be detenoined a priori only in tenos of the data
and
(3.6) bo = 2 + N(p _ 2) (~ _ 1: K.) .

From the range of K. and q as defmed in (3.5)-(3.7) of Chap. II, one checks that
bo ~ p. We may assume that
(3.7)
Indeed if not. we would have w $ A'wo for the choices
NK.
A = max
l~i~1
At.; and eo=-·
I bo
Remark 3.2. The proof shows that the numbers e and eo can be taken as
NK. eo(2-p)
eo=-, e= .
bo p
3-(i). About the dependence on lIull oo ,I1T

In the arguments below we will use the energy and logarithmic estimates ofPropo-
sitions 3.1 and 3.2 of Chap. II, for the truncated functions (u- k)±, over cylinders
contained in Q (R", CoR). When working with (u - k)_ we will use the levels
_ w
k=JJ +-2 for some i ~ 0,
8 +.,
I 0
and when working with (u - k) + we will take
for some i ~ O.
84 IV. Holder continuity of solutions of singular parabolic equations
These are admissible since

W
11(1.£ - k)±lIoo,Q(RJ>,coR) ::5 2so +i ::5 Do·
Let us fix Do as in (3.11) of Chap. II. Then. since w::5 2l11.£ll oo ,nT' (3.1) holds true.
within any subdomain of nT. if we choose So so large that
4P+8 C 2
(3.8) 280 = Co 1l1.£lloo,nT'
The a priori knowledge of the nonn 1I1.£lloo,nT is required through the number So. If
the lower order tenns b(x, t, 1.£, D1.£) in (1.3) satisfy (A3) of§5 of Chap. II. then, as
remarked there, the energy and logarithmic inequalities hold true for the truncated
functions (1.£ - k)± with no restriction on the levels k. Thus in such a case. So can
be taken to be one and no a priori knowledge of 1I1.£lloo,nT is needed.
The numbers A and Ai introduced in (3.7) will be chosen to be larger than
280 • In the proof below we will choose them of the type
i=1,2, ... ,
where fi ~ 0 will be independent of 1I1.£lloo,nT' We have just remarked that if the
lower order tenns b(x, t, 1.£, D1.£) satisfy (A3) of §5 of Chap. II. then So can be
taken to be one. We conclude that for equations with such a structure the numbers
Ai can be detennined a priori only in tenns of the data and independent of the
nonn 1I1.£lloo,nT'
4. Rescaled iterations
The following rescaled iteration technique applies to any subcylinder of fh and
it is crucial in both alternatives. Let m > 0 be given by
and consider the cube
Kd1R max IXil <

== { I$;i$;N dIR} ,
and the box
QR (mI' m2) == Kd!R x {_2 m2 (P-2) RP, o}.
Fix (x, i) E nT, and let R> 0 be so small that
[(x, i) + QR (ml' m2)] C nT·

Remark 4.1. If (x, i) == (0, 0) and 2m ! = A, m2 =0, then the cylinder
[(x, i) + QR (mI' m2)] coincides with Q (RP, coR). Analogously. ifm2 =0, mi =
So and l = 0, then. for a suitable choice of x the cylinder [(x, i) + QR (mIt m2)]
coincides with one of the boxes making up the partition of Q (RP, CoR).
4. Rescaled iterations 85
LEMMA 4.1. There exists a number Vo that can be determined a priori only in
terms o/the data and independent o/w, R and mt, m2 such that:
(I).I/u is a super-solution 0/(13) in [(x, l) + QR {m}, m2)] satisfying
essosc u<w
[(f,l)H2R(mlom 2)] -
and
I
meas {(x, t) E [(x, l) + QR (mll m2)] u{x, t) < J.t- + 2: }
:::; vo IQR{mllm2) I,
then either
or _ w
u{x, t) ~ J.t + 2m+1 '
where bo is defined in (3.6) and Ao is a constant depending only upon the data
and the numbers m}, m2. Analogously
(ll).I/u is a sub-solution 0/(13) in [(x, t) + QR (m}, m2)] satisfying
essosc u <w
[(f,l)+QR(ml,m2)] -
and
I
meas { (x, t) E [(x, l) + QR (mb m2)] u(x, t) > J.t+ - 2: }
:::; volQR (mt, m2) I,
then either
or
u(x, t) :::; J.t+ - 2:::+ 1'
PROOF: We only prove the statement regarding super-solutions. Assume (x, t) ==

(0,0) and construct the decreasing sequences of numbers
w w
kn = J.t- + 2m+! + 2m+}+n' n=O, 1,2, ... ,
and the families of nested cubes and cylinders

86 IV. mldcr continuity of solutions of singular parabolic equations
Consider (3.8) of Chap. n, written over the boxes Qn for (u - kn) _ and with the
choice of the cutoff functions Cn
0 < Cn(X,t):5 1, V(x,t) E Qn, andCn:l in Qn+1i
{ Cn = 0 on the parabolic boundary of Qni
IDr I < 2,,+2 (...!L) ~ 0 < r < 2(2-p)m2 2,,("+2) .

..n - R m 2 l , - ..n,t - R"
In this setting, (3.8) takes the fonn
(4.1) sup j(u - kn)~ C:(x,t)dx

_2(,,-2)m2 R" <t<O
,,- - K"
+j jlD (u - knL Cnl PdxdT

Q"
:5 ..,: (2: 1 f-Pjj(U - kn)~ dxdT

Q"
+ ..,: 2(2- p)m2j j(u - kn)~ dxdT
Q"
~
+.., { JIA;" ,dl R,. (T)I idT} r
-2(,,-2)"'2 Rt
Since
sup(u _ k) < ~-:w_
n - - m m 2 l+ 2'
Q"
the first two tenns on the right hand side of (4.1) are estimated above by
..,: G:f2(2-p)m2j/X.[(u-knL >O]dxdT.

Q"
To estimate below the two integrals on the left hand side, introduce the level
- 1
k n : 2 (kn + k n +1) < kn.
Then for all tE (_2(p-2)m 2 Rl:, 0)
j(u - kn)~ C:(x,t)dx ~ j(kn - kn )2-P (u - kn)~ C:(x,t)dx

Kn K"
~ G~a)2-P 2(p-2)(n+3)j(u - kn)~ C:(x,t)dx.

K"
Also we have
4. Rescaled iterations 87
!
Qn
!ID (u - knL (niP dxdr ~ !JID
Qn
(u - kn) _ (niPdxdr
- 'Y: (2:)2 2(2-p)m !!X [(U -

2 knL > 0] dxdr.
Qn
Put these estimates in (4.1), divide through by

( ; ) 2-1' 2(p-2)(n+3)
and in the resulting integrals introduce the change of variables
y= C:J ~
p x,
which maps Qn into
Setting also
v(y,z)=u (d1y,2(p-2)m2t) I (y,z)=«d 1y,2(p-2)m2t) ,
and
o
An(z)={y E KRnlv(y,z) < kn }, IAnl=! IAn(z)ldz,
-R~
we arrive at
II (v - knL (nll~p(Qn) ~ 'Y:;n (;r IAnl

~
+ ~ G~.r A.R'"K W -'{! IA.(z) I~ dz } •
By (3.7), AoRNICW-bo ~ I, and by Corollary 3.1 of Chap. I,

2-(n+3)p (.!:!-.)P
2m IAn+1 I
~ (kn - kn+t}P l(y,Z) E Qn+ll v(y,z) < kn+11

~ II (v - kn )+ 1I:,Qn+l
~ II (v - kn )+ <nll:,Qn
~ 'YIAnl~ II (v - k n )+ <nll~p(Qn)
"~ G~.r [:- 1A.I'+"r, + 1A.1"r, U 1A.(z)I' dz } "'P"] .

Thus setting
we have the recursive inequalities
Yn+l ~ -y4n " {y~+m:; + y~ Z!+K } ,

Zn+l ~ -y4n " {Yn + Z!+K}.
It follows from these and Lemma 4.2 of Chap. I that Yn and Zn tend to zero as
n -+ 00, provided
Yo + Z~+K ~ (2-y)-(l+K)/60 4-"(l+K)/6~ == vo,
where 80=min{~;K}.
Remark 4.2. The proof shows that the number Vo depends upon p but it is 'stable'
asp/2, i.e.,
as p --+ 2.
Remark 4.3. The conclusion of Lemma 4.1 continues to hold for cylinders of the
type
E=l
(4.2) QR (m,{3)==Kr x (-{3R" , 0) , r=~:) P R, {3 > 0,
provided {3 is independent of w and R. In such a case we take m = ml and Vo will

depend also upon {3.
5. The first alternative

Suppose that there exists a cylinder of the type of [(x, 0) + Q (RJ', doR)] making
up the partition of Q (R", CoR) for which (3.4) holds. Then we apply Lemma 4.1
with ml = 8 0 and m2 = 0 to conclude that
(5.1) u(x, t) ~ Il- + 28~+l ' 'I(x, t) E [(x,O) «

+ Q i)" ,dof)] .
We view the box [(x,O) + Q«i)" ,doi)] as a block inside Q (R", coR). Let
R(w) be the 'radius' introduced in (3.3). The location of x within the cube K'R.(w)
is only known qualitatively. We will show that the 'positivity' of (5.1) 'spreads'
over the full cube KcoR, for all times
(f)" ~ t ~ o.
In a precise way we will prove
5. The flfSt alternative 89
PROPOSITION 5.1. Assume(5.1)holdsforsomeXEK'R.(w). There exists positive

numbers Al and l1 that can be determined a priori only in terms of the data and
the number A in the definition of Q (RP, coR), such that either
(5.2)
or
As a consequence we may rephrase the first alternative in the following fonn.

COROLLARY 5.1. Assume that (3.4) holds for some cylinder of the type of
[(x,D) + Q (W, doR)] making up the partition ofQ (RP, CoR). There exists pos-
itive numbers A1 and s that can be determined a priori only in terms of the data
and the number A in the definition ofQ (RP, CoR), such that either
or
essosc
Q(pP,cop)
U:::;"'1 w, v P E (0, RI8),
where
"'1 == 1 - T(so+S).
We regard x as the centre of a large cube
which we may assume is contained in the cube KR1- •. Indeed if not, we would
have
16co > R- E, i.e., w < 16r-; AREo, £ 0 - pc
=--·
2-p
We will be working within the box
[(x,D) + Q ( iY, 8co R)]

and will show that the conclusion of Proposition 5.1 holds within the cylinder
[(x,0)+Q(ft,2c o R)] .
This contains Q ( f )P, CoR), regardless of the location of x in the cube K'R.(w).
5-(i). The p.d.e. in dimensionless form

Introduce the change of variables
x-x
x --+ 2co R'
90 IV. RUder continuity of solutions of singular parabolic equations
1- _
I
I
I
I
I
I
I
Figure 5.1
which maps l(x,O) + Q (i)P,8co R)] into Q4 == K 4 x (-41',0). Also introduce

the function
(5.4)
Denoting again with x and t the new variables, the function v satisfies the p.d.e.
(5.5) Vt - div i(x, t, v, Dv) + b(x, t, v, Dv) = 0, in 1Y(Q4),
where i: Q4xRN+1 -+ RN and b: Q4xRN+1 -+ R, satisfy the structure conditions
(5.6) i(x,t,v, Dv)·Dv 2: ~; (2;y-P IDvlP -!Po'
(5.8) Ib(x,t,v,Dv)1 ~ ~~ (~y-P (2~J IDvlP +ch.

Here Ci, i =0,1,2, are the constants appearing in the structure conditions (Al)-
(A3) of Chap. II. Moreover, setting
s. The flJ'St alternative 91
(5.9) 9 - + CPl-~ + CP2,

-= CPo -
the function 9 satisfies
(5.10)
where '"Y = '"Y(N,p,A,8 0 ,data) is a constant depending only upon the indicated
quantities and bo is defined in (3.6). The numbers" and q and q, f satisfy (3.5)-(3.7)
of Chap. ll. The infonnation (5.1) translates into
(5.11) v(x, t) >i a.e. (x, t) E Q(ho ) == {Ixl < ho } x {-41' < t :s; O},
where
(5.12) doR
ho = 8coR
1
=8 A
(28 )¥ < 1.
0
We regard Q( ho ) as a thin cylinder sitting at the •centre' of Q4. We will prove that
the relative largeness of v in Q(ho ). spreads sidewise(l) over Q2.
4 (0,0)
-------,
2ho
Figure 5.2
Proposition 5.1 will be a consequence of the following fact.

LEMMA 5.1. For every II E (0,1) there exists positive numbers A* > 1 and
6* E (0,1) that can be determined a priori only in terms 0/11, N,p and the data,
such that either
(5.13)
or
(1) For further comments on this phenomenon we refer to §l4-{i).

92 IV. H6lder continuity of solutions of singular parabolic equations
(5.14)
for all time levels t E [-2 P , OJ.

Remark 5.1. The key feature of the lemma is that the set where v is small can be
made arbitrarily small for every time level in [-2P , OJ.
5-0i). Proof of Proposition 5.1 assuming Lemma 5.1

In Lemma 5.1 we choose II = " o where " o is the number claimed by Lemma 4.1,
and determine 6· =6· (11o ) accordingly. We let m2 be dermed by
2- m2 = 6·(IIo }
and apply Lemma 4.1 with w = 1, J1. - = 0, R = 2, over the boxes
(0, t) + K2 X { _2 m2 (p-2)2 P , O} =(0, t) + Q2(O, m2}

as long as they are contained in Q2, i.e., for i satisfying
(5.15)
Since (5.14) holds true for all time levels -2 P 5 t 50, each such box satisfies
meas {(x, t) E [(0, t) + Q2(0, m2)J \ v(x, t) 52-m2} 5110\Q2(O, m2)\.
Therefore by Lemma 4.1 either (5.2) holds or
v(x, t) ;;:: T(m 2 +1) V(x, t) E Ql.
Returning to the original coordinates and redefining the various constants accord-
ingly proves Proposition 5.1.
6. Proof of Lemma 5.1. Integral inequalities

First we prove the lemma under the additional assumptions
(6.1)
These will simplify some of the calculations and will be removed later. The weak
formulation of (5.5) is
(6.2) J
K4
Vt'p(x, t}dx + J
K4
sex, t, v, Dv)·Dcpdx = - J
K4
b(x, t, v, Dv}cpdx
for all -4P < t < 0 and all testing functions

cp E C(Q4)nC (-4P ,0; W:,P(K4)).
6. Proof of Lemma 5.1. Integral inequalities 93
Let
(6.3) G(t) == (6k)-(1+.p) !

-4P
t
IIg(r) 11:14 dr,
where k and fJ are positive parameters to be chosen later, 9 is defined in (5.9)
and q is the number entering in the structure conditions (5.10). We define the new
unknow function
(6.4) w == v + G(t),
and rewrite (6.2) in tenns of w. Next, by the parabolic structure(l) of (6.2), the
truncation (k - w)+ is a subsolution of (6.2), i.e., for all testing functions r.p ~ 0
(6.5) !!
K4
(k - w)+ r.p(x, t)dx + !a
K4
(x, t, v, D (k - w)+) ·Dr.pdx
~ -!b
K4
(x, t, v, D (k - w)+) r.pdx - G'(t)! r.p(x, t)dx.
K4
In this fonnulation we take the testing function

(P
r.p== I'
[k - (k - w)+ + fJ k]P
where (== (1 (X)(2 (t) is a piecewise smooth cutoff function in Q4, satisfying
0~( ~ 1 in Q4, and ( == 1 in Q2;

{ (=0 on the parabolic boundary of Q4;
(6.6)
ID(11 ~ 1, 0 ~ (2,t ~ 1;
the sets {x E K4 I
(1 (x) > k} are convex Vk E (0,1).
We use the structure conditions (5.6)-(5.8), with the symbolism
C. -
_. = Ci
24p
(280A )2-P.
Set also
(6.7) ~ (w)
k
_!
-
(k-w)+
ds
[k _ s + 6kjP-1'
o
(6.8) [ k(1 + fJ) ]
!lik(w) = In k(l + 6) - (k - w)+ .
(1) See §l-(i) of Chap. II.

94 IV. Hl)lder continuity of solutions of singular parabolic equations
Then we obtain
! j ~1c(w)(1'dx + Co jIDtP1c (W)jP(1'dx

K4 K4
~ C1 j (IDtP1c (w)I()1'-l ID(ldx

K4
+ C2 22 (1'+1) (2~J jIDtP1c (WW(1'dx

K4
+p j(~1c(W»(P-I(tdx
K4
By the choice (3.8) of the number 8 0 , the second term involving IDtP1c(WW
is absorbed in the analogous term of the left hand side. The integral involving
IDtP1c(WW-1 is treated by means of Young's inequality and the resulting term
involving 1D!P1c(W)\P is absorbed in the analogous term on the left hand side.
The remaining term is majorised by an absolute constant depending only upon
Ci , i = 0, 1. Next, if we stipulate to take k in the interval (0, 1J, the integral in-
volving (t is majorised by 'Y/(2 - p), where 'Y is an absolute constant depending
only upOn p. Finally the sum of the last two integrals can be majorised by an ab-
solute constant. Indeed
We conclude that there exist constants .:yo and .:y depending only upon N, p, A, 8 0
and the data, such that
7. An auxiliary proposition 95
(6.9) ! j
K4
~1c(w)(Pdx +;Yo jIDtP1c (w)I P(Pdx ~ 2 ~ p.
K4
Next, since k E (0, i)

the function tP1c(W) vanishes for alllxl :5 ho • This follows
from the definition (6.4) of wand (5.11). Therefore we may apply the Poincare
inequality (2.9) of Proposition 2.1 of Chap. I, to minorise the second tenn on the
left hand side of (6.9). We summarise:
LEMMA 6.1. There exists two constants 'Yo and 'Y that can be determined a priori
only in terms of N,p, A, SO such that
(6.10) ! j ~1c(w)(Pdx + 'Yo j tP:(w) (pdx ~ 'Y,

K4 K.
where ~1c(W) and tP1c(W) are defined in (6.7)-(6.8).
Remark 6.1. The function G(·) introduced in (6.3) is defined through the numbers
k and 6 which are still to be chosen. By virtue of the structure conditions (5.10),
we have
G(t) ~ 'Y(6k)-(1+r-r) IIgll:;Q4

~ 'Y(6k)-(1+r-r) [RNICw-bo]o!r.
If we choose k6 = 6* E (0, 1) depending only upon the data, we may assume
without loss of generality that
(6.11) G(t) :5 'Y (6k)-(1+r-r) [RNIC w- bo ] r-r ~ 6*2.

Indeed, otherwise for such a selection of 6*
for some positive number (J depending only upon q and p and some 'Y depending
only upon the data.
The number 6* will be chosen Shortly only in tenns of the data. In view of
(6.11) we may regard the function w introduced in (6.4) as independent of k and
6.
7. An auxiliary proposition
Introduce the quantities
(7.1) Yn ==
-4P<t<O
sup j (P (x, t) dx, n=O, 1,2, ....
K4n[W( ·,t)<6")
The proof of Lemma 5.1 is a consequence of the following:

96 IV. Hiilder continuity of solutions of singular parabolic equations
PROPOSITION 7.1. The number II E (0,1) being fixed, we may find numbers
6, uE (0,1) depending only upon N,p the data and II, such that/orn=O, 1,2, ... ,
either
(7.2)
or
(7.3)
PROOF OF LEMMA 5.1: Iterating (7.2)-(7.3) gives
n=1,2, ....
Since Yo::; IK41, we have only to take n=no so large that
uno-I::; 112- N .
Then the lemma follows with 26· = 6no • Indeed, Yno ::; uno -1 Yo implies
I
meas {x E K2 w(x, t) < 26·} < IIIK21.
Recalling the definition (6.4) of wand the upper bound (6.11), this in tum yields
(5.14) and concludes the proof of the lemma.
7-(i). Proof of Proposition 7.1

In (6.10) we take k = 6n , n E N, where 6 E (0,1) is to be chosen. From the
definition (7.1) of Yn , it follows that for every e E (0, 1) there exists to E (-4",0),
such that
(7.4) / (" (x, to) dx ~ Yn +1 - e.

K4n[W(·,to)<6 n +1 ]
The numbers n E N and to E ( -4",0) being fixed, we consider the following two
cases:
either
(7.5) ! / ("
K.
(x, to) 4>6n (w (x, to» dx ~°
or
(7.6) ! / (" (x, to) 4>6n (w (x, to» dx < 0.

K.
In either case we may assume that Yn > II, otherwise the proposition becomes
trivial. Also, in (7.4) we may take e arbitrarily small within the range (0,11/2).
8. Proof of Proposition 7.1 when (7.6) holds 97
7-(ii). The case (7.5)

If (7.5) holds, it follows from (6.10) with k = 6n , that
(7.7)
We minorise this integral by extending the integration over the smaller set
On such a set
Therefore
(In 1 ~ 6) I' J (I' (x, to) dx ~ J (P(x, to)lfIln (w (x, to)) dx.
K.n[w(·,t o )<6 n +1] K.
From this, (7.7) and (7.4)
Yn+l ~c+C
1
( In U +6)-1' .
To prove the proposition in such a case, we choose 6 so small that
1+6)-"
C ( In-U
1/
~2'
Such a choice depends only upon the constants "Y, "Yo and 1/ and therefore it depends
only upon the data.
8. Proof of Proposition 7.1 when (7.6) holds
If (7.6) holds true, define
t. '" sup {t E (-4', t.)1 ~ 1(. (x, t)4'•• (w (x, t» <Ix <' o}.
By the definition oft.,
(8.1) J (P (x, to) ~6n (w (x, to)) dx ~ J (I' (x, t.) ~6n (w (x, t.)) dx.
K. K.
By the arguments of the first alternative

98 IV. mlder continuity of solutions of singular parabolic equations
J
K4
("(x, t.)!liln (w (x, t.» dx :5 C
and 'risE [0,1].
J (" (x, t.) dx :5 C [In +; _

1
1 6 s ]-" '
K4n[(6 n -wl+>s6 n ]
where C is an absolute constant depending only upon N and p and the data. By
the definition (7.1) of Yn • we have for all s E [0, I]
(8.2) J ("(x,t.) dx:5 min { Yn ; C [In 1 ~;~ s]-"}

K4n[(6 -wl+>s6 n ]
n
Yn if 0 :5 s < s.
={ C [In 1+1-.]
1 6 -"
if s. :5 s < I,
where s. is the root of the equation
[
Yn=CIn 1 6+ 6 ]-" .
1+ - s.
Solving it we find
e(c/Ynl1/P - 1
(8.3) s. = e(c/Ynl1/p (1 + 6).
Since Yn >11, we have
e(c/lIl l / P - 1
(8.4) s. < e(CM1/p (1 + 6) == 0"0(1 + 6).
Next we estimate the integral on the right hand side of (8.1). By the Fubini theorem
J("(X,t.)~6n (w(x,t.» dx
K4
8. Proof of Proposition 7.1 when (7.6) holds 99
Therefore (8.1) yields
(8.5) / (P (x, to) ~,s" (w (x, to» dx

K"
~ /1
n p
)P_l (
c5 (2-
[1 + 15 - s]
/ (P(X,t .. ) dx) ds.
o K"n[(,s,,-w)+>s,s,,)
The last integral on the right hand side of (8.5) is estimated by means of (8.2).
Taking into account the definition of s .. in (8.3), we have
/ (P(x,to)~,s" (w(x,to» dx
K"
Bo
c5n(2-p)
</ y. ds
- [1 + 15 _ sjP 1 n
o
1
c5n(2-p) [ 1 + 15 ] -P
+/ Gin ds
[1 + 15 - sjP-l 1 + 15 - s
So
1
c5n(2-p)
= / y. ds
[1 + 15 _ sjP-l n
o
- /1{y'-Gln[1l+c5-s
+ 15 ] -P}
n [1+c5-sjP-l
ds
c5n(2-p)
Bo
where
1
F(Yn , c5) =/ ds
[1 + 15 - sjP
1
o
_/1 (1 _£Y [In 1 +1+15 -15 s ] -P) [1 + 15ds- sjP-l .

n
So
100 IV. Holder continuity of solutions of singular parabolic equations
From Y n ~ v and (8.4) we have

1
F(Y. b) < /
n, - [1 + bds_ sjP-1
o
_ /1 (1 _Cv [In l+b-s

1+ b ]-P) [1+b-sjP-1
ds .
0"0(1+6)
These estimates in (8.5) give

1-6
bn(2-p)
(8.6) / (P (x, to) cJ6n (w (x, to» dx ~ Yn [1 - /(b)] / p-1 ds,
K. 0
[1 + b - sJ
where
1-6
ds
(8.7)
/(6) / [1 +6_ sjP-1 -
o
/1 (1 _Cv [In 1 +1+b -6s ]-P) [1 + bds- S]P-1

_ /1
[1
ds
+b- S]P-1 •
0"0(1+6) 1-6
Estimating below the left hand side of (8.6) we find
This and (8.6) yield
(8.8) Yn +1 - e ~ Yn (1 - /(b».
We estimate /(b) below. For this let 0"1 ~ 0"0 be defined by
(see also (8.4) ) .
Then integrating the first integral on the right hand side of (8.7) over the smaller
interval [0"1(1 + b), 1]. we derive the estimate
1 - 0"t}2- p
/(6) > -(1 -
( -
2b )2- P .
2 1+6
9. Removing the assumption (6.1) 101
We choose 6 so small that

1 2
/(6) > -(1 - 0'1) -P
4
and set
0' = 1- 4"1 (1 - O'I) 2 -p.
Since e E (0, v /2) is arbitrary, we obtain from (8.8)
This proves the Proposition if (6.1) holds.
9. Removing the assumption (6.1)
Inequality (6.10) holds in any case in the integrated form

t
f (P~kdx - f (P~kdx + 'Yo f f (Pwf(v)dx $ 'Yh,
K.(t) K.(t-h) t-hK4
for all t E [-4 P + h, OJ, h > O.
We divide by h and let h -+ 0 to obtain (6.10) where the term involving the
t-derivative is replaced by
(d~) - f (P(x, t)~k(W) dx

K,
==li~~~PX { f(P~k(W)dx - J(P~k(W)dx}dx.

K,(t) K,(t-h)
Define the set
S == {t E (-4 ,0)1 (d~) - f

P (P~k(w)dx ~ o},
K.(t)
and let to be defined as in (7.4). If to E S, we have
(9.1) f (Pwf(v)dx $'Y.

K,(to)
If to ¢ S but
sup {t < to I t E S} = to,
102 IV. ~lder continuity of solutions of singular parabolic equations
by working with a sequence of time levels tn E Sand {tn}-to, we see that (9.1)
continues to hold. If to , S and
T == sup {t < to I t E S} < to,
we derive the two inequalities
f
K,,(T)
(Pl]tf( w)dx :5 'Y,
f {P~k(w)dx:5 f (P~k(w)dx.
K,,(t o ) K4(T)
The remainder of the proof remains the same.
10. The second alternative

We assume here that (3.5) holds true for all cylinders [(x, 0) + Q (RP, doR)] mak-
ing up the partition of Q (RP, CoR). Since 8 0 ~ 1 we have
+ w w
,.,. - 280 ~,.,.- +280 '
so that we may rephrase (3.5) as
(10.1) \(X,t) E [(x,O) +Q(RP,doR)] I u(x,t) >,.,.+ - 2~o \

< (l-lIo)IQ(RP,doR) I,
for all boxes [(x,O) + Q (RP,doR)] making up the partition ofQ (RP, CoR).
Let n be a positive number to be selected and arrange that 2n ~ is an integer.
Then we combine 2~N
P of these cylinders to form boxes congruent to
coR
Figure 10.1
(10.2) Q (R", d.R) == Kd.R x (-R", 0) d. = ( 2sow+n )~ = do (2n) !.=.I!

p •
The cylinders obtained this way are contained in Q (RP , CoR), if the abscissa x of
their 'vertices' ranges over the cube K'R.l(W)' where
'R. 1 (w) = {A 2 -" - 2(Bo+n)~} w~ R

= {( --
2so+n
A)~ -1 }(- w- )~ R
2 o+n B
!.=.I!
where Ll == (~)
2 o+n
B
p - 1.
We will take A larger than 2B o+n and arrange that Ll is an integer. Then we re-
gard Q (R", CoR) as the union, up to a set of measure zero, of Lf pairwise disjoint
boxes each congruent to Q (RP, d. R). Each of the cylinders [(x, 0) + Q (RP, d. R)]
is the pairwise disjoint union of boxes [(x, 0) + Q (R", doR)] satisfying (10.1).
Therefore we rephrase (3.5) as
(10.3) I(X, t) E [(x, 0) + Q (R", d.R)] I u(x, t) > p.+ - 2~o I

< (1 - vo)IQ (R", d.R) I,
for all boxes [(x, 0) + Q (RP,d.R)] making up the partition ofQ (R",coR).
LEMMA 10.1. Let [(x, 0) + Q (R", d.R)] be any box contained in Q (RP, CoR)
and satisfying (10.3). There exists a time level
t· E (-R" , - 2 R") '
Vo
such that for all S ~ So + 1,

(10.4) Ix E [x + Kd.R] I u(x, t·) > p.+ - ;.1 < (II_-v:/2 ) IKd.RI·
PROOF: If (10.4) is violated for all tE (-R", -~R"). then
I(X, t) E [(x, 0) + Q (R", d.R)] I u(x, t) > p.+ - 2~o I

--1'RP
~ fix E [x + Kd.R] I u(x, t) > p.+ - ;.Idt
-RP
~ (1 - vo)IQ (R", d.R) I,
The next lemma asserts that a property similar to (10.4) continues to hold for
all time levels from t· up to O. The proof of the lemma will also detennine the
numbern.
LEMMA 10.2. There exists a positive integer n such that/or all t· < t <0.
(10.5) Ix E [x + KdoRll 1£(x, t) > p.+ - 2B~+n I< (1 - (~ r)iKdoRI.

PROOF: Modulo a translation we may assume that x== O. Consider the logarith-
mic estimates (3.14) of Chap. II, written over the cylinder K do R X (t· , 0), for the
function (1£ - k) + and for the levels
W
k=p.+ - - .
280
As for the number c in the definition (3.12) of the function IjI we take
w
c=--
28 0+n '
where n is a positive number to be chosen. Thus we take
(10.6)
where
Ht == esssup (1£- (p.+ - ~)) .
K"oRX(tO,O) 2 0 +
1be cutoff function x -+ (( x) is taken so that
( == 1, on the cube K(l-u)doR, 0' E (0,1),

{
ID(I:5 (O'd.R)-l .
With these choices, the inequalities (3.14) of Chap. II yield for all t· < t < 0,
o
(10.7) j 1j12(x, t)dx :5 j 1j12(x, t*)dx + (ud:R)P j j 1jI1jI~-PdxdT
KCI-")"oR K"oR tOK"oR
To estimate the various tenns in (10.7) we first observe that
IjI :5 n In 2, 1jI~-P:5 '-

\2BW
o+ n
)P-2 ' [ 1 + InHt '- W )-1] :51'n In 2.
\Fo+R
We estimate the first integral on the right hand side of (10.7). For this observe that
IjI vanishes on the set [1£ < p.+ - 2":0]. Therefore using Lemma 10.1,
jIjl2(x,t*)dX:5 n 2 ln2 2 (/--II:i2) IKdoRI·

K"oR
For the second integral we have
1'n
$ uP IKdoRI·
This estimation justifies the choice of the cylinders [(x, 0) + Q (RP, d.R)) over
the boxes [(x, 0) + Q (RP, doR)). Indeed the integrand grows like 2n (2-p) due to
the singularity of the equation. This is balanced by taking a parabolic geometry
!!.1!::.£l
where the space dimensions are stretched by a factor 2 P •
Finally the last tenn on the right hand side of (10.7) is estimated above by
where A2 = n2(so+n)bo and bo is defined in (3.6). Combining these remarks in

(10.7) and taking into account (3.7), we obtain for all t· < t < 0,
(10.8) f \li2 (x, t) dx $ n 2 102 2

K(l-l7)d o R
C I_-v: i2 ) IKdoRI + :: IKdoRI·
The left hand side of (10.8) is estimated below by integrating over the smaller set
I
{ x E K(l-u)doR u(x, t) > J.I.+ - 2s~+n } .
On such a set, since \li is a decreasing function of H:, we estimate
We carry this in (10.8) and divide through by (n - 1)2 102 2, to obtain for all t· <
t<O,
Ix E K(l-u)doR I 1.£(x, t) > J.I.+ - 2s~+n I

$ (n:lr (11_- v: i2 ) IKdoRI+ n:pIKdoRI·
On the other hand

106 IV. mlder continuity of solutions of singular parabolic equations
Ix EKdoR 1 u(x, t) > J.L+ - ~+

280 n I
:5 Ix E K(l-CT)doR I u(x, t) > J.L+ - 2s~+n 1+ IKdoR\K(l-CT)doRI
~ Ix E K(l-CT)doR I u(x,t) > J.L+ - 28~+nl + NuIKdoRI·
Therefore for all t· < t < 0,
Ix EK(l-CT)doR 1 u(x, t) > J.L+ - 28~+n I

~ [(n: lr (/--V:/2) + n;p + Nu]IKdORI.
To prove the lemma we choose u so small that uN $1 v~ and then n so large that
Remark 10.1. Since the number 110 is independent of w and R also n is indepen-
dent of these parameters.
11. The second alternative concluded

The information of Lemma 10.2 will be exploited to show that in a small cylinder
about (0,0), the solution u is strictly bounded above by
+ w for some m > So + n.
J.L - 2m '
In this process we also determine the number A introduced in §2 which defines the
size of Q (RP, coR). To make this quantitative let us consider the box
Q ({3RP, CoR) , {3 - 110

- 2'
We regard Q «(3RP, CoR) as partitioned into sub-boxes [(x,O) + Q «(3RP, d.R))
x
where takes finitely many points within the cube KR1(w) introduced at the be-
ginning of §10. For each of these subcylinders Lemma 10.2 holds.
LEMMA 11.1. For every liE (0, 1) there exist a number m dependent only upon
the data and independent of w and R such that for all cylinders
[(x,O) + Q «(3RP, d.R)] making up the partition ofQ «(3RP, CoR),
meas { (x, t) E I
[(x,O) + Q (,8RP, d.R)] u(x, t) > J.L+ - 2: }
< IIIQ({3RP,d.. R) I·
11. The se<:ond alternative concluded 107
PROOF: After a translation we may assume that (x, 0) == (0, 0). Set 81 =80 + n,
and consider the energy inequality (3.8) of Chap. n written for (u - k)+, where
8 = 81,81 + 1,81 + 2, ... , m - 1,
over the cylinder Q ({J(2R)1', 2d.R). Over such a box
(u(x, t) - (JL+ - ;,)) + :$;' a.e. (x, t) E Q ({J(2R)1', 2d.R) .
The cutoff function (x, t) - (x, t) is taken to satisfy

(== 1, on Q ({JRP, d.R),
{ ( = 0, on the parabolic boundary of Q({J(2R)1', 2d. R) ,
ID(I :$ d.kp, 0:$ Ct :$ IIJ~P.
We put these estimates in (3.8) of Chap. n and discard the first non-negative term
on the left hand side. This gives
(11.1) !!ID· (u -
Q(fJRP ,d. R)
(JL+ - ;.)) + dxdr:$11' (d.~)1' (;. )"IQ ({JR1',d.R)1
+; (;,y-1' G:tlQ ({JR1', d.R)! + 1'(d.R)N P
(lt
C
) R1' (1:,,) .
P
We estimate above the various terms on the right hand side of (11.1) as follows.
Since 8 ~ 81 == 80 + n,
( W
28
)2-1' < (~)2-1'
- 2 +
= ..!...
cf.
ao n
Therefore the sum of the first two terms is majorised by
(d.~)1' (;,r IQ({JR1',d.R) I,

for a constant l' dependent only upon the data. The last term is majorised by making
use of (3.6) of Chap. II and the definition (10.2) of d•. This gives
l' (d.R)N P (1:,,) R1' (1:,,) :$ ~pd~(~-1) RNICIQ ({JR", d.R) I

P
:$ (d.~)1'A3W-boRNICIQ({JR1',d.R) I,
where A3 = 2mbo and bo is the number introduced in (3.6). Combining these re-
marks in (11.1) we deduce that there exists a constant l' depending only upon the
data and independent of w and R, such that
(11.2)
!!ID (u -
Q(fJRP,d.R)
(JL+ - ;.)) +1"dxdr:$ (d.~)1' G:tl Q({JRP,d.R)I,
lOS IV. Hi>lder continuity of solutions of singular parabolic equations
for all 8 =81,81 + 1, 81 + 2, ... ,m -1. Next we apply Lemma 2.2 of Chap. lover
the cube K d• R for the functions
v = u(·, t),
and the levels
l- + W
- P. - 2.+1'
By virtue of Lemma 10.2
![u(.,t) < p.+ - ;']nKd•R! ~ (~YIKd.RI, Vt E (-fJRP,O).

To simplify the symbolism we set
o
I
A.(t) == {x E Kd.R u(x, t) > p.+ - ;.} , A.s = jIA.s(T)ldT.
-fjRP
Then, with these specifications, (2.7) of Chap. I yields
(;')IA.s+1(t)1 ~~ d.R jIDu(x,t)ldx, VtE (-fJRP,O).

A.(t}\A.+1 (t)
First integrate both sides in dT over (-fJJlP, 0), then take the p-power and ma-
jorise the right hand side by making use of the IJl)lder inequality and (11.2). We
obtain
(;. r IA.s+1I P ~ 'Y (d.R)P ( J j IDU1PdxdT)

-fjRPA.(T}
IA.\A.+1I P - 1
~ 'Y (;. r IQ (fJRP,d.R) IIAs \A.+1I P - 1

.
From this,
IA.+1I~ ~ 'YIQ(fJRP,d.R) l;!t IA.\A.+1I·
Adding these inequalities for 8=81, 81 +1,81 +2, ... , m - I,
To prove the lemma we have only to choose m so large that
( _'Y
m- 8 1
)~ -<v.
12. Proof of tile main proposition 109
Remark 11.1. This estimate deteriorates as p '\. l i.e., m / 00 as p '\.1. However

the choice of m is 'stable' as p /2.
To proceed we return to the box Q (PR!', CoR) and recall that it is the finite
union, up to a set of measure zero, of mutually disjoint boxes [(x, 0) + Q (PR!', d.. R)].
Therefore Lemma 11.1 implies
COROLLARY 11.1. For every vE (0, 1) there exist a number m dependent only
upon the data and independent of w and R such that
(11.3) meas {(X, t) E Q (PR'P, CoR) I u(x, t) > J.I.+ - ; . }

< vlQ (PR'" CoR) I.
We finally determine the size of the cylinder Q (PR!', CoR) as follows. First
in Corollary 11.1 select v = Vo and determine m accordingly. Then let m2 be given
by
(11.4) P -- Vo _
2 -
2m2 ('P- 2 )
,
and assume, by taking m even larger if necessary, that m ~ m2. Then determine
A from
(11.5)
With these choices the box Q (PR!', CoR) coincides with the cylinder QR (ml, m2)
introduced in §4. By Lemma 4.1, there exists a constant Ao dependent only upon
the data and independent of w and R such that either
wbo $ Ao RNK.,
or
w
u(x , t) < ,,+ - -
- r- 2m -
+1
where bo is the number introduced in (3.6). We summarise:
PROPOSITION 11.1. Suppose that (3.5) holdsforallcylinders [(x, 0) +Q (R!', doR)]
making up the partition ofQ (R'P, CoR). There exists a constant Ao dependent only
upon the data and indepetUknt of w and R such that either
(11.6)
orforaliO<p $ R/2,
(11.7) essosc u $
Q(fjpP ,cop)
"'0 w, where ~o == 1 - 2-(m+l).
12. Proof of the main proposition

The main Proposition 2.1 now follows by combining the two alternatives. Set
A == max{Ao j AI}, ." == min{."o j "'l},
110 IV. ftilder continuity of solutions of singular parabolic equations
and let 0 be dermed by
.!. = (.!) l/p = (~) l/p

0- 4P 22p + 1 '
Then setting Rl == RIO both alternatives can be combined into the following
statement: either
w"o < ARNie or e880SC u:5 'IW.
Q(pI',cop)
The process can now be repeated and continued as indicated in Proposition 2.1.
Indeed, by Remark 2.1, the process can be continued as long as (2.3) holds.
12-(i). Stability for p near 2

The proof of the first alternative is based on the integral inequalities (6.9) and
(6.10). Because of the right hand side of (6.9), Proposition 5.1 holds with constants
that deteriorate as p / 2. We briefly indicate how to prove Proposition 5.1 with
constants that are 'stable' as p / 2. First, using the information (5.11 )-(5.12) we
can show that there exists a positive number I. such that
meas {(x, t) E Q2 I v(x, t) < 2-1.} :5 lIo1Q21·
This is accomplished by the same technique as for Lemma 12.1. This technique
involves constants of the type 21.(2-p). We may select p. so close to 2 that
21.(2-p) :5 2, 'tip. :5 p:5 2.
With such a choice the method can be carried out as if the p.d.e. was not singular.
Next, an application of Lemma 4.1 implies that
v(x, t) ~ 2-(1.+1) a.e. (x, t) E Ql.
The application of Lemma 4.1 over the box Q2 involves again terms of the type
21.(2-p). As remarked before, these are majorised by an absolute constant for pE
IP.,2).
13. Boundary regularity

The proof of Theorems 1.2 and 1.3 regarding the regularity up to the lateral bound-
ary of aT, is similar to the proof of the interior regularity. The few changes needed
can be modelled after similar modifications presented in §11-13 of Chap. III for
the degenerate case p > 2. However the proof of regularity up to t = 0 exhibits
some differences.
13. Boundary regularity III
13-(i). Regularity up to t=O
Assume that U o is continuous with modulus of continuity say wo (·). Fix (x o , 0) E

fl x {O} and R > 0 so that [xo + K 2R ] C fl. After a translation we may assume
Xo = 0 and construct the cylinder
Qo (R", 2R) == K2R x {O, R"}.

Set
1'+ = esssup u, 1'- = essinf u, w = essosc u.
Qo(RP,2R) Qo(RP,2R) Qo(RP,2R)
Let 8 0 be the smallest positive integer satisfying (3.1) and construct the box
Qo (dR", R) == KR X {O, dR"}, d-

- -2m
( W)2-" '
where the number m > 1 is to be chosen. Notice that for all R> 0, these boxes are
lying on the bottom of flT. Also withous loss of generality we may assume that
2~ ~ 1 so that there holds
and osc u < w.

Qo(dRP,R) -
PROPOSITION 13.1. There exist constants '1, eo E (0, 1) and e, m > 1 that can
be determined a priori depending only upon the data satisfying the following. Con-
struct the sequences Ro = R, Wo =W and
1
Rn = en R, wn+1 = max {'1Wni e R:t }, n=I,2, ... ,
and the family of boxes
n=0,1,2, ....
and· essoscu~max{wni2essOSCUo}.
Q~n) KRn
We indicate how to prove the fIrst iterative step of the Proposition and show,
in the process, how to determine the number m. Set
and consider the two inequalities

_ w _
+
(13.1) I' - 280
w
< 1'0+ , I' + 2 > 1'0 ,
80
So ~ 2.
If both hold, subtracting the second from the fIrst gives

and there is nothing to prove. Let us assume for example that the second of (13.1)
is violated. Then for all 8 ~ 8 0 , the levels
w
k = IJ- + 28 '
satisfy the second of (4.16) of Chap. II. Therefore we may derive energy and log-
arithmic estimates for the truncated functions (u - k)_. These take the form
(13.2) sup /(U-k)~(x,t)(Pdx+ ffID(u-k)_CIPdxdT

O<t<dRp
- - KR
11
Qo(dRP,R)
~
:51' /fiu-k)"-'DC'PdxdT+'Y{iiBk.R(T)'~dT} r ,
Qo(dRP,R) 0
(13.3) sup /!li2(Dk",(U-k)_,c)C P(X)dx

O<t<dRP
- - KR
:5 l' / / !lil!liu (D;;, (u - k)_, c) 12- PIDCI PdxdT

Qo(dRP,R)
+; (1+ ~') ~B"R(T)11 dT f";"') ,

In {
where Dt and Bt,R are defined as in (4.2) and (4.4) of Chap. II.
LEMMA 13.1. For every vE (0, I), there exists a numberm>80~ 2 depending
only upon the data and independent of w and R such that either
(13.4) (b o defined in (3.6»
or
where
p=-
R
and
W
d= ( 2m
)2-P •
2
PROOF: Consider inequalities (13.3) written for k = IJ- 2":0. As a constant c ap-
pearing in the definition of !Ii (see (3.12) of Chap. II), we tak: c = 2~. Thus we
take
13. Boundary regularity 113
where
D; == lI(u - k)-lIoo,Qo(dRP,R)·
The cutoff function ( is taken to satisfy
on K p ,
vanishes for Ixl = R,
By considerations analogous to those developed in §12 we have the estimates
W )1'-2
2 - 1' < ( -
tJi 5 (m-s o }ln2, 1tJi.1
u - 2m '
and
'Y ff tJiltJi I
Qo(dRP,R)
u 2- 1' ID(I 1' dxdr :5 'YmIKpl·
Moreover the last tenn on the right hand side of (13.3) is estimated above by
'Ym\2ffl \2ffl)(2-1')~
CW )-2 CW . RNItIKpl:5 -ym2mbow-bo RNItIKpl,
r
where bo is the number introduced in (3.6). Combining these estimates in (13.3),

we deduce that if (13.4) is violated, then
(13.6) f
Kp(t)
tJi2 (D;, (u - k)_,c) dx 5 'YmIKpl, '<It E (0, dRP).
We minorise the left hand side of (13.6) by integrating over the smaller set
I
{ x E Kp u(x, t) < ~- + 2: }, '<It E (0, dJlP).
On such a set w
tJi2 ~ In2 2: 0
= (m - So - 1)21n2 2.
2"'"fT
These remarks in (13.6) give
I{x E Kp I u(x, t) < ~- + 2:}1:5 (m _:~_I)2I K pl, '<ItE(O,dR1'),
for a constant 'Y depending only upon the data. To prove the lemma we have only
to choose m so large that
(m - So - 1)2 < v.
Remark 13.1. The process described has a double meaning. It defines a level
+;' for the function u and the size of the box
p. -
d-
-
(-2W)2-"
m '
within which the set where u < p. - +;' is small.

To conclude the proof of Proposition 13.1, choose II = 110 , where 110 is the
number claimed by Lemma 4.1. Then derme m accordingly. By Lemma 4.1 and
inequalities (13.2)
u(x, t) > p.- + 2:+ 1' V(x,t) E Kf x (O,dR").
Notice that no shrinking occurs in the t-direction. This is due to the fact that in
(13.1) the cutoff function, can be taken independent of t.
14. Miscellaneous remarks
14-(i). Expansion of positivity
A crucial fact in the proof of Theorem 1.1 is the expansion of positivity of Propo-
sition 5.1. To focus on this phenomenon let us consider homogeneous equations
with measurable coefficients of the type
V E L~ (0, Tj L~oc(n)) nLfoc (0, Tj w,!:(n») , 1 <p< 2,

(14.1) {
Vt - (IDvl,,-2/lij(X, t) uZ;)Zj =0 in nT,
where the entries (x, t) - tlij(x, t) of the matrix (aij) are only measurable and
satisfy the ellipticity condition
for some A > O. In such a case, the various costants Ai and A appearing in the
proof of Proposition 5.1 are all zero and the function w introduced in (6.4) c0-
incides with v. TIle information (5.3) has been translated into the dimensionless
estimate (5.11 )-(5.12) (see Fig. 5.2).
Let us think of (14.1) as defined weakly in the cylindrical domain K4 x
(-4",0). The information (5.11)-(5.12) is that at the 'centre' of K4 x (-4P,O)
there is a thin cylinder KhD x (-4P, 0) where v > 1. The conclusion of the argu-
ments of §5-9 is that there exists a small positive number "Yo that can be determined
a priori only in terms of N,p and A, such that
v(x, t) ~ "Yo, V(x,t) E KIX(-l,O).

14. Miscellaneous remarks 115
Thus the 'positivity' of v over K ho spreads over a full cube K 1. Actually the infor-
mation (5.11)-(5.12) is only used to apply the Poincare inequality of Proposition
2.1 of Chap. I to derive the integral inequality (6.10). Precisely
j wf(v)(Pdx :5 meas {[Wk[V(X,

K4X{t}
J] = O]n[( = I]} jIDWk(V)IP(Pdx,
K4X{t}
for all t E (-4P, 0). Now to apply such an inequality it only suffices to have the
information
meas{[Wk[v(X,t)] = 0]nK2}~Qo > 0, for some Q o > 0, Vt E (-4P , 0).
In particular it is not necessary to know that the set [Wk(V) = 0] == [v ~ 1] is con-

centrated in a cylinder about the origin. We summarise:
THEOREM 14.1. Let v be a non-negative weak solution of(14.1) in the cylin-
drical domain Q4(46)==K4 x (O,46)for some 6>0. Assume moreover that
meas{x E K21 v(x,t) > ko} ~ Qo,
for some positive numbers ko and Q o and all t E (26,46). Then there exists a
number 'Yo ="10 (N,p, A, 6, Qo, ko ) that can be determined a priori only in terms
of the indicated quantities, such that
v(x, t) ~ 'Yo, V(x, t) E Kl x (36,46).
14-(;;). Extinction in finite time

Weak solutions of (14.1) may become extinct in finite time. We refer to §2-3 of
Chap. VII for a precise description of this phenomenon. The extinction profile is
a
the set [v =O)"'\aT. Theorem 14.1 implies that the extinction profile is a portion of
a hyperplane normal to the t-axis. Indeed if u(xo, to) >0 for some (xo, to) E nT •
by continuity we may construct a box about (xo, to) where the assumptions of
Theorem 14.1 are verified. It follows that the positivity of vat (xo, to) expands at
the same time to to the whole domain of definition of v(·, to).
14-(iii). Continuous dependence on the operator

We only remark that the comments on stability made in §14 of Chap. III. in the
context of degenerate equations. carry over with no change to the case of singular
equations.

Theorems 1.1-1.3 were established in [26J for the case when the principal part of
the operator is independent of t. This restriction has been removed in [27J and
entails the new iteration technique presented in §6-10. This technique differs sub-
stantially from the classical iteration of Moser [81.82.83J or DeGiorgi [33J. It ex-
tracts the 'almost elliptic' nature of the singular p.d.e. as follows from the remarks
in §14-(i). We will further discuss this point in Chap. VII in the context of Har-
nack estimates. The method is rather flexible and adapts to a variety of singular
parabolic equations. For example it implies the HOlder continuity of solutions of
singular equations of porous medium type. To be specific. consider the p.d.e.
Ut - div a(x, t, u, Du) + b(x, t, u, Du) = 0, in {h,

with the structure conditions
(At> a(x, t, u, Du)·Du ~ Co lul m - 1 lDul 2 - 'Po(x, t), mE (0,1),

(A 2 ) la(x, t, u, Du)1 ~ C 1 lul m - 1 IDul + 'PI (x, t),
(A3) Ib(x, t, u, Du)1 ~ C2 1D lul m 12 + 'P2(X, t).
We require that
u E L:: (0, Tj L~oc( ll») and lul m E L~oc ( 0, Tj W,!:( ll») .
The non-negative functions 'Pi, i = 0,1,2. satisfy (A4) - (As) of §I of Chap. I

with P = 2. Further generalisations can be obtained by replacing sm-l, S > 0 with
a function cp( s) that blows up like a power when s - 0 and is regular otherwise.
Results concerning doubly non-linear equations bearing singularity and/or degen-
eracy are due to Ivanov [52.53.54J and Vespri [l02J. A complete theory of doubly
singular equations. however. is still lacking.
v
Boundedness of weak solutions
1. Introduction
Let u be a weak solution of equations of the type of (1.1) of Chap. II in aT. We

will establish local and global bounds for u in flT. Global bounds depend on the
data prescribed on the parabolic boundary of flT. Local bounds are given in tenns
of local integral nonns of u. Consider the cubes Kp C K 2p . After a translation we
may assume they are contained in fl provided p is sufficiently small. For 0 ~ tl <
to < t ~ T consider the cylindrical domains
The local estimates are of the type.
(1.1) ~
lIulloo,Q. ,; ( 1 + If lUI'.) .. ,
where the numbers q" i = 0, 1, are detennined a priori in tenns of p and N and the
constant 'Y is detennined a priori in terms of the structure conditions of the p.d.e.
and Ql.
Unlike the elliptic theory, the estimate (1.1) discriminates between the degen-
erate case p > 2 and the singular case 1 l
{
div IDul,,-2 Du = 0, in fl.
118 V. Boundedness of weak solutions
These solutions satisfy the following estimate for any p

=
exists a constant 'Y -y (N,p, e), such that
> 1. For every e > °
there
Consider now the corresponding parabolic equation
U E Gloe (0, Tj L~oc(D)) nLfoe (0, Tj WI!;: (D») , p > 1,

(1.2) {
Ut - div IDul,,-2 Du = 0, in DT,
and the cylindrical domain Q~p == K 2p X (0, tl. Assume fmt that p > 2. Then for
all eE (0, 21 there exists a constant -Y='Y (N,p, e) such that for all tt~s~t
For the singular case 1 0. Then there exist a
constant 'Y='Y(N,p, r) such that for all tt<s=t there holds
When 1 < p < 2 such an order of local integrability is not implicit in the notion of
weak solution and it must be imposed. The counterexample of § 12 of Chap. XII
shows that it is sharp.
2. Quasilinear parabolic equations

Consider quasilinear evolution equations of the type
(2.1) Ut - div a(x, t, u, Du) = b(x, t, u, Du) in 'D'(DT ).
The functions a: nT x R N +1 -+ R N and b: nT x RN +1 -+ R, are measurable and

satisfy
2. Quasilinear parabolic equations 119
(B l) a(x, t, u, Du) . Du ~ GoIDul" - colul 6 - rpo(x, t),

(B 2) la(x, t, u, Du)1 :$ GlIDul,,-l + cllul6~ + rpl(X, t),
(B3) Ib(x, t, u, Du)1 :$ G2IDul"¥ + c2lul 6- l + rp2(X, t)
for p > 1 and a.e. (x, t) E ilT . Here Gi , C;, i = 0,1,2. are positive constants and 6
is in the range
N+2
p:$6<p~.
The non-negative functions rpi, i=O,I, 2. are defined in ilT and satisfy
(Bs)
where
-41 = (1-11: 0
p
)--
N +p'
11:0 E (0,1].
For fELl (ilT) and h E (0, T) we let !h denote the Steklov average of f. A
function u is a local weak sub(super)-solution of (2.1) in ilT if
(2.2) u E Gloc (0, Tj L~oc{il»)nLfoe (0, Tj WI!:(il») ,

and for every compact subset IC of il.
(2.3)
f {!
K:x{t}
uhrp+[a(x, T, u, DU)]h ·Drp- Ibex, T, u, DU)]h rp} dx:$ (?)O
for all 0 < t :$ T - h and all testing functions
(2.4) rp E Gloe (O,TjL2(1C»)nLfoc (O,TjWJ'''(IC», rp ~ O.

The statement that a constant 'Y = 'Y (data) depends only upon the data means
that it can be determined a priori only in terms of the numbers N, p, 4, 6,11:0, the
constants Gi , C;, i=O, 1,2. and the norms
~ 8-1
IIrpo, rpr ,rp;r 114,UT'
2-(i). The Dirichlet problem

Consider the boundary value problem
Ut - div a(x, t, u, Du) = b(x, t, u, Du), in ilT,

(2.S) { u(·, t)18U = g(., t), a.e. t.E (0, T),
u(·,O) = u o ,
We retain the structure conditions (Bd-(B6). and on the Dirichlet data 9 and 1£0
we assume
(2.6) 9 E L oo (BT),
(2.7) 1£0 E L2(n).
The notion of weak solution is in (2.5) of Chap. II.

Remark 2.1. Unlike the assumption (Uo ) in Chap. II. we do not assume here that
1£0 E Loo(n). Accordingly. our estimates of the norms 111£(', t)lIoo.n deteriorate as
t'\,O.
2-(ii). Homogeneous structures

Local and global sup-bounds. take an elegant form for solutions of equations of
the type
(2.8) Ut - diva(x, t,u,Du) = 0, in nT, P> 1
a{x,t,u,Du)· Du ~ CoIDuI P ,
(2.9) {
la{x, t, u, Du)1 ~ C1IDul p - 1,
for two given constants 0 < Co ~ Cl. The lower order terms are zero and the
principal part has the same structure as
(2.10) Ut - div IDul p - 2Du = 0 or Ut - (IU:r:iIP-2u:r:;),c; = O.

Because of the structural analogy with (2.10) we will refer to (2.8)-(2.9) as equa-
tions with homogeneous structure.
3. Sup-hounds
We let u be a non-negative weak subsolution of (2.1) and will state several upper
bounds for it. The assumption that u is non-negative is not essential and is used
here only to deduce that u is locally or globally bounded. If u is a subsolution.
not necessarily bounded below. our results supply a priori bounds above for u.
Analogous statements hold for non-positive local supersolutions and in particular
for solutions.
The estimates of this section hold for P in the range
(3.1) p> max {I j ::2}' i.e.• ~2 == N(P - 2) + 2p>0

The case \ < p ~ max { 1 j ;~2} will be discussed in §5. Let 6 and "'0 be the
numbers appearing in the structure conditions (Bl)-(B6) and set
3. Sup-bounds 121
N+2
(3.2) q=p--,
N
The range of 6 in (B4) is p:$ 6 < q. We will assume that
(3.3) max{pj 2} :$ 6 < q.

This is no loss of generality by possibly modifying the constants Ci and the func-
tions 'Pi, i = 0, 1, 2. We also observe that owing to (3.1), the range (3.3) of 6 is
non-empty. In the theorems below we will establish local or global bounds for s0-
lutions of (2.1). However precise quantitative estimates will be given only for the
case
(3.4) i = 0,1,2.
In this case we may take Ko = 1 in (B6) and K=p/N.
3-(i). Local estimates
THEOREM 3.1. Let (3.1) hold. Every non-negative. local weak subsolution U 0/
(2.1) in DT is locally bounded in DT . Moreover. if 'Pi E LOO(DT)' i = 0,1,2.
there exists a constant 'Y = 'Y (data) such that V [(x o, to) + Q (PP, p)] CDT and
VUE(O,I).
(3.5) sup U
[(zo.to)+Q(upP ,up»)
:$'Y((I-u)-(N+P)+IQ(PP,p)I)~ ( ffU6dXdT)~ 1\1.

[(zo,to)+Q(pp ,p»)
3-(ii). Global estimates: Dirichlet data
THEOREM 3.2. Let u be a non-negative weak subsolution o/the Dirichlet prob-

lem (2.5) and let (2.6) hold. Then u is bounded in Dx(e, T), Ve E (0, T). Moreover,
if'Pi E LOO(DT)' i=O, 1,2. there exists a constant 'Y = 'Y (data). such that/or all
O<t:$T.
(3.6) ~pu( .• t) $0:,"9 + 1 (ti + D,!. (/[U'dxdT) ~ AI

/fin addition the initial datum U o is bounded above. then
(3.6) ~u(.• t)" max {S:,"9; ~p ...} +1 (l!u'dxdr) ~ A1.

THEOREM 3.3 (THE WEAK MAXIMUM PRINCIPLE). Let u be a non-negative

weak subsolution of the Dirichlet problem (2.5) for equations with homogeneous
structure as (2.8)-(2.9). Then
(3.7) sup u :5 max {ess sup 9 i ess sup u o } .

aT ST a
Remark 3.1. The weak maximum principle holds for equations with homoge-
neous structure for all p> 1.
As a particular case, Theorems 3.1-3.3 give a priori sup-estimates for non-
negative weak solutions of
(3.8) Ut - div IDuI P - 2 Du = al U 6 - l + a2!p, at E R, i = 1,2,

where
N+2
1<c5<p--
- N
and
1 p
"4= N+p(I-lt o ), ltoE(O,I].
These conditions on the lower order terms are optimal for a sup-bound to hold as
it can be seen from the 'linear' case p = 2. Set p = 2 and a2 = 0 in (3.8). For a
local weak solution uE V,~c(flT) to be locally bounded, 6 must not exceed 2Nh2.
Likewise if al = 0, the forcing term II' must satisfy
A N+2
where q > -2-.
These are classical and optimal results for the linear case p=2 (see [67]).
4. Homogeneous structures. The degenerate case p > 2

Here we consider non-negative local or global subsolutions of equations with the
homogeneous structure (2.8)-(2.9). These structures reveal the basic difference
between the degenerate case p> 2 and the singular case 1 < p < 2.
THEOREM 4.1. Every non-negative. local weak subsolution u of (2.8)-(2.9) in

flT is locally bounded in flT . Moreover for all E E (0,2] there exists a constant
'Y depending only upon the data and E, such that V [(x o, to) + Q (8, p)]cflT and
VuE (0, 1),
4. Homogeneous structures. The degenerate case p> 2 123
(4.1)
Remark 4.1. If 9 = p",(4.1) is dimensionless but it is not homogeneous in 1.£.

In the linear case p = 2, (4.1) holds for any positive number e. In our case, e is
restricted in the range (0,2].
It is of interest to have sup-estimates that involve 'low' integral norms of the
solution. The next theorem is a result in this direction. Even though it is of local
nature, it will be crucial in characterising the class of non-negative solutions in the
strip RN x (0, T). (1)
THEOREM 4.2. Let u be a non-negative, local subsolution u 0/(2.8)-(2.9) in nT .

There exists a constant 'Y = 'Y (data), such that V [(xo, to) + Q (9, p)] c nT and
Vue (0, 1),
P/2
(4.2) sup
[(z .. ,t.. )+Q(a9,ap»)
1.£ :5
'Y";9/PP
(1 _ u)
N(p+l)+p
2
(
sup
t .. -9<T<t.. f 1.£(x, 'T)dx
[z .. +Kp)
)
1\ ( p")~
9 .
4-(ii). Global estimates for solutions of the Dirichlet problem
Consider a non-negative weak subsolution of the Dirichlet problem (2.5) for equa-
tions with homogeneous structure and let (2.6) hold. If the initial datum 1.£0 is also
bounded above, then the weak maximum principle estimate (3.7) holds true. If
however 1.£t is not bounded, it is of interest to investigate how the supremum of 1.£
behaves when t -+ 0.
THEOREM 4.3. Let 1.£ be a non-negative weak subsolution o/the Dirichlet prob-
lem (2.5) and let (2.6) hold. There exists a constant 'Y = 'Y (data), such that
Vte(O,T),
(4.3) ~ = N(P - 2) + p.
(1) See §7 flf Chap VI and §2 of Chap. XI.

Results of this kind could be used to construct solutions of the Dirichlet prob-
lem with initial data in L 1 (fl) or even finite measures. Indeed the regularity results
of Chap. III supply the necessary compactness to pass to the limit in a sequence of
approximating problems.
4-(iii). Estimates in L'T==RN x (0, T)
Consider a non-negative weak subsolution 1£ of (2.8) in the whole strip L'T. By

this we mean that 1£ is a local weak subsolution of (2.8) in flT for every bounded
domain fl C RN. To derive global sup-estimates, we must impose some control
on the behaviour of 1£ as Ixl-.oo. We assume that the quantity
(4.4) - . Jp>-Ie
1I1£II{r,t} = O<T<t
sup sup
p~r
1£(x,r)
P
-2) dx, ~= N(P - 2) + p,
Kp
is finite for some r > 0 and for all t e (0, T). The subsolution 1£ at hand is not
necessarily bounded. However it is locally bounded and and as lxi- 00 it grows
no faster than Ixl;;!,. This is the content of the next theorem.
THEOREM 4.4. Let 1£ be a non-negative subsolution 0/(2.8) in L'T, and assume
(4.4) holds. There exist a constant 'Y = 'Y (data), such that/or all te (0, T),
(4.5) sup 111£(·, t)lIoo,K ~ "'("Ii 1I1£II{~~}

p A t-~.
p~r pl'1(p-2)
PROOF: Apply Theorem 4.2 with the choices
(x o, to) == (0, t), (J = t, (1 = 1/2, P ~ r,
and p replaced by 2p. It gives
P/2
111£(·, t)lIoo,Kp
Ie
pi' P
-2) ~ "'("Ii ( O<T<tp~r
sup sup JP>-I(
Kp
1£(x, r) )
P
-2) At
_~
p- •
Remark 4.2. The assumption (4.4) is not restrictive. We will show in Chap. XI
that it is necessary and sufficient for a non-negative solution of (2.8) to exist in
L'T·
The right hand side of (4.5) blows up as t '\, 0 at the rate of at least t - ~ . Such
a rate is not optimal. However the advantage of Theorem 4.4 is that it does hold
for all t E (0, T). The purpose of the next theorem is two-fold. It gives an optimal
estimate of how the local sup-bound for 1£ may deteriorate as either Ixl - 00 or
t'\,O.
5. Homogeneous structures. TIle singular case 1 < p < 2 125
THEOREM 4.5. Let 1£ be a non-negative subsolution 0/(2.8) in ET and assume

(4.4) holds. There exists constants 'Y. and 'Y depending only upon N,p and the
constants Ci , i=O, 1 in the structure condition (2.9), such that
(4.6) for all °< t < 'Y.llull~;':} and/or all p ~ r

pp/(,,-2) pI>'
111£(·, t)lIoo,K p ~ 'Y tNI>' Ilull{r,t}' A = N(p - 2) + p.
Information of this kind are of interest in investigating the behaviour of the

solutions for t near zero and in studying the structure of the non-negative solutions
in ET. (1) The functional dependence in (4.6) is sharp as it can be verified from the
explicit Barenblatt solution
(4.7) B(x,') = d {1-~. (.l~~ t} ~, t>o
'Y" = (~) ,,-1

-L.
p; 2, p>2.
The function 8 solves the Cauchy problem

Ut - div IDul,,-2 Du = 0, in RN x (0, 00),
(4.8) {
8(·,0) = M60 ,
where 60 is the Dirac mass concentrated at the origin, and
M == 118(·, t)1I1,RN, 'TIt> 0.

The i~itial datum is taken in the sense of the measures, i.e., for every cp E Co(RN)
f
RN
8(x, t)cpdx --+ M cp(O), as t '\. 0.
°
For t > and for every p> we have °
5. Homogeneous structures. The singular case 1 max { 1; : : 2 } .
(1) See Chap. XI.

In this section we will show that weak solutions uEL'oc(nT), r~ I, are bounded
provided
p > max { 1; ;:r}.

Such integrability condition to insure boundedness is sharp. In §12 of Chap. XII
we produce a solution of the homogeneous p.d.e. (1.2)
that is unbounded.(I) Thus in the singular range 1 < p < max {I; JZ2}' the
boundedness of a weak solutions is not a purely local fact and, if at all true, it must
be deduced from some global information. One of them is the weak maximum
principle of Theorem 3.3 and Remark 3.1. Another is a sufficiently high order of
integrability.
A sharp sufficient condition can be given in terms of the numbers
(5.1) ~r = N(P- 2) +rp,
We assume that u satisfies
(5.2) u E L'oc(nT) , for some r ~ 1 such that ~r > o.
1be global information needed here is
u can be constructed as the weak limit in L,oc (nT ) of a

(5.3) {
sequence of non-negative bounded subsolutions of (2.8).
The notion of weak subsolution requires u to be in the class
By the embedding of Proposition 3.1 of Chap. I, we have
N+2
q=p--.
N
Therefore ifpis so close to one that ~q ~o, the orderofintegrability in (5.1)-(5.2)
is not implicit in the notion of subsolution and must be imposed.
(1) The notion of solutions that are not in the function class (2.2) is discussed in
Chap. XII
5. Homogeneous structures. The singular case 1 <p<2 127
THEOREM 5.1. Let u be a non-negative local weak subsolution 0/(2.8)-(2.9) in

flT and assume that (5.2) and (5.3) hold. There exists a constant 'Y = 'Y (data, r).
such that V [(xo, to) + Q (9, p)] CflT andVuE (0,1).
(5.4)
Remark 5.1. If 9 =pP, (5.4) is dimensionless but it is not homogeneous in u.
5-(;;). Estimates near t=O
Fix tE (0, T) and let us rewrite (5.4) for the pair of boxes
KTP x (ut, t), Kpx (0, t).
COROLLARY 5.1. Let u be a non-negative local weak subsolution of (2.8)-(2.9)

in flT and let (5.2)-(5.3) hold. There exists a constant 'Y = 'Y (data, r). such that
for all O<t~T and/or all uE (0,1).
'Yt-N/>'r
(5.5) supu(·, t) ~ ¥.E
K"p (1-0') r
Remark 5.2. Assume that (5.2) holds with r = I, i.e.,

2N
(5.6) p> N+1·
°
Then the behaviour of the supremum of u as t '\. is formally the same as that of
solutions of the Dirichlet problem for degenerate equations as in Theorem 4.3.
5-(;;;). Global estimates: Dirichlet data
A peculiar phenomenon of these equations is that, unlike their degenerate coun-

terparts, local and global estimates take essentially the same form. This appears,
for example, by comparing (5.5) with the next global estimate.
THEOREM 5.2. Let 1£ be a non-negative weak subsolution of the Dirichlet prob-

lem (2.5) and let (2.6) and (5.2)-(5.3) hold. Theruxists a constant "1 = "1 (data, r),
such that for all 0 < t $ T,
(5.7) s:r u(·.') ,; s~f 9+ ,Ni>. (It dxd-.-)"'

u' >.
6. Energy estimates
The proof of the sup-bounds stated in the previous sections is based on local and
global energy estimates similar to those of §3 of Chap. II.
6-(i). Local energy estimates

If [( X o , to) + Q (9, p) 1C !1T we let ( denote a non-negative piecewise smooth
cutoff function vanishing on the parabolic boundary of [( x o , to) + Q (9, p) I.
PROPOSITION 6.1. Let 1.£ be a non-negative local weak solution of (2.1) in
!1T and let (B.)-(B6) hold. There exist a constant "1 = "1 (data), such that
V [(xo, to) + Q (9, p}lc!1T andfor every level k > 0
(6.1) sup
to-8<t<to
/ (1.£ - k)! ("(x, t)dx + "1- 1 jrJriD (1£ - k)+ (I"dxdr
[zo+Kp) [(zo,to )+Q(8,p»)
$"1 //(1.£ - k)~ ID(I"dxdr +"1 //(1£ - k)! (,,-1(t dxdr

[(zo,t o )+Q(8,p») [(zo,to )+Q(8,p»)
to }7It.;(1+1C)
{
+"1 / / 1£6 X. [1.£ > 01 dxdr + "1 /IAt,p(r)ldr
[(zo,t o )+Q(9,p») to-8
where It = Ito Ii and

IAk,p(r)1 == mess {x E [Xo + Kp] I u(x, r) > k} .
In (6.1) the integral involving 1.£6 can be eliminated if
Ci = 0, i = 0,1,2, and b(x, t, 1.£, Du) == 0.

Moreover the last term can be eliminated if!pi == 0, i =0, 1, 2.
PROOF: The proof is very similar to that of Proposition 3.1 of Chap. II. First we
may assume that (xo, to) == (0, O) modulo a translation. Then in (2.3) we take the
testing functions
6. Energy estimates 129
cP = (Uh - k)+ (II,

where Uh is the Steklov average of u. All the terms are estimated as in §3 of
Chap. II, with minor modifications, except the integrals involving the lower or-
der terms b(x, t, u, Du). For these, we let h - 0 and use the structure condition
(B3) and Young's inequality to estimate
(6.2) I jlb(x, r, u, Du) (u - k)+ (1Ildxdr

Q(9,p)
:5 C2 I liD (u - k)+ IP¥ (u - k)+ (/dxdr

Q(9,p)
+C2 Ilu 6 - 1 (u-k)+(Pdxdr+ Ilcp2(U-k)+(PdXdr

Q(9,p) Q(9,p)
:5 ~o I liD (u - k)+ IP(Pdxdr + 'Y I Iu6 ')( [(u - k)+ > 0] dxdr
Q(9,p) Q(9,p)
+ 'Y I I cpr')( [(u - k)+ > 0] dxdr .

Q(9,p)
Thus we arrive at
sup I (u - k)! (P(x, t)dx + 'Y- 1 r riD (u - k)+ (IPdxdr

-9<t<O
Kp
JJ
Q(9,p)
:5'Y II(u - k)~ ID(IPdxdr + 'Y II(U - k)! (P-l(t dxdr

Q(9,p) Q(9,p)
+'Y I I u6 ')([(u - k)+ >0] dxdr + 'Y I I~')([(u - k)+ >0] dxdr,
Q(9,p) Q(9,p)
where
..r;.
~ = CPo +;;!:r+
CP1 6
CP2·
By the HOlder inequality and (B5)-(B6),
0 } ~(1+IC)
II~dxdr :511~lIq,nT {
IIA/c,p(r)ldr
Q(9,p) -9
Remark 6.1. Inequality (6.1) for the function (u - k)_ holds true for local super-
solutions of (2.1) and k:5 o.
Remark 6.2. Unlike inequalities (3.8) of Chap. II, the levels k here are not re-
stricted.
6-(ii}. Global energy estimates: Dirichlet data
PROPOSITION 6.2. Let U be a non-negative weak sub-solution of the Dirichlet

problem (2.5), let (2.6) hold and let k satisfy
(6.2) k ~ supgAO.
ST
There exists a constant 'Y = 'Y (data), such that for every non-negative function
t--+(t) eCl[O, T] andfor every O<t~T,
(6.3) sup j(u - k)! (T) dx + I liD (u - k)+ IP(T) dxdT

O<.,.<t 11
n n.
~'Y j(Uo-k)!(O)dx+'Y jj(u-k)!(t(T)dxdT
n n,
(t
+'Y If u6x [u > k] (T)dxdT+'Y ~IA1c'P(T)I(T)dT
)~(1+IC)
where 1£ =1£0 Ii, and

IA1c,p(T)1 == meas {x e U I U(X,T) > k}.
In (63) the integral involving u 6 can be eliminated if
£1=0, i=0,1,2, and b(x, t,u, Du) ==0.
Moreover the last term can be eliminated ifIPi==O, i=O, 1, 2.
A similar statement holds for the truncatedfunctions (u - k)_ provided
(6.2)' k < infgVO.
- ST
PROOF: If k satisfies (6.2), by Lemma 2.1 of Chap. I

(uhht) - k)+ E W!,P(U), VO<t~T-h.
Therefore the testing functions
are admissible in the weak fonnulation of the Dirichlet problem.

7. Local iterative inequalities 131
7. Local iterative inequalities

The common element in the proof of the sup-bounds stated in §3 and 4 is a set of
iterative inequalities. We will derive them, starting from the energy inequalities of
§6. Modulo a translation, we may assume that (xo, to) coincides with the origin.
Fix q E (0, 1) and consider the sequences
(l-q) (l-q)
Pn = qp + 2n p, (In = q(J + 2n
(J, n = 0,1,2, ... ,
and the corresponding cylinders QniEQ ((In, Pn). It follows from the definitions
that
Qo = Q «(J, p), and Qoo = Q (q(J, qp) .
Consider also the family of boxes
where for n=O,I,2, ...

_ . Pn + Pn+l 3(1 - q)
Pn = 2 =qp + 2n+2 p,
For these boxes we have the inclusion

Qn+lCQnCQn, n = 0,1,2, ....
Introduce the sequence of increasing levels
k
n
=k-~
2n '
where k is a positive number to be chosen. We will work with the inequalities (6.1)
written for the functions (u - kn+l)+, over the boxes Qn. The cutoff function (
is taken to satisfy
( :ani~hes_on the parabolic boundary of Qn,
{ (= 1 10 Qn,
ID'"I':. < 2"+2
- {1-a)p'
0 < <
- (t -
2"+2
(l-a)B"
With these choices, (6.1) yields
(7.1) sup
-8,,<t<O
j(u - kn+l)! (P(x, t)dx + {{ID (u - kn+l)+
JJ
(I P dxdT
Kp.. Q..
-y2np {{
~ (1- q)ppp JJ (u - kn+tl~ dxdT
Q..
+ (1 ~:)(J j j(u - kn+l)! dxdT

Q..
+-y jju6x [(u - kn+tl+ > 0] dxdT+-yIA n+llllh(1+ IC ),

Q..
where whe have set
IAn+11 == meas{(x,t) e Qn I u(x,t) > kn +1}'

The last two tenns can be eliminated for equations with homogeneous structure.
First we observe that for all 8 > 0
(7.2) II(u -kn)~

Qn
dxdT
Qn
-
~ II~u kn)~ X [u > kn+1]dxdT
~ (kn+1 - kn)' IAn+11
k'
= 2(n+1),IAn+ll.
Then we estimate
To estimate the integral involving u 6 , first write

2n+1 - 2
kn = kn+1 2n+1 _ 1 .
Then estimate below
(7.5) II(u -kn)~ dxdT ~ II(U -kn)~ [u > X kn+1] dxdT

Qn Qn
~ IIu ~:: =~)6

6 (1- X[u > kn+l]dxdT
Qn
~ 'Y: II n6 > u6 X [u kn+1] dxdT.

Qn
Finally
7. Local iterative inequalities 133
We combine these estimates into (7.1) to derive the following basic iterative in-
equalities
Moreover the last two tenns can be eliminated for equations with homogeneous
structure.
To proceed, construct a non-negative piecewise smooth cutoff function (n in
Qn, which equals one on Qn+l, vanishes on the lateral boundary of Qn and such
that
ID(nl $ 2n+2 /(1 - u}p.
Then the function (u - kn+l)+ (n vanishes on the lateral boundary ofQn and by
the multiplicative inequality of Proposition 3.1 of Chap. I,
(7.7) / /(u -
-
kn+l)~ (!dxdT $ 'Y (_sup
-Bn<t<OK
/ (u _ kn+l)! dx) i
~ ~
x (i/ID(U-kn+d+IPdXdT+ i/(U-kn+l)~ID(nIPdxdT).
Qn Qn
Remark 7.1. The estimates in (7.2)-(7.5) and the inequalities (7.6), (7.7) are
valid for any number
6 ~ max{pj 2}.
The structural restriction 6 = q does not play any role in the derivation of (7.6) and
(7.7).
8. Local iterative inequalities (p > max {1;&~2} )

Introduce the dimensionless quantities
(8.1) Yn = H
Qn
(u - kn)~ dxdr, n = 0,1,2, ....
We will derive an iterative inequality for Yn by estimating the right hand side of
(7.7) by (7.6). We assume first
(8.2) p > max { 1 j N2N}

+ 2 ' max{Pi 2} ~ 6 < q == PN+2
-r
and estimate
We estimate the last integral by (7.7) and in turn estimate the right hand side of
(7.7) by the inequalities (7.6) and (7.3). We arrive at the recursive inequalities
(8.3) Yn+l < -ybn ~ .A!* Y,!'+~*

- k~(q-6) (1 - O')P~
b ( PN(J)~*
n
+ -y k~(q-6)
1...2.
y'l+,N +"Vbn
n,
I.
(pN(J)'" (1
-y.
k6 n
)l+~"
,
where
and
(8.4) .A.. .. ( (;) kqz(P-'l + (~) f kqz<,-<»).

The last two tenns in (8.3) can be eliminated for solutions of equations with ho-
mogeneous structure.
9. Global iterative inequalities 135
9. Global iterative inequalities
We let u be any non-negative weak subsohition of the Dirichlet problem (2.5) and
assume that (6.2) holds so that u satisfies the energy estimates (6.3). Fix 0 < t ~ T
and introduce the sequence of increasing time levels
tn = ut (1 - 2: ), u E (0,1), n = 0,1,2, ... ,
and the cutoff functions
iftn+l~T~t
if tn < T < tn+l
Introduce also the sequence of increasing levels
k
kn = sup 9 + k - -2' n = 0, 1,2, ... , k > 0 to be chosen
ST n
and write (6.3) for the functions (u - kn+d + and the cutoff functions (n to obtain
(9.1) sup
t,,+1 <T;St
/(u - kn+l)! dx +
II
/
t
t,,+lll
"n
J" (u - kn+l)+ IPdxdT
t t
~ 'Y:: / /(U - kn+d! dxdr + 'Y / /u6x [u > kn+lJ dxdT

t"ll t"ll
where we have set
The last two terms of (9.1) can be eliminated for equations with homogeneous
structure. Moreover the last term can be eliminated if in the structure conditions
(B 1 )-(B3), !Pi ::0, i=O,I,2. If !Pi E LOO(nT), i=O,l, 2, then It= N'
Proceeding as in (7.2)-(7.5), we estimate
t t
(i) f f(U - kn+l)! dxdT ~ 'Y2:;~:2) f f(U - kn)~ dxdr,

t"a t"a
t t
(ii) f fU6 x. [u > kn+lJ dxdr ~ 'Y2n6 f f(U - kn)~ dxdT,

~a ~a
(iii) j IAn+l(r)ldT ~ 'Y:;6 jiU -kn)~ dxdT.

t" t"a
Combining these remarks in (9.1) we arrive at the recursive inequalities
Remark 9.1. The structure restriction 6 < q does not play any role in the derivation
of (9.2). This inequality holds for all 6 ~ max{p; 2}.
9-(i). Global iterative inequalities. The case p>max {I; J~2}

Next we assume that the numbers p and 6 are in the range (8.2). We apply the
multiplicative embedding inequality of Proposition 3.1 of Chap. I, and proceed
as in the case of the local inequalities. This process is indeed simpler. since
(u - kn)+(·,t)EW!'P(O) fora.e. tE(O,T). Setting
t
(9.3) Yn == f fa
t"
(u - kn)~ dxdr,
we obtain
(9.4)
10. Homogeneous structures 137
where b = 26(1+ t .. ).
In these. the last two tenos can be eliminated for solutions
of equations with homogeneous structure as in (2.8)-(2.9). Moreover the last teno
can be eliminated if. in the structure conditions (B 1 ) - (B 3 ). 'Pi == 0, i = 0, 1,2.
If 'Pi ELOO(nT), i=O, 1,2. then It=p/N.
Suppose now that the initial datum 1.1.0 in (2.7) is bounded above and let us take in
(6.3)
kn = max { S~! 9 j s~p 1.1.0 } + k - 2:' n = 0,1,2, ... ,

where k > 0 is to be chosen. Then the first integral on the right hand side of (6.3)
is zero and we may take ( == 1. In such a case. we arrive at an inequality analogous
to (9.1). where the first integral on the right hand side is eliminated and where the
integrals are all extended over the whole nt • Proceeding as above we find that the
quantities
Yn == H
nc
(1.1. - kn)~ dxdr
satisfy the recursive inequalities
-ybnlntl*~ 1+*~
(9.5) Yn +1:5 k~(9-6) Yn + -yb
n
Inti K! ( k16 Yn )1+K~
b = 26(1+~~).
For equations with homogeneous structure. all the tenos on the right hand side of
(9.5) are zero.
10. Homogeneous structures and 1 O.
The numbers Ar have been introduced in (5.1). We also assume that 1.1. can be
constructed as the weak limit in L[oc (nT) of a sequence of bounded subsolutions
of (2.8). By possibly working with such approximations we may assume that 1.1. is
qualitatively locally bounded. Below. we will derive iterative inequalities similar
to (8.3) but involving the L[oc -nonos as well as local sup-bounds of u.
If 1 <p< max { 1 j J~2}' we have q < 2. If (10.1) holds for some r E [1, 2).
then A2
> 0 and p > max { 1 j J~2}. Therefore it suffices to assume that (10.1)
holds for some r > 2. In such a case we have
(10.2) r>q,
138 V. Boundedncss of weak solutions
In (7.6) we discard the last two terms in view of the homogeneous structure of
(2.8) and, owing to Remark 7.1, set also 6 =r. We obtain
(10.3) sup
-9.. <e<o
J (u-k n +1)!(x,t)dx+ ffID(u-kn+d+IPdxdr
11 i
K~.. Q..
Define
Yn = H
Q..
(u - kn)~ dxdr, n= 0,1,2, ... ,
and estimate
Yn +1 ~ lIull:'~(9,p) H
Q..
u 9 dxdr.
We majorise the right hand side by means of (7.7) and in turn estimate the right
hand side of (7.7) by (10.3). We arrive at the recursive inequalities
(10.4) y.
n+1 -
"(bn
< (1 _ u)f(N+p) II u Il oo,Q(9,p)
r- 9 8 AIf y'1+f
n ,
where
(10.5) 8, '" { ( ; ) k-(.-p)"i' + (~) ~ k+')"i' }.

The recursive estimates (10.4)-(10.5) have a global version. Let u be a non-
negative weak subsolution of the Dirichlet problem (2.5) and assume that
u E L oo (0, Tj Lr(o» , for some r>2 satisfying ~r >0.
Then the quantities Yn defined in (9.3) with 6 = r satisfy
,,(bnlOel f r-9 1 1+f

(10.6) Yn +1 ~ k(r-2)~ lIulloo,O>cCe.. ,t) t~ Yn .
II. Proof of Theorems 3.1 and 3.2

The starting point in the proof of Theorem 3.1 is the inequality (8.3). We take (J = p"
and stipulate to choose k ~ 1. Recalling that max{pj 2} ~ 6 < q, the quantity .Ak
in (8.4) is majorised by 2. To simplify the presentation consider first the case
II. Proof of Theorems 3.1 and 3.2 139
i = 0,1,2,
so that we may take K. = N' With these choices, (8.3) yields
It follows from Lemma 4.2 of Chap. I that Yn - 0 as n - 00, provided k is chosen

to satisfy
k = max{ko i I},
where
Yo = H ulidxdT = ckiq-li)/f (1 - u)-(N+p) + IQ (PP,p) I) -1,

Q(PP,p)
for a constant C depending only upon the data. This in tum implies
The general case of K. E (0, N) is proved by a minor modification of these argu-

ments. It suffices to rewrite (8.3) as
Yn +1 :5 k~~::li) (1- u)-(N+p) + IQ (PP,p) I)~" {y~+~N + y~+~,,}

and follow the iteration process of Lemma 4.1 of Chap. I. To prove Theorem 3.2
we refer to the global recursive inequalities (9.4). As before, we take k ~ 1 and
consider frrst the case of 'Pi E VlO(flT ), so that K. = pIN,. Choosing u = 1 we
arrive at
'Ybnlfltl~N ( _l!.U) ~N Y l+~N
Yn +l:5 k~(q-li) 1+t p n .
It follows from Lemma 4.1 of Chap. I that Yn - 0 as n - 00, provided k is chosen

to satisfy
for a constant C depending only upon the data. This in tum implies that for all
O<t:5T
The general case of I( E (0, It) is proved by a minor modification of these argu-
ments. The second part of Theorem 3.2 is proved exactly the same way, by starting
from the recursive inequalities (9.5).
12. Proof of Theorem 4.1

We refer back to the iterative inequalities (8.3) and discard the last two tenns be-
cause of the homogeneous structure of the p.de. in (2.8)-(2.9). Since the resulting
inequalities hold for all p ~ 6 < p N12 , we take 6 = p and rewrite them as
(12.1) Yn+l =
'Y bn
~
m Ynl+m ,
-Jb AI:
(1- 0')" k
where Yn are defined in (8.1) and AI: are defined in (8.4) with 6 =p, i.e.,
We stipulate to take k so large that of the two tenns making up AI: the first domi-
nates the second, i.e.,
(12.2) k>
-
(-P")
(J
p!,
so that
(J
AA; ~ 204, 04= pp'
It follows from Lemma 4.1 of Chap. I that Yn - ° as n - 00 if we choose k from
Yo == H uPdxdr = CA- 1 (1 - u)(N+P) k 2 ,

Q(B,p)
where C is a constant depending only upon 'Y, b, N and p. For such a choice and
(12.2),
(12.3) esssup u
Q("B,,,p)
~ 'YVA
(1 - 0')
~( HUPdxdr) I" (~);6 .
Q(B,p)
This estimate proves the theorem for E = 2. Fix E E (0, 2) and consider the increas-
ing sequences
Po up, =
and for n= 1, 2, ...
12. Proof of Theorem 4.1 141
n n
(12.4) Pn = up+ (1- u)p L2- i, (In = u(J + (1 - u}(J L 2- i,
i=l i=l
and the corresponding cylinders Q(n) ==Q ((In, Pn). By construction
(12.5) Q(o) == Q (u(J, up) and Q(oo) == Q ((J, p) .
Set
(12.6) Mn = esssupu
Q(")
and write (12.3) for the pair of boxes Q(n) and Q(n+1). This gives
Mn $ "(2n~:: ffUPdxdr)! AA,.!-p

(
(1 - u) Q(,,+l)
<
-
M¥"(2n~ VA ( ffUP-2+Edxdr)! AA~
~1 ~ .
(1 - u) Q(fJ,p)
If1]E (0, I), the right hand side of this inequality is majorised by
~
d= 2 • ,
where
Combining these estimates we arrive at the recursive inequalities
n = 0,1,2, ....
From these, by iteration
n
Mo $1]nMn+1 +BdL(1]d)i, VnEN.
i=O
We choose 1] = fa
so that the sum on the right hand side can be majorised by a
convergent series and let n - 00 to obtain
,,(Af
sup u$ ~
Q(ufJ,up) (1 - u) •

The proof of the theorem is a consequence of the following:
PROPOSITION 13.1. Let u be a non-negative local sub-solution of (2.8)-(2.9) in .
nT, and let p > 2. There exists a constant "Y = "Y (data), such that
V [(x o, to) + Q (0, p)]CnT and Vu E (~, 1),
(13.1) H uPdxdT $ (1-"Y

[(zo,to)+Q(O",O'p»)
U
)Np (sup
to-'<T<to
fu(x, T) dx) P
[zo+Kp)
1\ ( PP)~
0 .
PROOF: We may assume that (x o, to) =(0,0), and having fixed uE (!, 1), con-
sider the increasing sequences {p,,} and {O}" introduced in (12.4) and the corre-
sponding cylinders Q("). Let (x, t) - ,,, be a non-negative piecewise smooth cut-
off function in Q(,,+1) that equals one on Q(") , vanishes on the parabolic boundary
of Q(,,+1) and such that
The function (u'"K,t) vanishes on aKp"+l' Therefore by the embedding in-

equality (3.1) of Chap. I applied with m= 1,
(13.2) / /(u,,,)p( ¥ )dxdT

Q"+l
The constant "Y depends only upon the data and it is independent of p, 0 and n. The
energy estimates for solutions of (2.8) give
H IDu("IPdxdT $ (1 2"P
u)PPP2 H 2
uPdxdT + (1 "Y_ "u)O H u2dxdT
Q .. +l Q"+l Q"+l
$ ~(1 ~;)'9 [(; )lt uPdzdTIl (~t],

where we have estimated the second integral by HOlder inequality. Without loss of
generality we may assume that
(;) HulldxdT > (~);;!" for all n = 0,1,2, ... ,

Q"+1
otherwise the Proposition becomes trivial. Combining these remarks with (13.2)
and setting
we obtain the recursive inequalities
By the interpolation Lemma 4.3 of Chap. I, we conclude that there exists a constant
'Y, depending only upon the data such that
H UlldxdT~
Q(tr9.trp)
'Y N
(1 - u) II
( sup
-9<T<O
K,.
fU(X,T)dx)1I

We may assume that the boundary datum is non-positive, by possibly replacing U
with
w =u-supg.
ST
We start from the global iterative inequalities (9.4) and discard the last two terms
on the right hand side since the p.d.e. has the homogeneous structure (2.8)-(2.9).
Taking 6 = p we obtain
y.
n+l ~
l tl m y'l+m
'Ybn f1
.!!±.I!......IL
n ,
(ut)N"+2" kWH
where Yn are defined in (9.3) and

(14.1)
It follows from Lemma 4.1 of Chap. I that Yn - 0 as n - 00 if k is chosen from
Yo = f/U PdxdT:5 C(CTt)~ Intl-1kP/p ,

at
for a constant C depending only upon 'Y, b, p and N. Thus for all CTt < T :5 t
(14.2) IIU(·.T)II~.n'; u(N+P~.tN" (lluPdzdT) pl.

Consider the decreasing sequence of time levels
t tn.
tn = - - -
2 4 ;=0
2-'
'
L
and apply (14.2) over the expanding domains n x {tn+ 1, t}, with CT taken from
i.e., CT = 1 -"~
LJ,-o
2-(;+1)
~ 2-(n+1).
1 + ,,~+1 2-(i+l)
LJ,=o
Setting also
Mn = sup lIu(·, T)lIoo,a,
t .. <-r<t
we obtain from (14.2)
where d=2(N+p)/",. By the interpolation Lemma 4.3 of Chap. I we conclude that
)..=N(p - 2) + p.

Even though the theorem is of global nature, our starting point is the recursive
inequality (12.1). We begin by observing that in the proof of Theorem 4.1 the
choice of k ~ 1 was made to guarantee that Ale could be majorised by a quantity

independent of k. Here we stipulate to choose k satisfying
1
k ~ "2supu("O),
K"p
and in (12.1)(1) replace Ale with the larger quantity
aA. = ( ; ) + (~) f (,"PK'~ U(.,O») (P-2)"t'

The numbers p and (1 being fixed, we let () be so small that, of the two terms making
up A .. , the second dominates the first, i.e.,
(15.1) (}<~( 2 )P-2

- sUPK"p u(·, 0)
The knowledge of such a () at this stage is only qualitative. It is part of the proof to
give an upper estimate for all the positive numbers () for which (15.1) is verified.
With these choices, the recursive inequalities (12.1) imply
y. _ 'Ybn k-~ Amy'l+m

n+l - .!'!±I!" n
(1 - (1)PN+2"
We proceed now as before and arrive at an analog of (12.3); namely, there exists a
constant'Y dependent only upon the data such that for all (1 E (0,1)
(15.2) sup
Q(ulJ,up)
u ~ 'Yff~ HUPdxdrf'A m~.
(1 - (1)
(
Q(IJ,p) )
If () and SUPK"p u(·,O) satisfy (15.1), it follows from (15.2) and the indicated
choices,
Supu(.,O)
(J N/2p
~ 'Y(PP/ )~ (Supu(.,O~
(2-p)~
N ( )1/2
l1 u p dxdr
K"p (1 - (1) \~"p 'l J(lJ,p)
J
Therefore for p.=N(p - 2) + p2,
P/IJ.
(15.3) supu(.,O)~ 'YL(N) (pp(})N/1J.

(
l1 uPdxdr )
K"p (1 - (1),. +p J J
Q(IJ,p)
(1) The inequalities (12.1) are written over the cylinders Q(6n , Pn) introduced at the
beginning of §7.
This inequality holds for all fJ,p,u for which (15.1) is verified. It also holds for
any pair of boxes
[(x o, to) + Q (fJ, p») and [(x o, to) + Q (ufJ, up)] ,
with arbitrary 'vertices' provided they are contained in ET • Fix any te (0, T) and
introduce the boxes
and Kp/2 x Ut, t}.
We rewrite (15.3) and (15.1) in terms of these cylinders, for which u= t.
LEMMA 15.1. Forallte(O,T)andp>Ojorwhich
-(,,-2)
( )
(15.4) t ~ 21'-1", sup u(x, t) ,
K,,/3
tMreholds
(15.5) ~~~ u(x, t) ~ (~)'Y N/IA (jf UpdxdT) p/IA

t/'lK"
For r > 0 introduce the quantity
(15.6) f(t)= sup {TN/Asupllu(.,T)lIoo,K,,}, ~=N(P-2)+p.

O<.,.<t p~.,. p'tbs
By possibly working within the time. interval (e, T) and then letting e '\, 0, we may
assume that f(t) is finite. This follows from Theorem 4.4. Let t* e (0, T) be the
largest time level for which
(15.7) t p/ A ~ 2" [f(t)]-(p-2) , VO < t ~ t*.
The knowledge of t* is only qualiwive. Shortly we will find a quantitative upper
bound for t*. Here we remark that owing to the definition of f (t) the condition
(15.4) holds for all p > r and all t e (0, t*). Consequendy (15.5) holds for all
t e (0, t*]. We estimate the integral on the right hand side of (15.5) as follows
ff
t
UPdxdT ~ p~ ft
(lI u(.,T)lI oo
p'tbs
,K,,)
1'-1
f U(X,T) dxdT
pN+'tbs
t/'lK,. t/2 K,.
_ (-
< 2)~ PP-
£:, { sup T
N/A
sup
lIu(.,Tlloo,K,.
.....IL
},,-1
t O<.,.<t p~r pP-'2
X { sup sup
o<.,.<t p~rK p
f ~:'1P-
dxdT}
"
~
= (~). p~ fp-l(t) Ilull{r,t}
16. Proof of Theorems 5.1 and 5.2 147
where the nonn 11·II{r,t} is defined in (4.4). Putting this estimate in (15.5) gives
.sup u(x, < "Y p-t!:rJ

t)- t N />..
!(t)P(P-l)/" IlluII P{r,t}
/" .
K
,,/2
We divide by (p/2)p/(p-2) and multiply by t N />... Then take the supremum for
p> r and use the fact that t E (0, t·) is arbitrary to deduce
'VO < t:S t*,
i.e.,
(lS.S) 'VO < t :S t·.
Thus it follows from (15.7) that (15.S) continues to hold for all 0 < t :S t., where
t. = "Y. II u 11 -(p-2)
{r,t"} .
16. Proof of Theorems 5.1 and 5.2

We first prove Theorem 5.1 for the case when the assumptions (5.1 )-(5.2) hold for
some 1:S r:S 2. In such a case we have p> max { 1; J~2}' and we may use the
iterative estimates (S.3). In these we discard the last two tenns and take 6 = 2. We
also stipulate to take
1
k> - sup u
- 2 Q(ulJ,up)
and arrive at
y.
n+l -
< "Y bn A~Ny'l+~N
! JII±,
!( 6) -"'tT n ,
6 = 2,
(1- u)P,-W-k, q-
where
(p-6)~ Ii.}
.Au = { (~) [ sup u] p + (~) p , 6=2,
p'P Q(ulJ,up)
and Yn are defined in (S.l). By Lemma 4.1 of Chap. I, Yn -+ 0 as n -+ 00, provided

we choose k from
Yo == if u 2d.xdr = C (1- u)N+p A;lk~(q-6),

Q(IJ,p)
for a constant C depending only upon the data. This implies

(16.1) sup
Q(fTlJ,fTp)
11.:5
'Y~~
~
(1 - u)"N"('9-Tf
(HuQ(',p)
2dxdT)
~
, 6=2.
We conclude the proof for the case r E [1, 2) by means of an interpolation process
similar to that of Lemma 4.3 of Chap. I; namely, consider the sequences Pn, (In
and the corresponding cylinders Q(n) == Q ((In, Pn), introduced in (12.4)-(12.5).
Define also the numbers Mn as in (12.6), and write (16.1) for the pair of boxes
Q(n) and Q(n+1). This gives
where
~ = { (;) M~-6)~ + (~) ~} .

Consider the two terms making up An. If for some n = 0, 1,2, ... the first term
dominates the second, we have
(J ) 1/(2-p)
(16.3) Mn< ( -
- pP
and there is nothing to prove. Otherwise, (16.3) fails for all n=O, 1,2, ... and
An:5 2 (~) ~ , n=O,l,2, ....
We deduce from (16.2)
Mn :5
'Y2np~ ~ (PP)~
~ Mn+1
(1 - u)P ,-
7i (H
Q(n+l)
11.
r
dxdT
)
~
The proof is now concluded as in Lemma 4.3 of Chap. I.

The proof of Theorem 5.1 for the case r > 2 is based on the fCCursive inequal-
ities (10.4). As before, we stipulate to take
1
k> - sup u.
- 2 Q(fT',fTp)
and majorise BIc by

B Ic <
_
k(2-r)~BfT'
where
17. Natural growth conditions 149
B. ~ { ( ; ) [Q(:~pA'-2)~ + (~) ~}
With these choices, we obtain from (10.4) the recursive inequalities
y; < '"'(bnB! II II r - q y;l+i

n+l - (1 _ u)i(N+p)k(r-2)!!p u oo,Q(8,p) n .
By Lemma 4.1 of Chap. I, Yn -+ 0 as n -+ 00, provided
y;o = jfurdXdT =
-
C(I- u)N+PB-lk(r-2)~llull(q-r)~
u oo,Q(8,p) '
Q(8,p)
for a constant C depending only upon the data. Thus
(16.4) sup u
Q(u8,up)
(r 2jfR+pj
< '"'( B(r 2jfN+pj lIull~~ ( jfurdXdT )

- (1 _ u);:!, u oo,Q(8,p)
Q(8,p)
Let Q(n) =Q(On, Pn) and Mn be defined as in (12.4)-(12.6). Then from (16.4)
where
B.~ {(;)M~-2)~ + (~)~}

The proof is now concluded as in the case r E [1, 2).
The proof of Theorem 5.2 is essentially the same. IfrE [1,2), it follows from
the recursive inequalities (9.4). If r > 2, we start from the global inequalities (10.6).
17. Natural growth conditions

Consider the Dirichlet problem
(17.1) { Ut - divlDulp - 2 Du = IDuI P , in nT, p> I,

ul r = I E LOO(r),
where r denotes the parabolic boundary of nT. The lower order term has the
'natural' or Hadamard growth condition with respect to IDul (see [48]). The notion
of weak solution is that of §2 of Chap. II. Here we stress that if we merely require
that IDul E V(flT), the testing functions must be bounded to account for the
growth of the right hand side.
The problem we address here is that of finding a sup-bound for a solution u.
It is known that weak solutions of (17.1) in general are not bounded, not even in
the elliptic case (see [15]). This is due to the fast growth of the right hand side with
respect to IDul. On the other hand the existence theory is based on constructing
solutions as limits, in some appropriate topology, of bounded solutions of some
sequence of approximating problems. The limiting process is possible if one can
find a uniform upper bound on the approximating solutions. Therefore the main
problem regarding sup-estimates for solutions of (17.1) can be formulated as fol-
lows. Assuming that a weak solution u of (17.1) is qualitatively bounded, find
a quantitative VlO (flT) estimate, depending only upon the data. In such a form,
the problem was fmt formulated by Stampacchia [93] in the context of elliptic
equations.
THEOREM 17.1. Let u be a bounded weak solution of (17.1). Then
07.2) lIulloo,aT :S F =IIflloo,r·
PROOF: By working with u+ and u_ separately, we may assume that u is non-
negative. Set
M = esssupu.
aT
If M > F, in the weak formulation of (17.1) we take the testing functions
(u - k)+, where k =M - E ~ F, for some E>O,
modulo a Steklov averaging process. These are admissible since they vanish on
the parabolic boundary of flT and are bounded. We obtain
esssup f (u - k)! dx + f f ID (u - k)+ I"dxdr

O<r<T
S1x{r}
JJ
aT
:s ffID(u-k)+I"(U-k)+dxdr
aT
:S E f f ID (u - k)+ I"dxdr.
aT
Thus if EE (0,1), we have (u - k)+ =0 in flT and
esssupu:S M - E.
aT
This contradicts the definition of M and proves the theorem.
COROLLARY 17.1. Assume f == o. Then (17.1) does not have any non-trivial
bounded weak solution.
17-(i). General structures

More generally we may consider the Dirichlet problem
u E C (0,T;L2(n))nLP (0, T; Wl,p(n)) ,
(17.3) { Ut - div a(x, t, u, Du) =
b(x, t, u, Du) in nT,
ul r = f E LOO(r),
where the p.d.e. satisfies the sbUcture conditions
(Bi) a(x, t, u, Du) . Du ~ ColDul P - <Po (x, t),
(B;) Ib(x, t, u, Du)1 ~ C21Dui P + <P2(X, t).
The lower order tenDs have the Hadamard 'natural' growth condition. Here Ci , i =
0,2, are given positive constants and the non-negative functions <Pi, i = 0, 2, sat-
isfy
(Bs)
where
(B 6 )
1 P
-: = (I-lI:o)-N .. 11:0 E (0,1).
q +p
THEOREM 17.2. Let u be a qualitatively bounded weak solution of(17.3) in nT.
There exists a constant C that can be determined quantitatively a priori only in
terms of the data. such that
PROOF: As before we may assume that u is non-negative. If M is the essential

supremum of u in nT, we may assume that M > 2I1flloo,r; otherwise there is
nothing to prove. In the weak formulation of (17.3), we take the testing function
(u - k)+, where
IIflloo,r ~ k < M.
This is admissible, modulo a Steklov averaging process, since it is bounded and it
vanishes in the sense of the traces on the parabolic boundary of nT • Calculations
in all analogous to those in §4-(ii) of Chap. II give
(17.4) sup
O<t<T
I
l1x{t}
(u - k)! dx + Co Jr [ ID (u - k)+ IPdxdT
2 J
aT
II
~ C2
aT
ID(u - k)+ IP(u - k)+dxdT
+ II
'Y {<Pox [u > k) + IP2 (u - k)+} dxdT.
aT
152 v. Boundedness of weak solutions
Here and in what follows we denote with 'Y a generic positive constant that can
be detennined a priori only in tenns of the data. Next choose k = M - 2e where
eE (0,1) is so small that M -2e ~ IIflloo,r, and
C2II ID (u - k)+ IP (u - k)+ dxdT ~ 2C2eII ID (u - k)+ IPdxdT

aT aT
~ ~o II ID (u - k)+ IPdxdT.
aT
Thus we may take
2e = min {lIflloo,ri ~ g: }.
Combining these calculations in (17.4), we arrive at
sup
O<t<T
I
12x{t}
(u - k)! dx + fr [ ID (u - k)+ IPdxdT
1
aT
~ 'Y II
aT
"oX [u > k] dxdT.
By RUder inequality and (B5HB6 ) the last tenn is majorised by
where 'Y is a constant depending only upon the data and
Ale == {(x,t) E nT I u(x,t) > k}.

Consider the sequence of increasing levels
e
kn =M - e - 2n ' n = 0,1,2, ... ,
and the corresponding family of sets
An == {(x,t) E nT I u(x,t) > len}.

TIlese remarks imply that for all n E N
sup
O<t<T
I
12x{t}
(u - kn )! dx + 11[[ ID (u -
aT
kn )+ IPdxdT
_ 'Y IAn 14
< (1+11:) ,
for a constant 'Y depending only upon the data. From this and the multiplicative
inequality of Proposition 3.1 of Chap. I,
i.e., for all n=O, 1, 2, ... ,
-p~ n
IAn+l I _< 'Ybn E IA 11+1<
,
It follows from Lemma 4.1 of Chap. I that IAn 1- 0 as n - 00 if
In this case we would have
u ~M - e a.e. aT
which contradicts the definition of M. Now
i.e.,
IAol ~ (!) P j1ulPdxdr.

n
If the right hand side is less than 'Y. we have a contradiction. Thus
To prove that lIullp,nT is bounded above only in terms of the data, we may assume,
modulo a shift that involves the supremum of the boundary data, that u is a bounded
non-negative weak solution of (17.3) vanishing on r in the sense of the traces. In
the weak formulation of (17.3), take the testing function
where Q is a positicve parameter to be chosen. We may also assume without loss

of generality that Ut E L2 (aT). We obtain
j! aI(j
o 0
(eOS -1)ds) dxdr + Q III DulPeoudxdr
at
:S O2 IltDulPeOUdxdr + 'Y 11(1 + CPo) eoudxdr,
at at
for a constant 'Y = 'Y (data) and for all t E (0, T). We choose =202 and set
Q
to obtain
IIwllt-p(aT) :S 'Yo + 'Yl II (1 + CPo) wPdxdr,
aT
for two constants 'Yi = 'Yi (data) , i=O, 1. Next by (Bs) - (B6 ),
11(1 +CPo) wP-dxdr

aT
Moreover since w(·, t) vanishes om on for a.e. t E (0, T), by the embedding of
Proposition 3.1 Chap. I,
IIwll:~.aT :S 'Yllwllt-p(aT)' 'Y = 'Y (data) .

Combining these remarks in (17.5) we conclude that there exist two constants
Co ,C1 , depending only upon the data such that
IIwll:~.aT :S Co +Clln,.I~lIwll:~.aT·
If T is so small that, say
then
IIwll:~.aT :S 2Co •
For arbitrary T> 0, the argument can be repeated up to covering the whole flT in
a finite number of steps.
The sup-bounds of §3 are essentially due to Porzio [87]. They follow a parabolic
version of DeGiorgi iteration technique (see [67]) and remain valid even in the
'linear' case p = 2. An effort has been made to trace the dependence of the var-
ious constants upon the size of the domains where the estimates are derived. We
have also computed how the various estimates deteriorate when t -+ O. In the
case of homogeneous sbUctures for degenerate equations (see §4), the interpo-
lation estimate (4.1) is of particular interest. It reveals a behaviour dramatically
different from the linear case p= 2. An estimate of this kind (i.e., for small e) had
been proved by Moser [83] for solutions of linear parabolic equations with mea-
surable coefficients. Generalizations to quasilinear equations with 'linear' growth
p = 2 are in [7,97]. The global estimates in ET of §4-(IV) are taken from [41].
We have given a different and simpler proof. For the porous medium equation
with power-type non-linearities, estimates of the same nature have been proved
by BtSnilan-Crandall-Pierre [10]. Analogous estimates for general non-linearities
appear in [4]. Still in the context of the porous medium equation, rather precise
local sup-estimates have been recently obtained by Andreucci [3]. For equation
with singular structure (1 < p < 2), the theory of local and global boundedness
has started only recently in [42] and [43]. Improvements to equations with general
structures are in [87]. The results of Theorems 5.1-5.3 are sharp. They will play
a central role in the Harnack estimates of Chap. VII. The integrability condition
(5.2) is sharp as shown by the counterexample in §13 of Chap. XII. The arguments
of §17 appear in [101].
VI
Harnack estimates: the case p > 2
1. Introduction
We will establish a Harnack-type estimate for non-negative weak solutions of de-
generate parabolic equations of the type
U E Gloc (0, Tj L~oc(ll}) n Lfoc (0, Tj WI!;:(ll)) , p> 2,

(1.1) {
Ut - div IDulp-2 Du = 0, in llT.
Since the equation is invariant by the scaling x - hx, t - hPt, h > 0. it may seem
plausible that the Harnack estimate of Hadamard [50] and Pini [86],(1) would hold
in the geometry of the cylinders
(1.2)
This is not the case, as one can verify for the explicit solution (x, t) - 8(x, t) in-
troduced in (4.7) of Chap. v. Let (xo, to) be a point of the free boundary {t= Ixl>'},
and let p> 1. Then if to is sufficiently large, the ball Bp(x o ) taken at the time level
to - pP intersects the support of x - 8 (x, to - PP) in a open set. Therefore
sup 8(x,to - PP) >

Bp(zo)
° and
This reveals a gap between the elliptic theory and the corresponding parabolic
theory. Indeed non-negative weak solutions of
(1) See (2.2) in the Preface.

2. The intrinsic Harnack inequality 157
div IDul p - 2 Du = 0, uE w,!:(n), p > 1,

satisfy the Harnack inequality, (2) whereas solutions of the corresponding parabolic
equation (1.1) in general do not.
Let u be a non-negative local solution of the heat equation in nT. Then for all
e> 0 there exists a constant 'Y depending only upon N and e, such that for every
cylinder Qp(xo, to) C nT and for every uE (0, 1),
(1.3) sup
Q .. ,,(Xo,to)
u <
-
'Y
(1 - u).;¥
N 2 ( ff u dxdr ) ~ ,
E
Q,,(xo,to)
where Qp(xo, to) is defined by (1.2) with p=2. This local sup-bound of the solu-
tion in tenns of the integral average of a small power of u, is a key fact in Moser's
proof of the Harnack estimate. An estimate of this kind does not hold for solutions
of (1.1) and it is replaced by the more structured inequality (4.1) of Chap. V. A
study of [83) however reveals that (1.3) continues to hold for sufficiently smooth
solutions of
(1.4)
With this in mind one may heuristically regard (1.1) as it were (1.4) written in
a time scale intrinsic to the solution itself and, loosely speaking, of the order of
t [u(x, t)]2- p. Next we observe that (2.2) in the Preface is equivalent to
(1.5)
The Harnack estimate of Krylov and Safonov [64) for non-divergence parabolic
equations is given precisely in this fonn.
This suggests that the number [u(xo, t o)]2-Pis the intrinsic scaling factor and
leads to conjecture that non-negative solutions of (1.1) will satisfy the Harnack
inequality with respect to such an intrinsic time scale.
2. The intrinsic Harnack inequality

The following theorem makes rigorous the heuristic remarks of the previous sec-
tion.
THEOREM 2.1. Let u be a non-negative weak solution off1.1). Fix any (xo, to) E
fl.r and assume that u(xo, to) > o. There exist constants 'Y> 1 and C > 1, depend-
ing only upon N and p, such that
(2.1)
(2) See [82,92,96J.

158 VI. Harnack estimates: the case p> 2
where
(2.2)
provided the cylinder
(2.3) Q4p(9) == {Ix - xol < 4p} x {to - 49, to + 49}
is contained in nT •
t o+ 0
I. I
p
4p
Figure 2.1
Remark 2.1. The values u(xo, to) are well defined since u is locally Wider con-
tinuous in nT.
Remark 2.2. The constants "'( and C tend to infinity as p - 00. However they are
'stable' asp'\.2, i.e.,
lim ",(N,p), C(N,p) = "'(N, 2), C(N, 2) < 00.

",,"2
Therefore by letting p _ 2 in (2.1) we recover. at least fonnally, the classical

Harnack inequality for non-negative solutions of the heat equation. Such a limiting
process can be made rigorous by the C,~ (nT ) estimates of Chap. IX.
In Theorem 2.1 the level 9 is connected to u(%0' to) via (2.2). It is convenient
to have an estimate where the geometry can be prescribed a priori independent of
the solution. This is the thrust of the next result which holds for all 9 > O.
THEOREM 2.2. There exists a constant B > 1 depending only upon N and p.
such that
3. Local comparison functions 159
(2.4) V (xo, to) E nT. Vp, (J > 0 such that Q4p«(J) c nT,
u(xo, to) ~ B { (~) ~ + (;) NIp [B!?!o) u(·,to + (J)f/J],

where
(2.5) A = N(P - 2) + p.
Remark 2.3. Inequality (2.4) holds for all p E (2,00), but the constant B is not
'stable' as p '\,2, i.e.,
lim B(N,p) = 00.
p'\.2
In (2.4) the positivity of u(xo, to) is not required and (J > 0 is arbitrary so that
Theorems 2.1 and 2.2 may seem markedly different. In fact they are equivalent,
i.e.,
PROPOSITION 2.1. Theorem 2.1 <=> Theorem 2.2.
In view'of Remark 2.3, the equivalence is meant in the sense that (2.1) implies
(2.4) in any case and (2.4) implies (2.1) with a constant 'Y = 'Y( N, p) which may
not be 'stable' as p'\,2. A consequence of Theorem 2.2 is
COROLLARY 2.1. There exists a constant B > 1 depending only upon N and p,
such that
(2.6) V(xo, to) E nT, Vp, (J > 0 such that Q4p«(J) c nT,
ju(z,t.)dx S B{ (~t + ( ; t P [U(Z.,t.+9)J VP }

B,.(zo)
2-(i). Generalisations
All the stated results remain valid if the right hand side of (1.1) contains a
forcing term f, provided
(2.7) q> (N +p)/p
and / is non-negative. We will indicate later how to modify the proofs to include
such a case.
3. Local comparison functions

Let p> 0 and k > 0 be fixed and consider the following 'fundamental solution' of
(1.1) with pole at (x, t):
~
(3.1) _ kpN { ( Ix - xl ) ;f-r } P-
Blc,p (x, tj x, l) == SN/>'(t) 1 - S1/>'(t) +'
where A is defined in (2.5) and

(3.2) S(t) = b(N,p)kP- 2pN(p-2)(t -l) + p>', t ~ f,
P-l
b(N,p) = A ( p~ 2 )
By calculation, one verifies that Blc,p (x, tj x, l) is a weak solution of (1.1) in RN x
{t > l}. Moreover for t = f it vanishes outside the ball B p (x) and for t > f the
function x- Blc,p (x, tj x, l) vanishes, in a C1 fashion, across the boundary of the
ball {Ix - xl < S1/>'(t)}. One also verifies that
Blc,p (x, t; x, l) ~ k,
and that forf~t~t*, the support of Blc,p (x,t;x, l)
D* == {IX - xl ~ Sl/>'(t)} x [f, t*],

is contained in the cylindrical domain
Q* == BS1/~(t.) (x) x [f, t*].

If u is a non-negative weak solution of (1.1) in Q* satisfying
u(x,l) ~ k for Ix - xl < p,

then
u(x, t) ~ Blc,p (x, t; x, l) , 'v'(X,t)EQ*.
This is a consequence of the following comparison principle.
LEMMA nT satisfying
3.1. Let u and v be two solutions of (1.1) in
u,v E C (O,T; L2(n») n £P (O,T; W1,p(n» n C (liT)
{
u ~ v on the parabolic boundary of nT.
Then u~v in nT .
PROOF: We write the weak form of (1.1) for u and v in terms of the Steklov-
averages, as in (1.5) of Chap. II, against the testing function
[(v - U)h]+ (x, t) =[ *! t+h 1

(v - u)(x, T)dT +'
h E (0, T), t E [0, T - h).
Differencing the two equations and integrating over (0, t) giv\ 'S
3. Local comparison functions 161
j[(v - U)hJ! (x, t)dx - j[{V - U)hJ! (x,O)dx

n n
= -2//n. [lDvl p - 2 Dv -IDuI JI - 2 Du] h·D [{v - U)h]+ dxdr.
As h -+ 0 the second tenn on the left hand side tends to zero since (v - u) + E
C (liT). Applying also Lemmas 3.2 and 4.4 of Chap. I we arrive at
/{v - u)~{x, t)dx

n
= -2 // (IDvIJl-2 Dv -IDuI JI - 2DU) ·D{v - u)dxdr :5 O.
n.n(v>u)
3-(i). Local comparison junctions: the case p near 2

The next comparison function is a subsolution of (1.1) for p > 2 and for p < 2
provided p is close enough to 2. For definiteness let us assume p E [2,5/2J and
consider the function
(3.3)
(3.4) t ~ t,
where the positive numbers II and ~(II) are linked by
(3.5) ~(II) = 1- v(p - 2) .

P
Introduce the number
(3.6) p(v) = 4(1 + 2v)/{1 + 4v),
and observe that
1 1
(3.7) 4 :5 ~(v) :5 2' for p E [2, p(v)J .
LEMMA 3.2. The number v> 1 can be determined a priori only in terms of N
aTul independent ofpE [2, 5/2], such that Qk,p is a classical subsolution of
!Qk,P - div (lDQk,pIJl-2 DQk,P) :5 0 in RN x {t > t}.
PROOF: For (x, t) ERN x {t>t}, set

IIzll == t(~) ~~) , :F == (1 - IIzll;!r ) + ' a == (p ~ 1) 2.
Then, by calculation,
(3.8) £* (g/c,P) = -v:F;!r + NaP-1:F

- -LaP-1Ilzll;!r + ~(v)a:F;!r IIzlI;!r.
p-l
Introducing the set
1( N(P-l»)]
IIzlIFT ~ 2 1+ N(P-l)+p
--Z..-
£1 == [ ,
we have
:F < p in £1,
- 2[N(P - 1) + p]
and therefore by (3.5)
r: (g/c,P) :5 -v:F;!r + NaP- 1 + ~(v)a:F;!r

- aP - 1 (N + -L)
p-l
IIzll;!r
:; ...-1 [-\(v) (N(P -"1) + p) o!t - 2(P~ 1)1

:;; .p--l[H N(P + ~2(P"":'~~I)1
-"1) p t-
:5 aP- 1
(~ 2(P~ 1») < O.
-
Within the set
1( N(P - 1) )]
£2 == [ IIzll ;!y < 2
p- 1 + N(P _ 1) + p ,
we have
:F> P
- 2[N(P-l)+p]
It follows from (3.5) and (3.7) that
p
£* (g/c,p) :5 -v ( 2[N(P _ 1) + p]
);!r + aP - 1 N + ~(v)a
p );!r a -2
:5 -v ( 2[N(P _ 1) + p] + p [NpaP + 1] .
Choosing
(3.9) V== max ~ [NpaP-2+1] (2[N(P-l)+P])~,

pe[2,5/2) P P
we have in either case
One verifies that for t = t
(hc,p (XI tj X, f) $ k,
and that for t $ t $ t*, the support of (hc,p,
'R,* == {Ix - xl < L'''(II) (t) } x[t I t*],

is contained in the cylindrical domain
C* == {Ix - xl < L'''(II)(t*)} x[t I t*].
Therefore if u is a solution of (1.1) in C* such that
U(X, f) ~ k
then
U(X, t) ~ (h:,p (x, tj ft, f) in C*.
Remark 3.1. The same proof shows that gkr is a sub-solution of 0.1) also for
P< 2, providedp is close to 2. Precisely ifpE (4 - p(v),2) ..

Let (xo, to) E nT and p > 0 be fixed, assume that U(XOI to) > 0 and consider the
box
where C is a constant to be detennined later. The change of variables
X-Xo
x----
P I
maps Q4p into the box Q == Q+ U Q- , where

Q+ == B4 X [0, 4C), Q- == B4 X (-4C, 0].
(0,0)
I
,
I
•
--------------------~
c/o 1
4 -4
Figure 4.1
We denote again with x and t the new variables. and observe that the rescaled
function
v(x,t)
tpp)
= U (x o,1 t) U ( xo+px, t o+ [U (xo,to )IP-2
0
is a bounded non-negative weak solution of
{
Vt - div (lDvlp-2 Dv) = ° in Q
v(O, 0) = 1.
To prove the Theorem it suffices to find constants 'Yo E (0, 11 and C > 1 depending
only upon N and p such that
ill! v(x, C) 2: 'Yo·

Construct the family of nested and expanding boxes
T E (0, I],
and the numbers
M.,. == sup v, N.,. == (1- T)-fl, T E [0,1),
Q...
where {J > 1 will be chosen later. Let To be the largest root of the equation M.,. = N.,..
Such a root is well defined since Mo = No. and as T /1. the numbers M.,. remain
bounded and N.,. /00. By construction
Since v is continuous in Q there exists within Q.,." at least one point. say (i, t).
such that
v (i, t) = N.,." = (1 - To)-fl.
The next arguments are intended to establish that within a small ball about i and
at the same time-level f the function v is of the same order of (1- To)-fl. For this
we make use of the R))der continuity of v and more specifically of Lemma 3.1 of
Chap. III.
Set
R = 1- 'To
2 '
and consider the cylinder with 'vertex' at (x, t)
By construction [(x, t) + Q (RP, R)] c Q!:tfa and therefore
sup v ~ N~ = 213 (1- 'To)-p == w.

[(z,i)+Q(RP ,R»)
If A is the number determined by Proposition 3.1 of Chap. Ill, we may choose

{J> 1 so large that (213 fA) > 1. Therefore the cylinder
[(x, t) + Q (aoRP, R)] , :0 == (~r-2 = [213 (1 ~'To)-pr-2 > 1

is contained in [(x, t) + Q (RP, R»), and
osc v < w.
[(z,i)+Q(ooRP,R») -
It follows that [(x, t) + Q (aoRP, R)] can be taken as the starting box in Lemma
3.1 of Chap. III. We conclude that there exist constants "( > 1 and 0:, Co E (0,1)
such that for all rE (0, R].
We let r =u R and then choose u so small that for all {Ix - x I < u R},
(4.1) v(x, t) ~ v (x, t) - 211+1"((1 - 'To)-pu Ot
= (1 - 2p+1,,(uOt ) (1 - 'To)-p
1
= 2(1 - 'To)-p.
The various constants appearing in Proposition 3.1 and Lemma 3.1 of Chap. III, in
our context, depend only upon N and p and are indePendent of v, (1) therefore the
number u can be determined a priori only in terms of N, p and (J. We summarise:
(1) See §3-(I) of Chap. III.

LEMMA 4.1. There exist a number u E (0,1) depending only upon N,p and P
such that
(4.2)
Remark 4.1. The location of (x, t) and the number 'To (and hence R) are deter-
mined only qualitatively. However in view of (4.1) the number u is quantitatively
determined as soon as P> 1 is quantitatively chosen.
4-(i). Expanding the positivity set

We will choose the constants P> 1 and C> 1 so that the qualitative largeness
of v(·, t) in the small ball BcrR(X) turns into a quantitative bound below over the
full sphere Bl at some further time level C. This is achieved by means of the
comparison functions of§3. Assume first thatpE [2, p(II)], where II is the number
determined in Lemma 3.2, and consider the function (ik,p introduced in (3.3). with
the choices
1 _
(4.3) k = -(I-or)
2 0
~
'
p=uR.
At the time level t=C the support of x-(ik,p (x, Cj x, t) is the ball
where
'Y = 'Y(u, II ) = 21(U)"¢l
'2 .
Choose
II
P = A(II) and
Since Ixl < 1 and t E (-1,0]. these choices imply that the support of x -
(ik,p (x, Cj x, t) contains B2. and by the comparison principle
inf vex, C) ~ inf (ik,p (x, Cj x, t)

ZeBI ZeBI
~ 2-(1+2&1) (i) "¢l {1 - (~ ·)6}~

p-
== 'Yo·
The various constants depend only upon N and p and are 'stable' as p'\. 2.
Turning to the case p ~ p( 1/ ), we consider the comparison function Bk,p (x, t; x, l)

introduced in (3.1)-(3.2), with the choice of the parameters k and p as in (4.3). At
t =C the support of Bk,p (., C; x, l) is the ball
I_ - ;;1'< {b[~(I- TO)--r-'(aR)N(P-') (C - i) + (aR)'}

= {b-yP-2 (1 - T o )(N-.BHp-2) (C -l) + (O'R)>-},
where
-y(N, (3) ="21 (0')

2 N and b=~ -p- ( )P-l
p-2
o
Choosing
3>-
(4.4) {3=N and C = b-yp-2'
we see that the support of Bk,p (-, C; x, l) contains B2, and by the comparison
principle,
(4.5) inf v(x,C) ~' inf Bk,p(X,C;x,l)

ZeBI ZeBI
~ (2)-(1+'i') Gt {1-
== -Yo·
Gtt'
Remark 4.2. These estimates involving the comparison function Bk,p hold for
all p > 2. However as p'\. 2, the constant -Yo in (4.5) tends to zero. The purpose
of introducing an auxiliary comparison function gk,p for p near 2 is to have the
constants under control as p approaches the non-degenerate case p = 2. We also
remark that gk,p is a subsolution of (1.1) only for p close enough to 2.

Let (x o, to) E nT, p> 0 and (J> 0 be fixed so that the box Q4p((J) is contained in
nT. We may assume that (x o, to) coincides with the origin and set u. == u(O,O).
If C and -y are the constants detennined in Theorem 2.1, we may assume that
(5.1)
Indeed otherwise
B == (2C);f-J ,
and there is nothing to prove. By Theorem 2.1 and (5.1)
u* ~ 'Yu(x, t*),
Consider the 'fundamental solution' Bk,p with pole at (0, t*) and with k='Y-1U*.
By the comparison principle, at the level t=(J, we have
(5.2)
where
u«,6);' ;t.; {1- (Sl~~I(t)t r ~
"Ixl < p,
Here .x and b are defined in (2.5) and (3.2) respectively. It follows from (5.1) that
('Y~~2 + 1) p>' ~ S(t) ~ ('Y:- 2 + 2~ )u~-2 ( ; ) p>'.
Therefore (5.2) gives
( pp)N!>.
u(x,(J)~u~!>'"9 'Y1, 'Y1 == 'Y1(N,p),
and the theorem follows with
B = max { 'Y-;>'!p; (20) ~ } .

We have shown that Theorem 2.1 implies the estimate of Theorem 2.2. To prove
the equivalence of Proposition 2.1, assume that (2.4) holds true for all (J> 0 such
that Q4p«(J) c nT . Choose
Then if Q4p«(J) c nT, (2.4) gives
u(xo, to) ~ 2B N (p-2)! >. in(f ) u(·, to + (J).

Bp Zo
5-(i). About the generalisations

The only tools we have used in the proof are the HOlder continuity of the so-
lutions of (1.1) and the comparison principle. The integrability indicated in (2.7)
6. Global versus local estimates 169
guarantees the local HOlder continuity.(l) Moreover the comparison principle re-
mains applicable since J ~ o.
6. Global versus local estimates
The assumption that· the cylinder Q 4p( 8) be contained in the domain of definition
of the solution is essential for the Harnack estimates of Theorems 2.1 and 2.2 to
hold. Indeed the function (x, t) - 8(x, t) introduced in (4.11) of Chap. V does
not satisfy (2.4) for Xo = 0 and to arbitrarily close to zero. This is not due to the
pointwise nature of (2.1) and (2.4). A Harnack inequality, with to arbitrarily close
to zero, fails to hold even in the averaged form (2.6). To see this let 1£ be the unique
weak solution of the boundary value problem
1£t - (I1£z IP-21£z)z = 0 in Q:= (O,I)x(O,oo),
('P)
{ 1£(0, t) = 1£(1, t) = 0 for all t ~ 0,
1£(,,0) = 1£0 E C:'(O, 1)
1£o(x) E [0,1], "Ix E (0,1) and 1£o(x) = 1 for x E U, i)·
We claim that
-1 1£
(6.1) 1£t > - - - in V'(Q).
- p-2 t
Let us assume (6.1) for the moment. Since 0 ~ 1£ ~ 1, by the comparison principle
(6.1) implies that
_ 1
- (11£.IP 21£_) < t> O.
• • z - (p - 2)t'
At any fixed level t, the function x-1£(x, t) is majorised by
-yx6
v(x,t) = ~, OE ( P-l
-p-,1 ) , (-y0)P-1 (1 - 0)(P - 1) ~ ~2'
ti=I p-
Indeed
1
- (Ivz IP- 2vz)z ~ (p _ 2)t and v(O, t) = 0, v(l, t) > O.
Therefore for every 0 E ( 7' 1) there exists a constant C =C (0), such that
C(O)
1£ (!, t) ~ t 1/(p-l)'
(1) See the structure conditions in §1 of Chap~ II, Theorem 1.1 of Chap. III and Theorem
3.1 of Chap. V.
Now assume that (2.6) holds for to=O, x o =!, 9=t and p= 1. Then for t>I
1 ~ canst (cp!-, + c*) -- 0 as t - - 00.
The proof of (6.1) is a particular case of the following
6-(i). Regularising effects

PROPOSITION 6.1. Let u E V (0, Ti WJ,P(I1» be the unique non-negative
weak solution of
Ut - div IDul p - 2 Du = 0, in I1T' p> I,
(6.2) {
u(·, 0) = U o E L2(11), U O ~O.
Then ifp>2.
-1 u
(6.3) Ut> - - - in 1)'(11) a.e. t > 0,
- p-2 t
andifI<p<2.
1 u
(6.4) Ut < -- - in 1)'(11) a.e. t > O.
- 2-p t
PROOF: We only prove (6.4). By the homogeneity of the p.d.e., the unique solu-
tion v of (6.2) with initial datum
v(·,O) = kp!-,uo, k > 0,

is given by
(x, t) - - v(x, t) = kp!-, u(x, kt).
If k ~ 1, v(·, 0) ~ U o and v(·, t) ~ u(·, t) in 11, Vt E (0, T). Fix t E (0, T) and let
k = (1 + ') for a small positive number h. Then
u(x, t + h) - u(x, t) = u(x, kt) - u(x, t)
= k~kp!-,u(x,kt) - u(x,t)
= k~v(x,t) - u(x,t)
~ (k~ - I)u(x, t).
By the mean value theorem applied to ( k ~ - 1),

h =.! u(x, t)
(6.5) u(x,t + h) - u(x,t) ~ 2 _ p (1 +e)2-li-t-
for some ( e (0, ~).If h <0, and Ihl « I, we have k < 1, v(·,O) ~uo and (6.S)
holds with the inequality sign reversed. Divide by h and in (6.5) take the limit in
'D'(I1) as h-+O.
7. Global Harnack estimates 171
Remark 6.1. In the proof of the proposition, the homogeneity of the operator and
the positivity of the initial datum, are essential.
7. Global Harnack estimates
The averaged Harnack estimate (2.6) holds with to arbitrarily close to zero for
non-negative local solutions of (1.1) in the strip ET == R N X (0, Tj, i.e.,
(7.1) {u E Cl~ (0, T; L~oc(RN)) n L:oc (0, T; Wj!;:(RN)) ,

Ut - dlV (/Du/ p - 2 Du) = ° 10 ET •
THEOREM 7.1. Let u be a non-negative solution 01(7.1) in E T . There exist a
constant B> 1 depending only upon N and p. such that
(7.2) 'V (xo, to) E ET , 'V p, (J > °such that to + (J < T,

f
Bp(XD)
u(x, to)dx~B{ (~)~ + ~)N/P[B!?L) u(·, to + (J)] AlP}.
Inequality (7.2) is more general than (2.6) in that the value u(Xo, to + (J) is
replaced by the infimum of u over the ball Bp(x o) at the time level to + (J.
In (7. 1) no conditions are imposed on x-u(x, t) as /x/- 00 and no reference
is made to possible initial data. The only global information is that the p.d.e. is
solved in the whole strip ET. Nevertheless (7.2) gives some control on the solution
u as /x/- 00, namely,
COROLLARY 7.1. Every non-negative solution 01 (7.1) in ET satisfies
(7.3) 'VxoERN, 'Vr>O, 'VeE (O,T)
sup sup ! ~dx~UXT B

--.L
[
1+ (-)
T --.L
p::-2 1Alp
u(xo,T-e)
O<.,.S;T-E p?r P P e~ P r
Bp(XD}
PROOF: Apply (7.2) with to=TE (0, T-e), divide by p~ and take the supre-
mum of both sides for p ~ rand T E (0, T - e).
8. Compactly supported initial data

The proof of (7.2) will be a consequence of the following:
PROPOSITION 8.1. Let v be a non-negative solution 0/ the Cauchy problem

V E C (R+j L2(RN)) n V (R+'i W1,P(R N )) ,
{ Vt -divIDvl p - 2 Dv=0 inEoo ==RNxR+,
(8.1)
( ) EC(Br) /orsomer>O,
v',O = Vo ~ oand { _ . RN,-B
=0 In r'
There exists a constant B=B(N,p) > 1. such that for all 8>0,
Inequality (8.2) can be regarded as a special case of (7.2) when additional

infonnation are available on the initial datum.
Basic facts on the unique solvability of (8.1) are collected in §12. We assume
the Proposition for the moment and proceed to gather a few facts about v.
LEMMA 8.1. For each t E R+, the function x--+v(x, t) is compactly supported
in RN, i.e.,
(8.3) V T E R +, 3 R = R(T) > 0 such that

supp{v(·,t)} C BR(T), Vt E (O,T).
Moreover the 'mass' is conserved, i.e.,
(8.4) !
aN
v(x,t)dx = !vo(x)dx,
Br
vt ~O.
PROOF: Consider the function 8k,p introduced in (3.1)-(3.2), with p = 2r and
+-m
(x, f) == (0,0). For t=O and Ixl <r,
8 ..... (.,0; 0, 0) ~ p } l=l

~ supVo,
Br
provided k is chosen sufficiently large. By the comparison principle, v 5 8k,p' The

second statement follows from the first by integrating the p.d.e. over RN x (0, t).
Remark 8.1 (Existence of solutions). In view of (8.3), a solution of the Cauchy
problem (8.1) can be detennined by fixing any T > 0 and solving the p.d.e. in
8. Compactly supported initial data 173
the bounded domain nT == B R(T) X (0, T) with homogeneous boundary data

on Ixl = R(T) and the same initial conditions as in (8.1). It follows from the
comparison principle of Lemma 3.1 that the solution is unique. This construction
and Proposition 6.1 also give the regularising inequality
-1 v
(8.5) Vt > ---
-p-~t'
in V' (Eoo) ,
and the estimate

(8.6) supv(·, t) $ SupVo'
RN -Br
COROLLARY 8.1. The quantities
(8.7) -
IIvll r = sup sup
tER+ p?,r
J v(x,t)
p>'-!(
P
-2) dx, A = N(p - 2) + p,
Bp
(8.8) f(t) = sup {7'N!~sup IIv(.,r)lIoo,B p }
O<T<t p?,r pp!(p-2)
are finite. Moreover there exists constants 'Y. and 'Y depending only upon N,p.
such that
(8.9) for all 0 < t < 'Y. Ilv"I~-P, andfor all p ~ r,

pp!(p-2)
IIv(.,t)lIoo,Bp $'Y tN!~ Wv"I~!~, A=N(p-2)+p.
PROOF: The estimate (8.9) is the content of Theorem 4.5 of Chap. V.
8-(i). Proof of Theorem 7.1 assuming (8.2)

It suffices to prove (7.2) for (xo, to) == (0,0) and (J E (0, T). Fix p = r > 0
and consider the Cauchy problem (8.1) with initial datum
if x E Br,
N-
if x E R \B
r•
By the results of Chaps. III and V, the solution u is locally bounded and locally
HOlder continuous in E T • Therefore up to the translation that maps (xo, to) into
the origin, u(·, 0) is continuous in B r . By (8.3) the comparison principle of Lemma
3.1 can be applied over the bounded domain B R(T) X (0, T) to yield v $ u. Then
(7.2) follows from (8.2).
174 VI. Harnack estimates: the case p > 2
9. Proof of Proposition 8.1
LEMMA 9.1. There exists a constant 'Y ='Y(N, p) such that
(9.1) for all 0 < t < 'Y.llvll~-P and for all p ~ T,

t
I I IDvlp-1dxdT ~ 'Yttp1+;!J Ilvlll!+~.

o Bp
PROOF: The calculations below are fonnal in that they require v to be strictly
positive. They are made rigorous by replacing v with v + e and lelling e - O. Let
x - (x) be a non-negative piecewise smooth cutoff function in B2p that equals
one on Bp and such that ID(I ~ 1/p. By the HOlder inequality
t
I jlDvIP-1(P-1dxdT
o Bap
t
~I I(T7IDvIP-IV-27(P-l) (T-7 v2 7)dxdT

o Bap
(i (i T-~v2~
c! 1
~ I Tl/PIDVIPV-2/P(PdxdT) P I dxdT) P
o Bap 0 Bap
== [J1 (t)] ~ [J2(t)]; .

To estimate J1(t), in the weak fonnulation of (8.1) we take the testing function
I{J == tl/pvl-2/p(p to obtain
t
P It;v2~(PdX + P - 2 IIT;IDvIPv-2/P(PdXdT
2(P-l) p
B2p 0 Bap
t
=p I IT;IDvIP-2vl-2/P(P-IDV.D(dxdT
o Bap
t
+ 2(P 1-1) If.1 7"P - 1V2c!
p (PdxdT
.
o B 2p
In the estimates below 'Y denotes a generic positive constant that can be detennined
a priori only in tenns of N and p and that might be different in different contexts.
By Young's inequality and the structure of the cutoff function (
9. Proof of Proposition 8.1 175
P It
11'; IDvlp-2Vl-2/p(P-1 Dv· D(dxd1'

o Bap
We conclude that there exists a constant "Y="Y(P) such that
Estimating L i , i = 1, 2, separately we have
L1 :::;"Ypl+~/t 1''4!-1 (1'N/>. IIv(" 1')IIoo'Bap)(P-a~P+l)

\ (2p)~
LI v(x, 1') dx) d1'
p~
o h
:::; "Ypl+~ It
1'£t!-1 [/(1')] (p-a~Ptl) Ilvll r d1'

o
:::; "Y pl+~ t£t! [/(t)] (p-a~Ptl) Ilvll r .
By Corollary 8.1
I(t) :::; "Y Ilvll~/'\

Therefore for all such t,
L1 :::; "Y pl+~ tf (tlllvll~-2) f IIvll!+~

). 1. 1+ E.::.!
:::; "Y p1+-;=f tx Ilvll r ---x-.
Next
L2 :::; "Ypl+~ It1'*-1 (1'N/>' IIv("(2p)~

o
1')lIoo,Ba ~ p ) LI h
v(x,1') dx) d1'
p~
:::; "Y pl+~ t* Ilvlll!+~ .

On the other hand J2(t) == L2 and the Lemma follows.
Remark 9.1. The estimates above show that 'Y ='Y( N, p) /00, as p \. 2.
Remark 9.2. The proof is independent of the fact that the initial datum is of com-
pact support and that v is a solution in the whole 1:00 • The lemma continues to hold
for every non-negative solution in 1:T for some T > 0, provided the quantities
lIu(·, T)lIoo,B~
sup sup 2
O<?<r-E p>r ppl(p-)
are finite for all E E (0, T). The conclusion will hold for all times
(9.1)' o < t :5 'Y.llullr,T-E.

Remark 9.3. Lemma 9.1 is independent of the homogeneous structure of the
p.d.e. Indeed it continues to hold, in the same form, for equations with homoge-
neous structure as in (2.8)-(2.9) of Chap. V, provided the analog of Corollary 8.1
is in force. A version of this Corollary can be proved, by essentially the same tech-
nique, for solutions of equations with general structure such as (2.1) of Chap. V.
Remark 9.4. The functional dependence upon t on the right hand side of (9.1) is
optimal, as shown by the following example. The family
IX I ...L..}~
{l-'Y
p-l
.>. -NI.>.
(9.2) Br(x,t)={t+r) p ( 11'>') ,
(t+r'>']
+
'Yp = ( ~1)~ -p-'

p-2
p > 2; t, r > 0,
solves (8.1) with initial data Brh 0) supported in the ball B r. By calculation we
have, for all p ~ r
!
t
!IDBrIP-ldxdT = 'Y(N,p) [t + r'>'] 11'>' - r,

o B2(1
where 'Y(N, p) is an explicit constant independent of rand t. The assertion follows

by letting r -+ O.
Let Eo denote the integral average of the initial datum, i.e.
(9.3)
By the conservation of mass (8.4)

10. Proof of Proposition 8.1 continued 177
Ilvll r =sup sup p-~fv(x,t)dx

teR+ p?r
B"
$ sup sup p-~ f v(x, t) dx

teR+ p>r
- RN
--r-~Eo·
Therefore Lemma 9.1 can be rephrased as
LEMMA 9.2. There exists a constant "Y="Y(N,p) such that
rP
(9.4) for all 0 < t < "Y. EC- 2 ' andfor all p ~ r
t 1
(9.5) ~ fflDvlP-ldxdT $"Y (:p)X (;)~ E~+~.

OB"
1O. Proof of Proposition 8.1 continued
Let ( be the standard cutoff function in B2r that equals one on B r . In the weak
formulation of (8.1) take x -+ (P (x) as a testing function and integrate over B 2r X
(0, to), where
(10.1)
and e is a small positive constant to be chosen. Making use of (9.5) we obtain
to
f
B2..
v(x, to) dx ~ 2- N
B..
f Vo dx - ~ f
OB2 ..
f IDvl p - 1dx
~ 2- N Eo - "Y (e"Y.)l/~ Eo
-- 2-(N+l) E 0,
for the choice
We summarise:
LEMMA 10.1. There exist a constant c. that can be determined a priori only in
terms of N and p. such that
f v(x, to) dx ~ 2-(N+l) Eo,

B2r
Since v is continuous, there exists some x E B2r such that

(10.2)
Next we apply the Harnack estimate of Theorem 2.1. For this, construct the cylin-
der
Q_ B (-) { 4C (6r)" 4C (6r)" }
= 46r X X to - [v(x, t o)],,-2 ' to + [v(x, to)]" 2 '
where C is the constant determined in Theorem 2.1 and 6 > 0 is to be chosen. Such
a box is contained in Loo if
t > 4C(6r)"
o -
v x,to )],,-2·
[(_
Using (10.1) and (10.2) we see that this is the case if

r" 4C(6r)"2(N+l)(p-2)
E'II
,. -->
E~-2 - g-2
Therefore Q C Loo for the choice
It follows from Theorem 2.1 that

_ 4C(6r)"
"{Ix - xl < 6r} at the time level t = to + ,,-2
[v(x, to)]
(10.3) v(x, t) ~ -y-1 v(x, to)

~ 2-(N+1)-y-1 Eo
== coEo·
Therefore we have located a ball or radius 6r about x and at the time level f where v
is bounded below by eoEo. In view of(10.1) and (10.2) the time level tis bounded
above by
(l0.4)
11. Proof of Proposition 8.1 concluded 179
11. Proof of Proposition 8.1 concluded

Let (J > 0 be fixed. We may assume that
(11.1)
Indeed otherwise
Eo:5 B ( 8);;!-'
rP ,
and (8.2) becomes trivial. We will expand the bound below on v given by (10.3),
up to the time level 8 over the ball B r • Consider the 'fundamental solution' 8/c,p
introduced in (3.1)-(3.2), with pole at (x, t) and
p=6r,
By the cQmparison principle,
(11.2) vex, t) ~ 8 co E o .6r (x, tj x, t) , V x E R N , Vt ~ f.
Let us estimate below the right hand side of (11.2) at the time level t=8. First by
(10.4) and (11.1)
- 1
8-t>-8
- 2 '
therefore the support of X-+8co E o .6r (x, 8j x, t) will cover the ball B 4r about the
origin if
8(8) = b(eoEo)p-2 (6r)N(p-2) (8 - t) + (6r)'\

~ ~ (c oE o)P-2 (6r)N(p-2) 8
~ (8r)".
This will occur if
(11.3)
We may assume that (11.3) is in force and estimate above
8(8) :5 b [eo6Nt-2 E:- 2 r N(p-2) 8 + 6'\r N(p-2) r P
:5 {b [eo 6N t- 2 + ~:} ~-2 r N(p-2) 8

== 61 E:- 2 r N(p-2) 8.
We return to (11.2). These estimates imply that for all x E Br for t =8
Therefore
Eo == f Vo dx :$ 6;>'lp (0 ) NIII [
rP W! v(·, 6}] >'111 ,
Br
and the Proposition follows with B=max{(2Bl}~; BF; 6;>'11I}.

12. The Cauchy problem with compactly supported
initial data
1be proof of (7.2) is based on comparing u with the unique solution of the Cauchy
problem
V E C (R+; L2(RN}) n V (R+; Wl'II(RN» ,

{ Vt - div IDvllI-2Dv = 0 in !Joe == RN xR+,
(12.1)
EC(Br} forsomer>O,
v (,,0) = Vo ~ 0 and { _ . RN\B
= 0 10 r.
Such a problem plays a role also in the theory of Harnack estimates for non-
negative weak solutions of (1.1) in the singular case 1 1 will be studied in
Chaps. XI and XII. To render the theory of Harnack inequalities self-contained
we briefly discuss the unique solvability of (12.1) for all p > 1. First, the notion of
solution is:
(a) For every compact subset JC c RN and for every T > 0, u is a local solutions
of the p.d.e. in JC x (0, T), in the sense of (1.2)-(1.4) of Chap. II.
(b) v(·, t} -+ Vo in L2(RN}.
PROPOSITION 12.1. There exists a unique solution to (12.1) for all p > 1.
PROOF: For n = 1, 2, ... let Bn be the ball of radius n about the origin and con-
sider the boundary value problems
Vn E C (R+; L2(Bn» n V (R+; W:'II(Bn»

(12.2) { Vn,t - div IDvn lp-2 DVn = 0 in Bn x (0, n),
vn(-,O} = Vo E L2(Bn}.
12. The Cauchy problem with compactly supported initial data 181
The functions Vn vanish in the sense of the traces on Ixl = n. We regard them as
defined in the whole Eoo by extending them to zero for Ixl > n. The problems
(12.2) can be uniquely solved by a Galerkin(l) procedure and give solutions Vn
satisfying
(12.3)
The sequence {v n }n6N is equibounded(2) in E oo , and uniformly Holder continu-

OUS(3) in Ex (e, 00) for all e > O. In the weak formulation of (12.2), we take the
testing function (v n + e )p-2Vn , modulo a Steklov average. Letting e - 0 gives
(12.4) VneN.
Therefore
Vn e LP (R+; W1,P(R N )) uniformly in n.
A subsequence can be selected and relabelled with n such that Vn - v uniformly
on compact subsets of Eoo and weakly in LP (R+; Wl'P(RN)). The limit v is
in the function space specified by (12.1), it is HOlder continuous in Ex (e, 00)
for all e > 0, and it satisfies the p.d.e. weakly in Eoo. To prove this we select a
compact subset IC c RN and some T > O. Then if n is so large that IC C B n , we
write (12.2) weakly against testing functions supported in ICx (0, T). The limiting
process can be carried on the basis of the previous compactness and the non-linear
term is identified by means of Minty's Lemma. (4) It remains to show that v takes
the initial data Vo in the sense of L2(RN). Let 1/ e (0, 1) be arbitrary and let VO,'l
be a mollification of Vo such that
!lvo - VO,'l!l2,RN --+ 0 as 1/ '\. O.

In the weak formulation of (12.1), take the testing function Vn - vo,'l modulo a
Steklov average. If n is so large that supp[VO,'l] C B n , we obtain
t
jlvn - VO,'l12(t)dx :::; !lvo - VO''lIl~,RN + 'Y j jIDVo''lIPdxdT, "It> 0,

RN ORN
for a constant 'Y depending only upon p. Letting n - 00,
t
IIv(·, t) - voll~,K: :::; 211vo - VO''lIl~,RN + 'Y j jIDVo''lIPdXdT, "It> 0,

ORN
(1) See J.L.Lions [73] or Ladyzhenskaja-Solonnikov-Ural'tzeva [67].

(2) By the weak maximum principle of Theorem 3.3 of Chap. V.
(3) By the HOlder estimates of Theorem 1.2 of Chap. III and Theorem 1.2 of Chap. IV.
(4) See G. Minty [78].
for all compact subsets IC eRN. From this,
]~ IIv(" t) - Vo1l2,K: = 211vo - Vo,,,1I2,RN, for all '1 E (0,1).
To prove uniqueness we first write the p.d.e. satisfied by the difference w =VI - V2
of two possibly distinct solutions originating from the same initial datum vo , i.e.,
wE C (R+;L2(RN») nLP (R+;Wl,P(RN »),
{
(12.5) Wt - div (IDvIIP-2 DVI - IDV2l p - 2DV2) = 0, in 1:00 ,
w(',O) = 0, in L~oc(RN).
In the weak formulation of (12.5), take the testing function w(, modulo a Steklov
average, where x --+ (x) is a non-negative piecewise smooth cutoff function in
the ball B2R that equals one on BR and such that ID(I ~ 1/R. This gives, for all
t>O,
!
t
~!IWI2(t)dX + !<IDVIIP-2DVI -IDV2I,,-2Dv2,Dvl - DV2) (dxd1'

Bil OB2Il
!
t
=- !<IDVIIP-2DVI-IDV2I,,-2DV2,D()Wdxd1'.
OB21l
1be second integral on the left hand side is non-negative(l) and it is discarded.
1berefore
! IwI2(t)dx ~ IIwll p,l;ao (IIDVlllp,l;; + IIDV2l1p,l;ao)P-1

Bil
Uniqueness follows letting R --+ 00.

A similar argument proves the following weak comparison principle.
LEMMA 12.1. Let Vi, i = 1,2, be two weak solutions to (12.1) originating from
bounded and compactly supported initial data Vo,i, i= 1, 2, satisfying Vo,l ~Vo,2'
Then VI ~ V2 in 1:00 ,
PROOF: In the weak formulation of the difference w = VI - V2, take the testing
function w+( modulo a Steklov average.
(1) See Lemma 4.4 of Chap. I.

In the classical work of Moser [81,82,83], ~e HOlder continuity is implied by the

Harnack estimate. Conversely we use the HOlder estimates of Chaps. III and N
to establish a Harnack inequality. This point of view, even though not explicitly
stated, is already present in the work of Krylov and Safonov [64]. The results of
§2 have been established in [40]. A version of these holds for non-negative weak
solutions of the porous medium equations
u E C' oc (0, Tj L~oc(fi») , urn E L~oc (0, Tj W,!;;(fi») ,

(13.1) {
! u - Llurn =0 in fiT, m > 1.
In particular the intrinsic Harnack estimate takes the form
THEOREM 13.1. Let u be a non-negative weak solution 0/ (13.1). Fix any
(xo, to) E fiT and assume that u(xo, to) > O. There exist constants 'Y > 1 and
C> 1, depending only upon Nand m, such that
(13.2)
Q4p(9) == {Ix - xol < 4p} x {to - 49, to + 49}

is contained in fiT.
A version of Corollary 2.1 for (13.1) appears in [6]. For the remaining results
we refer to [40]. The 'fundamental solutions' Blc,p are due to Barenblatt [8]. The
comparsion function (ilc,p for p close to 2, is introduced in [40]. The technical
device of the family of expanding cylinders Q.,. in §4 appears in Krylov -Safonov
[64]. The regularising effects of Proposition 6.1 are due to Benilan and Crandall
[9].
VII
Harnack estimates and extinction
profile for singular equations
1. The Harnack inequality

We will investigate the local behaviour of non-negative solutions of the singular
p.d.e.,
(l.1) {
u E C,oc (0, T; L~oc(n») n Lfoc (0, T; W,!:(n)), 1 <p<2
Ut - div IDuIJI-2 Du = 0, in UT.
Weak solutions of (1.1) exhibit an intriguing behaviour. Even though in general
they are not locally bounded,(I) they might become extinct after a finite time. It
turns out however that the Harnack inequality of Theorem 2.1 of Chap. VI contin-
ues to hold provided p satisfies the further restriction
2N
(1.2) N + 1 <p < 2.
We will show that such a range of p is optimal for a Harnack estimate to hold. The
extinction in finite time, the Harnack inequality and the LOO-estimates are linked
by the range (1.2) of the parameter p.
THEOREM 1.1. Let u be a non-negative weak solution of (1.1 ) and let (1.2) hold.
Fix any (x o, to) E nT and assume that u(xo, to) > O. There exist constants 'Y> 1
and cE (0, 1). depending only upon Nand p. such that
(1.3)
(I) See §5-(IV) of Chap. V.

1. The Harnack inequality 185
where
(1.4)
(1.5) Q4p(0) == {Ix - xol < 4p} x {to - 40, to + 40}
is contained in nT.
Remark 1.1. The statement of Theorem 1.1 is the same as that of Theorem 2.1 of
Chap. VI except that now the constant c is 'relatively small'; that is, the positivity
of u(xo, to) spreads over the ball Bp(xo) but is preserved only for the 'relatively
small' time c [u(xo, t o)]2-Ppp.
Remark 1.2. As p '\. J~l' the constant "Y tends to infinity and c tends to zero.
However these constants are 'stable' as p /' 2. i.e.,
lim "Y(N,p), c- 1 (N,p) = "Y(N, 2), c- 1 (N,2) < 00.
p/2
Therefore the classical Harnack inequality for non-negative solutions of the heat
equation can be recovered by letting p /' 2 in (1.3). The limiting process can be
made rigorous by the C,7/:
(nT ) estimates of Chap. IX.
t o+ a
1 I
p
4p
Figure 1.1
Fix (xo, to) E nT, assume that u(xo, to) >0 and construct the truncated
'paraboloid' of two sheets
"{s.,,} (x o, to) == { s ~ It - tol > c [u(xo, to)]2-PIx - xol PIJP } ,

where c is the number claimed by Theorem 1.1 and IJ and s are positive parameters.
A consequence of (1.3) is the following:
186 VB. Harnack estimates and extinction profile for singular equations
COROLLARY 1.1. Let u be a non-negative local weak solution of (1.1) and let
p be in the range (1.2). There exist constants c E (0,1) and 'Y > 1 that can be
determined a priori only in terms of N and P. such that
°
V(Xo, to) E nT, v 8 > such that P{s,4} (XO, to) C nT,
(1.3') u(xo, to) ~ 'Yu(x, t), V(x, t) E P{s,l} (xo, to).
In particular for solutions of the Cauchy problem, we have

COROLLARY 1.2. Let u be a non-negative local weak solution of (1.1) in RN x
R + and let p satisfy (1.2). There exist constants c E (0, 1) and 'Y > 1 that can be
determined a priori only in terms of N and p, such that (XOI to) ERN X R + .
(1.3")
Figure 1.2
l-(i). Harnack estimates of 'elliptic' type

The p.d.e in (1.1) is singular in the sense that the modulus of ellipticity of its prin-
cipal part becomes infinite at points where IDul = O. At these points the 'elliptic'
nature of the diffusion dominates the 'time-evolution' of the process; that is, the
positivity of u at some point (xo, to) 'spreads' at the same time level over the
1. The Harnack inequality 187
full domain of defmition of x -+ u(x, to). This is the content of Theorem 14.1
of Chap. IV and holds for non-negative solutions of (1.1) for the whole range
1 < p < 2. When p is in the range (1.2), such a property can be made quantitative
and takes the form of an elliptic Harnack inequality.
THEOREM 1.2. Let u be a non-negative weak solution off1.1) and let (1.2) hold.
Fix any (x o, to) e nT and construct the cylinder
Q4p(O) == {Ix - xol < 4p} x {to - 40, to + 40},
(1.6) {
0= c [u(x o , t o )]2- p pP, c> 0,
where c is the constant of (1.4). There exist a constants.'Y > 1, depending only upon
N and p, such that
(1.7)
provided Q4p(O) c nT . The constant'Y /00 as eitherp'-.. J~l or p/2.

Remark 1.3. The strict positivity of u(xo, to) is not required and the Harnack
estimate (1.7) holds at the same time level.
Remark 1.4. While Theorem 1.1 is •stable' as p /2, this is not the case of Theo-
rem 1.2. Indeed (1.7) fails for solutions of the heat equation. To verify this consider
the heat kernel in I-space dimension
r(x,t) = ~e-~, xeR,

v41rt
and apply (1.7) for the sequence of points (x o, to) == (n, 1), n e N. If Theorem
1.2 were to hold for p= 2, we would have for some p> 0
r(n, 1) ~ 'Y r(n + p, 1).

Letting n -+ 00 we get a contradiction.
1-(U). Generalisations
The theorems generalise to the case when the right hand side of ( 1.1) contains
a forcing term I(x, t, u) provided
(1.8) o ~ I(x, t, u, Du) ~ lo(x, t) + F u,

for a constant F and a function fo satisfying
N+p
(1.9) q>--.
P
188 VII. Harnack estimates and extinction profile for singular equations
2. Extinction in finite time (bounded domains)
PROPOSITION 2.1. Let n be a bounded domain in R N and let u be the unique

non-negative weak solutior. of
u E C (R+; L2(n)) n LP (R+; wJ,p(n)) , 1<p<2,

(2.1) { Ut - div IDul p - 2 Du = 0, in nT,
u(·,O) = Uo E LOO(n), Uo ~ O.
There exists a finite time T· depending only upon N, p and u o , such that
(2.2) u(·, t) == 0, forall t ~ T·.

Moreover
'1 .11 Uo 11 22 -nP Iurll N!p-2l+2p

2l1i if max {I ;Z2} <p<2
j
(2.3) O<T· < { '
- "', •• 11 U o 11 s,n,
2- p 8 = N(2-p)
P if I<P$;Zl' N~2
where '1. and '1•• are two constants depending only upon Nand p.
Remark 2.1. There is an overlap in the range of p in the two estimates of (2.3).
For 1 lder continuous in x [e, 00) for all e > O. Assume first that p ~ ;~l. In
keeping with the notation of Chap. V we let
Ar == N(P - 2) + rp, Vr> 1 and A == N(p - 2) + P for r = 1.

In the weak formulation of (2.1) take u as a testing function, modulo a Steklov
average. This gives
(2.4)
By the U>lder inequality and the embedding of Corollary 2.1 of Chap. I, we have
These remarks in (2.4) yield the differential inequality
!lIu(.,t)lb,n +'Yll1u(.,t)II~:r: $ 0 in V'(R+),
where
By integration
2. Extinction in finite time (bounded domains) 189
(2.5) IIu(·, t)II~1f

, :::; IIuoll~1f
, - (2 - p)'Ylt,
as long as the right hand side is non-negative. From this,
(2.6)
and
0< r* :::; 1'''lnl ii lIuoll~:6.
Remark 2.2. The estimate (2.6) is 'stable' as P '\. J~l' i.e., as ~ '\. O. As p /
2, the boundary value problem (2.1) tends to the corresponding boundary value
problem for the heat equation(l) for which there is no extinction in finite time.
Accordingly, letting p / 2 in (2.6) gives
lIu(·, t)II2,n :::; lIuo ll2,n e-th2InI2/N,

where l' is the constant of the embedding of Corollary 2.1 of Chap. I. Next we take
p in the range
2N
I<P<N+l' N~2.
In the weak formulation of (2.1) we select, modulo a Steklov average, the testing
function u s - 1 , where
2-p
(2.9) s=N-- > 1.
P
This gives
where
1'2 == (s-l) (s+ ~_ 2»)"

By the embedding of Corollary 2.1 of Chap. I and the specific choice of s, we have
(1) If u(p) are the solutions of (2.1) and u is the solution of (2.1) with p = 2, the
convergence takes place in the sense
U (p) ,U"'i
(p) --+ U, U"'i .
In Co< [-n X (E,OO )] ,vE > 0 ,
\oJ i = 1,2, ... ,N,
and
uniformly in [0, TI, "IT> o.
Estimates of u~) in Co< (ax (E, 00») uniform in p>2N/(N +2) will be given in Chap. X.
190 Vll. Harnack estimates and extinction profile for singular equations
We conclude that
d
dt lIu(·, t)II.,n + '13I1u(·, t)II::t: ~ 0 in Z>'(R+), '13 == '1-P '12.
From this, by integration
u t < u {
II (., )1I.,n - II olls,n - 1
( )
2-P'13 t }
~
lIuolI!1 +
This in tum implies (2.3).
Remark 2.3. These estimates deteriorate as p /' JZl
and are 'stable' as p '\.1.
However we cannot infer the convergence of (2.1) to a boundary value problem,
in some reasonable topology, since the Hinder estimates of Chap. IV deteriorate as
p'\.I and (2.1) only gives IDul EL1(nT) uniformly in p.
Remark 2.4. Proposition 2.1 holds for solutions of variable sign. The only modi-
fication in the proof occurs in the case 1 <p< JZ1'
N ~ 2. For this it suffices to
take the testing functions lul s - 2 u.
2-(i). The Harnack inequality and the rate of extinction

1be extinction profile is defined as the set 8[u>O] n noo. By Theorem 16.1 of
Chap. IV the extinction profile of the solution of (2.1) is the portion of hyper-
plane n x {t =T*}. The Harnack estimate cannot hold in a 'parabolic geometry'
independent of u, say, for example, within a cylinder of the type
Qp(xo, to) == Qt(xo, to) u Q;(xo, to), Q;(xo, to) == Bp(xo) x {to ± PP}·
Indeed if (x o, to) belongs to the extinction profile and p is so small that Qp(x o, to) C
n oo , the solution u of (2.1) is positive in Q;(xo, to) and it vanishes identically in
Qt(xo, to).
1be intrinsic geometry of the Harnack inequality (1.3) implies an estimate of the
rate of extinction of u( ., t) as t /' T*. We let
M = lIull oo ,nco •
LEMMA 2.1. Let u be the unique non-negative weak solution 0/ (2.J) and let p be
in the range (1.2). There exists a constant '1 depending only upon Nand p, such
that/or all (x, t) E n x (~. , TOo)
3. Extinction in finite time (in a,N) 191
(2.7) u(x,t):S 'Ymax { M 2 -p; [dist{x,8n}]p

T.}~ (T*
~
-t)~ .
PROOF: Fix x En and ~. :S t :S T*, assume that u(x, t) > 0 and set
(2.8)
T*
4p == min { dist{x,8n}; ( 2M2-p
)l/P} .
We apply (1.3) over the ball Bp(x) and the cylinder
Q4p(X, t) == B4p(X) X {t - [u(x, t)]2- P(4p)P, t + [u(x, t)]2-P(4p)P} .

By virtue of the choice (2.8) such a cylinder is contained in noo and (1.3) holds
for it. We must have
T* - t ~ c[u(x,t)]2- p pp,
otherwise, by the Harnack estimate, u(x, t) = 0 against the assumption. This in
tum implies the lemma.
3. Extinction in finite time (in RN)
PROPOSITION 3.1. Let u be the unique non-negative weak solution ofthe Cauchy
problem
(3.1)
Then, if
2N
(3.2) l<p< N+l, N~2,
there exists a positive number T* depending only upon N, p and U o such that
u(·, t) == 0, "It ~T*.
Moreover
2-p
(3.3) o < T* < 'V·*llu 11 2s,B.,.'
- I 0-p 8=N--,
P
for a constant 'Y** depending only upon N and p.
PROOF: The solution of (3.1) can be constructed as the uniform limit in Eoo of
the sequence {un}nEN of the solutions of the problems in bounded domains(l)
(1) See §12 of Chap. VI.

Un E C (R+; £2 (Bn)) n lJ' (R+; WJ,P(Bn ») ,

{
! Un - div IDun l p - 2DUn = 0 in Bn x (0, n)
un(-,O) = Uo E £2(Bn ).
If p is within the range (3.2). by Proposition 2.1. the extinction time of Un T:
is estimated independent of meas Bn. Moreover since Un ~ Un+l' we also have
T: ~ T:+l' This proves (3.3).
Remark 3.1. The proposition holds for data of variable sign. Also U o need not be
of compact support and it would suffice to assume
8=N~.
The proof is the same except for making precise in what sense the solutions of
(3.1)n converge to the solution of (3.1). (2)
3-(i). The range (1.2) is optimal for a Harnack estimate to hold

Fix (xo, to) E RN X (0, T·), where to is so close to T· to satisfy
(3.4) T. C
- to < 4P to,
where c is the constant appearing in (1.4). Now choose p>O so large that
(3.5)
By the choice (3.4), the box
Q4p(Xo, to) == B4p(Xo)

x {t o- [u(xo, t o)]2- p (4p)P, to+[u(xo, t o)]2- p (4p)p}
is contained in Loo. If (1.3) were to hold for 1 < p < J~l' N ~ 2, for some
constants c and 'Y independent of p, it would give
o < u(xo, to) ~ 'Y inf

zEB.. (zo)
u(x, T·) = O.
Remark 3.2. The choice (3.5) is possible in the whole Loo.

The same arguments imply that if p is in the range (1.2), no extinction in finite
time can occur, for solutions of (3.1) . Within such a range, the Harnack estimate
holds. Therefore if a finite extinction time T· were to exist, the choices (3.4)-(3.5)
would give u(x, T·) >0.
(2) We have Un E U(L'oo) uniformly in n. However such an order of integrability is not

sufficient to guarantee that the solutions Un are bounded. The number ).8 = N(P- 2) + sp
is zero and the condition (5.1)-(5.3) of Chap. V are violated. Questions of convergence
for general initial data in Lloc (RN) will be discussed in Chap. XII.
4. An integral Harnack inequality for all I < p < 2 193
4. An integral Harnack inequality for all 1 < p < 2

A weak integral form of (1.3) holds for any non-negative weak solution of (1.1)
for p in the whole range 1 < p < 2, and it is crucial in the proof of the pointwise
estimate (1.3). In the estimates to follow we denote with 'Y = 'Y(N,p) a generic
positive constant, which can be determined a priori only in terms of N and p and
which can be different in different contexts.
PROPOSITION 4.1. Let u be a non-negative weak solution of (1.1) and let 1 0 such that B4p (x o) c fl, 'Vt > to
(4.1)
Since 1 < p < 2, the number A = N(P - 2) + p might be of either sign. The
proposition can be regarded as a weak form of a Harnack estimate, in that the
Ll-norm of u(·,t) over a ball controls the Ll-norm of U(·,T) over a smaller
ball, for any previous or later time. It could be stated over any pair of balls Bp(xo)
and Bqp(xo) for q E (0, 1). The constant 'Y ='Y(N,p, q) would depend also on q
and 'Y(N, p, q) / 00 as q / 1.
Remark 4.1. The proof shows that the constant 'Y(N,p) deteriorates as p /2.
The proof depends on some local integral estimates of the gradient IDul which we
derive next.
4-(;). Estimating the gradient of u

PROPOSITION 4.2. Let u be a non-negative weak solution of (1.1) and let 1 <
p< 2. There exists a constant 'Y='Y(N,p) such that
'V(xo, to) E floo, 'Vp > 0 such that B4p(Xo) c fl, 'Vt > to, 'Vv > 0, 'Vq E (0,1),
there.. holds
t
(4.2) j j(T - to); (u + v)-~ IDulPdxdT

toB.... (zo)
< 'YP
- (1 - q)p
[1 + (t - to) vP-2] (~);.
pP p>'
!tf.::!l
x { sup ju(x,T)dX +VPN } " ,

to::5T::5t
B .. (zo)
t
(4.3) ~j fiDUIP-ldXdT
toB.. p(zo)
(4.4) ~j jlDulP-1dxdT
toB.. p(zo)
Remark 4.2. The estimates (4.1)-(4.4) have been stated 'locally'. However they
=
continue to hold for to 0, i.e. for cylinders Bp(xo) x (0, t) carrying the 'initial
data'.
Remark 4.3. The constant 'Y( N, p) in (4.2)-(4.4) tends to infmity as p /2.

PROOF OF PROPOSITION 4.2: We translate the coordinates so that (xo, to)
coincides with the origin and will work with non-negative weak solutions of
(4.5) Ut - div IDul p - 2 Du = 0, in B 4P x (0, (0), p E (1,2).
<
Fix 0' E (0, 1) and let x -+ (x) be a non-negative piecewise smooth cutoff function
in Bp that equals one on Blip and such that ID(I ~ 1/(1 - O')p. In the weak
formulation of (4.5) take the testing function
v> 0,
modulo a Steklov averaging process. We obtain for all t > °

4. An integral Harnack inequality for all I < p < 2 195
(4.6) 2; P j j T~ (u + v)-~ IDuIP(PdxdT

OBp
2(P P_ 1) t" .If (u + v) ~

" (x, t)(Pdx
Bp
t
+ 2(P ~ 1) j j T~-l (u + v) 2(,,;1) (PdxdT

OBp
t
+p j j T~ IDul p - 2 Du·D((P-l (u + V)l-~ dxdT.

OBp
We estimate the various tenns on the right hand side in tenns of the quantity
8 == sup ju(x,T)dx.
. O<.,.<t
-: - Bp
Since pE (1, 2) we have 2(P;1) < 1. Therefore by the HOlder inequality
(I·) p
2(P-l)"
t.lj(u+v)~ I"Pdx
" ..
Bp
J.
'5:,y t" p "
~ ( -
sup ju(x, T)dx + v pN
O<.,.<t
- Bp
)
~
='Y P ( p>'t)~ (8 + vpN) ~

" .
:5 'Y sup j (u + v) ~
j T" -ldT o<.,.<t
J. " (x, T)dx
o . - - Bp
Next, by Young's inequality

(iii) p III .,.; IDulp- Du·D(,p-l2 (U + V)l-a dxd.,.1

OBp
t
~ ~¥ II .,.; (u + V)-a IDuIP(Pdxd.,.
OBp
t
+ (1 2~;ppI' II.,.; (u + v)p-2 (u + v) 2('1'-1) dxd.,..

OBp
This last integral is estimated above by
p "Y(N,p)
(1 - u)p
(.!.-)
pi'
Vp - 2 (.!.-) ; (8 +
p>'
VpN) 2(,;1) •
Combining these estimates in (4.6) proves (4.2). To prove (4.3), write (4.2) with
(x o, to) == (0,0) and select v 2 - p = (t/ PP). Then by th~ HOlder inequality
t
(4.7) IIIDUIP-1dxd.,.
OB"p
t
{f H=1
= JJ 'T'P I' (u+v)-"12=1 IDulp-1.,.-p.12=1 (u+v)"12=1
I' " dxd.,.
I'
OB"p
2=1 .1
,; (ijT*(U+ v)-iIDUIPdzdT) " (il.,.;_l(U + v)~ dxd.,.) I'
~OB"p OB"p
The last integral is estimated above by
(a) "YP p>' ( t)f;{ 8+ pi' (t)~pN}~

~ "YP (p>.t );!1c.ll
8
( t.) ~
I' +"YP p>' .
The first integral on the right hand side of (4.7) is estimated by (4.2) with the
indicated choice of v and it is majorised by the same quantity on the right hand side
of (a), apart for a factor (l-U)I- P. Combining these remarks in (4.7) proves (4.3).
Finally (4.4) follows from (4.3) by a further application of Young's inequality.
4. An integral Harnack inequality for all 1 < p < 2 197
4-0i). Proof of Proposition 4.1

Assume that (Xo, to) coincides with the origin and consider the family of
expanding concentric balls
n
Bn == {Ixl < Pn} , Pn = P L 2- i, n = 0,1,2, ....

i=O
We have Bo == Bp and Boo == B2p' Introduce also the 'intermediate' spheres
and let x -+ 'n (x) be a piecewise smooth non-negative cutoff function in Bn that
equals one on Bn and such that ID'nl ~ 2n+2/ p. In the weak formulation of (4.5)
take 'n as a testing function to obtain
for any two time levels 'Tl and'T2 in [0, tJ. We take as 'T2, a time level in [0, tJ such
that
We also set
Sn == sup ju(x, 'T)dx.
O<T<t
- - Bn
Since'Tl E [0, tJ is arbitrary, (4.8) implies
Next we apply (4.3) over the pair of balls Bn and Bn+l for which (1 - u) ~
2-(n+2). This and Young's inequality give
2n;2
t
j jIDuIP-1dxd'T ~ ",bn ( ; )
.1
P (Sn+l) 2(,;1) +",bn (;l) 6
-P
0-
Bft
~eSn+l+",(N,p,e)bn ( t)J!;
pl '
valid for every eE (0, I), for some constant ",(N,p, e) depending only upon N, p
and E. Combining these estimates, we conclude that for every E E (0,1) there exists
a constant 'Y( N, p, E) such that
s.. !> <Sft+! +1(N,p,<) {z +.(~) r-.} b".
11le Proposition now follows from the interpolation Lemma 4.3 of Chap. I.
5. Sup-estimates for &~2 < p < 2

We now combine the Ll:: -estimates of §5 of Chap. V with the integral inequality
(4.1). If p is in the range (1.2), we may take r = 1 in (5.1)-(5.4) of Chap. V and
rewrite the latter as
(5.1) lIu(·, t)lIoo,Bp(zo) ~ 'Y (t - to)-~ ( sup

to~.,.~t
J T)dx~
u(x, pI)..
B Sp / 2 (zo)
+'YC~to)~ .
This and Proposition 4.1 imply
LEMMA 5.1. Let u be a non-negative local weak solution of (1.1) in noo and let
(1.2) hold. There exists a constant 'Y(N,p) such that
V(xo, to) E noo , Vp> 0 such that B4p(Xo) c n, "It> to
(5.2) sup u(x, t)

zeBp(zo)
~ 'Y (t - to)-~ (inf
to~"'9
J T)dx~
u(x, pI"
B2p(Zo)
+'Y C~to)~.
The constant 'Y( N, p) tends to infinity as either p '\, J~1 or as p / 2.
Remark 5.1. The lemma continues to hold also for to = 0, i.e., for cylinders
Bp(xo ) x R+ carrying the 'initial' data.
The peculiar feature of this estimate is that the supremum of the solution
over a ball at some time level is bounded above by the Ll-nonn of u over a
larger ball at either the same time level or some 1uture' time. This is in contrast
with the behaviour of non-negative solutions of the heat equation. Accordingly,
the constant 'Y(N, p) deteriorates as p / 2.
6. Local subsolutions 199
5-(;). A special/orm 0/(5.2)

We will use this fact in the following form. Let u be a non-negative weak
solution of the p.d.e. in (1.1) in some space-time domain and let p be in the range
(1.2). Let R>O and assume that the cylinder
Q4R == B4RX{-4, O}
is all contained in the domain of definition of u. Then
6. Local subsolutions
As in the degenerate case, the proof of Theorem 1.1 is based on expanding the
positivity set of the solution u by means of suitable comparison functions. Let
b, k, IJ. be positive parameters satisfying
(6.1)
Consider the cylindrical domain with annular cross section
(6.2) Q(8) == {:-1 k P- 2 < IxlP< I} x{O, 8},

and the function
(6.3)
LEMMA 6.1. Assume that p is in the range (1.2), i.e.,

A == NCp - 2) + p > O.
Then the constant b=b(N,p) can be chosen a priori only dependent upon Nand
p, so that
'rIk > 0, 'rI IJ. > 0 > satisfying (6.1),

(6.4) Wt - div(lDwl p - 2 Dw) ~ 0 a.e. in Q(8),
8=min{ (2~)P-l IJ.j k2 - P}.

Remark 6.1. The proof below shows that the constant b> 1 is •stable' as p /,2.
200 YD. Harnack estimates and extinction profile for singular equations
Figure 6.1
PROOF OF LEMMA 6.1: The function x-!li(x,t} is radial and decreasing with
respect to Ixl. so that writing (6.4) in polar coordinates we have
C(!Ii} == !lit - div(ID!liI,,-2 D!Ii}

= !lit + ( N ; 1) (_!li,},,-l _ (p _ 1)( _!li,),,-2!1i",
where
p=lxl, !Ii' = ~!Ii

dp
;
We write
(6.5) ( _!li,)2-" C(!Ii) = (_!li,)2-"!lit + 'R.(!Ii) ,

'R.(!Ii) == N - 1 (-!Ii') - (p - 1}!Ii" ,
p
and calculate 'R.(!Ii) as follows. First we set
~ (IXI")~
IIzll =kFIb -t ; F = 1 + IIzll
k
W=--l 11 = (1 -lxI 2 );!r.
FF-;
Then by direct calculation

6. Local subsolutions 201
(6.6)
=
We calculate the expressions 1/1' w'v + wv' and 1/1" = w" v + 2w'v' + wv" from
(6.6) and combine them into 'R.(I/I) to obtain
(6.7) 'R.(I/I) :5 -P-(I - p2);;!-r

2-p
(w) M {N -
p2 :F
1
- (p - 1) [ - P -IIzll + 1] + p-1 }
2-p:F :F
+ 2pw(I- p2)~ {(N - 1) _ ~ M + I}.
p-I 2-p:F
Rewrite the first factor in braces on the right-hand side of (6.7) as
P-
{ ... } = ( N -
2- p :F
M).
We will impose on /I z II to be so large that
(6.8) N-_p_M<o.
2 -p :F
This is possible since N(p - 2) + p == A> O. The second term in braces on the
right hand side of (6.7) is negative if we choose IIzll to satisfy (6.8). If N = 1,2,
this is a direct consequence of (6.8). If N ~ 3,
(P_I){N-I _ ~M
p-I 2-p:F
+I} = (N- _P_
2-p :F
M)
+ -p- (3 - 2p)M + (p- 2).
2-p :F
The first term is negative in view of (6.8) and the second is negative since p >
;:~l > ~ if N ~ 3. We drop the last negative term on the right hand side of (6.7)
and estimate
(6.9) R.(!li) :::; _P-

2-p
v (w) 1l:.ll [N __P_1l:.ll] .
p :F2 2-p :F
We return to (6.5) and estimate above the term ( -!li,)2-"!lit. First using (6.6)
Also
!lit = _1_ vw ~. 1l:.ll
2-p :F t
Therefore
(-!li')2-"!lit :::; ~ (~)2-" vwM~.
2-p p :F t
We combine this with (6.9) into (6.5) and set
C* (!li) = (-!li')2-"[!lit _ div(ID!liI,,-2 D!li)] (2 - p):Fp2

vwllzll
to obtain
C*(!li) :::; "YW2-"pP
t
+ P[N - -p-
2-p :F
M] .
From the definition of w and II z II
w 2-"pP = (
t
IIzll
1 + IIzll
),,-1 1-" <- 11'-1
6 _I_
and
.c*(!li) < ...::L + -p-
- 11'-1 2- p
[-A + E-]
:F
.
We will choose II z II so large that ~ :5 ~,and then select 6 from
"y p A
-----=0
11'-1 2- P 2 .
We will have ~ == I41T :5 ~ if for example IIzll > ¥, i.e. if

k 2 -" V-I Ixl" (~) ,,-1' > t > O.
From the construction of the cylinder Q(fJ) in (6.2), we have
k2 -" V-I Ixl" ~ ",.

Therefore to prove the lemma it suffices to take
o < t :::; fJ = ( 2p
A ),,-1 ",.
7. Tune expansion of positivity 203
7. Time expansion of positivity

The next subsolution of (1.1) will be employed to expand the set of positivity of u
in the direction of increasing t. Set
(7.1) k~
g;(x;t) := Re(t)
{
1-
( IX I,,);2:r}2
R(t) +'
R(t):= kP - 2 t + pP, F:= l-lIzlI;2:r, IIzll:= t l:).

e
Here k and p are positive parameters and > 1 is a number to be chosen indepen-
dent of k and p. For t = 0 the function g;(., 0) is supported in the ball B p and for
t > 0 the support of x -+ g;( x, t) is the 'expanding' ball
We will consider g; only within the domain
The function g; is continuous in 1){k,e}. vanishes on the 'lateral' boundary of

V{k,e}and it is of class Coo in the interior ofV{k,e}.
LEMMA 7.1. The number e= e( N, p) can be determined a priori only in terms
of N and p so that
Remark 7.1. The constant eis 'stable' as p / 2.

PROOF OF LEMMA 7.1: By direct calculation within 1){k,e} we have
ek,,-lppe 2 2 k"-I~ -L
g;t = - Re+l(t) F + p-l ~+l(t)Fllzllp-l,
2p k~ (IXI);2:r x
Dg; = - p _ 1 ~(t)F R(t) lxi'
IDg;I,,-2 Dg; = _ ( ~ )
,,-1 [ kPPe F ],,-1 -=---
p- 1 Re(t) R(t) ,
_ div 1vg;1,,-2 Dg; = (~)"-1 _1

p-l R(t)
[k~ F]"-1
~(t)
(N _pIlZII;2:r) .F
Setting
204 VU. Harnack estimates and extinction profile for singular equations
the previous calculations give
(7.3) C· (4)) = {- f.F + _2_lIzll~

p-l
~)P-l [~ ]p-2( _ IIZIl~)}
+ (p _ 1 R£.(t) F N P F .
Introduce the two sets
£1 == {(X,t) E V{k.£.} I:F < 6}, £2 == {(X,t) E V{k.£.} IF ~ 6}.

Here 6 is a small positive number to be determined so that. within £1. the last term
on the right hand side of (7.3) is negative. i.e.•
With such a choice we have in £1
(7.4) c- (4)) ~ _2_

p-l
+ (~)p-l
p-l
(N _~)6 ,
where in estimating the term containing R(t) on the right hand side of (7.3) we
have used the fact that 1 < p < 2. We determine 6 so that the right hand side of
(7.4) is non-positive and observe that such a choice can be made independent of
f.. Next. having determined 6. within £2 we have
(7.5) c- (4)) ~ -f.6 + _2_ + N (~)p-l [nt.(t) ~] 2-1'

p-l p-l pPt. 6
Within the range (7.2) of t we estimate
nt.(t) 1]2-1' ( 1)£.(2-1') -2 (e)2-p

[--- < 1+ - 61' < - .
pPt.6 - e -6
We substitute this estimate in (7.5) and choo~e eso large that the right hand side
is non-positive.
8. Space-time configurations
Locally bounded weak solutions of (1.1) are locally R;lder continuous in the in-
terior of their domain of definition. "ripE (0,1). This is the content of Theorem 1.1
of Chap. IV. The proof consists of controlling the essential oscillation of a local
solution over a family of nested and shrinking cylinders. Such a control is estab-
lished in Proposition 2.1 of Chap. IV. by working with cylinders whose 'space
dimensions' are rescaled in terms of the solution itself. As observed in Remark 2.2
8. Space-time configurations 205
of Chap. IV, such a geometry is not the only possible. A version of Proposition
2.1 holds for an intrinsic parabolic geometry where the scaling occurs in the 'time
dimension'. We restate the proposition for such a geometry in the context of (1.1)
and in a form convenient for the proof of the Harnack inequality. Let 1.£ be a local
weak solution of (1.1). Fix (x, t) E nT and suppose that we can fmd a cylinder of
the type
(8.1) [(x, t) + Q (aoR", R)] == {Ix - xl < R} x {l- aoR"} , ao == (~) 2-",
where A is an absolute constant, R is so small that [(x, t) + Q (aoR", R)] c nT,
and w is any positive number satisfying
(8.2) sup 11.£1 :5 w.

[(z,l)+Q(ooRr> ,R)]
PROPOSITION 8.1. There exist constants eo, 1/ E (0,1) and C, A > 1 that can
be determined a priori depending only upon N and p, satisfying the following.
Construct the sequences Ro = R, WO = W
Rn =C-nR, Wn+l = TJWn, n = 1,2, ... ,
and the boxes

_
Q(n) == {Ix - xl < Rn} x {l- anR~, l}, an - ( Wn)2-"
A .
and essosc 1.£ :5 W n ·

Q(n)
A consequence is the HOlder continuity of 1.£ at (x, t). A particular case is

LEMMA 8.1. There exist constants "'( > 1 and a E (0, 1) that can be determined a
priori only in terms of the data, such that for all 0< p :5 R
essosc u(x, t) :5 "'(Wo

Ix-zl<p
(RP)Q .
Remark 8.1. This is a version of Lemma 2.1 of Chap. IV, stated for a 'fixed time'
l.
Remark 8.2. The constants A and C depend only upon N and p and are indepen-
dent of u. Moreover they are 'stable' as p /2. This follows from the remarks of
§3-(I) of Chap. IV.
9. Proof of the Harnack inequality
We let u be a non-negative local weak solution of (1.1) in nT and let p be in the

range (1.2). Let (xo, to) E nT, assume that u(Xo, to) > 0 and consttuct the cylinder
Q4p(Xo, to) == {Ix - xol < 4p}

x {to - [u(x o•t o)]2-" (4p)", to + [u(xo, t o)]2-" (4P)"}.
where we assume that p is so small that Q4p(X o, to) c nT . The change of variables
x-xo
x~--,
p
Q+ == B4 x [0,4"),
Denoting again with x and t the new variables, the rescaled function
is a bounded non-negative weak solution of
{
Vt - div IDvl,,-2 Dv =0 in Q,
v(O,O) = 1.
To prove the theorem it suffices to determine constants c and "Yo in (0, 1), depend-
ing only upon N and p such that
(9.1) inf v(x, c) ~ "Yo.

zEBl
9-(i). Locating the sup of u in Q

For TE (0,1) consttuct the family of nested expanding cylinders
and the numbers
M.,. == supv,
Q..
Here 6 E (0, 1) is a small number to be chosen later and has the effect of rendering
'/lat' the boxes Q.,..
9. Proof of the Harnack inequality 207
Remark 9.1. This construction is similar to that in the proof of Theorem 2.1 of
Chap. VI. The cylinders QT however are 'thin' in the t-dimension. Also the ex-
ponent of (1 - T) in the definition of NT is fixed and depends on the singularity
of the p.d.e.
For T = 0, we have Mo = No. Moreover as T / ' 1
and
since v E L~(Q). Therefore the equation MT = NT has a largest root, say To,
which satisfies
Since v is HOlder continuous in Q, it achieves the value MTo at some point (x, f) E
QTo and
(9.2) sup v(x,f) $ 2~(1- To)-~.

Iz-zl<l-{p
LEMMA 9.1. There exist a positive number e that can be determined a priori only
in terms of N and p, such that
OJ -> ~(I-'T.
v(x , f' 2 0
)-~ , 'v'lx - xl < e(1 - To).
Remark 9.2. The proof employs the estimates of Lemma 5.1 in the form (5.3).
Therefore e "\. 0 as p /' 2.
PROOF OF LEMMA 9.1: Construct the box
- I-To
(x, f) + Q4R == {Ix - xl < 4R} x {t - 4, fl, where 4R = -2-'
Apply to such a box the estimate (5.3) with the appropriate change of variables to
obtain
sup v(x, t) $ "Y( !v(x, f)dx)"/>' + "YR-~, 'Vf - 1 $ t $ t.

Iz-zl<R
B2R
In view of (9.2) and the definition of R
sup v(x,t) $ "Y1(1- To)-~, 'v't -1 $ t $ t,

Bi!.=;al
where "Y1 ="Y1 (N, p) is a constant that can be determined a priori only in terms of
N and p. Next consider the cylinder
By virtue of such a construction we have
sup v ~ "Yl(1- TO)-~'

QRo
The 'vertical size' of QRo is larger than
Therefore QRo satisfies the space-time configuration of (8.1 )-(8.2). We conclude

that
'v'0 < p < Ro, 'v'lx - xl < p, at the level f
vex, f) ~ v (x, f) - "Y"Yl(1 - To)-r-; (;J °
Since v (x, f) = (1 - To)-~, by taking p='TIRo. 'TIE (0,1) we find
vex, f) ~ (1 - To)-r-; (1- "Y"Yl'1°),

~
'v'lx - xl ~ '1Ro == '1(8"Yl p )-1(1_ To)
and the lemma follows by taking '1 so small that enough (1 - "Y"Y1 '1°) = ! and
then choosing
9-(i;). Time-expansion o/positivity

The previous arguments are independent of the number 6. We will now de-
tennine 6.
LEMMA 9.2. There exist small positive numbers Co, 6 that can be determined a
priori only in terms of N and p, such that
(9.3) vex, t) ~ co(1 - To)-r-;, 'v'lx - xl < e(1 - To), 'v'6 ~ t ~ 26.
PROOF: Consider the comparison function ~ (x - x; t - f) in the domain V{k,(} (x,

defined in (7.1)-(7.2) with the choices,
p = e(l- To).
The function ~ is a subsolution of (1.1) for a time interval
For t = f by virtue of Lemma 9.1, v ~ ~ (x - x; 0). Therefore by the comparison

principle
9. Proof of the Harnack inequality 209
(x-i)"-R(t.l)
~ __+-______________-,2
·&to
Figure 9.1
v ~~ in {Ix - xl P < R(t -l)} x {O < (t -l) < 36}.

In particular for 6 < t - t < 36 and Ixl :5 e(l - To),
vex, t) ~
~(l-TO)-~
[lie + 1](
{
1-
(ae
ae +
);;!T}2+
1
== co(l - To)-~.
The location of t in the box Q".o is only known qualitatively. However, as (t -l)
ranges over [6, 36], the intervals [t + 6 < t < t + 36] have the common intersection
[6 :5 t :5 26] and the lemma is proved.
Remark 9.3. The number 6"" 0 as p / 2. This follows from Remark 9.2 and the
choice of 6 above.
9-(iii). Sidewise expansion of positivity

We will expand the positivity set of v over the ball {Ixl < I} at the time level
t=26. For this we will prove that there exist a constant 'Yo ='Yo(N,p) such that
vex, 26) ~ 'Yo, 'v'lx - xl < 2.
Consider the comparison function
(9.4)
I/t
( X-x.~)
3'3P'
introduced in (6.3), in the annular cylindrical domain
{e(l- To) < Ix - xl < 3}x {6,26}.

The number k is given by
k = co(l- To)-r-;,
where Co is determined in Lemma 9.2. The parameter JJ here can be chosen by
imposing
11'-1 e1'
i.e.• u<---
r- - 2-1' 31'.
Co
Wechoose
1 11'-1 e1' }
JJ = min { 4; ~-1' 31' '
and pick (J according to the second of (6.4). By further restricting either JJ or the
number 6 of Lemma 9.2 we may assume that (J =6. The function t[I in (9.4) vanishes
for Ix - xl=3 and fort=6. Moreover for Ix - xl =e(l - To) and 6 <t~26.
t[I (
X - x
-3-' 3P
t - 6) 5 Co(l - To)-r-p
-I!.-
5 v(x, t),
by Lemma 9.2. Therefore by the comparison principle. we have for t = 26 and
Vlx-xl<2
9-(iv). Proof of Theorem 1.1 for p near 2

The proof is very similar to that of Theorem 2.1 of Chap. VI for p close to 2. We
only indicate the main differences.
As before. construct the family of expanding cylinders Q.,. == {Ixl < T} x
{-T, O} and the numbers
where f3 is a positive number to be chosen. The definition of the numbers N.,.

differs from that in §9 since f3 is arbitrary. Let To E [0, 1) be the largest root of the
equation M.,. = N.,.. so that
If (x, t) is a point in QTo where v achieves the value M To ' we have

P I
( ) ~2 p( I-To )-; I - To - I - To
(9.5) vX,t x-xl<-2-; t - -2-<t<t.-
Let
and consider the box
From the definitions of QT and Ro we have Q 0 (x, t) c Q!:t;a, so that by (9.5),
IIvlloo,Qo(z,l) ~ 2P (1 - To)-P
Therefore Qo (x, t) satisfies the space-time configuration (8.1)-(8.2). It follows
that
\fIx - xl < Ro, Iv(x, t) - v (x, t) I < -y2P(1 - To)-P (~) Q
By taking p=eRo and then e sufficiently small we have

LEMMA 9.1'. There exists a small positive number e E (0, I) that can be deter-
mined a priori only in terms of N, p such that
I _
f\ >
v(x ,OJ 2 - ~)
- -(I 0,
P \fIx - xl < e(1 - To) P~ 1
P + .
Remark 9.5. The constant e depends upon f3 but it is 'stable' as p / 2 since no

use has been made of (5.3).
The proof can now be completed by expanding the positivity set of v with the
aid of the comparison function g,.,p introduced in §3-(i) of Chap. VI.

Fix a point (x., t.) E aT assume that 1.£ (x., t.)
cylinder
> ° and for R> 0, construct the
Q8R (x., t.) == {Ix - x.1 < 8R}

x {t. - c [1.£ (x .. , t .. )]2-" 8R", t. + c [1.£ (x .. , t.)]2-" 8R"} ,
where c=c(N,p) is the number appearing in Theorem 1.1, and R is any positive
number such that Q8R (x., t.) is contained in the domain of definition of u.
We first establish an auxiliary proposition, then we will prove that it implies
the theorem.
PROPOSITION 10.1. There exists constants C=C(N,p) and,,=,,(N,p) that

can be determined a priori only in terms of N and p, such that
(10.1) C- 1 sup u(·,to) ~ u(xo, to) ~ C inf u(·,to),

B"ReZ.) B"Re Z.)
where xo=x. and
(10.2)
PROOF: The change of variables

x-x. t - t.
x- - - t -
R ' [u (x., t.)] 2 PRP
maps QSR (x., t.) into Q =

Bs x (-8,8). Denoting again with x and t the new
variables. the rescaled function
v(x, t) =u (1x., t. ) u (x. + Rx, t. [u (x., t.)]2- PtRp)

is a bounded non-negative solution of
{
Vt - div IDvl p - 2 Dv =0 in Q.
v(O,O) = 1,
We first prove that there exist a quantitative constant C = C (N, p). such that
1 -
(10.3) C ~ v(x, c) ~ C, v Ixl < 1.
By the Harnack inequality (1.3)
(10.4) v(x, c) ~ 'Yo, Vlx/ < 2,
for a quantitative constant 'Yo = 'Yo (N, p). This proves the estimate below in (10.3).
For the estimate above we require the following lemma.
LEMMA 10.1. There exists a quantitative constant" E (0,1) depending only
upon N and p, such that
(10.5) v(x, -c) ~ 2/'Yo, Vlx/ < 2".

PROOF: For x ranging over the ball B4 consider the closed truncated 'paraboloid'
t + c ~ c[v(x. _c)]2- p Ix - xiI', -c ~ t ~ O.
By the Harnack estimate
(10.6) v(x, -c) ~ 2- v(x, 0), Ix - xiI' < [v(x, _C)]p-2 ,

'Yo
and in particular v(O, -c} = lho. Since v is HOlder continuous, the set
{x I u(x,-c} <2ho}
is non-empty and contains a ball about the origin. We claim that in particular it
contains the ball B 2 f/' where
(277}P = (~) 2- p •
If not, there would exist some x E B2f/ such that v(x, -c} = 2ho. It follows that
the ball
Ix - xlP < [v(x, _c}]p-2 = (277t
covers the origin, and (10.6) for x=O gives
-2 = v(x,-c) $ 1
-.
~o ~o
The contradiction proves the lemma.

To prove the estimate above in (10.3), we combine the quantitative bound
(10.5) with Proposition 4.1. This gives
We return to the original coordinates and write the estimate above in (10.3) as
u(x,t o} $ Cu(x.,t.}, 'v'lx - xol < 77R, XO == x •.

Since, by Corollary 1.1, u (x., t.) $~u(xo, to} the left estimate in (10.1) is proved.
The estimate below in (10.3) reads
On the other hand the estimate above in (10.3) for x = x. gives u(xo, to} $
Cu (x., t.). Combining these last two estimates proves the bound above in (10.1)
and the proposition follows.
1D-(i). Proof of Theorem 1.2

Fix (x o, to) E fh and p > O. Let 77 be the constant claimed by Lemma 10.1
and let ~ be the constant of the Harnack estimate (1.3). Set
and construct the cylinder

QSr(Xo, to) == {Ix - xol < 8r}

x {to - c [u(xo, t o)]2- p 8rP, to + c [u(xo, t o)]2- p 8r P} .
Wthout loss of generality, we assume that QSr(X o, to) C [}T. First fix the time
level
and choose R> 0 from
to - t. = c[u(x.,t.)]2- p RP,
The defmitions of t. and R give
c [u(xo, t o)]2- p r P = to - t. = c [u (x., t.)]2- p RP.
By Corollary 1.1, u (x., t.) ~ ')'u(xo, to). Therefore
2..=l
R ~ ')' p rP == p/T/.
Applying Proposition 10.1 with such a choice of the point (x., t.) and radius R
proves the theorem.
Theorem 1.1 and its proof is taken from [44]. The form of Theorem 1.2 was con-
ceived by Nash [84], who believed it to be true for solutions of the heat equation.
Moser [83] pointed out that (1.7) is not dilation invariant for solutions of the heat
equation. It becomes scalar invariant in a specific intrinsic geometry. The results
adapt to equations of porous medium -type and its generalisations (see [44n. In the
context of the plasma equations estimates of the rate of extinctions were derived
by Berryman-Holland [13,14]. Proposition 3.1 is due to &nilan and Crandall [9].
The estimates of §§ 4 and 5 are taken from [42]. The subsolution tV of §6 appears
in [44]. The subsolution ~ of §7 is a modification of a subsolution introduced in
[4]. It is natural to ask whether an intrinsic Harnack estimate continues to hold for
non-negative solutions of p.d.e.'s with full quasilinear structure. This is the case if
p = 2 and it remains an open issue for degenerate (p> 2) and singular (1 < p < 2)
equations. A step in this direction is in [29]. It is shown that Theorem 1.1 holds
true for non-negative weak solutions of
Vt - (IDvIP-2aij(X,t)u:J:JI:', = 0,
where (x, t) --+ aij (x, t) are only bounded and measurable and the matrix (aij) is
positive definite.
VIII
Degenerate and singular parabolic
systems
1. Introduction
We turn now to quasilinear systems whose principal part becomes either degener-
ate or singular at points where IDul =0. To present a streamlined cross section of
the theory. we refer to the model system
u == (UI, U2, ... , Urn), mEN,

(1.1) { Ui E C'oc (0, TjL~oc(n))nLP (0, Tj w,!,:{n)) , i=l, 2, ... ,m,
Ut - div IDul p - 2Du =0 in nT'
The solutions are meant in the weak sense
(1.2) I
n
Uirpi{X,T)dXI
t2
tl
+ II
t2
tin
{-Uic,oi,t + IDuIP-2Dui·Dc,oi} dxdT=O,
for all intervals [tl, t2] C {O, T] and all testing functions cP == (cpt. CP2, ••. , rpm)
satisfying
(1.3) CPi E W,!;; (0, T; L2{n)) n Lfoc (0, T; wJ,p{n)), i = 1,2, ... , m.
For these we derive local sup-bounds on the modulus of the solution lui and its
space gradient IDul and establish the estimate
(1.4) Ui, zj EC1:,c{nT), i=I,2, ... ,m, j=1,2, ... ,N,

216 VIn. Degenerate and singular parabolic systems
for some a E (0, 1). This is the focal point of the theory. Weak solutions of elliptic
systems in general are not continuous everywhere within their domain of defini-
tion. We refer to [48J for counterexamples and an account of the theory. Solutions
of (1.1) are regular everywhere in nT because of the special nature of the sys-
tem. If u solves (1.1), then the function IDul 2 is a non-negative subsolution of
a parabolic p.d.e. (1) It is precisely such a property, which for elliptic systems is
called 'quasi-subharmonicity' ,(2) that permits one to prove (1.4) everywhere in
nT.
These estimates can be extended up to t = 0 if the system in (1.1) is associated
with a smooth initial datum 110. They also carry over to the lateral boundary of
nT if (1.1) is associated with homogeneous either Dirichlet or Neumann data on
ST == an x (0, T). If the data are not homogeneous, the theory is fragmented and
incomplete. In the case of non-homogeneous Dirichlet data, we will show that
Ui EC6 (nx (e, T» for arbitrary 6 E (0,1), 'rIe E (0, T),
provided p > max {I; J~2}' However the key estimate (1.4) is not known to
hold in such a case, and it is a major open problem in the theory.
The C 1 ,o regularity (1.4) requires a preliminary estimation ofthe type
(1.5) IIDulloo,K: :5 const, IC a compact subset of nT.
1be degenerate case p> 2 and the singular case pE (1, 2) are rather different with
respect to such an estimate. The function class in (1.1) implies that(3)
(1.6)
If P > 2, such integrability suffices to establish (1.5). If 1 O.
This is analogous to the condition imposed in Theorem 5.1 of Chap. V. It implies
(1.4) and in addition
(1.8)
l-(i). Aboutthe singular case l<p<2

In the degenerate case p > 2, the behaviour of the solutions of (1.1) is entirely
a local fact. In particular the sup-bound (1.5) and the estimate (1.4) are a sole con-
sequence of u being a weak solution of (1.1). If 1 < p < 2 due to the singular
(1) See (3.3) in the Preface or (1.8) of Chap. IX.

(2) We refer to Meier [77] for some sufficient conditions for an elliptic system to be
quasi-subharmonic.
(3) See Proposition 3.1 of Chap. I.
1. Introduction 217
nature of the p.d.e. some global infonnation is needed. This is not related to sys-
tems. Indeed it occurs also in Theorem 5.1 of Chap. V to establish a sup-bound for
solutions of a single equation. Since our estimates involve u and DU the global t
infonnation needed regards both the solution and its space gradient. Let r ~ 2 sat-
isfy (1.7) and let U be a local weak solution of (1.1) for p E (1,2). We assume
that
U can be constructed as the weak limit in L[oc(f1T) ofa

(1.9) {
sequence of bounded subsolutions {Un her of (1.1)
satisfying in addition IDunl E L~oc(f1T)'
We stress however that all our estimates will depend only upon the quantities
lIull r ,K;, IIDull"K;, x:; a compact subset of f1T .
Such an assumption is not restrictive in view of the available existence theory(l)

and the special fonn of (1.1).
1-(ii). General structures

We will develop the theory for the homogeneous system (1.1). The same re-
sults however continue to hold for the following general class of quasilinear sys-
tems
(1.10) ~1J,'
at - I
div A (i) (x , t , Du) = B(i) (x , t , u , Du) in nT,
i = 1,2, ... ,m,
where the functions

A(i)=(A(i)
-
A(i) A(i»)..
l ' 2 , ... , N
rl
uT
xR Nm --+RN ,
B(i) : f1T xRxR Nm --+ R, i = 1,2, ... ,m,

satisfy the structure condition
(S3)
(1) See Lions [73).

218 vm. Degenerate and singular parabolic systems
m
(85 ) 2: IB(i)1 ~ CllDul p- l + !P2,
i=l
where Ci , i =0,1. are given positive constants and !Pi, i = 0,1,2, are given non-
negative functions satisfying
N +2
!Po + !PI#r + !P22 E L q,oc (n )
uT, q > -2-'
Remark 1.1. The structure condition (82 ) is somewhat fonnal since there is no
stipulation that Ut.",,,,,,; have meaning at all. More correctly it should be written
with Ut,,,,u:; and DUi''''i replaced by tensors ~t,k,j. Neverthless we prefer the for-
mal but suggestive fonn of (82 ).
We will develop the main points of the theory for the model system (1.1) and
indicate later how to modify the arguments to include (1.10).
2. Boundedness of weak solutions

We will use the notation of§3 of Chap. II. Thus Q (0, p) is the cylinder with 'vertex'
at the origin. Its cross sections are the cubes Kp and its height is O. The cylinder
[(xo, to) + Q (0, p)] has the 'vertex' at (xo, to) and is congruentto Q (0, pl. With (
we denote a piecewise smooth non-negative cutoff function in Q (0, p) vanishing
on the parabolic boundary of Q (0, p).
THEOREM 2.1 (THE CASEp>2). Letubealocalweaksolutionof(l.l),and
let p > 2. Then for all e E (0, 21 there exists a constant 'Y depending only upon
N,p, mand e, such tlultforeverycylinder [(xo, to) + Q (O,p)] c nTandforever
CTE(O,I),
(2.1) sup
[(so,t o )+Q(0'9,O'p»)
lui <
-
'Y
(1 -
(Ojpp)lIE
CT)(N+p)/E
(f! lu IP - 2+£dXdT)
(so,to)+Q(9,p»)
liE
A (~);!J .
THEOREM 2.1 (THE CASE l<p<2). Letubealocalweaksolutionof(l.l)
for 1 0,

2. Boundedness of weak solutions 219
and that (1.9) holds. There exists a constant 'Y depending only upon N, p, m and
r such thatforevery cylinder [(x o, to) + Q (9, p)] c nT andfor every C1E (0, I),
(2.3)
2-(i). An auxiliary proposition

The arguments are similar to the proof of local boundedness of solutions of a
single equation and are based on local energy inequalities which we derive next.
We set
(2.4) lul=w.
PROPOSITION 2.1. Let u be a local weak solution of the system (1.1) in nT,
and let f(·) be a non-negative, bounded, Lipschitz function in R+. There exists a
constant 'Y='Y(N,p, m), such that
(2.5) V(x o, to) E nT Vp, 9> 0 such that [(xo, to) + Q (9, p)] c nT
sup I (1~f(8)d8) ("(x, t)dx
to-9~t~O 0
(zo+Kp)
+ IIIDw l" f(w)("dxdr + IIIDUI,,-2IDwI2wf'(w)("dxdr

(zo,to)+Q(9,p») [(zo,to)+Q(9,p»)
~ 'Y I I w" f(w)ID(I"dxdr + 'YI I (fo~f(8)d8) (,,-l(t dxdr.

(zo,to)+Q(9,p») (zo,t o)+Q(9,p»)
PROOF: The weak fonnulation (1.2) can be rewritten in tenns of Steklov aver-
ages,as
(2.6) I { ! Ui,htpi + [I Dul,,-2 DUi]h·Dtpi} dxdr = 0, Vh E (O,T),

n
VO< t~T - h, VCPi E W~'''(n) n L2(n) i = 1,2, ... ,m
Since ItUi,h E L?oc( nT ), this implies
(2.6)' ! Ui,h - div [lDul,,-2 DUi] h = 0 a.e. in nT·

220 VID. Degenerate and singular parabolic systems
Without loss of generality we may assume that (x o, to) coincides with the origin.
In (2.6), take the testing function
We add over i = 1,2, ... , m and integrate in dt, over the interval -() $ t $ 0, to
obtain
j! f(t;/(3)ds) ,'dxdT
-8 Kp
t
+ J J [lDulp-2Dudh·Dui,h!(luhl) (PdxdT
-8K p
t
+ J J [IDU 1P- 2 a~l Ui,h] h Ui'hr~~IIUhl) U;,h a~l u;,h(PdxdT

-8K p
t
= -p J J [lDulp- 2DUi] h Ui,h! (luhD (p-l D( dxdT.

-8K p
We perform an integration by parts in the rust integral and then let h - t O. The
various limits are justified since IDul E Lfoc(lh) and lui EC,oc (0, T; L~oc(l1T».
This gives
(2.7) sup
-8<t<O
- -
J(
Kp
10f~J(S)dS) (P(x, t)dx
+ J JIDulP !(w)(PdxdT + JJIDuIP-2IDwI2W!'(w)(PdxdT
Q(8,p) Q(8,p)
'$ PJfiDuIP-IW!(W)(P-IID(ldxdT + p !!(1wS!(S)dS) (P-l(t dxdT.

Q(8,p) Q(8,p)
By Young's inequality for every 1] > 0
! !IDuIP-1w!(w)(P-1ID(1 dxdT $ 1] !!IDuIP !(w)(PdxdT

Q(8,p) Q(8,p)
+ -Y(1]) J J wP !(w)ID(IPdxdT.
Q(8,p)
Next by Schwartz inequality

2. Boundedness of weak solutions 221
N m N m
IDwl2 = w- 2 L (Ut Ut.z;)2 $; w- 2 L u~ L L U~.z; == IDuI2.
;=1 l=1 ;=1 t=1
Therefore IDulP ~ IDwI P • Combining these estimates in (2.7) proves the propo-
sition.
COROLLARY 2.1. The integral inequality (2.5) continues to hold/or non-negative.
non-decreasing functions / in R +. satisfying
sup /'(8) <00, /orall k > 0,
O~s~k
provided
(2.8)
PROOF: °
Fix k> and write (2.5) for the truncated functions
/(8)
fk(S) == { f(k)
for ° $; 8 $; k
for S ~ k.
Letting k -+ 00 gives (2.5) for such an f. The limit of the various terms on the left
hand side follows from Fatou's Lemma and the limit of the terms on the right hand
side is justified by virtue of (2.8).
2-(ii). Proof of Theorems 2.1

The starting point is the energy estimate (2.5) where we assume, up to a trans-
lation, that (xo, to) coincides with the origin. Fix uE (0, 1) and consider the family
of nested cylinders Qn ==Q (6n , Pn), where
Pn = up + (1;: u) p, n = 0, 1,2 ... ,

(2.9) { (1 _ u)
6n = u6 + 2n 6.
It follows from the definition that
(2.10) Qo = Q (6, p) and Qoo = Q (u6 up) .
Consider also the family of boxes
(2.11)
where forn=O, 1,2, ...

_ Pn + Pn+l 3(1 - u)
{ Pn = 2 = up + 2n+2 p,
(2.12)
8 = 6n + 6n+l = 6 3(1 - u) 6
n 2 u + 2n +2 •
222 VIll. Degenerate and singular parabolic systems
For these boxes we have the inclusion
Qn+l C Qn C Qn n = 0,1,2, ....

Introduce the sequence of increasing levels
k
(2.13) kn = k -2-
n
where k is a positive number to be chosen. We will work with the inequalities (2.5)
written for the functions (u - kn+l) +, over the boxes Qn. The cutoff function (n
is taken to satisfy
(n vanis~es ~n the parabolic boundary of Qn
(2.14) { (n == 1 m Qn
2n +2 2n +2
ID(nl ~ (1 _ (1)p' 0 ~ (n,t ~ (1 - (1)6.
Set
if (8 - kn+l) ~ e
(2.15) if 0 < (8 - kn+l) < e
if (8 - kn+d ~ 0,
and as a function few) take f~ [(w - kn+l)+]. We put these choices in (2.5) and
neglect the non-negative term involving IDul p - 2 since f;(8) ~ O. Letting e - 0
we obtain
We estimate the two integrals on the right hand side as in (7.2)-(7.5) of Chap. V.
This gives the inequalities
(2.16) sup j(W-kn+d!(x,t)dx+ ffID(w-kn+1)+IPdxdT

-8.. <t<O
Kp..
JJ
Q..
~ (1'Y~n;)p (PPkI6 _ p + 6k!-2) ff(w - kn)~ dxdT,

Q..
valid for all 6 ~ max {Pi 2}. If P > 2, the proof is now concluded as in the proof
of Theorem 4.1 in §12 of Chap. V. If 1 < P < 2, we may take 6 = r in (2.16) and
3. Weak differentiability of IDulEy! Du and energy estimates for IDul 223
obtain the analog of the recursive integral inequalities (10.3) of Chap. V. The proof
of Theorem 2.1 for the singular case 1 < p < 2 is now concluded as in the proof of
Theorem 5.1 in §16 of Chap. V.
3. Weak differentiability of IDuI P ;2 Du and energy

estimates for IDul
The main tool in investigating the local behaviour of the of the space-gradient of
the solutions of (1.1) are certain local energy estimates for Ui,zj' These are derived
by first differentiating' (1.1) and then by taking testing functions roughly speaking
I
of the type
ipi = Ui,zj f(lDul),
up to some localising cutoff function. Here f(·) is a non-negative Lipschitz func-

tion in R+. In this section we discuss a rigorous way of carrying the indicated
calculations.
PROPOSITION 3.1 (THE DEGENERATE CASE p> 2). Let u be a local weak
solution in fiT of the degenerate system (1.1). Then
IDul2j!Ui,z; EL1oc(o, Tj W1!;;(fi») , i=I,2, ... ,m, j=l, 2, ... , N,
and there exists a constant 'Y='Y(N,p), such that
(3.1) j jIDuIP-2ID2UI2dxdT
[(zo,to)+Q(119,l1p)]
$ (I..? 0")2 [p-2 + 0-1J j J (1+IDulfI) dxdT

[(zo,t o)+Q(9,p)]
where
m N
ID 2 ul 2 == LL
i=1 j,k=1
u~,z;z.·
Moreover
(3.2) Ui,zj ECloc(O, Tj L1oc(n)) , i=l, 2, ... , m, j= 1, 2, ... , N.
PROPOSITION 3.1 (THE SINGULAR CASE 1 < p < 2). Let u be a local weak
solution of the singular system (1.1) in fiT and let the approximation assumption
(1.9) hold. Then
2 r. ~ .
IDul E.jl Ui.x;ELloc\O,TjW,~(il»),
12 .
'=1,2, ... ,m, J=1,2, ... ,N,
and there exists a constant 'Y='Y(N,p), such that
(3.3) IIIDuIP-2ID2uI2dxdr
[(xo ,to )+Q( 1711,17 p»)
5 (1 : q )2 [p-2 + 6- 2] (1 + M~) 11(1 + IDuIP) dxdr,

[(xo,to)+Q(II,p»)
where
Moreover
Ui,x, E Lfoc (0, Tj w,!;:(n)) ,
and there exists a constant 'Y = 'Y( N, p) such that
(3.4) IIID 2 u 1P dxdr
[(Xo ,to )+Q( 0'11,0' p l)
5 (1:q)p [p-P + 6- P] (1 + M:) 11(1 + IDuIP) dxdr.

[(xo,to)+Q(II,p»)
Finally
(3.5) Ui,x, EC,oc(O, T; L?oc<n») , i= 1, 2, ... , m, j= 1, 2, ... , N.
This local regularity pennits to derive local energy estimates for Du. To simplify
the symbolism we set
(3.6) v=IDul·
Given a cylinder [(x o, to) + Q (6, p)] c ilT we let' denote a non-negative piece-
wise smooth cutoff function in [(x o, to) + Q (9, p)] that vanishes on the boundary
ofthe cube [xo + K pl. In particular we are not requiring in general that , vanishes
for t=to-6.
3. Weak differentiability of IDul ~ Du and energy estimates for IDul 225
PROPOSITION 3.2 (LOCAL ENERGY ESTIMATES). Let u be a local weak so-

lution of (1.1) for p > 1. In the singular case 1 < p < 2 assume in addition that the
approximation assumption (1.9) be in force. Let also / (.) denote anon-negative,
nOR-decreasing Lipschitz function in R +. There exists a constant 'Y ='Y( N, p) such
that
(3.7) 'v' (xo, to) E nT, 'v' [(Xo, to) + Q (6, p)] c nT
f(
t
sup r:/(S)dS) (2 (X, t)dx
to-9~t~O [zo+K~)Jo t
0-
9
+ ff vP- 2 1D2 u1 2 /(v)(2dxdr

[(zo,to)+Q(9,p»)
+ ff vP- 1 1Dv 12 /,(v)('2dxdr
[(zo,t o )+Q(9,p»)
+ (p - 2) t ff vP-3IDv.DuiI2/'(v)('2dxdr
1=1 [(zo,t o )+Q(9,p»)
$ 'Y ff vI' /(v)ID(1 2 dxdr + 'Y

[(zo,to)+Q(9,p»)
ff (l:/(S)dS)
[(zo,to)+Q(9,p»)
"t dxdr .
COROLLARY 3.1. The integral inequalities (3.7) continue to holdfor non-negative,

non-decreasing functions / in R + , satisfying
sup /,(s) <00, forall k > 0,

O~.~k
provided
(3.8)
PROOF: Analogous to that of Corollary 2.1.
3-(i). Taking discrete derivatives of (1.1)

For a function FE Lfoc(nT) and TJER\{O}. we introduce the discrete derivative
with respect to the Xj variable
CjF(x,t)==TJ-l{F (Xl, ... ,X; + 11, .. "xN)-F(Xb'" ,x;, .. "XN)}'
This is defined for
X E nl'7l == {x E nl dist(x,an) > ITJI} ,
where we let 1111 be so small that n l'7l is not empty. We also let cF denote the
discrete gradient of F ,i.e.,
The discrete derivative of (2.6)', with respect to Xj, takes the fonn
(3.9) ~6
at '·U· h -
I,
div [6, ·IDuI P- 2Du·]
I h
= 0'
i = 1,2, ... , m, a.e. nl'll x (0, T - h).
In transforming the term [6j !Du!P-2Dui] , we only specify the Xj variable for
simplicity of symbolism. We have
(3.10) 6j !DuIP-2 DUi

1
=~! d~ {luDu(x; +,,) + (1 - U)DU(Xj)r- 2
o
x (UDUi(X; +,,) + (1- U)DUi(Xj»)}du
!
1
= D6jUi luDu(Xj +,,) + (1 - u)DU(Xj)I P - 2 du

o
1
+(p - 2)6jUl,Z.!luDu(x; +,,) + (1- u)Du(x;)r- 4

o
x (UUl,Z/c(Xj + 11) + (1- U)UI,z/c(Xj»)
x (UDUi(X; + 11) + (1- U)Dui(Xj) )00.
To simplify the symbolism we let ~P) (u) denote the N-dimensional vector
~P)(u) = UDUi(Xj + 11) + (1- U)DUi(X;)

and let ~ (j) (u) be the N x m matrix
~(j)(u) =uDu(x; +,,) + (1 - u)Du(xj).
Having fixed the point (xo, to) E nT, if!(xo, to) + Q (9, p)] c nTwe may assume,
up to a translation, that (xo, to) coincides with the origin, and then by choosing 111!
and h sufficiently small we may assume that Q (9, p) c nl'll x (0, T - h). We
multiply (3.9) by the testing function
where, is a standard non-negative cutoff function that vanishes on the boundary of

K p' We integrate over ( -9, t) for arbitrary -9 < t ~ 0, and add over i = 1, 2, ... , m
and j = 1, 2, ... , N. This gives
3. Weak differentiability of IDul~ Du and energy estimates for IDul 227
/.(J."";/(B)da) ,'(%,t) <I:. ~.

t
+I I [DjIDul,,-2 DUi] h ·DDjUi,h! (lc5uhl) (2dxdr

-9Kp
t
+ II [c5jIDul,,-2Dui]h·c5jUi,hD! (lc5uhl) (2dxdr

-9Kp
t
= -2 I I [c5j IDul,,-2 DUi] h'DjUi,h! (lc5uhl) (D( dxdr

-9Kp
+2 jJ(f.""';/(B)da)
-9Kp
(C. dztlr.
In this equality we first let h '..... 0, while I'll > 0 remains fixed. The various limits
are justified since IDul eLfoc(nT) and ueC, oc (0, T;Lfoc(n»). Making use also
of (3.10) we obtain
16 1 ) t
(3.11) sup I ( rus!(s)ds (2(x,t)dx
-9<t<0
Kp
Jo -
9
+ l
I I (foiLl(j) (u)I,,-2dtr ) IDDj U 2 !(lc5ul)(2dxdr
Q(9,p)
+(p - 2) I I (follil(j)(U)I,,-4ILl(j)(U).Dc5jUI2dtr) ! (IDuD(2dxdr

Q(9,p)
+ I I (foiil(j) (u)I,,-2du ) IDIDuflc5ull' (l6ul) (2dxdr

Q(9,p)
+(p - 2) I I (foiil(j)(U)I"-4Ll(j)(U)'D6jUil~j)(U)6jUidU)
Q(9,p)
x DIc5ul!' (lc5ul) (2dxdr
~ 2(P - 1) I I (foiil(j) (u)I,,-2dtr ) ID6j U116ul! (16ul) (ID(ldxdr

Q(9,p)
+2 II(foI6~1!(S)dS) "tdxdr.
Q(9,p)
228 VID. Degenerate and singular parabolic systems
First we observe that the sum of the fmt two integrals over Q «(J, p) on the left
hand side, is bounded below by
min{Ij (P -I)} ff (foiJ1(;) (U)I P- 2d,q) ID6;u1 2f (16ul) (2dxdr.

Q(9,p)
If p > 2, this is obtained by discarding the coefficient (p - 2). If 1< P < 2, we

estimate below
(p - 2) ff (111J1(;)(U)IP-41J1(;)(U).D6;uI2 d,q ) f (16ul) (2dxdr

Q(9,p)
2! (p - 2) ff (Li
Q(8,p)
J1 (j) (u)IP-2d,q ) ID6;ur f (l6ul) (2dxdr.
Next by Young's inequality, for all e > 0,
ff (l
Q(8,p)
iJ1 (j) (u)IP-2d,q ) ID6;u116ul f (l6ul) (ID(I dxdr
$ eff Q(9,p)
(liJ1(;) (u) IP-2d,q ) ID6;u1 2f (l6ul) (2dxdr
+ ff
'YE ( liJ1(;) (u)I P- 2d,q) 16ul 2f (l6ul) ID(1 2dxdr.
Q(9,p)
These remarks in (3.11) give the integral inequality involving discrete derivatives
3. Weak differentiability of IDul1j! Du and energy estimates for IDul 229
(3.12) sup
-9<t<0
Kp
I(10fI6~f(s) dS) (2(x, t) dx t
-
9
+ [min{I; (p -I)} - e) II (liil

Q(9,p)
(j) (a)IP-2d,q)
x ID6jUl2 f (16ul) (2dxd-r
+ II (liil
Q(9,p)
(i) (a)IP-2d,q ) IDl6ufl6ulf' (l6ul) (2dxd-r
+(p - 2) II (liil
Q(9,p)
(j) (a)IP-4 (il(i)(a).D6j u) .1~j)(a)6jUid,q)
x DI6ull' (16ul) (2dxd-r
5')' II (li
Q(9,p)
.1(i) (a)IP-2d,q) 16ul 2f (16ul) ID(1 2dxd-r
+7 JJ (l"'1
Q(9,p)
f (8)d}C, dxd7,
for a constant ,),=,),(p, e).
3-(ii). Weak differentiability oflDullj!ui,zi

In (3.12) take f == 1 and select a cutoff function that vanishes on the parabolic
boundary of Q(0, pl. In particular, (., -0) = O. We discard the first tenn and
observe that the integrand in the remaining integral on the right hand side is non-
negative. Therefore letting" - 0 with the aid of Patou's Lemma gives
(3.13) II vP-2ID2UI2,2dxd-r 5 ')' II (vPID(/2 + v2((t) dxd-r

Q~~ ~~~
for a constant')' = ')'(P). If p > 2, the inequality (3.1) follows from (3.13) by
choosing (, a cutoff function that equals one on Q (aO, a p) and such that
1 1
ID(/5 (1 - alp' 05 (t 5 (1- a)O·
To prove (3.3) for the singular case, we transfonn the last integral in (3.13) by
means of an integration by parts as follows.
IIv 2CdxdT = II Du.Duv1!j! v!TCdxdT

Q(9,p) Q(9,p)
= II UiD [v2:f1 DUi] v!T CdxdT

Q(9,p)
+ II UiVEj! DUiDv!jR CdxdT

Q(9,p)
+ II UiVEj! DUiv!jR DC dxdT

Q(9,p)
~'Yllulloo,Q(9,p) II (vP-2ID2uI2C2) t v!jR dxdT

Q(9,p)
+'Yll u ll oo,Q(9,p) I I (I + IDuI P) IDCI dxdT.

Q(9,p)
Finally (3.4) follows from (3.3) and RUder inequality, since
lfiD2ulPdxdT = II(vP-2ID2uI2)P/2v~dxdT.
Q(9,p) Q(9,p)
Since v1!j! ID 2uI E L~oc(nT). the energy inequality (3.7) follows from (3.12) by
letting '1--+0.
3-(iv). Continuity of Ui,:I:;(t) in L~oc(n) and energy estimates

By virtue of (3.1) and (3.3) the system in (1.1) can be written in the differentiated
fonn
(3.14) &t 1,:1:; - div (IDuIP-2 Duo)

!!.U· 1 :t&;
= 0 in 1J'(flT)'
i = 1,2, ... ,N, j = 1,2, ... ,m.
Moreover (3.12) implies that
(3.15)
These two facts imply that t --+ Ui,:t&; (t) is weakly continuous in L~oc(n). Indeed
let cP E L2 (K p) and let {'Pn} be a sequence of functions in C~ (Kp) such that
IIcp - 'Pnll2,Kp --+ 0 as n --+ 00.
Taking CPn as a testing function in (3.14) and integrating over Kp x (tl, t2) gives
4. Boundedness of IDul. Qualitative estimates 231
! !!
t3
[Ui,z; (t2) - Ui,z; (td] CPndx = (IDul,,-2 DUi) Dcpn,z;dxdr

~ ~~
for almost all -0 < tl < t2 $ O. Therefore
limsup ![Ui'Z;(t 2) - ui,z;(td] CPndx = O.

It 3- t d-..o
Kp
From this and (3.15)
I
lim sup f[Ui'z; (t2) - Ui,zj (td] CPdxl
It 3- t d-..o
Kp
$ limsup
I t 3- t lf-<O
Kp
! [Ui,z;(t2) - ui,z;(td] CPn dx
+ 2 sup IIUi,z; 112,Kp(t)lIcp - CPn1l2,Kp.

-8$t$O
To prove that Ui,z; is strongly continuous in L~oc(n} it suffices to prove that
(3.16) limsup (lIui,z;(1I2,Kp(t2) -lIui,z;(1I2,Kp(tl») --+ 0,

f-<O
It 2- t l
where, is a piecewise smooth cutoff functions in Kp vanishing on oKp. In (3.14)

take the testing function
and integrate over Kp x (tl' t2). By calculations similar to those leading to (3.12)
we obtain
If
Kp
[.'(!,) - .'(',)J "dxl
4. Boundedness of IDul. Qualitative estimates
Using the weak differentiability of IDul2j1ui,z; we first prove that IDul is in

Lloc(nT ) for all q ~ 1. If K-o C K-l are compact subsets of n T • we will show that
the norm IIDullq,K:o is bounded only in terms of q, dist {K-o; K- l } and the norm
IIDullp,K:l. We will do this in a qualitative way and with no precise specification
of the functional dependence. We will use such qualitative infonnation to prove
still qualitatively that /Du/ E Ll:c {!1T ), with bounds only dependent on local
V-nonns of /Du/. Finally, in the next section, we will tum such qualitative in-
fonnation into precise quantitative estimates of IIDulloo,K: o over compact subsets
lCoc{h.
LEMMA 4.1. Let u be a local weak solution of(1.1). Moreover in the singular
case 1 2. Let Q (6, p) c {h and let (
be a standard non-negative cutoff function vanishing on the parabolic boundary of
Q (6, p). Thus, in particular, (., -6) =0. In (3.7), take J(v) =vP, where P?O is
to be chosen. Proceeding fonnally we obtain
(4.1) sup Iv +pe (x, t)dx $ 'Y jrJf (1 + vP+

-9<t<O
2 lt ) dxdr
- - Kp . Q(9,p)
o
(4.2) IIIDv£¥r dxdr $ 'Y II (1 + vP+ P) dxdT,
-9K p Q(9,p)
where 'Y = 'Y (N,p, p, (t, D(). These are rigorous if the right hand side is finite.
We apply the embedding Theorem 2.1 of Chap. I to the functions
x -+ (v£¥() (x, t), a.e. t E (-6,0),
over the cubes Kp. It suffices to consider the case N > 2. Indeed if N = 1, 2, we
may consider u as a vector field defmed in RN N ? 3, up to a localisation, and
deduce inequalities (4.1 )-(4.2) for it. Let 6 be a positive number to be chosen. Then
by Corollary 2.1 of Chap. I and HOlder's inequality
We integrate over (-6,0), to obtain

4. Boundedness of IDuI. Qualitative estimates 233
JJ....•..
Q(8.p)
"dzdT $ JJIv.'¥ 'I'dzdT C:~r<o
Q(8.p) ~
J.'If ,'(x, t)JJ
- - Kp
1N
Choosing 6 = 2¥1 and combining this with (4.1)-(4.2) gives the recursive in-
equalities
(4.3) I I tr.8~+i (2dxdr :5 'Y 11(1 + vJ'+.8) dxdr,

Q(8.p) Q(8.p)
for a constant 'Y = 'Y(N,p,/3,(t,D(). The right hand side is finite for /3 = o.
Therefore IDul E Lr;4/N ({IT). We may now again apply (4.3) with /3 =4/N and
proceed in this fashion to prove the lemma.
We now tum to the singular case 1 < p < 2. In (3.7) assume that (x o, to) ==
(0,0) and choose a cutoff function ( that vanishes on the parabolic boundary of
Q «(J, p). Take also f (v) = v.8, where /3 ~ 0 is to be chosen. By working with
the approximations claimed by (1.9) we will use the qualitative infonnation that
IDul E L~oc(nT). Our estimates however will be only in tenns of IIDull".Q(8.p).
Proceeding fonnally we obtain from (3.7)
(4.4) Illvl!.±f=.! D 2U(r dxdr :5 -y { II tr.8dxdr + II V2+.8(dxdr} ,

Q(8.p) Q(8.p) Q(8.p)
where -y=-y (N,p, {3, (t, D(). Also by a fonnal integration by parts
/lv.8+ 2 (dxdr = //V~+f-2 Du.DuvP+~-P(dxdr

Q(8.p) Q(8.p)
= IluD(V~DU) v~(dxdr
Q(8.p)
+ // u vP.±f=! Du Dv~ ( dxdr

Q(8.p)
+ II u vP.±f=! Du v~ D( dxdr
Q(8.p)
:5 'Yllulloo.Q(8.p) / /l v l!.±f=3 D2ul( v~ dxdr

Q(8.p)
+ 'Yll u lloo.Q(8.p) II v.8+1 dxdr.

Q(8.p)
We combine this with (4.4) and make use of the Schwartz inequality to arrive at
(4.5) JJ v P+2Cdxd'T $ 'Y

Q(6,p)
!J( +
Q(6,p)
1 v P+(2-p) + vP+l) dxd'T,
for a constant
'Y='Y (N,p, (j, Ct, DC, lI u ll oo ,Q(6,p») .
This inequality is indeed rigorous as long as the right hand side is fmite. We apply it
fIrSt with (j =p-l to deduce that IDul E L~l (nT ). with bounds only dependent on
II Dullp,Q(fI,p) . Then we apply it again with {j =p to deduce that IDul E L~2 (nT ).
Proceeding this way proves the lemma.
,
LEMMA 4.2. Let u be a local weak solution of(1.1). Moreover in the singular
case 1 2. Let Q (6, p) c nT and let Qn
and Qn be the family of cylinders introduced in (2.9)-(2.12). Let also k,. and Cn
be respectively the increasing levels defined in (2.13) and the cutoff functions in
Qn introduced in (2.14). We put these choices in the energy estimates (3.7) and as
a function f (v) take
f(v) :: (v - kn+l)r 2 •
By virtue of Lemma 4.1 and Corollary 3.1. such a choice is admissible. The teon
involving D 2 u is estimated below by
2 22
I I v"- ID uI f(v)C2dxd'T ~ (~) P-jIID (v - 12
kn+l)! C!dxd'T.
Qft Qft
1bese choices yield the inequalities
(4.6) -fl:{:~O 1[(v-kn+1)lCn]\x,t)dx

K pft
+ kP - 2 !!ID
Qft
[(v - kn+1)l Cnr dxd'T
$ (12!;2p2 JJV2(P-l)X[(V - kn+l)+> 0] dxd'T

Qft
for a constant 'Y='Y(N,p). To simplify the symbolism let us set
(4.7) Yn :: !!(V-kn)~dXd'T.
Qft
4. Bc..undedness of IDul. Qualitative estimates 235
Then we have(l)
(4.8) / / X[(v - kn+l)+> 0] dxdr ~ 'Y 2k7 / / (v - kn)~ dxdr

Qn Qn
== 'Y2np k -p Yn.
By Proposition 3.1 of Chap. I with m=p - 2 and q= 2(N + 2)/N.
(4.9) Yn+1 ~/ /[(V - kn+d,:/2

Qn
(nr dxdr
$ (tfi(-- (.]'''# dxdT)

k.H )'." w'h
(if>IJ- - oJ dxdT)
x "-+1)+> .to
~ (1'Y~:)2 k-~Yn~ {p-2 / /V2(P-I)x[(V - kn+l)+>O] dxdr

Qn
+ (J-I //vPX[(v - kn+l)+> 0] dxdr} I

Qn
where.\2 == N(P - 2) + 2p. These are the key recursive inequalities needed to
derive a quantitative sup-bound for IDul. We will use them first in a qualitative
way as follows. First let A denote a lump constant depending upon (J I (I, P and the
quantities
IIDuIl2,Q(B,p) II Dullq,Q(B,p) q = (N + 2)(P - 2) + p.

Then we estimate
/ / v 2(p-I)X[(v - kn+l)+>O] dxdr

Qn
= / / vwhvP~x[(v - kn+d+> 0] dxdr

Qn
(I) See §7-(i) of Chap. V and, in particular, estimate (7.2).

236 VIll. Degenerate and singular parabolic systems
We have also(1)
(4.10) II
Q..
vPX[(v - kn+d+>O] dxdT ~ -y2np Yn·
Therefore
II v 2(P-1)x[(v - kn+1)+> 0] dxdT ~ A2npYn~.

Q..
1bese remarks in (4.9) give the recursive inequalities
Yn+1 ~ A k-m bny;+wh , n = 1,2, ... , b = 4"+1
where we have also used the choice k ~ 1 and the inequality
y;+wh ~ A y1+wh.
It follows from Lemma 4.1 of Chap. I that Yn -+ 0 as n -+ 00 if
Yo = (Ak- m ) -(N+2) b-(N+2)2.
1berefore
IIDulloo,Q(a9,ap) ~ max{l; k}
~ 1 + A¥b(N+2)2/p IIDullp,Q(9,p).
We now tum to the singular case 1 2
if r = 2,
where Ie (.) is the Lipschitz approximation to the Heaviside graph, introduced in
(2.15). After we let e -+ 0, the first term on the left hand side is bounded below for
all t E ( -fJ, 0) by the quantity
K
f(l tJ
k"+!
s2- p s (s - kn +1r- 2 dS) (!(x, t)dx
+
p"
~ r~1 (~r-Pf[(V-kn+1)12(nr dx.

K p"
(1) See for example estimate (7.5) of Chap. V.

4. Boundedness of IDul. Qualitative estimates 237
The term involving D 2 u is estimated below by
IIID2U l2 (v - kn+1)~-2 x[(v - kn +1)+>O] (~dxdr
r(~dxdr.
Qn
~ r~ IIID (v - kn+d12
Qn
Combining these estimates in (3.7) we arrive at
(4.11) k 2- p sup I [(v - kn+1>12 (n(x, t)] 2 dx

-B<t<O
- - KPn
+ IIID (v - kn+1)1 2 (n1 2dxdr

Qn
where 'Y='Y(N,p, r). To simplify the symbolism we set
(4.12) Sn= JI(v-kn)~dxdr,

Qn
and combine (4.11) with the embedding of Proposition 3.1 of Chap. I, with m =
p=2 and q=2(N + 2)/N. This gives
'Y 2(r+2)n 2 (2N"~+r wh
(4.13) Sn+1 ~ (1-0')2 k- + Sn +
X {p-2 Ilvrx[(v - kn+d+> 0] dxdr

Qn
+ 6- 1 II v r+(2- p )x[(v - kn +1)+>O] dxdr} ,

Qn
where we have used a version of (4.8). These are the key inequalities needed to
derive a quantitative bound of IDul in the singular case 1 < p < 2. As before, we
will use them fIrst in a qualitative way. We choose k ~ 1 and let A denote a lump
constant depending upon p, 6, 0' and the norms
II vIl2,Q(B,p) , IIvll q,Q(B,p) , q=(N + 2)(2 - p) + r.

Then we estimate
238 vm. Degenente and singular parabolic systems
ff
Q..
l1r+{2- p)X[(v - kn+d+>0] dxdr
= f/11(2-p)+m l1r~x[(V - kn+l)+>O] dxdr

Q..
~ ~
$ (IfoftdxdT) (If v',f.(v - ~,)+>ol dxttr)
In deriving the last inequality we have used a version of (4.10). These estimates in
(4.13) yield the recursive inequalities
Sn+l <Abnk-msl+rtr
_ + n ,
Here we have used the choice k ~ 1 and the inequality
S!+~ ~ A S!+rtr .
The proof is now concluded as in the degenerate case.
5. Quantitative sup-bounds of \Du\
THEOREM 5.1 (THE CASEp>2). Letubealocalweaksolutionofthedegen-

erate system (1.1). There exists a constant 'Y='Y(N,p) such that
V(xo,to)eflr, V[(x o,to)+Q(9,p)]cflr, Vo-E(O,l),
(5.1) sup
.
IDul ~ (
'YV(9/p2)
)(N+2)/2
()~
ff IDul dxdrI'
[(zo.to)+Q{...8 ....p») 1- 0-
(zo.to )+Q(8.p»)
"9(p2)i6 .
THEOREM 5.2 (THE SINGULAR CASE 1 o.

Then there exists a constant 'Y='Y(N,p, r) such that
5. Quantitative sup-bounds of IDul 239
(5.3)
Remark 5.1. The constant -y(N,p, r) in (5.3) tends to infinity as vr-O.
5-(i). Proof of Theorem 5.1

We start from the recursive inequalities (4.9) and estimate the first integral on the
right hand side as follows:
//v 2(P-l)X[(v - kn +1 )+> 0] dxdr

Q..
~ (s~~v) 2-'lJv"X[(V - kn +1)+> 0] dxdr

Q..
~ 2 (s~~ v) 2-jJ( v - kn)~ dxdr.

n"
Q..
Therefore (4.9) yields
(5.4) Yn +1 ~ (1 ~b:)2 k--ih {(s~~vr-2 p-2+(r Y~+~, 1} b=4".
If for some n= 0,1,2, ... we have
there is nothing to prove. Otherwise we rewrite (5.4) as
Yn +1 ~
-ybn
2
(
sup v
)"-2 _~k N+2
l+~ .
Yn
[(1 - u)p] Q(9,p)
It follows from Lemma 4.1 of Chap. I that {Yn } neN - 0 as n - 00, if k is chosen
from
(N+2)/2 (N+2~(p-2)
k>'2/ 2 == ( 'Y
b(N+2)/2)
2
( )
sup v JrJrv"dxdr.
[(1 - u)p] Q(',p) Q(',p)
240 vrn. Degenerate and singular parabolic systems
We conclude that there exists a constant ,,(=,,(N,p) such that

1-4/>'2
(
(5.5) sup V < "( sup v )
Q(a9,ap) - [(1 - 0")p]2(N+2)/>'2 Q(9,p)
2/>'2
X
(
IlvPdxdr )
Q(9,p)
If O"E (0,1) is fixed. consider the family of boxes Q(n) :;:: Q (On, Pn). where
Po:;:: O"p
and for n= 1, 2, ...
n n
Pn = O"P+ (1- O")p LTi On = 0"0 + (1- 0")0 L2-i.

i=1 i=l
By construction.
Q(O) :;:: Q (0"0, O"p) and Q(oo) == Q (0, p) .
Set
Mn = esssupv
Q(n)
and write (5.5) for the pair of boxes Q(n) and Q(n+1). This gives
(5.6)
where
B :;:: ,,(>'2/ 4
[(1 - 0")p](N+2)/2
(f!vPdxdr) 1/2
'
d 2(N+2)/2.
=
Q(9,p)
The proof is now concluded by the interpolation Lemma 4.3 of Chap. I.
5-(ii). Proof of Theorem 5.2

We start from the recursive inequalities (4.13) and estimate
IIvr+(2- )x.[(V - kn+d+> 0] dxdr

p
Qn
~ (s~~v) 2-pII vrx[(v - k n +1)+> 0] dxdr
~ 2 (s~~v
nr
r- Qn
p
Sn.
5. Quantitative sup-bounds of IDul 241
We may assume that
Qn v - - !!...
>
sup 2 p tior all n = 0 , 1 , 2 , ....
p2'
Otherwise there is nothing to prove. Taking this into account, we rewrite (4.13) as
S < 'Ybn k"Nh(r+2-p)

(sUPQ(8,p) v )2-P Sl+wh
n+l - (1 _ 0-)2 (J n
where b = 4r+l. By an argument analogous to that in the degenerate case, this

implies
---.l!.±L
sup v< 'Y (
SUP
Q(8,p)
V2-P) 2(,+2-p)
(5.7)
Q(<T8,<Tp) - (1 - 0') r~t!p (J
1!(r+2-p)
( )
X jjvrdXdT
Q(8,p)
The proof is now concluded with an interpolation process as in the degenerate case.
This is possible if the power of the term sUPQ(8,p) von the right hand side of (5.7)
is less than one. Since
(2 - p)(N + 2) =1 _ Vr ,
2(r + 2 - p) 2(r + 2 - p)
this occurs if (5.2) holds. We also remark that the interpolation process applied to
(5.7) generates a dependence of the type of l/vr in the constant 'Y(N,p, r) appear-
ing in (5.3).
5-(iii). Interpolation inequalities

The inequality of Theorem 5.1 can be interpolated. For example consider
(5.1) for (xo, to) == (0, 0) and rewrite it as
sup
Q(<T8,<Tp)
v <
-
'Y ..[(iTiJ)
(1 - 0')(N+2)!2
(sup
Q(8,p)
v) ~ (n Q(8,p)
Vp-2H dXdT)1!2
....1....
A( ~y-3
Such an inequality can be interpolated as long as e E (0, 2] and proves the follow-
ing:
THEOREM 5.1' (THE CASE P > 2). Let u be a local weak solution of the
degenerate system (1.1). Then for every E E (0,2]. there exists a constant 'Y =
'Y(N,p,E) such that
V(xo, to) E nT, v [(x o, to) + Q (8, p)] c nT, Vq E (0,1),
)
l/~
(5.8) sup
[(zo,t o)+Q(1J'9,lJ'p»)
IDul $ (
'Y
1-
(8/,r) l/~
q
)(N+2)/~
(
H IDul
,,-2+~
[(zo,to)+Q(9,p»)
dxdT
A ( ,r)~
8 .
Remark 5.1. The constant 'Y = 'Y( N, p, E) / 00 as E '\. O.
Also, (5.3) can be interpolated. We rewrite it for (xo, t o):= (0, 0) and in the fonn
'Y (,r /8) N/vr !t!:.=.U

"r ( )2/ Vr
sup
Q(1J'9,lJ'p)
V <
- (1- q)2(N+2)/vr
sup
(Q(9,P) )
V
H
Q(9,p)
vqdxdT
A (!..)"!;
p2 .
This can be interpolated as long as qE (0, r] satisfies 2(::q) < 1. This occurs if
(5.9) IIq :=N(p - 2) + 2q > O.
The interpolation process gives
THEOREM 5.2' (THE SINGULAR CASE 1 < p < 2). Let u be a local weak
solution of the singular system (1.1) and let the approximation assumption (1.9)
be in force. Moreover let r ~ 2 satisfy (5.2). Then for every q E (0, r] satisfying
(5.9) there exists a constant 'Y='Y(N,p, r, q) such that
(5.10)
Remark 5.3. The constant 'Y(N,p, r, q) in (5.10) tends to infinity as IIq -0.
6. General structures 243
Remark 5.4. Estimate (5.10) is fonnally equivalent to (5.3), the only difference
being that q is not required to be larger or equal to 2. The only condition is that
(5.9) be verified. In particular, (5.10) holds for q=p provided
2N
(5.11) p> N +2'
COROLLARY 5.1. Let 1 <p<2. Then ID2Ul eL1oc(nT ).

PROOF: From (3.3), for every [(zo, to) + Q (8, p)] c nT,
! !I D2U I2dxdT = !!IDuI2-PIDUIP-2ID2UI2dxdT

[(zo,to)+Q(8,p») [(zo,t o)+Q(8,p»)
:::; sup IDu1 2-p !!IDUIP-2ID2UI2dxdT < 00.

[(zo,t o)+Q(8,p») [(zo,t o)+Q(8,p»)
6. General structures
Let U be a local weak solution of the non-linear system (1.10) subject to the struc-
ture conditions (81 )-(86 ), The local boundedness of U can be established as in the
proof of Theorem 2.1. The main modification occurs in the handling of the 'per-
turbation terms' l(Ji, i = 0, 1, 2. These contribute to the energy inequalities (2.5)
with an extra tenn of the type
! !{1(J0 (I(w) + wl'(w» + (1(J1ID(1 + 1(J2) wl(w)} dxdT.

[(zo,to)+Q(8,p»)
Given the choice (2.15) of 1(')' these terms are estimated as in the sup-bounds
established in Chap.V for general equations. (1) The weak differentiability of the
tenn IDulp-2 Du follows from the structure conditions (81 )-(82), We proceed as
before by working first with the discrete derivatives. All the tenns involving the
'derivatives' 6j Ui,zc' are dominated by the tenns arising from the right hand side
of (82 ), (2) Following the same process of §3 yields local energy estimates similar
to (3.7) with constants 'Y ='Y( N, p, Co, C1 ) and with the right hand side augmented
by the extra integral
! !{1(J0 (I(v) + vl'(v» + (1(J1ID(1 + 1(J2) vl(v)} dxdT.

[(zo,to)+Q(8,p»)
(1) See for example Theorem 3.1 of Chap. V and its proof.
(2) See also Remark 1.1.
These energy estimates imply that IDul E L~(nT) be the same iterative tech-
niques of §4. The 'perturbation terms' are dealt with as in Chap. V.
In the case of a single equation the estimate (1.5) up to ST has been established
by Lieberman [68]. Estimates in the norm cl,a up to ST for Dirichlet data, are
not known even for elliptic systems. Results for a single elliptic equations are
due to Lieberman [69] and Lin [72]. The general structures of §1-(l1) have been
introduced first by Tolksdorff [95]. The arguments of finite differences to prove
that IDul,-2Ui,Xj is weakly differentiable were introduced by Uhlenbeck [99] in
the context of elliptic systems. The sup-bound of Theorem 2.1 for the degenerate
case p > 2 is new. The same theorem for the singular case 1 ~~2'
and in Choe [31] for all p > 1 provided (1.7) holds. Even though some quantitative
estimates of the gradient appear in a variety of forms in [27,36,37], Chen [25] and
Choe [30], the precise form of Theorems 5.1 and 5.2 as well as their interpolated
version in §5-(lII}, seems to be new.
IX
Parabolic p-systems: Holder continuity
of Du
1. The main theorem

The space gradient Du of local weak solutions of the quasilinear system (1.10) of
Chap. VIn are locally HOlder continuous in nT provided the structure conditions
(St}-(Ss) are in force. We will show this first for the homogeneous system (1.1)
and then will indicate how to extend it to the general systems (1.10). The estimates
of this chapter hold in the interior of nT and deteriorate near its parabolic boundary
r. If /C is a compact subset of nT we let dist(/Cj r) denote the parabolic distance
from /C to the parabolic boundary r of nT, i.e.,
dist (/Cj r) == inf

(.. ,·lelC
(Ix - yl + ~) .
(1I,·ler
THEOREM 1.1. Let u be a local weak solution 0/(1.1) o/Chap. VI/l. Moreover
if 1 < p < 2 let the approximation condition (1.9) be in /orce. Then
(x, t) -Ui,zj (x, t) E Ct!c(nT), for some a E (0,1),
for all i = 1, 2, ... ,m and all j = 1, 2, ... ,N. Moreover/or every compact subset
/C OinT. there exist constants a=a(N,p) E (0,1) and-y=-y (N,p, II Dull oo,K:) >
1. such that
(1.1) IUi,Zj (Xl, tt} - I

Ui,Zj (X2' t2) ~ -y ( IXI-X21+ltl-t211/2)Q
dist (/C; r) ,
/or every pair o/points (XI,tl), (X2,t2)E/C.

Remark 1.1. The constants -y and a are independent of dist (/C; r). They how-
ever deteriorate as p'\. I, i.e.,
246 IX. Parabolic p-systems: ltilder continuity of Du
lim inf 'Y (N, p, II Dull 00 K) , a-I (N,p) - + 00.

p'\,l '
Remark 1.2. The functional dependence of'Y upon IIDulloo,K will be given in
§§3 and 4.
l-(i). Some notation and the two basic propositions

The proof of Theorem 1.1 is based on estimating the essential oscillation of
Ui,%j in cylindrical domains of the type [(x o, to) + Q (6, p)] c {IT. After a trans-
lation we may assume that (x o, to) coincides with origin. Let IJ and R be positive
numbers and consider the cylinders
Qn(lJ} == Knx {-1J 2 - P R 2 ,0}, satisfying

{
(1.2) sup IDul:5 IJ.
QR(")
The geometry of Qn(lJ) is intrinsic in that the t-dimension is 'stretched' by a

factor, loosely speaking, of the order of IDuI 2 - p • Let us assume for the moment
that such boxes can be constructed. Then Theorem 1.1 is a consequence of the
following two propositions.
PROPOSITION 1.1. There exist numbers II, It, 6 in (0, 1) that can be determined
a priori only in terms of N and p. such that if
there holds
(1.4) //IDU - (Du)"+l12dxdr :5 1t6N +2/ /IDU - (Du},,1 2dxdr,

Q,,,+lR (,,) Q,,, R(")
for all n= 1,2, ... , where
(Du)" = ff Dudxdr.
Q'''R('')
PROPOSITION 1.2. There exists numbers j <u<t']< 1 and such that if
there holds
1. The main theorem 247
(1.6) IDul(x, t) ~ rn~,
These two facts will be used to establish the following:
THEOREM 1.2. Assume that the cylinder Q R (/J) satisfies (1.2) for some /J > 0.
There exist constants "Y> 1 and a E (0, 1) that can be determined a priori only in
terms of N and p, such that
(1.7) ~~ Ui,z; ~ "Y/J (~r, VO<p~R,

for all i=I,2, ... ,mandall j=I,2, ... ,N.
1-(;;). Constructing QR(/J)

°
Assume first that p > 2. The number R> being fixed, let /Jo be the smallest
value of the parameter /J such that QR(/Jo) C {IT. If IIDulloo,QIlC",o) ~ /Jo, then
QR(/Jo) satisfies (1.2). Otherwise we take as /J the largest root of the equation
IIDulloo,QIlC"') = /J.
Such an equation has finite roots since IIDulloo,QIlC",o) > /Jo, and
/J-IIDulloo,QIlC",) remains bounded as /J-OO.

These arguments are based on the fact that, since p > 2, the 'vertical size' of Q R(/J)
decreases as /J increases. In the singular case we consider instead boxes of the type
QR(/J) == {Ixl ~ /J~ R} x {-R2,0}, satisfying

(1.2)' {
sup IDul ~ /J.
QIl("')
As /J increases, the cubes {Ixl < /J ~ R} shrink. Therefore there exist some /J for
which (1.2)' holds.
Remark 1.3. The previous propositions could be stated and proved in the geom-
etry of the boxes (1.2)'. Indeed setting /J~ R=r permits one to recast the 'space
scaling' of (1.2)' in terms of the 'time scaling' of (1.2).
248 IX. Parabolic p-systems: HOlder continuity of Du
1-(;;i}. More about the intrinsic geometry

Take formally the xj-derivative of (1.1) of Chap. VIII and multiply the ith
equation of the system so obtained. by 'Ui.z j ' Adding over i = 1,2, ... , m and j =
1,2, ... ,N. and setting W = IDul 2 we arrive at the formal differential inequality
(1.8) ! W - (al,kw2j!wZk) Zl ~0 in nT,

where
(1.9) at,k == { 6t ,k + (p - 2) U'i;;:j;Zk } .

The matrix (at,k) is positive definite and w is a non-negative weak solution of
an equation of the porous medium type. This is a parabolic version of the quasi-
subharmonicity. The degeneracy of (1.8) is of the order of w 2j! and it is overcome
by the choice ofthe parabolic geometry of QR(I-').
2. Estimating the oscillation of Du

We assume Propositions 1.1 and 1.2 for the moment and proceed to prove Theorem
1.1. Let QR(I-') be a cylinder satisfying (1.2) and define the two sequences
(2.1) {
1-'0 = 1-', Ro = R and for n = 1,2, ... ,
I'n+l = TJl'n, Rn+l = CoRn,
where TJ and 0' are the numbers claimed by Proposition 1.2 and
(2.2)
Since TJ E (!, 1). we have Co E (0,1) for all p > 1. Suppose the assumption (1.5)
bolds with R = Ro and I' =1'0' Then
sup IDul ~ TJ 1-'0 == 1-'1'

Q.. Ro("'o)
From the definitions (2.1) and (2.2) it follows that Rl < Ro and
R21 a 2..,p-2
'r R20 _ (0')2 R20
I-'f-2 = - 4 - TJP-21-'~ 2 = '2 1JP-2 •
This implies that the cylinder QRl (I-'d is contained in QtTRo (1-'0) and
sup IDul ~ 1-'1.
QR1(",d
Therefore QRl (I'd satisfies (1.2) an4 if the assumption (1.5) of Proposition 1.2 is
verified again for such a box we have
2. Estimating the oscillation of Du 249
sup IDul ~ 1-'2.

QR2(~d
Proceeding in this fashion, suppose the assumption of Proposition 1.2 is verified

for the cylinders
QRn (I-'n) , n = 0,1,2, ... ,no - 1 for some positive integer no·
Then
(2.3) sup IDul ~ I-'n == '1 n l-'o, n = 0,1,2, ... , no.

QRn(~n)
From the definitions (2.1) and (2.2) it follows that
One verifies that Co < '1 for all p > 1, and consequently al E (0,1). We rewrite
(2.3) as
(2.4) sup IDul ~ 1-'0 (Rn)Ql ,for n = 0, 1,2, ... , no·

D
QRn(~n) no
Suppose now that the assumption (l.5) of Proposition l.2 fails for no. We call
Rno the switching radius. Then for the box QRno (I-'nJ the assumption (1.3) of
Proposition 1.1 holds and we conclude that
(2.5) H IDu - (DU)i 12 dxdr ~ KiH IDu - (Du)o 12 dxdr

Q ,iRno (~no ) Q Rno (~no )
< i 2
_KI-'n . 1 2
o ' t=, , ...
Writing
(2.6) I(Du)i+ 1 - (DU)iI 2 ~ 21Du - (DU)i+l1 2 + 21Du - (DU)ir
and taking the integral average over Q6i+1Rno(l-'no) gives
'Y = 2 (K + cS-(N+2») .
Therefore {(DU)iheN is a Cauchy sequence whose limit we denote with Du( x o , to).
To motivate this terminology we recall that our arguments are carried over an ar-
bitrary cylinder [(xo, to) + QR(I-')] with vertex at (xo, to). Therefore if no is the
switching radius of the box [( x o , to) + QR(1-')], the limit ofthe averages,
HDUdxdr,
[(zo,tO)+Q'iRno (~no)]
250 IX. Parabolic p-systems: Hl)lder continuity of Du
n
is Du(x o• to) for almost all (x o• to) E T • It follows from (2.6) that
2. 2
1Du(xo•to) - (Du), 1 ~ 'Y 1t'lJno ' i = 1.2•....
Fix 0 < p< Rno and denote with (Du)p the integral average of Du over Qp(lJnJ.
Let i be a positive integer such that
(2.7)
and estimate
(2.8) I(DU)p - (DU),r ~ if IDu - (DU),r dxd-r

Qp(""o)
~ 'Y6-(N +2) It'1J!0'
Therefore
(2.9) IDU(xo• to) - (DU)pr ~ 2IDu(Xo. to) - (DU),r
+ 21 (Du)p - (DU),r
::; 'Y(6) It'1J!0'
It follows from (2.7) and (2.9) that
Let 2Qo =min{Ql; Q2}. Then combining (2.9) and (2.4) we conclude
LEMMA 2.1. There exist constants 'Y> 1 and Q o E (0. 1) that can be determined
a priori only in terms 0/ Nand p. such that/or almost all (x o•to) E nT such that
[(xo. to) + QR(IJ)] c nT. and/orallO<p~R. there holds
(2.10)
Moreover
(2.11) if IDu - (DU)pr dxd-r ~ 'Y IJ! (~) 2

Q
o •
Qp(""o)
Remark 2.1. The lemma holds also in the geometry of the boxes [(xo. to) + QR(IJ)]
introduced in (1.2)'. Indeed we may set 1J2.j! R=r and work within the cylinder
[(xo. to) + Qr(IJ)]. We arrive at a version of (2.10) that reads
(2.10)'
3. RUder continuity of Du (thecasep>2) 251
Returning to the geometry of [(%0. to) + QR(P)] proves the assertion. Analogous
considerations bold for (2.11).
3. HOlder continuity of Du (the case p > 2)

We assume that IDul ELOO(nT ) and set
P = II Dull OO,DT •
This is no loss of generality. by possibly working with another compact set
K:,' satisfying K:, C K:,' c nT. and
dist (K:,j K:,') ~ Id.

We will prove the mlder continuity of Du in the time and space variables sepa-
rately.
3-(i). HOlder continuity in t

Fix two points (%0. t.) E K:,. i = 0.1. with the same 'abscissa' Xo' We let
tl > to and construct the cylinders
[(xo.t.) + QR(P)] ={IX - xol < pEj! R} x {t. - R2. t.}.

The box [(xo. h) + QR(P)] intersects [(xo. to) + QR(P)] at the point (xo, to). if
(tl - to) < R2. Moreover they are contained in nT if
(3.1)
LEMMA 3.1. Let (3.1) hold. There exist constants 'Y > 1 and Q E (0.1) that can
be determined a priori only in terms 0/ Nand p such that/or all pairs (x o • t.) E
K:" i=O.l.
PROOF: Let Rn, be the switching radii of the cylinders [(xo. ti) + QR(P)] and
introduce the two boxes
Qi = [(xo. t.) + QR", (Pn,)]
= {Ix - x.1 < PRy Rn, } x {t. - R~,. ttl .
Assume first that
(3.3)
252 IX. Parabolic p-systems: H6lder continuity of Du
and for i=O,l construct the two cylinders
Ci == [(xo, ti) + Q"'2('I-'O) (I'n;)]

= {IX - xol < I'::T J2(tl - to) } X {ti - 2(tl - to), ti}.
By virtue of (3.3) we have the inclusions Ci C Qi, i = 0, 1. Moreover Co and C1

intersect in a box satisfying
(3.4) meas [Co n Cl ] ~ min {I'no ; Jl.nl} (tr - t o)(N+2)/2.
Set
(Du)c; == H
c;
Dudxdr, i = 0,1,
and estimate
IDu(xo, tl) - Du(xo, to)1 ~ IDu(xo, t.) - (DU)Cl I

+ IDu(xo, to) - (Du)C o I
+ I (Du)c, - (Du)c I· o
By (2.10) and Remark 2.1 we have, for i =0, 1.
IDu(xo, ti) - (Du)c; I ~ 'Y I' ( "'tlR- to)a

o
To estimate the last term we add and subtract Du(x, t) where (x, t) E Co n Cl , and
then take the integral average over such intersection, i.e.,
I (Du)co - (Du)co I ~ HI (Du)c, - Du(x, t)ldxdr

conel
+ H IDu(x, t) -. (Du)c o Idt.
conel
Without loss of generality we may assume that min {I'no ; I'nl} = I'nl. Then we
estimate the first integral by extending the integration over the larger set Cl . Taking
into account the definition of C1 , (3.4) and (2.11), we obtain
HI (DU)Cl - Du(x, t)1 dxdr ~ 'Y HI (DU)Cl - Du(x, t)1 dxdr

conel Cl
~'YI' ( ~)ao
R
To estimate the second integral. let fJ be a small positive number to be chosen and
assume that
3. .Hl)lder continuity of Du (the case p> 2) 253
(3.5)
Then using again (2.11) and (3.4),
H
Co nel
IDu(x, t) - (Du)co Idt
N(P-2)/2H
:5 "I ( ~:: ) IDu(x, t) - (Du)Co Idt
Co
< (I'no )N(P-2)/2 (~)QO

- "I I'nl I' R
Qo-{JN(p-2)/2
< ( V(tt - to) )
-"II' R
Therefore if (3.3) and (3.5) hold, the assertion (3.2) follows by taking {3 =
Qo/ N (p - 2) and then choosing Q =Qo/2. If (3.5) is violated,
( ~) Q /N(p-2) o
IDu(x o , to) - DU(Xl, tdl :5 21' R 0 ,
and the assertion follows by suitably modifying the defmition of Q.

We consider next the case when (3.3) is violated, i.e.,
2(tl - to) > min {R!o j R!1}'

If R!, :5 2(tl - to) for i=O, 1, then by (2.4)
IDu(x o, tt) - Du(xo, to) I:5 I'no + I'nl :5 "II' (v'fl=t;;)

R
Q
Therefore we may assume that, say,
R!1 :5 2(tt - to) < R!o'

We conclude the proof by reducing this case to the situation (3.3). Let n. :5 nl be
a positive integer satisfying
R!. ~ 2(tl - to) ~ R!.+l'

and introduce the cylinders Qo and Q •• where
Q. == (x o, ttl + QR". (I'n.) == {Ix - xol < I':f Rn. } x{tl - R!.,tl}'
Since we have
2(tt - to) :5 min {R!o j R!.} ,
the box Q .. will now play the same role as the cylinder Ql in the case (3.3). The
proof is now concluded as before, observing that for Q.. the two inequalities (2.10)
and (2.11) hold true.
2S4 IX. Parabolic p-systems: H6lder continuity of Du
LEMMA 3.1'. There exist constants 1> 1 and Q e (0,1) that can be determined a
priori only in terms 0/ N andpsuch that/or every pairo/points (x"to)elC, i=
0,1,
PROOF: If JI.'~ 1 we take R=dist (lC;r) in (3.1). Otherwise we take IJ¥ R=

dist (/C; r).
3-(U). Holder continuity in x

Fix two points (xo, to) and (Xl, to) in /C, at the same time level to. and let
[(Xi, to) + QR{IJ») == {Ix - Xii < R} x {to - 1J2- P ~, to}, i = 0,1,
be two boxes satisfying (1.2). The box [(xo, to) + QR(P)] will intersect Xl iflxo-
xII < R. Moreover they are contained in nT if
(3.6)
LEMMA 3.2. Let (3.6) hold. Thereexistconstants1> 1 andQE (0, 1) that can be
determined a priori only in terms 0/ Nand p. such that/or all (x" to) E /C, i =0, I.
(3.7) I
IDU(XI, to) - Du(xo, to) $ 11J (Ixo ~ XII) Q •
PROOF: Let R,." be the switching radii corresponding to the two boxes
[(Xi, to) + QR(IJ»). and construct the two cylinders
. Qi == [(x" to) + QR", (IJn.)]
== {Ix - xii < R,.,,} x {to -IJ!;-P R!" to} •
Consider separately the following two cases:
(3.8)
(3.8)'
If (3.8) holds. construct the two boxes
C, == [(Xi, to) + Q21%o-%11 (1Jn')]

== {Ix - x,1 < 21xo - xII} x {to -1J!;-P4Ixo - xll2, to}.
By construction these are contained in Qi and they overlap in a box satisfying
3. ltilder continuity of Du (the case p> 2) 255
Set
(Du)c, == if
c,
Dudt,
and estimate
IDu(Xt. to) - Du(xo, to)1 :5; IDu(xI, to) - (DU)Cl I

+ IDu(xo, to) - (Du)c I o
+ I(DU)Cl - (Du)c I· o
The proof now proceeds as for the HOlder continuity in t with minor cbanges.
LEMMA 3.2'. There exist constants "y > 1 and Q E (0, 1) that can be determined a
prior; only ;n terms of N and p such that for every pair ofpoints (Xi, to) E /C, i =
0,1.
PROOF: If JJ ~ 1. in (3.6) we take R = dist (/C; r). If JJ < 1. we take R =

JJ2.f! dist (/C; r). Then (3.7) reads
(3.7)'
If
( IXl - Xo I )
0/2
2.f!
JJ :5; dist (/C; r) ,
there is nothing to prove. Otherwise (3.7)' gives
Thus (3.10) follows by suitably redefining the number Q.
3-(iii). A version of Theorem 1.1

Combining Lemmas 3.1' and 3.2' gives the following form of Theorem 1.1:
256 IX. Parabolic p-systerns: Holder continuity of Du
THEOREM 1.1' (THE DEGENERATE CASE p> 2). Let u be a weak solu-
tion in nT of the degenerate system (1.1) of Chap. VIII, and assume that I-' =
IIDulloo.oT < 00. There exist constants 'Y > 1 and Q E (0,1) that can be deter-
mined a priori only in terms of Nand p such that, for every compact subset fC of
nT •
(1.1') IDu(xo, to) - Du(xl. t 1 )1
< (Ixo - Xli + max{I;I-'2j!}v'lto- t11)0<

- 'YI-' dist (fC; r) ,
for every pair of points (Xi, til E fc, i=O, 1.

Remark 3.1. The constants 'Y and Q are independent of dist (fc; r) and 1-'.
The fonn of (1.1)' suggest we reduce the system (1.1) of Chap. VIII to another
=
for which I-' 1. Introduce the change of variables
Vi
Ui
== -, t
.
= 1, 2 , ... ,m, and T = tl-'p-2.
IJ.
Vt - div IDvlp - 2 Dv = 0, in nT == nx (0, T I-'p-2) ,

and IIDvll oo,~6T
~ = 1. We write (1.1)' for v in the variables (x, T) and return to the
original coordinates. This gives
for all pairs (Xi, til E fC, i =0, I, where
IJ. - dist (fc; r) == inf

( .. ,tIEIC
(Ix - yl + 1J.2j!~)
(v,-IET
is the intrinsic parabolic distance from fc to r.
4. HOlder continuity of Du (the case 1 <p< 2)

In the singular case 1 < p < 2, the role of the parabolic geometry is reversed. As
before, we investigate separately the HOlder continuity in the space and in the t,
variables. The arguments are similar to those in the degenerate case and we only
indicate the main differences.
4. Hl)lder continuity of Du (the case I to
and construct the cylinders
[(Xo, ti) + QR(I-')] == {Ix - xol < I-'Ej! R} X {ti - R2,ti}' i = 0,1.
The box [(xo, t l ) + QR(I-')] intersects [(xo, to) + QR(I-')] if (h -to) < R2. More-
over they are contained in flT if
(4.1) max {1-'2jl R; R} :$: dist (J(,j r) .
Proceeding as in the case p> 2 we have

LEMMA 4.1. Let (4.1) hold. There exist constants 'Y > 1 and a E (0,1) that can
be determined a priori only in terms of N and p such that
Next if 1-'?I, we take R=d in (4.1), and if 1-'< I, we rewrite (4.2) as
(4.2)'
Arguing as in the proof of lemma 3.2' and bY.possibly redefining the constants 'Y
and a, we obtain
LEMMA 4.1'. There exist constants 'Y > 1 and a E (0,1) that can be determined a
priori only in terms of Nand p such that for every pair of points (xo, ti) E J(" i =
0, I,
4-0i). HOlder continuity in x

Fix two points (xo, to) and (Xl, to) in J(" at the same time level to, and let
[(Xi, to) + QR(I-')] == {Ix - Xii < R} x {to -1-'2-p R2, to}, i = 0, 1,
be two boxes satisfying (1.2). The box [(xo, to) + QR(I-')] intersects Xl if Ixo -
XII < R. Moreover they are contained in flT if
(4.4)
We proceed as in the case p> 2 and establish

258 IX. Parabolic p-systerns: Holder continuity of Du
LEMMA 4.2. Let (4.4) hold. Thereexistconstants'Y> 1 andaE (0, I) that can be
determined a priori only in terms of N andp,such thatforall (Xi, to)EK:, i=O, 1.
4-(iii). A version of Theorem 1.1

Combining Lemmas 4.1 and 4.2 gives the following fonn of Theorem 1.1
THEOREM 1.1" (THE SINGULAR CASE 1 1 and a E (0,1) that can be
determined a priori only in terms of N and p such that. for every compact subset
K",ofnT.
(1.1") IDu(Xo,to) - DU(Xlotl)1
max{l j /J¥}lXo - xII + vito - tll)Q

~ 'Y/J ( dist (K",j r) ,
for every pair of points (Xi, ti) EK"" i=O, 1.

Remark 4.1. The constants 'Y and a are independent of dist. (K:j r) and /J.
Arguing as in §3-(III), the HOlder continuity of Ui.:J:j can be expressed in
tenns of the intrinsic parabolic distance /J-dist (K:j r).
5. Some algebraic lemmas

We let QR(/J} c nT be a cylinder satisfying (1.2) and consider the system
(5.1) ! Ui - div IDulp - 2 DUi = 0, in QR(/J), p > 1,
and the one obtained by taking the derivative with respect to Xj' i.e.,
(5.2) ! Ui,:J:; - div (IDUIP- 2 DUi,:J:; + a:; IDuIP- DUi) = 0,

2
in QR(/J), i=I,2, ... ,m, j=I,2, ... ,N.

We let V denote a vector in R Nxm satisfying
(5.3)
We also let 'Y='Y(N, p) denote a generic positive constant that can be detennined
a priori only in tenns of the indicated quantities.
5. Some algebraic lemmas 259
LEMMA 5.1. There exists a constant 'Y = 'Y(N,p). such that for every vector
V E RNxm. and/or all p > 1.
(5.4) {lDul + IVI)Y IDu - VI:5 'YIIDuIY Du -IVIYVI·
LEMMA 5.2. Let 1 < p < 2. There exists a constant 'Y = 'Y(N,p) such that/or
every vector V E R Nxm.
(5.5) IIDuIP-2 Du - IVlp-2VI:5 'YIDu - ViP-I.

Moreover if the vector V satisfies (5.3). then
(5.6) IIDuIP-2 Du -IVlp- 2VI:5 'Y,",p- 2IDu - VI,

(5.7) IIDuIP-2 Du _IVIP-2vI2IDuI2-p :5 'Y,",p- 2IDu - V12.
Remark 5.1. These lemmas are algebraic in nature and could be stated for any
pair of vectors U and V, provided (5.3) is replaced by
(5.3)'
Also in (5.4) the number (1'-2)/2 could be replaced by any number and in (5.5)-
(5.7) the number (1'-2) could be replaced by any negative number.
PROOF OF LEMMA 5.1: By calculation,
IIDulY Du -lvIYVIIDu - VI
2: 1(IDuI'i'Du-IVI'i'V, DU-V) I
/ f1d
= \10 ds ISDu + (1 - s)VI
Y (sDu + (1 - slY) ds, Du - V
)
= 10f~IsDu + (1 - s)VI
Y IDu - VI 2ds
+ p; 21~ 0 IsDu + (1- s)VI

y I{sDu + (1- s)V, Du - V)I 2 ds
~ min{l; (p - 1)}IDu - VI 210f~IsDu + (I - s)VI ~ ds.

If1<p<2,
1o~IsDu + (I - s)VI
Y ds ~ {lDul + IVI) Y ,
and the lemma follows in this case. If p > 2, assume for example that IDul > IVI.
Then
260 IX. Parabolic p-systems: Hi)lder continuity of Du
1o~IsDu + {I - s)V/ ~ ds ~ 111/2

{sIDul- (I - s)IVI) ~ ds
1 1!=.!
~ -p IDul--'-- .
PROOF OF LEMMA 5.2: For 1<p<2. write
/IDul,,-2 Du - IVI,,-2V/ ~ IVI,,-2IDu - VI

+ /IDul,,-2 -IVI,,-2/IDul
~ IVI,,-2IDu - VI
IDu~-1 )
+ {2 - p)IVI,,-2IDu - VI ( eIVI,,-1 + {I _ e)IDul,,-l '
e
for some E [0,1]. Interchanging the role of Du and V gives
(5.8) /IDul,,-2 Du - IVI,,-2vl
~ IVI,,-2IDu - VI { 1 + (2 - p) eIVI,,-1 ~~71~-;)IDUI"-1 } •

e
for some E [0, 1]. and
(5.8') IIDul,,-2 Du - IVI,,-2vl

IVI,,-I}
~ IDul,,-2IDu - VI { 1 + (2 - p) '1I VI,,-1 + (1 _ '1)IDul,,-1 '
for some 'IE [0,1]. To prove (5.5) assume rust that
1
(5.9) IVI> "2 IDu - VI·
This implies
These inequalities in (5.8) prove (5.5). If (5.9) is false. its converse gives the two
inequalities
These in (5.8)' imply that the term in braces on the right hand side is bounded
above by an absolute constant. Moreover
IVI,,-2IDu - VI ~ IDul,,-2IDu - VI,,-IIDu - VI 2-".
The two inequalities (5.6) and (5.7) are an immediate consequence of (5.8) and the
assumption (5.3).
5. Some algebraic lemmas 261
Let H be the vector in R Nxm defmed by

(5.12) Hi == IDul p- 2DUi -IVlp- 2Vi -IVl p- 2 (DtI.i - Vi)
- (p - 2)IVlp- 4 Vt,k (Ul,:/:. - Vt,k) Vi, i= 1, 2, ... , m.
We will estimate IHI for all p> 1. For this we first set
(5.13) Wet) == tDu + (1 - t)V, for t E [0,1],
and rewrite (5.12) in the form
1
Hi = j ~ {ltDu + (1 - t)VIP-2 (tDui + (1 - t)Vi)} dt

o
-IVl p- 2 (DUi - Vi) - (p - 2)IVlp-4 Vt,k (Ul,:/:. - Vt,k) Vi
1
= (DUi - Vi) j{IWIP-2 -IVI P- 2} dt

o
1
+ (p - 2) (DUl - Vt) j{IWIP-4WlWi -IVI P- 4 VtVi }dt

o
== HP)
I
+ H~2),
I
where we have dropped the t-dependence from W. From (5.13)

(5.14) W - V = t (Du - V) ,
and for every sE [0,1].
(5.15) sW + (1- s)V = V + st(Du - V).
LEMMA 5.3. There exists a constant "'( = "'(N,p) such that/or every constant
vector V ERNxm satisfying (5.3), and/or all p> 1,
(5.16)
Remark S.2. The lemma holds for every pair of vectors U and V satisfying (5.3)'.
PROOF OF LEMMA 5.3 (p>2): Assume first thatlVI ~ 21Du - VI. Then
1
IHI ~ ",(p)IDu - VI j(IW(t)I P- 2 + IVIP-2) dt

o
~ "'(IDu - VIP-l ~ IZI (lDul + IVj)p-2IDu - V12.
Therefore (5.16) follows in this case since V satisfies (5.3). If

262 IX. Parabolic p-systems: Hl)lder continuity of Du
(5.17) IVI > 21Du - VI,

then by the mean value theorem and (5.14)-(5.15),
IHI ~ 'Y(P)IDu - VI 2 flsW + (1 - s)VIP- dt,

3
for some s E [0,1]. By (5.15) and (5.17) we have
and this implies the lemma.

PROOF OF LEMMA 5.3 (1<p<2): Itsufticestoprovethat
(5.18)
Assume first that
(5.19) IVI ~ IDu - VI,

and let t* E [0, 1] be defined by
t* = IVI
IDu-VI
Then
1
IHI ~ 'Y(P) fltlDu - VI_IVII,-2IDu - VI dt + 'YIDu - VIIVI,-2

o
"~ {l~ l'IDu - VHVIIP - 1 dt + i~ IIIDu - VHVII P - 1dt }

+ 'YIDu - VIIVI,-2
~ 'Y (lVIP-l + IDu - VIP-l) .
Therefore taking into account (5.19) and (5.3), the lemma follows in this case.
Consider now the case when (5.19) is violated, i.e.,
(5.19)' IVI > IDu - VI·

Then fori=l,2, ... ,m,
6. Linear parabolic systems with constant coefficients 263
1 1
Hi = (DUi - Vi) I I :alsW + {1 - s)VIP- 2 dsdt

o0
1 1
+ (Dut - Vi) I l:a {lsW + {1 - S)VIP-4 {sWt + (1- s)Vi)

o0
X {sWi + (1 - s)Vi) }ds dt.
Therefore
1 1
(5.20) IHI :5 'Y{p)IDu - VI 2 I It IsW + {1- s)VIP- 3 dsdt.

o0
Next, by (5.15) and (5.19)'
IsW + (1 - s)VI = Iv + st(Du - V)I
~ IIVI- stlDu - VII

~ IVI{1 - st).
This in (5.20) gives
1 1
IHI :5 'YIDu - V1 21V lp- 3 Ilt{1 - st)P- 3 ds dt.

o0
Since 1 < p < 2. the last integral is finite and the Lemma follows.
6. Linear parabolic systems with constant coefficients
Let V be any vector in RNxm satisfying (5.3). To the system (5.1) we associate
its linearised version
(6.1)
a
at Vi - (IVIP-2 vi ,xI + (p - 2)IVlp-4Vj,k Vj,xt Vi,l) XI '
in QR{/J), i = 1,2, ... , m.
Let
v == (Vl,V2, ••• ,vm )
and for 0 < p:5 R we let (Dv) p denote the integral average of Dv over Qp (/J).
264 IX. Parabolic p-systems: mlder continuity of Do
THEOREM 6.1. There exists a constant'Y='Y(N,p). such thatforall O<p:SR.

and for every constant vector WE R Nxm.
(6.2) H IDv - (Dv)p 12 dxdT :S 'Y (~) 2 H IDv - W1 2 dxdT.

~w ~w
To prove the theorem we introduce the change of variables
v(x, t) - V (x, tp.2-,,) .

This transfonns Qp(p.) into Qp(l) == Qp for all 0 < p:S R. and tranfonns (6.1)
into a system for which
In a precise way, the transfonned vector v is a solution of
8 ( .. )
(6.3) at Vi - a~:~ Vi,z. Zt = 0,
where the coefficients
a i,j = IVI,,-2 {flijt,lI: + (p _ 2) \-j,ll:

t,ll: - IVI2 Vi,t }
satisfy the ellipticity condition
(6.4) Co (N,p)leI 2 :S a~~lI:eiltej,ll: :S C1 (N,p)lel 2 , Ve E RNxm,
for two given constants Co < C 1 depending only upon N and p. Therefore it will
suffice to prove Theorem 6.1 for p. = 1. In the remainder of the section we let v be
a solution of (6.3) in QR and let (6.4) hold. Let a denote a multiindex of size lal.
i.e.,
N
a==(al,a2, ... ,aN), ajENU{0},j=l,2, ... ,Nj lal=Laj,
j=1
and for f E Coo (Q R) let
For non-negative integers m and n we also set
ID;'fl == L ID:II, D~f== ~f, ID:fl = ID~II = III·

lal=m
LEMMA 6.1. There exists a constant 'Y ='Y(N,p) such that for all non-negative
integers m, n and all 0 < p:S R,
(6.5) II ID,:+l D~VI2 dxd1' ~ 'Yp-211ID': D~VI2 dxd1',

Qp/2 Qp
(6.6) 1!ID~+lD':VI2dxd1' ~ 'Yp- 4 !!ID':D~VI2dxd1'.

Qp/2 Qp
PROOF: The system (6.3) is also solved by the vectors W == D';Div. Let (be
a non-negative smooth cutoff function in Qp vanishing on the parabolic boundary
of Qp and such that
Multiply the system (6.3), written for w, by the testing function W(2
and integrate
over Qp. to arrive at (6.5). To prove (6.6). mUltiply the same system by Wt(2
and
integrate over Qp. This gives
(6.7) !!lwtI2(2dXdT+ !! (a~,,{Wj,x/c! Wi,X t ) (2dxdT

Qp Qp
= -2 I! a~:{ Wj,x/c Wi,t( (Xt dxdT

Qp
+~ !!IWtl2(2dXd1' +;!!IDwI 2dxd1'.

Qp Qp
The integral involving a~:{ on the left hand side of (6.7) equals
These remarks in (6.7) give
!!IWtI2dxdT ~ 'Yp- 2!!IDw I2 dXdT.

Qp/2 Qp
The lemma now follows by applying (6.5) and suitably modifying the scale of the
radii P and p/2.
266 IX. Parabolic p-systems: H61der continuity of Du
LEMMA 6.2. There exists a constant 'Y='Y(N,p) such that/or all 0 < p~ R
(6.8)
PROOF: It suffices to prove the lemma for 0 < p ~ R/2 N +2. Let ( be the standard
cutoff function in QR/2N +1 that equals one on QR/2N +2 and such that
ID(I ~ 2N+2/R and 0 < (e ~ (2N+2/R)2.

IfO<p<R/2 N+2, we have
(6.9) !!lvI2 dxdT ~ 'YpN+2"v"~,Qp ~ 'YpN+2"v("~,QIl/2N+1.

Qp
On the other hand for all (x, t) eQR/2N+1,
t
Iv(l(x, t) = I! De (v() (x, T)dTI
_R2/2 N+1
~ 'Y !ID: Dt(v()1 dxdT
QIl /2N+1
Combining this with Lemma 6.1 we obtain the estimate
"V("~,QIl/2N+1 ~ 'YR-CN+2>!!lvI2dxdT.
QIl
This in (6.9) proves the lemma.

PROOF OF THEOREM 6.1: Since the vectors vz",z. solve (6.3) for h,s =
1,2, ... , N, we have from Lemma 6.2
(6.10) !!ID2v I2 dxdT ~ 'Y (~)N+2!!ID2vldxdT'

Qp QIl/2
for a constant 'Y='Y(N,p) and for all 0<p~R/2. To estimate the right hand side
of (6.10) observe that the vectors v z. solve the system
(6.11)
Let W be any constant vector in R Nxm and multiply (6.11) by the testing function
(Wi,z. - Wi,.) (2. where ( is the standard cutoff function iO Q R that equals one
in QR/2. This gives
IIID2vI2d3:dr :5 'Y R- 2 IIIDv - WI 2d3:dr.

QIl/2 QIl
To estimate the left hand side of (6.10) set
f
(Dv)p(t) = Dv(x, t)dx,
K,.
VO<p:5R/2, _p2 :5t:50.
Then x - (Dv,(x, t)-(Dv,)p(t» has zero average over Kp. and by the embed-
ding Theorem 2.1 and Remark 2.1 of Chap. I.
IIIDV - (Dv)p(r)1 2d3:dr:5 'Yp211ID2vI2d3:dr.

Qp Qp
Write
(6.12) IIIDV - (Dv)pI2 d3:dr :5 'Y (~) NHIIIDV - Wl 2 d3:dr

Qp QIl
o
I
+ 'YpN / (Dv)p - (Dv)p(r)j2 dr,
-p2
and estimate the last term by

o
pN/I(Dv)p - (Dv)p(r)1 2dr:5'YpN+2 sup I(Dv)p(t) - (Dv)p(r)r·
-p2
_..2<t,,.<O
,,- - -
Next integrate (6.11) over K p x (r, t) and divide by meas{ K p} to obtain
I(Dv)p(t) - (Dv)p(r)j :5 'YP-NIIID 2v 1d3:dr

Qp
$. p-N P"P (£fID'VI'dzdr) 1/'
$. ~p-N/' (£fIDv _WI'dzdr) 1/.

Therefore the last term on the right hand side of (6.12) is estimated by
'Yp2 (~)N+2fIID2vI2d3:dr:5 'Y (~)N+'l/IDV - WI 2 d3:dr.

QIl/2 QIl
268 IX. Parabolic: p-systems: mlder continuity of Du
7. The perturbation lemma
LEMMA 7.1. There exists a constant 'Y = 'Y(N,p) such that/or every constant
vector V in RNxm satisfying (5.3),
PROOF: Let ( be a cutoff function in QR(P.) that equals one on QR/2 (p.). and
such that
In the weak formulation of (5.2) we take the testing functions
modulo a Steldov time average. We obtain
(7.2) sup flDU - VI2(2(X, t) dx + f fIDul,,-2ID2uI2(2dxdT

-,.2-PR2<t<O JJ I
- - QR(,.) QR(")
$;'Y ffIDU-VI 2((t dxdT + J,

QR("')
where
J='Y f fODUI,,-2DUi -IVI,,-2Vi,;)z; (Ui,z; - Vi,;) (D(dxdT.

QR("')
If p > 2. we have
J $; ~ ffIDUI,,-2ID2UI2(2dxdT + 'Yp.;~2 fflDU - VI 2 dxdT.

QR(,.) QR(,.)
Putting this estimate in (7.2) proves the lemma in the degenerate case. To estimate
J in the singular case 1 <p< 2. we fU'St integrate by parts in the variable Xi. This
gives
7. The perturbation lemma 269
(7.3) J ~ 'Y ffIIDUIP-2DU -IVIP-2VIID2UI(ID(ldxdT

QIl(,.)
+ ff II DuIP- 2 Du - IVIP- 2 VIIDu - VI(ID 2(ldxdr

QIl(,.)
==Il+h
By (5.7) of Lemma 5.2 and Schwartz inequality
II ~ ~ f fIDUIP-2ID2UI2(2dxdr + i£;~2 f flDU - Vl 2dxdr,

~W· ~w
and by (5.6)
12 ~ 'Yp;~2 fflDU - Vl 2dxdr.
QIl(")
Combining these estimates in (7.2) proves the Lemma.
Let 8pQR/2 (p) denote the parabolic boundary of QR/2 (p). Consider the
boundary value problem
Vi,t - (IVIP-2 vi ,x; +~ - 2) IVlp- 4 Vt,k vi,xlo Vj,i) Xj , in Q R/2 (JL)

(7.4) {
Vi L')pQIl/2(,.)=Ui, &= 1,2, ... ,m.
The existence of a unique solution to (7.4) can be established for example by a
Galerkin procedureP) The solution v == (vt, V2, ••• , 11m) of (7.4) is 'regular' in
the interior of QR/2 (p), in the sense of Theorem 6.1. The next lemma compares
U and v.
LEMMA 7.2. There exists a constant 'Y='Y(N,p), such thatfor all 0< p< R/2
and for every vector V satisfying (5.3),
(7.5) fpDU - Dvl2dxdr ~ 'Y (p_2f! IDu - Vl2dxdr)1I

Qp(") QIl(")
f flDU - Vl 2dxdr,
QIl(")
where a=mint!; i}.

PROOF: Write the system (5.1) in the form
! Ui - (IVI P- 2U i,Xj + (p - 2)IVl p - 4 Vt,k Ui,:t:1o VjJ) Xj = div Hi,

i = 1,2, ... ,m,
(1) See Lions (73).
where the vectors Hi are introduced in (S.12). From this. subtract (7.4). and in the
weak fonnulation of the system so obtained, take the testing function Ui -Vi. This
is admissible since it vanishes on lJp QR/2 (",). Adding over i= 1, 2, ... , m. gives
",p-1/ IDu - Dvl 2 dxdT ~ ..,/ / IHIIDu - DvldxdT,

Q 11/2 (,,) Q11/2 (,,)
where we have taken into account the fact that V satisfies (S.3). Using Schwartz
inequality on the right hand side and then Lemma S.3 to estimate IHI2. we arrive
at
(7.6) / / IDu - Dvl 2 dxdT

QIl/2(")
~ ..,,,,-2(P-l)// (IDul + IVI)2(P-2) IDu - V1 4 dxdT.

QIl/2(")
To estimate the right hand side of (7.6) assume first that N ~ 4 so that
a = min{!'
- 2' N
~} = ~
N'
To simplify the symbolism we let r =",2-p R2 /4. We have
n.7) // (lDul + IVI)2(p-2) IDu - VI 4 dxdT

QIl/2(")
~ l(;.~~Do' + IVIl2(p-2) IDo _ VI'U) i

By Lemma 7.1
(7.8) ~*(P-l) sup

-r<t<O
(fIDU
J I
_V 12dt) i
- - KR/2
To estimate the last factor in (7.7) we majorise the integrand by means of Lemma
5.1. It gives
(lDul + IVI)2(p-2) IDu - VI 4 = {(IDuI + IVI) zy! IDu _ VI} 4

~ 'YIIDulZY! Du _IVIZY!vI 4
~ ~P~IIDulZY!DU-IVIZY!vl~·
Let x -+ {(x) be a non-negative piecewise smooth cutoff function in KR that
equals one on K 3R / 4 and such that ID{I ~ 4/ R. Then for a.e. tE {-r, O}, by the
embedding Corollary 2.1 of Chap. I, we have
N-2
~ 'Y~P¥ (![lIDUIZY! Du -IVIZY!VI{] ~ dx)-,;r

K3R/4
~ 'Y ~p¥ ! ID [IDulZY! Du -IVIZY!V] {1 2dx

K3R/4
Here in estimating the last term we have used the algebraic inequality
272 IX. Parabolic p-systcms: H5lder continuity of Du
which follows from (5.6) of Lemma 5.2 with p replaced by (p + 2)/2. Therefore
the last factor on the right hand side of (7.7) is estimated by
1V.~~DoI
N-2
+ IVI)2lP-2) IDo - VI'.J") -,,- dT
:5 'Y I'P~ { jjlDulP-2lD2ul2dxdT

Q3R/2C",)
j
+ I'p-2 R- 2 jlDU - V 12 dxdT}
QRC",)
:5 1'2CP -l)-1f R- 2 j jlDU - V1 dxdT,

2
QRC",)
where we have also used Lemma 7.1. We now combine these calculations in (7.7)
and then in (7.6) to obtain
jjlDU - Dvl 2 dxdT
QpC",)
provided N ~ 4. If N = 2, 3, we transform the integral on the right hand side of

(7.6) by HOlder's inequality as follows.
(7.9) II ODul + IVn 2(p-2) IDu - Vl 4 dxd'T

QR/2(")
o
= I IIDU - VI (IDul + IVn 2(p-2) IDu - Vl 3 dxd'T
-rKR / 2
1
~ 1(l~-V'2dzr
x (/ODU + IVI)4(P-2) IDu - V 16 dx) ! d'T
KR/2
'
~ p.2j! sup
-r<t<O
(fIDU
J I
- V12dx) !
- - KR/2
1
xl(J.!;IDuI + IVI)'" IDu -vi]' dz) ·d.
By Lemma 7.1
We estimate the last tenn on the right hand side of (7.9) separately for N =3 and
N=2.
The case N=3
Let, be defined as before. Then for a.e. t E ( -r, 0).

274 IX. Parabolic p-systems: II)lder continuity of Du
(J[(IDul + IVI) E? IDu - Vr dx ) 1

KR./2
~ 'Y JL f Rl (J[ (lDul + IVI) E? IDu _ Vf dx) 1

KR./2
,; 7,,1 RI (L!!IDuI'i'Du-IVI'i'VI,r <Ix) I

~ 'Y JLf Rl JID [IDulE? Du -IVIE?V] '1 dx. 2
K SR./4
1berefore
l(J.!~,Du' + IVI)'i' IDu -Vi]" <Ix) I dT
~ 'Y JL2(P-l)-f R-i JJIDU - V1 dxdT. 2
QR.(")
Combining these estimates in (7.9) and then in (7.6) proves the lemma for N =3.
TbecaseN=2
We apply the embedding Theorem 2.1 of Chap. I with q = 6, Q = 2/3 and
B =1. This gives for a.e. t E (-r, 0)
( J[(IDuI + IVI)'i'IDu -Vi]" J I

KR./2 )
,; 7 (f.~~DuI'T' Du -IVI'T'VI'j' <Ix) !

~ 'Y JID [IDulE? Du -IVIE?V]
KSR./4
,r dx
8. Proof of Proposition 1.l-(i) 275
1berefore
l(J.!~IDuI + IVI)"'IDu - VI]' dz) ! dT
~ 'Y~fR fflD [IDul~ Du-IV\2j!v] (1 dxd.,.. 2
QR(")
We estimate these integrals by means of Lemma 7.1 and combine the calculations
in (7.9) and in (7.6) to conclude that (7.5) holds with a=~.
8. Proof of Proposition 1.1-(i)
LEMMA 8.1. There exist constants ~, 6, E E (0, 1) that can be determined a priori
only in terms of N and p. such that ijVo is a constant vector in R Nxm satisfying
(8.1)
(8.2) H\DU - V l dxd.,. ~ E~2,

o 2
QR(")
then there exists a constant vector V t E R Nxm such that
(8.4) ffiDU - V l dxd.,. ~ ~6N+2ff\DU - V l dt,

t 2 o 2
Q'R(") QR(")
(8.5) HIDU-Vt I2 ~E~2.

Qu(,,)
PROOF: Let v be the unique solution of (7.4) and set
Vt == HDvdt,
Qu(,,)
where 6 E (0,1) is to be chosen. The perturbation Lemma 7.2 with V =V o • the

triangle inequality and (8.2) give
276 IX. Parabolic p-systems: Jl)lder continuity of Du
I PDU - V l l2 dxdT :5 "Yeo IIIDU - V ol2dxdT

Qu(,,) QIl(")
+ I fiDV - V l l2 dxdT.
Qu(,,)
By Theorem 6.1
lfiDV - V l l2 dxdT:5 "Y6 N +4IIIDv - V ol2dxdT,

Q,Il(") QIl/2(")
and again by Lemma 7.2 with V =V 0 and (8.2)
II IDv - V ol2dxdT :5 "Y (1 + EO) I IIDU - V ol2dxdT,

QR/2(") QR(")
for a constant "Y="Y(N,p). Combining these inequalities we obtain
I fiDU - Vll2dxdT:5 "Y (6 NH +eO)IIIDU - V ol2dxdT, 6:5 1/2.

Q,Il(") QIl(")
To prove (8.4) choose EO =6NH , and then 6 so small that
2"Y62 :5 ".
Inequality (8.5) follows from (8.4) and the s1I1Illiness assumption (8.2). To prove
(8.3) write
Vl - Vo =H (Dv - Vo)dxdT
Q,R(")
=H {(Dv - Du) + (Du - Vo)}dxdT

Q,R(")
and
IVl - V o l2 :5 2 H IDu - Dvl2dxdT + 2 H IDu - V ol2dxdT.

QIIl(") Q,R(")
By Lemma 7.2 and the indicated choices of E and 6
H IDu - Dvl2dxdT :5 " H IDu - V ol2dxdT.

~RW ~RW
Therefore using again (8.2)

8. Proof of Proposition 1.l-(i) 277
(8.6) IV1 - V ol2 ~ 2 (K, + 6-(N+2») H IDu - V o l2dxdT

Qa(,,)
~ 2 ( K, + 6-(N +2) ) dl 2(NH) p.2 ~ 2K,p.2.
By choosing K, sufficiently small we may insure that
and
LEMMA 8.2. There exist constants K" 6, EE (0,1) that can be determined a priori
only in terms of N and p. such that if V 0 is a constant vector in R Nxm satisfying
(8.1) and (8.2). then there exists a sequence of constant vectors {Vi} el in RNxm.
satisfying
(8.8) H IDu - Vil2dxdT ~ Ep.2,

Q6I a(")
(8.9) ! !IDU - Vi+!1 2dxdT ~ K,6 N+2!/IDU - V i l 2dxdT,

Q6l+ 1 a(") Q6 I a(")
for i = 1, 2, .. "
PROOF: The sequence is constructed inductively by using the procedure of the
previous lemma. To prove that IVil are in the range (8.7), we refer back to (8.6),
H
i.e.
IVi +! - V i l2 ~ 2 (K, + 6-(N+2») IDu - V i l2dxdT.
Q6I a(")
We iterate over i and use again the smallness assumption (8.2) to obtain
IVi+l - V i l2 ~ 2 (K, + 6-(N+2») K,i H IDu - V o l 2dxdT

Qa(,,)
~ 2p.262(NH) (K, + 6-(N+2») K,i.
From this by taking roots and adding over i

00. ,fK.
IV i+l - Vol ~ P.6L.,fit ~ p. 1- ,fK.'
i=l
278 IX. Parabolic p-systems: Ifi)lder continuity of Du
where we have used the specific choice of 6 in tenns of It. Choosing now It suffi-
ciently small proves the Lemma.
9. Proof of Proposition 1.1-(ii)

The number 11 in the assumption (1.3) can be chosen to insure the existence of a
constant vector Vo E RNxm satisfying (8.1) and (8.2). This is the content of this
section. Set IDul =v and, for all 0 (1-1I)1-'},
(9.2) B; == {(x, t) E Qp(l-') Iv(x, t) < (1 - II)I-'} .
We will choose 11 E (0, l) and rewrite (1.3) as
(9.3) IBRI ~ IIIQR(I-')I, 11 E (0,1)·
LEMMA 9.1. There ex;stsa constant."Y ="Y(N, p) such thatforall uE (0,1)
(9.4) jr fIDul,,-2ID2uI2 dxdT < "Y 1-'211 RN.

P - (1- u)2
A: 1t
PROOF: Consider the differentiated equation (S.2) and in its weak fonnulation
take the testing function
Ui,z; (v 2 - k2)+ (2, k = (1- 211)1-',
modulo a Stelclov averaging process. Here ( is a non-negative piecewise smooth
cutoff function in QR(I-') that equals one on QtT R (I-') and such that
1 1-',,-2
ID(I ~ (1- u)R' 0 ~ (t ~ (1- u)W'
After we add over i = 1, 2, ... m and j I =1, 2, ... , N, we arrive at
(9.S) sup, /(,:,2 - k2): (2(x, t) dx
-",3-PR2<t<O
- - Kit
+ / /v'P-2IDv212(2x. [v> k] dxdT

QIt(,,)
m N
+ ?:?: //IDul,,-2IDUi ,z.:/ 12 (V 2 - k2)+ (2dxdT
1=1 .1=1 Q"It(")
~ "Y / /v'P-2IDv21 (v2 - k2)+ (ID(I dxdT

QIt(,,)
+ "Y //(v 2 - k2): ((tdxdT

QIt("')
9. Proof of Proposition l.l-(ii) 279
for a constant "Y="Y(N,p). By the Schwartz inequality
"Y jjvP- 2IDv2I (v 2 - k2)+(ID(ldx.dT

QR(p)
~ jjvP-2IDV212(2X[V > k]dxdT

QR(p)
+"Y2 j j vP- 2 (v 2 - k2): ID(1 2dxdT.

QR(P)
We put this in (9.S) and in the resulting inequality we discard all the non-negative
terms on the left hand side except the integral containing DUi,Zi' This gives
(9.6) fjlDulP-2lD2ul2 (v 2 - k2)+ (2clxdT

QR(P)
~ "Y j j(vP-2ID(12 + (t) (v 2 - k2): dxdT.

QR(p)
Since (v 2 - k2)+ ~ 411J.1.2,
jj(v 2 - k2): (t clxdT ~ (1"Y~:;~~2 J.l.P- 2IQR(J.I.)1

QR(P)
where we have used the structure of (and the intrinsic geometry of QR(J.I.). Also
f r fvP-2 (v 2 _ k2)2 ID(1 2dxdT < "Y 1I2 J.1.4. RN.

j'
QR(P)
+ - (1-0')2
This is obvious if p > 2. If 1 (I - 211)J.I.. We estimate below the integral on the left hand side
of (9.6) by extending the integration over the smaller set [v> (I - II )J.I.]. On such
a set, (v 2 - k2)+ ~ IIJ.1.2. These remarks in (9.6) prove (9.4).
Set for all O<p~R and all tE [-J.l. 2- p p2,O]
(Du)p (t) == f
Kp
Du(x, t) dx.
LEMMA 9.2. There ex;sts'(l constant "Y="Y(N,p). such that/or 'all O'E (l,l)
PROOF: Fix UE(O, 1). and foraH tE [-1£2- P (uR)2,O]. set
Vet) == f IDulEj! Du(x, t) dx.

K"R
We apply the multiplicative embedding of Theorem 2.1 of Chap. I to the functions
x -+ IDulEj! Du(x,t) - Vet), 'Vt E [-1£2-p(uR)2,O],
which have zero average over KtrR. For the choice of the parameters
N
Q = N + 1' q = 2, s = 1,
we obtain
IIIDulEj! Du - V(t)1 2dx ~ "YI(lDuIP-2ID2uI2) wtr dx
~R ~R
x V!IDuI'i'Du-V<tlldz) ~
The last integral is majorised by ''/#£ rn R-Af:r . Therefore integrating this inequal-
ityover [-1l 2- P (uR)2,O] gives
(I£rn RHr) -jII IDulEj! Du - V(T)1 2dxdT

Q"R(/J)
~ "Y II (lDuIP-2ID2uI2) Jfh dxdT

Q..RC/.')
="YI!(IDUIP-2ID2uI2) wtr dxdT

A: R
+II(lDuIP-2ID2uI2) wtr dxdT
B;R
9. Proof of Proposition l.l-(ii) 281
where A~ and B: are defined in (9.1)-(9.2). We estimate the first integral by

Lemma 9.1 and the second by using the 'smallness condition' (9.3) and Lemma
7.1. We conclude that there exists a constant 'Y='Y(N,p) such that
(9.8) f1r [I
Q..R(p)
IDul
¥ 12 'Y 1J2 Vl/(N+l)
Du - VCr) dxdr ~ (1 _ u)2N/(N+l) R
N+2
.
Introduce the vectors wet) by
Vet) == Iw(t)l¥w(t),
and observe that
Iw(t)1 ~ IJ, '<It E (-1J 2 - P (uR?, 0] .

By the algebraic Lemma 5.1.
(9.9) IIIIDu l¥ Du - v(r)1 2dxd7'

Q"It(p)
~ II(lDu l + Iw(r)I),,-2IDu - w(7')1 2dxd7'.

Q"It(p)
We treat separately the cases p> 2 and 1 2
We minorise the left hand side of (9.9) by extending the integration over the
smaller set A~R' On such a set.
(lDul + Iw(t)l)P-2 ~ IDul,,-2 ~ 22 -"IJP - 2•
This with (9.8) yields
[ [ 2 'Y1J 2 v 1/(N+l)
(9.10) 11 IDu - W(7')1 dxd7' ~ (1- u)2N/(N+l) IQR(IJ)I·
A~1t
Next write
IIIDU - w(t)1 dxdr = IIIDU - w(7')1 2dxd7' + IIIDU - w(7')1 2dxd7'.

2
Q"It(p) A~1t B~1t
The first integral is estimated in (9.10) and the second is majorised by 21J2 Q R (IJ) vi I.
in view of the 'smallness' condition (9.3). We conclude that
282 IX. Parabolic p-systems: HOlder continuity of Du
for a constant "( = "((N,p). The minimum on the left hand side is achieved for
V == (Du)aR (t). This proves the lemma if p> 2.
The singular case 1 < p < 2
Since Iw(t)1 $ JJ, we have (IDul + Iw(t)l)P-2 ~ 2P- 2JJP- 2. Putting this in
(9.9) and combining it with (9.8) gives
jJr[ IDu - w(t)1 2dxdT "( JJ2 vl/(N+l)

$ (1 _ u)2N/(N+l) IQaR (JJ) I·
Q"R(/J)
The proof is now concluded by a minimization procedure.
10. Proof of Proposition 1.1-(iii)

Let (Du) p denote the integral average of Du over Qp(JJ), i.e.,
(Du)p == I f DudxdT.
Qp(/J)
LEMMA 10.1. There exists positive constants "(, a, b that can be determined a
priori only in terms of N and P. such that for all u E (i, 1).
(10.1) I f IDu - (Du)aR 12dxdT $ "( JJ2 {(1 :au)b + (1- u)}.
Q"R(/J)
PROOF: By Lemma 9.2
2 "( JJ 2 v1/(N+l)
I f IDu - (Du)aR! dxdT $ (1 _ u)2N/(N+l)
Q"R(/J)
+ If! (Du)aR - (Du)aR (T)!2 dxdT

Q"R(/J)
and
(10.2) H! (Du)aR - (Du)aR (T)!2 dxdT

Q"R(/J)
$ sup
-"l-P(aR)l<t 8<0
r- - 1 -
lfKtlR
(Du(x,t) - DU(X,8») dx12.
Let (f= (1 +u)/2 and denote with x--+((x) anon-negative smooth cutoff function
in K&R that equals one on KaR and such that
10. Proof of Proposition l.1-(iii) 283
- 2
ID(I ~ (1-i1)R == (l-u)R'
4 1D2-1( ~ (l-u)R'
16
Write
!
K"R
(Du(x, t) - Du(x,s») dx = !
aKR
(Du(x,t) - Du(X,7'»)(2dx
- !(Du(x,t) - Du(x,s»)(2dx.
K.R\K"R
The last integral is estimated above by 'Y( 1 - u )p.RN . To estimate the flJ'St integral
we integrate the differentiated system (5.2) over (7', t). multiply by (and integrate
over KaR. This gives
(10.3) !
K.R
(Ui,Zj (t) - Ui,zj (s) )(2 dx
= Iif ('
aK.R
div ( .,..-' Du;.., + 0:;' Do;) ""dBl·
Thecasep>2
The right hand side of (10.3) is estimated by
To estimate the last integral write
! !IDul~ ID2Uldxd7' = ! !IDUI~ ID2uldxd7'

~RW ~R
+! !IDul~ ID2uldxd7'
S:;R
~ IQR(P.)!! (f!IDuIP-'ID'U1'' ' tt.! !

Au )
+ IBill! (!!IDuIp-2ID'U1'''''tt.!!
Qu(,,) )
284 IX. Parabolic p-systems: mlder continuity of Du
The frrst integral is estimated by Lemma 9.1 and the second tenn is estimated
by the 'smallness' condition (9.3) and Lemma 7.1. Combining these estimates in
(10.2) proves the lemma.
The case l<p<2
Estimate the right hand side of (10.3) as follows.
II( Ui,:J:; (t) - Ui,:J:; (8) )(2 dx

~u
= V/
sK.R
D(-' {IDol""" Du - I(Du)'R (s)l.-2 (Du)'R (sn., dzdTl
t
~ I IID2(IIiDuI P- 2Du -I (DU)uR (8)11'-2 (DU)uR (8)1 dxdr.

sK;;R
By the sttucture of the cutoff function ( and (5.5) of Lemma 5.2, this is majorised
by
t
(1 _ !)2 R2 IllDu - (Du)uR (s)l"-1 dxds

SK.R
We estimate the last integral by Lemma 9.2 and combine it with (10.3) to prove
the lemma.
LEMMA 11.1. Let e E (0, 1) be the number claimed by Lemma B.l. There exists
a number v E (0, i) such that if (9.3) holds, then
(11.1) HIDU -
QR(")
(DU)RI 2dxdr ~ ep.2,
(11.2)
PROOF: Write
II. Proof of Proposition 1.1 concluded 285
H /Du - (Du)RI dxdT = O'N+2 H

2
IDu - (DU)aRI 2dxdT
Q RC,,) Q.RC,,)
+ IQR(/J)1-1jjIDU - (Du)Rr dxdT

QR(,,)\Q.RC,,)
+ O'N+2H I(Du)R - (DU)aRI 2dxdT.

Q.R(")
The flrst integral is estimated by Lemma 10.1 and the second is bounded above by
")'(1 - q)/J2. To estimate the last integral write
(DU)R-(Du)aR = IQaR (/J) 1-1 {O'N+2jjDudxdT - j j DUdxdT}

QRC,,) Q.RC,,)
= IQaR (/J) 1- 1{ (O'N+2 - lifj DudxdT + j j DUdxdT} .

QR(,,) QRC,,)\Q.RC,,)
This implies that
and
H IDu - (DU)RI 2 dxdT :s ")' /J2 { (1 :°O')b + (1 - 0') } .
QR(")
To prove (11.1) choose 0' so close to one that ")'(1- 0') :s

~e. and then v so small
that ")'vO(1 - 0')-" :s
~e. To prove (11.2) we flrst observe that the deflnitions
(9.1)-(9.2) imply
Then by the 'smallness' assumption (9.3)
jjlDu I2 dxdT ~ jjlDU I2 dXdT ~ /J2(1- v)3IQR(/J)I.

QRC,,) Ail
Using now (11.1)
H IDuI 2 dxdT - H (Du)~dxdT = H IDu - (DU)Rr dxdT

QR(") QRC,,) QRC,,)
:s e/J2.
From this
286 IX. Parabolic p-systems: fR)lder continuity of Du
I(Du)RI 2 ~ H IDul 2dxdr - E",2 ~ {(I - 11)3 - E} ",2.

QR(")
PROOF OF PROPOSITION 1.1: Let E, 6 Ie E (0,1) be fixed as in Lemma 8.1.

We start the iteration process of Lemma 8.2 with Vo == (Du)R, and let {Vi} ~
be the corresponding sequence of constant vectors in RNxm satisfying (8.7). It is
apparent that, by an application of the triangle inequality, the vectors Vi can be
replaced by (DU)i' by possibly modifying the number Ie.
12. Proof of Proposition 1.2-(i)
We assume that the smallness condition (9.3) does not hold, i.e.,
(12.1)
LEMMA 12.1. Let (12.1) hold. There existssome t ••
(12.2)
such tMt
1-11
(12.3) mess {x E KR I v(x, t.) > (1 - II)",} < 1 _ 11121KR1, v = IDul·
PROOF: Indeed if not,

_,,2-P(~/2)R2
IAR/ ~ f mess {x E KR / v(x, r)

_,,2-PR2
> (1 - II)",} dr
~ (1 - II)/QR("')/,
We will work with the function w == IDuI 2, which satisfies (1.8) within the
cylinder KR x (t., 0). Introduce the change of variables
r = -tit., (= xlR, w«(,r) = w(Re,-t.r)
and the convex function of w
%
w I}
== max { ",2 i '2 .
Then KR x (t., 0) is mapped into Ql == K 1 X (-1,0) and, denoting again with
(x, t) the transformed variables, % satisfies
(12.4) %t-(At,k%Zt)o;,,:50 in Ql and 0<%:51,
12. Proof of Proposition 1.2-(i) 287
where the matrix (At,Al) is uniformly elliptic with eigenvalues bounded above and
below independent of p. Indeed it follows from (1.9) and the range (12.2) of t.
that
for two constants co(N,p, v) $ Co(N,p, v). The information of Lemma 12.1 in
terms of z implies
I-v
(12.6) meas {x E Kl I z(x, -1) > (1 - v)} $ 1 _ 1I/2IK11.
Without loss of generality we may assume that z satisfies (12.4) in a slightly larger
box, say Q2. This can be achieved by starting for example with Q2R (p). Propo-
sition 1.2 is a consequence of the following:
THEOREM 12.1. Let z E C (-2, OJ L2(K2» nL2 (-2,0; W 1 ,2(K2») be a sub-
solution off12.4)-(12.5), and let (12.6) hold. There exists 11 =lI(N,p, II) E (0, 1),
such that
meas {(x,t) E Q! I z(x,t) > (1-1I)} = O.
In view of (12.4), the proof of the theorem uses techniques typical of a single
equation. Even though these methods have been presented in various forms in
Chapters n and m, we reproduce here the main points, to render the theory self-
contained.
12-(i). Some energy estimates/or z

LEMMA 12.2. Let 0 < 110 < v and consider the function
!li(z) = In+ { II }
- II - (z - (1 - 11»+ + 110 •
There exists a constant"( = ,,(N,p, v) such that for all t E (-1,0) andfor all
0<0'< 1,
(12.7) J !li2(z) dx $
K"x{t}
J !li2(z) dx + (I!0")2
KIX{ -I}
JJ
Ql
!li(z) dxdT.
PROOF: Let x - (x) be a cutoff function in Kl that equals one on Ka. and in
the weak formulation of (12.4) take the testing function !lilli' (2 , modulo a Steklov
averaging process. Then (12.7) follows by estimates analogous to those in Propo-
sition 3.2 of Chap. II.
288 IX. Parabolic p-systems: fR)lder continuity of Du
LEMMA 12.3. ForO<p~ 1 andO<u< lIet (x, t)-((x, t) beacutofffunction

in Qp that equals one on Qap and vanishes on the parabolic boundary 01 p. There
exists a constant 'Y='Y(N,p, v) such that lor all k ~ 1/2
(12.8)
The proof of (12.8) is analogous to the proof of the energy estimates of Propo-
sition 3.1 of Chap. II. The spaces Vm,P( Qp) for m, p ~ 1 are introduced in §3 of
Chap. I.
LEMMA 13.1. There exists a constant 110 E (0, v) depending only upon N,p, v
such that lor all t E ( -1,0)
(13.1) meas {x E Kl I z(x, t) > (1 - 110)} < (1 - 1.12/4) IKll·
PROOF: We will use the logarithmic inequality of Lemma 12.2. Since !If(z) van-
ishes on the set [z < (1 - v)], by virtue of (12.6), the first term on the right hand
side of (12.7) is majorised by
I-v 2(1.1)
1- 1.1/2 In ~o IKll·
The second term is majorised by
We estimate below the right hand side by extending the integration to the smaller
set [z(·, t) > (1 - 110)]' On such a set
!If(z) ~ In (1.1/2110)'
Combining these estimates in (12.7) gives
Also
13. Proof of Proposition 1.2 concluded 289
meas {x E KI I z(x, t) > (1 - '10)}

~ meas {x E Ku I z(x, t) > (1 - '10)} + (1 - O')IKII
< 1 - v IKllln2 (v/'1o)
- 1 - v/2 In 2 (v/2'10)
'"t In (v/'1o)
+ (1 _ 0')21Kllln2 (v/2'10) + (1 - O')IKII·
Choose 0' so that (1 - 0') ~ v 2 /8 and then '10 so that
'"t In (11/'10) < 112.

(1 - 0')2ln2 (11/2'10) - 8
By choosing '10 even smaller if necessary. we may insure that
1- IIIn 2 (11/'10) < 1 _ 112.

1- 11/2 ln 2 (11/2'10) - 2
Having determined '10. let So be the largest positive integer such that2- So ~
'10' For s ~ So. set
o
A.(t) == {x E KI I z(x, t) > (1 - 2- S ) } , As == jIAs(T)ldT.
-I
Then Lemma 13.1 implies that

(13.2) 'It E (-1,0).
LEMMA 13.2. For every v. E (0, 1) there exists a positive integer s. > So such
that
(13.3)
PROOF: Apply Lemma 2.2 of Chap. I to the functions x-+z(x, t) fortE (-1,0).
and for the levels
l = 1- 2-(·+1),
Taking into account (13.2). we obtain
2- S lAsH I ~ IKI \~s(t) I j IDzl dx

A.(t)\A.+1(t)
"~(N,p, v) (j ID (z - (1- 2-'))+ I'dx) I

x (lA.(t)I-IA s+1(t)l) i .
290 IX. Parabolic p-systems: Ht;lder continuity of Du
We square both sides of this inequality, integrate in dt over ( -I, 0) and estimate
the resulting integral on the right hand side by the energy inequalities (12.8) written
over the pair of cylinders Ql and Q2. This gives
4- s A~H :5 "Y4- s (As - AsH) .

Divide through by 4- s and add these inequalities for 8=80'So + 1, ... ,S. - 1 to
obtain
s.
(s. - So - I)As. :5 "Y E (As - AsH) :5 "YIQ11·
Therefore
PROOF OF THEOREM 12.1: Consider the family of nested boxes
and the increasing levels

n=O,I, ... ,
and set
Yn == meas{(x,t) E Qn I z(x,t) > k n }.
Write the energy inequality (12.8) over the boxes Qn for the functions (z - k n )+,
where ( is the standard cutoff function in Qn that equals one on Qn+l' By the
embedding Proposition 3.1 of Chap. I, with m =p = 2,
II (z - kn )! dxd-r:5 II [(z - kn )+ (]2 dxd-r

Q"+1 Q"
x (if [(z - k,.)+ <) if< dzd_,y-It. Ynwb

:5 II (z - kn)+ (II~2.2(Q")Ynwh
:5 4S • Y~+wh .
On the other hand
YnH :5 "Y4n+s • II (z - kn )! dxd-r.

Q"+1
Therefore
y.n+l < ny'1+1i1h
_ "Y4 n , n=O,I,2, ....
It follows from Lemma 4.1 of Chap. I that {Yn } - 0 as n - 00 provided
(13.4)
To prove the theorem we have only to pick 8. by the procedure of Lemma 13.2 so
that (13.4) is satisfied and then set 'I = 2-(8.+1).
14. General structures

Consider the general non-linear system (1.10) of Chap. VIII subject to the struc-
ture conditions (81)-(~). The proof of Propositions 1.1 and 1.2 for these systems
is analogous to that in §§6-11. The corresponding 'linear' system about a point
(x o, to) E flT is
For this, the linear analysis of §6 can be carried with minor changes. The analog of
the 'algebraic' lemmas of §5 are a direct consequence of the structure conditions
(81 )-(~). In the proof of Propositions 1.1 and 1.2, when working within cylinders
[(x o, to) + Q (6, p)], the 'perturbation terms' <Pi, i =0,1,2, contribute terms that
are infinitesimal with p of higher order with respect to those generated by the
principal part. This is due to the integrability condition (86). Further details on the
estimation of the lower order terms can be carried out with an analysis similar to
the estimation of the lower order terms in Chaps. II-V.

The content of this Chapter is essentially taken from [36,37]. The estimation of
the oscillation of Du in §§2 and 3 builds on [37] but it is essentially new. The
algebraic Lemmas of §5 are scattered in the literature mainly without proofs. We
have attempted to rephrase them in the context of p-systems. The theory of linear
parabolic systems of §6 is taken from Campanato [23]. The rest of the Chapter
follows [36,37].
X
Parabolic p-systems: boundary
regularity
1. Introduction
We will establish everywhere regularity up the boundary for weak solutions of the
parabolic system
U::(UlIU2, ..• ,Um), meN,
{ UieC (e,T; L2(O»nV(e,T; Wl,P(O» ,
(1.1)
Ui,t -div IDulp-2 DUi =Bi(X, t, U, Du),
in 0 x (e, T),
ee(O,T), i=I,2,,,.,m, p>max{liJ~2},
associated with Dirichlet boundary data
(1.2)
in the sense of the traces on ao, of functions in Wl,p(n). The basic assumptions
on ao, the boundary data g and the forcing term B
g:: (91t!I2, .. · ,9m),

are the following:
(A l ) ao is of class Cl,~ for some oX e (0,1), in the sense
of (1.2) of Chap. I. Thus the norm IlaolhH is finite.
(A2) The functions gi, i = 1,2, ... , m, are restrictions to ao

of functions 9i' dermed in the whole OT, and satisfying
1. Introduction 293
(1.3) -
9i,z; E C (nT),
.\-
9i,t E LOO(nT),
i = 1,2, ... , m , j = 1,2, ... , N.
We set(l)
m N
(1.4) /11/1 == L L {/l9i /loo,nT + /l9i,tlloo,nT + [9i'Z;].\,nT}·
i=1 j=1
(A3 ) IB(x,t, u,Du)1 ::; Bo (1 + IDuIP-1), a.e. in nT,
for some given constant Bo. We say that a constant 'Y = 'Y (data) depends only
upon the data if it can be determined a priori only in terms of
(data) == (N, p, B o, IIOnll1+.\, /11/1) .

THEOREM 1.1. Let u be a weak solution of (1.1 )-(1.2) in n x (e, T), and let
(A I )-(A3) hold. Then
fl.i E C l - OI (nx (e, Tj) , for every aE (0, 1), i = 1,2, ... , m.
Moreover for every aE (0,1) and every eE (0, T), there exists a constant
'Y = 'Y (a,e, /lDullp,nx(~,T),data),

such that
(1.5)
The constant 'Y tends to infinity as either e '\. or as a'\. 0. °

Remark 1.1. The constant 'Y is •stable' as p -+ 2.
THEOREM 1.2 (HOMOGENEOUS BOUNDARY DATA). Letubeaweaksolution
°
of (1.1 )-(1.2) with 1== and let (At) - (A2 ) hold. For every e E (0, T) there
exist constants
'Y = 'Y (e, /lDullp,nx(~,T),data) >1 and a=a(data) E (0, 1)

such that
[fl.i'Zi]OI,nx[~,T) ::; 'Y, i = 1,2, ... , m, j = 1,2, ... ,N.

The constant 'Y / 00 as e '\. 0.
We will only carry the proof of Theorem 1.1. The proof of Theorem 1.2 fol-
lows exactly the same arguments, where in the various estimates the contributions
coming from /11/1 are discarded.
(I) For a smooth function r/J, the norm [r/Jh,K is defined in (1.3) of Chap. I.
294 X. Parabolic p-systems: boundaJy regularity
2. Flattening the boundary

Let e E (0, T) be fixed. We will estimate the oscillation of Ui about each point
(x o• to) E an x (e, T). For this we first introduce a change of coordinates that
maps a small portion of an about (xo, to) into a portion of an hyperplane. After
a translation we may assume that (xo, to) coincides with the origin. We will work
within the cylinder
Q'R, == K'R, x {-'R., O} , 2'R. = min{po; e},
where Po is the number that determines the structure of an as in (1.2) of Chap. I.
The portion of the boundary annK'R, is represented by
XN = 4>(x), x == (Xl, X2, ••• , XN-l) ,
where 4> is a function of class C l ,>. in the (N -1) -dimensional ball8'R,. satisfying
(2.1)
The last condition can be realised by taking a smaller Po if necessary. With respect
to the new variables
Xi = Xi, i = 1,2, ... ,N-I;

the portion annK'R, coincides with the portion of the hyperplane XN =0 within
K'R,. We orient XN so that, say, nnK'R, C {XN >O} and set
Q~ == Q'R,n{XN > O}.
Denoting again by X the transformed variables x and with Ui, B i , 4>, etc., the trans-
formed functions. the system (1.1) takes the form
(2.2)
(2.3)
(2.4) ( ) _ ( IN-l -D4>(X))

A X = -DcI(x) (1 + 1D4>12(x)) ,
where IN-l is the (N -1) x (N - 1) identity matrix. To reduce (2.2) to a system
with homogeneous boundary data on Q'R,n{ X N = O} set
Wi=U;-.9i, i=I,2, ... ,m,
and rewrite (2.2) in the form
(2.5) ! Wi - div Ai (x, t, Dw) = Bi + a~l A" in Q:k,

2. Flattening the boundary 295
Figure 2.1
(2.6) Ai,t (x, t, Dw) = at,k (x, Dw + Di) Wi,x" ,

(2.7) Bi=Bdx,t,w+g,Dw+Di)- !9i,
(2.8) At = at,k (x, Dw + Di) 9i,x".
Using the assumptions (Al)-(A3) we find the following structure conditions and
regularity properties on the various tenns of (2.5):
Ai,t(x, t, DW)Wi,XI ~ "YolDw + Dil,,-2IDwI2
(2.9) {
Ai,t(x,t,Dw)wi,xt $ "YllDw + Dil,,-2IDwI2,
for two positive constants "Yo $ "Yl depending only upon the data. Moreover for all
i=I,2, .. . ,m and k=I,2, ... ,N,
(2.10) IAi,k(x, t,e) - Ai,k(y,T,e)1 $ "Y (1 + lel,,-l) (Ix - yl + It - TI)~

Ve E R Nxm , and for a.e. (x,t), (y,T) E Q*,
296 X. Parabolic p-systems: boundary regularity
(2.11) IBi (x, t, Dw) I :::; 'Y (1 + IDw l,,-I) ,

(2.12) Ili.l(X,t,w,Dw) I:::; 'YIDul,,-2.
From (2.1) and the definitions (2.3)-(2.4) and (2.6), it follows that
(2.13) . = I~ + b 1,,-2 ~i,1c 61c,l,

Ai,l (0, O,~) V ~ E R N xm,
(2.14) b == (Di) (0,0).
2 -(i). Comparison Junctions

Consider cylindrical domains of the type
[(X o, to) + Q (R2+'I,R)] ,

where fJ E ( -1, 1) is to be chosen, Xo E K x n { x N =O} and the faces of the cubes
(xo + KR] are parallel to the coordinate'axes. These boxes are contained in Qx if
(2.15)
which from now on we assume. The proof of Theorem 1.1 is based on comparing
w in a neighborhood of each point (xo, to) E QX/2 n {x N ~ O}, with the solution
of
V == (111.112"'" 11m) , mEN,
(2.16) { l1i,t-div IDvl,,-2 Dl1i =0, in [(x o, to) + Q (R2+'I, R)] nQ~,
l1i = Wi on 8" [(xo, to) + Q (R2+'I,R)]nQ~,
where 8"Q denotes the parabolic boundary of a cylindrical domain Q. The exis-
tence of a unique weak solution of (2.16) can be established by a Galerkin proce-
dure. (1) Denote by
(f,XN) , f==(Xl,X2, ... ,XN-l),

the coordinates in K x. Then since w vanishes for x N =0, we also have v(x, 0, t) =
O. We let v and wdenote the odd extensions of v and w in the cylinder
[(x o, to)+Q(R2+'I,R)ln{zN~o}, i.e.,
in [(x o, to) + Q (R2+'I, R)] n{XN ~ O}

in [(x o, to) + Q (R2+'I, R)] n{XN :::; O},
w == { W(:,XN' t), in [(xo, to) + Q (R2+'I, R)] n{XN ~ O}
-W(X,-xN,t), in [(Xo, to) +Q (R2+'I,R)]n{xN :::;O}.
Then, by the reflexion principle v is the unique solution of
(1) See, for example, (73).

3. An iteration lemma 297
V == (iit,V2, ... , Vm), mEN,

(2.17) { vi.t-divIDvl,,-2Dvi=O, in [(X o ,to )+Q(R2+'7,R)] ,
Vi = Wi on 8" [(xo, to) + Q (R 2+'7, R)] .
It follows from the interior estimates of Theorems 5.1 and 5.2' of Chap. VIII that
IDvI is bounded in the interior of [( x o, to) + Q (R2+'7, R)] and it satisfies the
sup-bounds (5.1) and (5.3). We restate these bounds for the special geometry of
[(x o , to) + Q (R 2+'7, R)].
THEOREM 2.1 (THE DEGENERATE CASE p> 2). Let v be the weak solution
0/(2.17). There exists a constant 'Y='Y(N,p) such that/or all O<p'5.!R
(2.18) sup
[(zo.to )+Q(p2+'I.p»)
IDvl '5. 'Y (R'7 ff IDvl" dxd1") )
1/2 + R- ~.
[(zo.to )+Q(R2+'I.R)
THEOREM 2.2 (THE SINGULAR CASE max { Ii A~2} < p < 2). Let v be
a weak solution 0/(2.17). There exists a constant'Y = 'Y(N,p) such that/or all
O<p'5.!R
(2.19) sup
[(zo.t o )+Q(p2+'I.p»)
IDvl '5.'Y (R- '7 ff IDvl" dxd1")l/V + Rr-.,
N
P
[(zo.to)+Q(R2+'I.R)
where 1I,,=N(p - 2) + 2p.

Remark 2.1. Theorem 2.2 is a restatement of Theorem 5.2' of Chap. VIII with
=
q p. By Remark 5.4 of the same Chapter, such a choice is admissible.
3. An iteration lemma
LEMMA 3.1. Let 8"'" rt'( 8) be a non-negative non-decreasing junction defined in

[0, I] and satisfying
(3.1) rt'(p) '5. A (1/ rt'(R) + ~A(R.8-V"+"R-"), VO K. and II E (0, 1).
Then for every
(3.2) 0'5. 6 < K. (1 ~ ~):): /3) ,
there exists a constant'Y depending only upon A, /3, II and 6, such that
P )(j-tf,+6
(3.3) cp(p) ~ 'Y ( R' (cp(R) + 1)
(1 - v)1\:
VO<p~R~ 1, where q = 1 + f3 .
PROOF: Choose Ro < 1 and define the sequence Rn+l = ~, n = 0, 1,2, ....
Then
cp(Rn+1) ~ A~l-II)tf,cp(Rn) + ~A ( n!;'~" + ~+~/,)

= A~l-II)"cp(Rn) + A~+~/'.
Iteration of these inequalities gives
Now
and
Therefore if A ~ 2
cp(Rn+1) ~ An+1 ( ~1 ) (j cp(Ro) + 2An+1l(+~/'.

Fix 6 in the range (3.2) and set
E = I\: (1 ~ ~):): f3) - 6; f3 - ~ = f3 - I\: + 6 + E.

Then
n+1
The first coefficient in (3.3) is independent of n if n is so large that AR~m ~ 1.

Let no be the smallest integer satisfying
4. Comparing w and v (the case p > 2) 299
qno+l InA
-->--.
no+ 1 - pnR~1
It follows from (3.3) that if n ~ no,
If Ra < 1 is fixed, for every p E (0, Raj there exist some n E N such that
Rf+l ~p~ Rtf. Therefore the equation p = R!,q" has a root (J E [I, q]. Starting
the process with Ra replaced by R!, gives
Remark 3.1. The lemma continues to hold for 11=0. The constant 'Y on the right
hand side of (3.3) is 'stable' as 11'\.0.
4. Comparing w and v (the case p> 2)

We start by comparing w solution of (2.S) with the solution v of (2.16). Having
fixed (xo, to) E Ql'R n {XN = O}, we may assume, after a translation, that it
coincides with the origin. Setting
the vectors w and v satisfy
(4.1) ! (Vi - Wi) - div (IDvIP-2Dvi -IDwIP-2Dwi)

= - div(Ai(x,t,Dw) - Ai(O,O,Dw»
- div (Ai(O,O,Dw) -IDwIP-2Dwi)
-Bi(X,t,Dw)- 1:>8 fil(X,t,w,Dw), in+Q1t,

uXl '
(4.2) Vi - Wi = 0 on the parabolic boundary of +Q1t.
From (2.10) it follows that
IAi,l(X, t, Dw) - Ai,t(O, 0, Dw)1 ~ 'Y R).. (1 + IDwIP-l),

for i=I,2, ... , m and 1.=1,2, ... , N. Moreover from (2.13) and (2.14)
IAi(O, 0, Dw) -IDwIP-2 DWil ~ 'YIIDw + bl p - 2-IDwIP-21IDwl

~ l' (1 + IDwIP- 2 ) .
In the weak fonnulation of (4.1) take the testing function Vi - Wi modulo a Steklov
average, and add over i = I, 2, ... , m. We estimate the tenns on the right hand side
by the remarks above and the left hand side by making use of the algebraic Lemma
4.4 of Chap. I to obtain
//IDW - DvlPdxdr ~ 'YR). //(1 + IDwIP-1) IDw - Dvl dxdr

+Qlc +Qlc
+1'//(1 +IDwIP-2) IDw - Dvl dxdr

+Qlc
+1'//(1 +IDwIP-1) Iw - vldxdr

+Qlc
= [(I) + [(2) + [(3) .
In the estimates below we integrate over the boxes Q'k, rather than +Q'k. In doing
50 we think of v and w as defined in the whole Q'k through an odd extension as
indicated in §2. By the Schwartz inequality
[(I) ~ ~//IDW - Dvl Pdxdr+'YR).P!Y //(1 + IDwIP) dxdr,

Q~ Qlc
[(2) ~ ~//IDW-DvIPdxdr+'Y//(I+IDwIP)~ dxdr.

Q~ Q~.
Since v - w vanishes on the lateral boundary of Q'k, by the Sobolev embedding, (1)
1(') S
z.::.!
~R (If (1+ JDwJP) kdT) , (I/,Dw -DvIP dzdT r .1
~ ~//IDW-DvIPdxdr+'YRP!Y //(1 + IDwIP) dt.

Qlc Q~
Combining these estimates gives
(1) Corollary 2.1 of Chap. I.

(4.3) //IDW - DvlP dxdT ~ 'YR).t=r, //(1 + IDwIP) dxdT

Q~ Q~
+"'1//(1 + IDwIP)~ dxdT.

Q~
From this we deduce two inequalities. First, since
p- 2 <1 and (1 + IDwIP)~ ~ (1 + IDwIP),

p-l
we have
(4.4) If IDvlPdxdT :5 "'1 If (1 + IDwI P) dxdT.

Q~ Q~
Second, for 0: > 0 set
(4.5) :F (0:, "I, R) == Rap If (1 + IDwIP) dxdT,

Q~
and observe that
//(1 + IDwIP)~ dxdT ~ 'YRN+2+,,-ap~ [:F(O:'''I,R)l~.

Q~
This implies that VO<p~R~!n
(4.6) //(1 + IDwIP) dxdT ~ 'YR).t=r, //(1 + IDwlP) dxdT

Q: Q~
+//IDvIPdXdT+'YRN+2+,,-ap~ [:F(O:'''I,R)l~.
Q:
By taking R sufficiently small and by interpolation, the first term on the right hand
side of (4.6) can be eliminated. This is the content of the following lemma:
LEMMA 4.1. There exists a constant "'1 ='Y(data) such that for a/IV 0 < p ~ ! R ~
In
(4.6') //(1 + IDwIP)dxdT ~ 'Y!!IDvIPdxdT
Q: Q;p
+'YRN+2+,,-ap~ [:F(O:'''I,R)l~.
PROOF: It suffices to prove the lemma for R ~ R" where Ro is so small that
'Y (2Ro)'\;!-r ~ ~.
Let 0 < P ~ ~ Ro and consider the sequence of radii
Pn ==p+2-"p, n=O,I,2, ....
Write (4.6) for R= Pn-l and P= Pn,n~h and set
Yn ==/ PI + IDwIP) dt
Q:n
z == 'Y/ /IDvIPdXdT + 'YRN+2+"-QP~ [F (a, 1], R)l~ .

Q~"
Then by iteration from (4.6),
Yoo ==//(1 + IDwI P) dxdr

Q:
We return to (4.6)' and estimate the integral involving Dv in terms of Du.

!
Let 0 < P~ R. Then by Theorem 2.1 and (4.4)
/ /IDvlPdxdr ~ 'YpN+2+"II Dv ll:a,Q:

Q:
,; ~pN+2+' { Jl!lPI' (U IDvIPdzdT) pl'+ R"-,!. }
,; ~pN+""'{ Jl!lPI' (U IDvIPdzdT)

x (UIDvIPdzdT) + R-'-,!.}'
Next choose
1] = a(p - 2), for some a> O.
Then
Ir',/2 (u IDvl'do:dr )
~
" ~ [.1" (Q, ., RlI'" ,

and by suitably modifying the constant 'Y we deduce from (4.6)' that for all 0 <
p~R~!,R.,
(4.7) //(1 + IDwI P) dxdr
Q: ~ 'Y [.1' (a, 71, R)l

~ (p)N+2+'I f f
R )} (1 + IDwlP)dxdr
Q1r
+'Y[F(a'71,R)l~ RN+2+'I-a~ +'YpN+2+'I R-ap,
for a constant 'Y = 'Y (data). Set
(4.8) F(a) == sup

(:Co,to )EQR/2
O<pSRSR/2
{pap H + IDwI
(1
[(zo,t o )+Q:l
P) dt}
and
(4.9)
We summarise:
PROPOSITION 4.1. Let a > 0 and 71 = a(p - 2). There exists a constant'Y =
'Y (data), independent 0/ a, 71, p, R, such that
/orall (xo,to)EQ:k/2 and/orall O<p~R~'R./2,
(4.10) //(1 + IDwlP)dxdr

[(zo ,to)+Q;J
~ 'Yg(a)(~)N+2+'I /f (1 + IDwlP)dxdr
[( Zo ,to )+Q7tJ
+ 'Yg(a)RN+2+'I-ap~ + 'YpN+2+'I R-aP.
PROOF: The previous arguments prove the proposition for those points
(xo, to) EQ!Rn{XN =O}.
The estimate is obvious for boxes [(x o, to) + Q7tl c Q~, by interior estimates. If
[( xo, to) + Qkl intersects {x N =O}, then either
(4.11) [(Xo,to)+Q1R]CQR or [(Xo,to) + Q1R]n{x N =O}#0.

!
In the first case we may establish (4.10) with R replaced by R. The general case
follows by suitably modifying the constant 'Y. If the second of (4.11) holds, we let
(4.12) X. == (Xo,l' Xo,2, . .. ,Xo,(N-l), 0)
and observe that
[(X., to) + QiR] C I(xo, to) + Qkl·
We carry on the process leading to (4.10) for such a new box, for all 2X o ,N :5 p <
! !
R. This implies that (4.10) holds for all Xo,N < p:5 R. If p:5 Xo,N, we consider
the cylinder [(Xo, to) + Qt.N]' which satisfies the inclusion
[(Xo, to) + Ql z o,N] c Q:k.

Then by interior estimates, (4.10) holds with R replaced by Xo,N. Combining the
two cases and suitably modifying the constant 'Y we conclude that (4.10) holds for
-+ 1
all (XO, to) E Q!"R and all O<P:5 R:5 2'R.
5. Estimating the local average of IDwl (the case p> 2)
LEMMA 5.1. For every 0: E (0,1) there exists a constant 'Y = 'Y (0:, data), such
thot
jorall (xo,to) E Q:k/2 and/orall 0<p:5R:5'R/2,
(5.1) H (I, + IDwI P ) dxdr:5 'Y(o:,data) p- oP , '1 = o:(p- 2).

[(zo,to)+Q:l
PROOF: Define the sequences 0: 0 = (N + 2)/2 and for n= 1, 2, ... ,
We will prove inductively that
(5.2)
Since IDwIELP(nT),
H (1 + IDwI P ) dxdr :5 'YP-¥ P (1 + IIDwll:,nT) .

[(z.,to)+Q~O 1
Therefore
6. Estimating the local averages of w (the case p > 2) 305
Suppose the lemma holds for Q n and let us show that it continues to hold for Q n +1.
If F(Qn):'5 1'(Qn), the quantity Q(Q n ) introduced in (4.9) is bounded and we may
use (4.10) with Q = Q n and TI = TIn. We apply the iterative Lemma 3.1 to the
function
tp(p) = JJ(1 + IDwI P) dxdr,
[(zo,to)+Q~l
with the choice of parameters
p-2 -
(5.3) v = --,
p-l
6 = 6n Q np.
We obtain
If (1 + IDwI P ) dxdr :'5 1'(Qn+d p-On+lP (tp(R) + 1).

[(zo,to)+Q~n 1
Let 0 < P:'5 'R/2 be fixed and consider the point (xo, to) == (0, 0). Without loss of
generality assume that
p'r/n+l / p'r/n is an integer,
and partition the cylinder [(x o, to) + Q~n+l] into s = p'r/n+l-'r/n adjacent boxes
with 'vertices', say (0, td, (0, t2), ... , (0, t s ). Then
If (1 + IDwI P ) dxdr:'5 p'r/

n
p'r/n+l
~
~
If (1 + IDwIP ) dxdr
Q:n+l 3=1 [(o,tj)+Q~nl
:'5 1'(Qn+l) p-On+lP.
We may treat analogously the other points of Q'R./2 and the inductive inequality
(5.2) follows. To prove the lemma it suffices to prove that {Q n } -. 0 as n -. 00.
The sequence {Q n } is deacreasing. We claim that {Qn} -. O. Indeed if not,
lim
n--+oo
Q n = Q o > 0,
and the definitions of {Q n } and {6n } would imply
Therefore Qn+l :'5 Qn(I-6o). This in tum implies {Q n } -.0.

Remark 5.1. The constant l' on the right hand side of (5.1) is 'stable' as p ~ 2.
This follows from the choice (5.3) of the parameter v and Remark 3.1.
6. Estimating the local averages of w (the case p > 2)

We return to cylinders bearing the natural parabolic geometry, i.e., Q p == Q (p2, p)
and will work within the boxes
Since w vanishes for XN =0, we regard it as defined in the whole QR, by an odd
extension across {x N =o}. Let
(w)o,p == HW(X,t)dxdr
[(zo,to)+QpJ
denote the integral average of w over I(xo , to) + Qp]. If (x o, to) coincides with
the origin, we let (w)o,p==(w)p' Also let
(w)o,p (t) == f w(x, t)dx, t E (to -l, to).

{zo+KpJ
We observe that if Xo E {XN =O}, we have (w)o,p (t) =0 for all t E (to - p2, to).
since w is odd across {XN =O}, and in particular (w)o,p =0.
LEMMA 6.1. For every a E (0,1) there exists a constant "( = "( (a, data), such
lhat
(6.2) H Iw - (w)o,p IPdxdr :5 "((a)pp(l-OI),

[(zo,to)+QpJ
for all cylinders satisfying (6.1 ).

PROOF: We first observe that from Lemma 5.1 and its proof it follows that for
every cylinder satisfying (6.1)
(6.3) H (1 + IDwIP) dxdr :5 "((a) p-OIP.

[(zo,to)+Qp]
If Xo E {XN =O} by the Poincare inequality and (6.3).
H Iw - (w)o,p IPdxdr :5 "((a) pp(l-OI).

{(zo,to)+QpJ
Consider next the case
(6.4) [(xo, to) + Qp+ap] C Q~/2' U E (0, l) to be chosen.
By a translation we may assume that (xo, to) coincides with the origin. We have
6. Estimating the local averages of w (the case p > 2) 3C17
H Iw - {w)pIPdxdT :5 H Iw - {w)p {T)I PdxdT

Qp Qp
+ HI {W)p (T) - (w)p IPdxdr

Qp
== ](1) + ](2).
By the Poincare inequality and (6.3)
](1) :5 ')'{a) {I'(I-a).
Next
(6.5) ](2) =H /H [w{x, t) - w{x, r)] dxdrr dxdt.

Qp Qp
We estimate the integrand on the right hand side of (6.5) by making use of the
equation (2.5), over the cylinder Q P+tT p. Let x -+ ({ x) be a non-negative piecewise
smooth cutoff function in K p+tTP that equals one on Kp and such that ID(I :51/up.
In the weak formulation of (2.5), take ( as a testing function and integrate over
K p+tTP x [T, t] to obtain
/!
Kp+"p
([w{x, t) - w{x, r)] dx/
:5 t !!IAi{X,w, Dw).D(+At(xt+Bi(ldxdT
1=1 Qp+"p
:5 u'Yp! /(1 +
Qp+"p
IDwIP-1) dxdT
By the properties of ( and (6.3) with a suitable choice of a we conclude that
1/[w{x, t) - w{x, T)] dxl :5 ~pN+(I-a)

l(p
+ / /[w{x,t) - w{x,T)]dxl.
Kp+"p\Kp
Let {w)P+tTP denote the integral average ofw over the Qp+tTP, i.e.,
(W)P+ITP == H
Qp+"p
w(x, r)dxdr.
Then
Ij[W(X,t) - w(x,r)] dxl

.1
~ IKp+ITP\Kpl~ { ( jIW(X,t) - (w)P+ITP 1PdX) p
r}
Kp+"p
.1
+ L!.~W(X' (W)..,., I'

T) - do:
Combining these estimates in (6.5) gives
[(2) ~ ;P pp(l-a) + 'YUp- 1 HIW -

Qp+"p
(W)p+ITPI Pdxdr.
We conclude that for every a E (0, 1) there exists a constant 'Y = 'Y( a) such that for
every uE (0,1) and for every pE (0, fR)
(6.6) H
Qp
Iw - (w)pI P dxdr ~ 'Y~~) p(l-a)p
+ 'Y(a)uP- 1 H
Qp+ .. p
Iw - (w)p+ITPI Pdxdr.
This implies the lemma, in the case (6.4) holds, by the interpolation process of
Lemma 4.3 of Chap. I. This process yields the choice of u E (0, !).
Finally, having
fixed u E (0, !), consider the case when
[(Xo, to) + Qp+lTp]n{XN = O} i: 0.

Letx. E {XN =O} be defined as in (4.12) and observe that the box [(x., to) + Q2p]
'centered' at (x., to) contains [(xo, to) + Qp]. Therefore, by the Poincare inequal-
ity, since the average ofw over [(x., to) + Q2p] is zero,
H Iw - (w)o,p IPdxdr
[(zo,to)+Qp)
~ 'Y H IwlPdxdr
[(z.,to)+Q3p)
~ 'Y(a)pp(l-a).
7. Comparing w and v 309
6-(i). Proof of Theorem 1.1 (the case p>2)

The proof is a consequence of Lemma (6.1) and the averaging theory of
Campanato-Morrey spaces [22,23,33,79]. It can also be proved directly, starting
from (6.2), by arguments similar to those in §§2 and 3 of Chap. IX.
7. Comparing w and v (the case max {I; &~2} <p<2)
Assuming that (x o, to) == (0, 0), the functions w and v satisfy

8
(7.1) at (Vi - Wi) - div (IDvIP- 2 DVi -IDwlp- 2 DWi)
= - 8 (al,k(x, DU)Ui,Zk -IDulp-2Ui,Zk Ot,k)
!3
uXl
- div (lDulp-2 DUi - IDwlp- 2DWi)
- Bi(x, t, w, Dw), in +Q'k
Vi - Wi =0, on the parabolic boundary of +Q'k.
The boxes Q'k are formally identical to those introduced in the degenerate case
p > 2. In the singular case we will take TJ E ( -1, 0). In writing (7.1) we have used
the definitions (2.3) and (2.8). From (2.3)-(2.4) we derive the estimate
lal,k(x, DU)Ui,ZI -IDulp-2 Ui ,ZI! $ -yR)..IDulp-l,

i=1,2, ... ,m, l,k=1,2, ... ,N.
Moreover by Lemma 4.4 of Chap. I,
IIDuIP-2Dui -IDwlp-2Dwil $ -ydD(u - w)I P- 1$-y,

since the boundary data g are regular. In the weak formulation of (7.1) we take the
testing functions Vi - Wi modulo a Steklov averaging process, integrate over +Q'k
and add over i = 1, 2, ... , m. Using the remarks above to estimate the correspond-
ing terms on the right hand side gives
(7.2) //(11ID (sv + (1- s)w) IP- 2dS) IDw - Dvl2 dxdr
+Qk
$ -yR).. / /IDuIP-1IDW - nvl dxdr

+Q~
+ -y/ /IDW - Dvl dxdr + -y //IBIIW - vi dxdr.

+Q~ +Q~
In carrying the estimates below, we think of v and w as defined in the whole Q'k
by an odd reflexion across {XN =o}. By the Poincare inequality and (2.7)
2::!
$ ~ ([[<1+ IDwIP) dxdT) ·

2::! .1
$ ~R ([[<1 + IDwIP) dxdT) · ([[IDv - DwIPdxdT) •

Introduce the two sets
£1 == {(x, t) E Q'k I IDw - Dvl ~ IDwl} ,
£2 == {(x,t) E Q'k IIDw - Dvl < IDwl}·
11lD (sv + (1- s)w) IP- 2 ds ~ ~IDw - DvIP- 2.

Therefore from (7.2) it follows
!!IDw - DvlPdxdr+ !!IDWIP-2IDw - Dvl2dxdr

£1 £2
:5 'YR'>'{!! IDuIP-IIDw - Dvldxdr

£1
+ ! ! IDuIP-IIDw - Dvldxdr}
£2
~
+ { [[ IDw - Dvldxd-r + [/ IDw - DvldxdT }
2::!
+ ~R ([[<1 + IDwIP) dxdT) ·

"{Jf iDw - Dvl'dxdT+ ff IDw - DvI'dxdT}
\ £1 £2
IIp
In t~l~" inequality we absorb the integrals of IDw - vi extended over £1 into the
analogous term on the left hand side by means of Young's inequalitj. Using also
the definition of £2 we arrive at
7. Comparing w and v 311
(7.3) !!IDW - DvlPdxdr+ !!IDwIP-2IDW - Dvl2dxdr

£1 £2
$ 'YR >..JJ(l + IDwIP) dxdr + 'Y!J(l + IDwl) dxdr.

Q~ Q~
We estimate the right hand side of (7.3) by
'Y R + +'1+>"-OP {nap H(1 + IDwIP) dt}

N 2
[(Zo.to)+Q~l
1
+ 'Y RN +2+'1- 0 {nap H(1 + IDwIP) dt} P
[(zo.to)+Q~l
$ 'Y R N +2+'1- o p>"o [.r(a) A .r1/P(a)] ,

where
~o=max{!;
P
1-~}
ap
E(O,l)
and where as before we have set
Therefore
(7.4) JJ IDw - DvlPdxdr + JJ IDwIP-2IDw - Dvl2dxdr

£1 £2
$ RN +2+'1- o p>"o [.r(a)A.r1/P(a)] .
Rewrite the integrand in the second integral on the left hand side of (7.4) as
Observe also that on the set E2 • IDvl $ 21Dw I. so that

IDwIP-2IDvI2 $ 22-PIDvIP.
These remarks in (7.4) prove the following:
312 X. Parabolic p-systerns: boundary regularity
LEMMA 7.1. There exits a constant 'Y = 'Y (data), such that
forall (xo,to) E Q:k/2 andforall 0<p5,R5,'R/2,
(7.5) IllDwlP dxdT 5, 'YR N+2+'1- op>,0 max {F(Q); F1/P(Q)}

[(xo,to)+Q~1
+ 'Y III Dvl PdxdT,

[(xo,to)+Q~1
(7.6) III Dvl P dxdT 5, 'YRN+2+'1-op>,o max {F(Q); Fl/p(Q) }

[( Xo ,to) +Q1t1
+ 'Y IllDwlP dxdT ..

[(x o ,t o )+Q1t)
To estimate the last integral on the right hand side of (7 .5) we make use of Theorem
2.2. the defmition of F(Q) and (7.6). Assuming that (xo, to) coincides with the
¥l
origin, we have for all 0 'o [F(Q) " F1/P(Q)]
+ R-N'I H (1 + IDwl P) dxdT }

PIlip
,
Q1t
where lip = N(p - 2) + 2p > O. Choose
(7.8) '1=Q(p-2), QE (0; ::2] >0.
Then the first term on the right hand side of (7.7) is estimated above by
'Y pN +2+'1 R-oP.
Setting also
Q(Q) '= max {pilip; Fillip; F N (2-p)/lI p ; I} ,

8. Estimating the local average of IDwl 313
the second term on the right hand side of (7.7) is estimated by
-yQ(a) (~)N+2+'1II(1 + IDwI P ) dxdr

Q1t
+ -yQ(a) pN+2+'1 R-(N'1+ 2a p>.o)p/"p.
Using the dermition (7.8) of 1] we have
p ap2
-(N1] + 2apA o ) -
lip
= -a + -(1
lip
- Ao).
Since Ao E (0, 1) we estimate R-(N'1+ap>.o)p/"p ~ R-a P, and summarise:

LEMMA 7.1. Let a and 1] be chosen as in (7.8). There exists a constant -y =
-y(data), independent of a, 1], p, R, such that
forall (xo,to) E Q:k/2 andforall 0<p~R~n/2,
(7.9) 11(1 + IDwlP)dxdr

[(xo,to)+Q~J
~ -y Q(a) (~) N+2+'1 11(1 + IDwIP ) dxdr

[(x o ,t o )+Q1tJ
+ -yQ(a) pN+2+'1R- a P.
8. Estimating the local average of IDwl
LEMMA 8.1. For every a E (0,1) there exists a constant -y = -y (a,data), such
that
forall (xo,to) E Q:k/2 andforall 0<p~R~n/2,
(8.1) H + IDwI
(1 P) dxdr ~ -y(a,data)p-ap •
[(xo,to)+Q~J
PROOF: Derme sequences ao=(N + 2)/2 and forn=O, 1,2 ... ,
1]n = an(p - 2),

It is apparent that {on}, {l1n}, {On} -+ 0 as n -+ 00. We will prove by induction

that
(S.2) F(on) :5 ')'n (on. data) , n = 0, 1,2, ....
Since IDwleLP(lh),
H+ (1 IDwI P ) dxdr:5 ')'p_P(~+3) (1 + IIDwllp,SlTt.

[(:r:o,to)+Q~o I
Therefore
F(oo) :5 ')'0 == (1 + IIDwllp,SlTt·
Assume now that (S.2) holds for some n and let us show that it continues to hold
for n + 1. We apply the iterative Lemma 3.1 with the choice of the parameters
(3 == N + 2 + l1n, " = onP, 0 = on, 1/ = 0,

to the function
<p(p) = II (1 + IDwI P ) dxdr,
1(:r:o,to)+Q~n I
which satisfies (7.9), to conclude that for all (xo, to) e Q:k/2 and for all 0 < p:5
R:5'R/2,
In particular, p being fixed, (S.3) must hold for radii p. satisfying
Without loss of generality we may assume that

"',,-'Intl
2+'In+1 _ D
p. = (;. is an integer.
Then we may regard the cube Ixo + K pI as the disjoint union, up to a set of measure
zero, of (f..)N cubes Ix; + Kp.1 centered at points x; of [xo + Kpl. Similarly
we regard the cylinders [(x o• to) + Q~n+l] as the disjoint union, up to a set of
measure zero,of(i.)N cylinders [(x;, to) + Q::]. We write (S.3) for each of these
cylinders and add up for j =1, 2, ... ,i. to obtain
II (1 + IDwI P ) :5 ')' (i.)N p~+2+"n-anp+6n

1(:r:o,to)+Q:n+l1
Therefore
pQn +1P H + IDwI

(1 P) dxdr5,'Y and F(an H,1Jn+t} 5, 'YnH'
[(:l:o,to)+Q:n+1]
8-(1). PROOF OF THEOREM 1.1 (THE CASE max {I; ~~2} < p < 2)
By a cube decomposition technique similar to the one outlined, Lemma 8.1
can be rephrased in tenns of the parabolic cylinders Qp=Q(p2, p).
LEMMA 8.1'. For every a E (0, 1) there exists a constant 'Y = 'Y (a, data), such
that
(8.1)' H (1 + IDwI P ) dxdr 5, 'Y(a,data)p-QP.

[(:l:o,to)+Qp]
With this lemma at hand the proof is now concluded as in the degenerate case.
First we may establish a version of Lemma 6.1 and then the HOlder continuity of u
follows from the arguments of [22,23,32,79] or by those in §§2 and 3 of Chap. IX.
The proof of Theorem 1.1 is in [27]. The iteration Lemma 3.1 is in the same spirit of
similar results of Campanato [22,23]. The technique is indeed a degenerate version
of [22,23]. Techniques of this type near the boundary appear in Giaquinta-Giusti
[49]. The boundary behaviour of solutions of (1.1) is essentially not understood. In
the case of a single equation some results appear in Liebennan [69] and Lin [72].
XI
Non-negative solutions in ET-
The case p>2
1. Introduction
Non-negative solutions of the heat equation in a strip ET == RN X (0, T) are some-
what special in the sense that they grow no faster than
(1.1) a < 1/4T, as Ixl-oo.

Let I' be a 0' - finite Borel measure in R N with no sign restriction. We say that p.
bas the growth (1.1) if
(1.2) /e-~ldlJl < 00, .
RN
where IdILI is the variation of 1'. Then the Cauchy problem

Ut - Llu = 0, in ET ,
(1.3) {
u(·,O) = 1',
is uniquely solvable within the class of functions satisfying (1.1). The •initial mea-
slUe' is taken in the sense
(1.4) /U(x,t)CPd/-L~ /cpd/-L, as t'\.O, 'v'cpEC~(RN).

RN RN
2. Behaviour of non-negative solutions as lxi- 00 and as t '\. 0 317
Conversely every non-negative solution of the heat equation in ET verifies (1.4)

for some u-finite non-negative Borel measure Jl satisfying the growth condition
(1.2). The measure Jl is unique and it is called the initial trace of u. In tum the
initial trace of u determines u uniquely. These are the basic elements of a classical
theory developed by Tychonov [98], Tacklind [94] and Widder [105]. A perhaps
rough summary of the theory is that the structure of all non-negative solutions of
the heat equation is determined by the heat kernel
1 _~
r(x, t) = (411't)N12 e t, t > O.
Consider now non-negative local weak solutions in ET of
u E C loc (O,T; L~oc(RN»)nLroc (0, T; WI!;:(R N)) , p>2,

(1.5) {
Ut - div (lDulp-2 Du) =0 in ET.
The analog of r(x, t) for the degenerate p.d.e. (1.5) is the Barenblatt explicit so-
lution
~
(1.6) B (x, t )
_ t -NI>.{ 1 -
= 'Yp (IXI)P!Y}"-+ '
til>' t > 0,
_....l....p-2
"yp == A ,,-1 - - , A = N(p - 2) + p.
P
We call this a 1undamental solution' only in the sense that
B(x,t) --+ (411') NI2 r(x,t) pointwisein ET, asp'\.2.
Solutions of (1.5) cannot be represented as convolutions of initial data with B( x, t).
Nevertheless the sup-estimates of Chap. V and the global Harnack estimates of §7
of Chap. VI permit a precise characterisation of the class of non-negative solutions
of (1.5) in the whole ET • with no reference to possible initial data. Such a charac-
terisation essentially says that all non-negative solutions of (1.5) behave as t '\. 0
like the 'fundamental' solution B(x, t), and as lxi- 00 they grow no faster than
Ixl p/ (p-2). For these solutions we will establish the existence of initial traces and
prove their uniqueness when the initial datum is taken in the sense of Lloc<RN).
2. Behaviour of non-negative solutions as Ixl-+ 00 and as

t'\.O
Let u be a non-negative local weak solution of (1.5) in ET • For e E (0, T) and r > 0
set
-
IIlulllr.T-~ = sup sup
O<T:$;T-e p>r
J
u(x,r)
>'I( -2) dx,
P I'
A=N(p - 2) + p.
Kp
318 XI. Non-negative solutions in ~T. The case p>2
THEOREM 2.1. There exists a constant 'Y='Y(N,p) such that/or all eE (0, T),
;!, }>./P
(2.1) IIlullr,T-~ $ 'Ye-p!-, { 1 + (~) p- u(O,T-e)
Moreoverforall tE (0, T-e), and all p~r,
pp/(p-2) pI>'
(2.2) lIu(·, t)lIoo,K p $ 'Y tNt>. IIlullr,T-~'
t
(2.3) j jlDulP-IdxdT $ 'Y tl/>'pl+~ Ilull!~~,

OKp
p2/(p-2) 2/ >.
(2.4) IIDu(', t)lIoo,K p ~ 'Y t(N+l)/>' Ilullr,T_~'
Moreover (x, t) -+ Du(x, t) is Holder continuous in Kp x (e, T -e) with HOlder

constants and exponent depending only upon N, p, 'Y, p, e and IIlullr,~.
Remark 1.1. The functional dependence of these estimates is optimal as it can be
verified for the explicit solution 8(x, t).
Remark 1.1. The estimates (2.2)-(2.4) hold for solutions of variable sign. pro-
vided we assume (2.1).
PROOF OF THEOREM 1.1: The estimate (2.1) is the content of Corollary 7.1 of
Chap. VI. whereas (2.2) follows from Theorem 4.5 of Chap. V. The gradient bound
(2.3) is Lemma 9.1 of Chap. V and Remade 9.2. Inequalities (2.2)-(2.3) hold for a
time interval
(2.5)
for a constant 'Y. ='Y.(N,p). In view of (2.1) they can be considered valid for all
tE (0, T-e). Indeed working within ET • we may state them for every substrip
RN x [tl' t2j, O$t! <t2 ~T-e, t2 - tl ~ 'Y.. llullr,T-~'

In the proof of (2.4) we will work in the time interval (2.5). We begin with a qual-
itative information.
LEMMA 2.1. For every e E (0, T) and for every r > 0 the quantities
IIDu(·, t)lIoo,K p
sup
p>r P2/(P -2)
arefinitefor all 0< t~ T-e.
PROOF: By the interpolation Theorem 5.1' of Chap. VIII with e= 1, q =1/2 and
9=!t we deduce
3. Proof of (2.4) 319
for all r E (it, t). Estimating the right hand side by (2.3) we obtain
IIDu(·,r)lIoo,Kp <
p2/(p-2) - "Y
(till UnUlp-2
r,T-£:
)1/>'111 ullIII r,T-£: +C~ •
Next we will turn such infonnation into the quantitative estimate (2.4).
3. Proof of (2.4)
Let t>O and p>r be fixed and consider the box Qo==K2pX [it, tjcE T • Let the
radii {Pn} and time levels {t n } be defined by
n=0,1,2, ...
and introduce the corresponding family of nested shrinking cylinders
Qn == K pft x{tn,t},
with vertex at (0, t). We will estimate the quantity IIDu(·, t)lIoo,K p ' by using the
techniques developed in Chap. VIII. The starting point is the the iterative inequality
(5.4) in that chapter, which we rewrite here in the context of the cubes Qn as
y.n+l _< "Y bnk-~'LIy'l+~

+ "n , ~2 == N(p - 2) + 2p,
where
Yn == ! !ODu l - kn)~ dxdr,
12ft
(3.1) 'It == {sg: IDul p - 2 p-2 + t- 1 },

and k is a positive number to be chosen. By Lemma 4.1 of Chap. I, the sequence
{Yn } tends to zero as n-+oo if k is chosen to satisfy
We conclude that there exists a constant "Y = "Y( N, p) such that for all it:5 r:5 t,
(3.2) IIDu(.,r)lIoo,Kp :5 "Y'It~ (j !IDulpdxdr) 2/>'2
t/2 K2p
320 XI. Non-negative solutions in E T • The case p>2
To proceed we introduce the non-decreasing function of t
A>( ) _ ¥- IIDu(·, T)lIoo,K p

(3.3) ..... t = O<r$!
sup T sup
p>r p
2/( -2)
P
,
By Lemma 2.1 and (2.2) this quantity is well dermed. In the estimates below we
write ~ == ~(t) if the dependence upon t is unambiguous. We estimate the quantity
1t introduced in (3.1) by
and deduce from (3.2) that for alllt $ T $t
and
(3.4)
Estimating G1(t) we have

3. Proof of (2.4) 321
To estimate G 2 (t) we refer back to the p.d.e. in (1.5). Let ( be a non-negative

piecewise smooth cutoff function in K4pX{it, t} that equals one on K2pX{ !t, t}
and such that ID(I ~ 2/ p and (t ~ 4/t. Taking u(P in the weak formulation of
(1.5) we obtain
In estimating G2 (t) we use the estimation (2.2) and the range (2.5) oft.
Combining these estimates in (3.4) gives for all 0 < t ~ 'Y.lllulll ~,7-E'
322 XI. Non-negative solutions in Er. The case p>2
for a constant 'Y='Y(N,p). It follows that 4>(.) is majorised by the solution of
V'(t) ~ 'Y t- (N+l¥P-2) Vp-l(t),

{
V(O) = 'Ylllull~:;_E' 0 < t ~ 'Y.mull~:;_E·
Solving this explicitly gives
1berefore choosing t so small that
2/>.}-1/(P-2)
{1- 'Y (t l~ull~:T2_E) ~ 2,
we will have
t¥ IIDu(·, t)lIoo,K < 2 IlluI1 2/>'
p
p2/(p-2) - 'Y r,T-E
for all such t and all p> r.
4. Initial traces
THEOREM 4.1. Let u be a non-negative local weak solution of(1.5) in :E T . There

exists a unique Radon measure IL such that
(4.1) lJ$ f
RN
u(x, t)cpdx = f
RN
cpdIL, 'v'CPEC~(RN).
Moreover. as Ixl- 00, IL 'grows' at most as IxI P/(p-2). Precisely.
(4.2) sup
p>r
f
Kp
dIL
/( -2)
pP P
< 00, 'v'r > O.
PROOF: The existence of a Radon measure IL satisfying (4.1 )-(4.2) follows from
the global Harnack estimates of §7 of Chap. VI. Indeed by Corollary 7.1 of that
Chapter, for every cube [x o + K pj C RN and all cp E Cr;' (Kp),
If
_ Kp
u(x, t)cp(X)dXI ~ 'Y (N,p, p, T, U(Xo, T-e» IIcplloo,K p '
5. Estimating lDur- 1 in Er 323
for all 0 < t :$ T -E and all E E (0, T). Therefore {u(·, t)}O<t<T-€ is a net of
equibounded linear operators in Cgo(RN), and for a subnet, indexed with t',
for a Radon measure IJ. The uniqueness of such a measure is a consequence of the
following:
LEMMA 4.1. Let u be a non-negative local weak solution of (1.5) in ET. Then
'Vp>O, 'VO'E(O, I}, 'VO<r<t <T/2,

(4.3) f u(x, t)dx ~ f u(x, r}dx- ;,(t - r}l/'\p~ Ilull~;'~.
K(1+")p Kp
PROOF: Fix 0 < r < t and 0' E (0, I), and let x ...... (x) be a non-negative piecewise
smooth cutoff function in K(1+cr)p that equals one on Kp and such that ID(I :$
2p/0'. In the weak fonnulation of (I.S) take ( as a testing function. Integrating
over (r, t) gives
t
f u(x, t) dx
K(1+")p
~ f
Kp
u(x, r) dx - 0'2p f
T
PDuI P- 1 dxds.
K(1+")p
To prove (4.3) we estimate the right hand side of this inequality by (2.3).
We now prove the uniqueness part of Theorem 4.1. Suppose that out of the
net {u(x, t)}O<t<T-€ we may select two subnets indexed with r' and t' such that
for all 'P E ego (RN) and IJ :/= II. Then we let r ...... 0 along r' in (4.2) and then let
t ...... 0 along t'. This gives
Interchanging the role of IJ and II proves the Theorem since 0' E (0, I) is arbitrary.
5. Estimating IDu Ip-l in ET
Local integral estimates of IDul p - 1 are crucial both in the global Harnack estimate
of §7 Chap. VI and in the theory of initial traces. The inequality (2.3) of Theorem
2.1 is local but holds for all p > r. Therefore it implies some control on the be-
haviour of IDul as Ixl -+ 00. This behaviour can be given an integral form, by
means of the weights
(5.1)
where a is a positive number satisfying
.x
(5.2) ap= --2 +u, for some u > o.
p-
THEOREM 5.1. Letu bea non-negative local weak solution of(1.5) in ET. Then
for every u > 0, there exists a constant 'Y = 'Y( N, p, u) such that for all r > 0 and
all EE (0, T),
(5.3) sup
O<t<T_
ju(x, t)Aa(x) dx ~ 'Ylllullr,T-E'
- aN
t
(5.4) j jlDulP-l Aa(x) dxdr ~ 'Ytl/>'l~ull~;'~.
oaN
Remark 5.1. The constant 'Y(N,p, u) /00 as u '\,0.
PROOF OF (5.3): Without loss of generality we may assume that r = 1. Then
foralIO<t~T-E,
fu(x, t)Aa(x) dx ~ f u(x, t)Aa(x) dx + f: f u(x, t)Aa(x) dx

aN {Izl<l} n=O {2n<lzl<2n+l}
00
~ mU~lr.T-E + 2~ L 2- an mullr,T-E.
n=O
PROOF OF (5.4): It will suffice to establish the estimate for t in the interval (2.5).
We will use this fact with no further mention. First we observe that the inequality
(5.5)
holds for all x E RN and all 0 < t ~ T-E. This is obvious if Ixl ~ r with the
constant 'Y depending also upon r. If Ixl > r, we apply (2.2) to the cube K 21zl.
Let T/ E (0, T -£ ) and in the weak formulation of (1.5), take the testing function
l/p I_a (Al/p r)P

(t - T/ )+ U P a+ 1 .. ,
P
where x -+ «(x) is the usual cutoff function in Kp. After a Steklov averaging
process and standard calculations, we obtain
5. Estimating IDulp - t in Er 325
t
IP A 1. (Pdxdr
(5.6) f (r - "l)l/pjIDU
u 2 / p "'+,.
" Kp
t
~ 'Y j(r - "l)I/P j u¥up-IID (A~:*()IP dxdr
" Kp
t
+ 'Y pr - "l)~-lj u¥ A1/puA",dxdr = J~l) + J~2).
" Kp
As for J~2) we have
t £=l
J(2) < "'f(r_'TI)t- 1jr N <:A 2 ) lu(x,r)I" u(x r)A (x)dxdr
p - /./ (1 + Ixl p)1/p ,'" ,
" Kp
so that by (5.5) and (5.3),
J~2) ~ 'Y(t - "l)!-,AWull!;'~.

We estimate J~I):
t
J~I) ~ 'Y j(r-"l)* jU¥UP-1A",+;ID(IPdXdr

" Kp
t
+'Y pr - "l)l/pju~uP-IIDA~:*IPdxdr
" Kp
= J~l,l) + J~I,2) .
Since
,. ~ 'YIA~+:!7+1.IP
IDA '"I/+P1.1 ,. ,. ,. ~ 'YA", A I/ p At,
by (5.5), (5.3) and the range (2.5) of t,
t
J~1,2) ~ 'Y j(r - "l)~ j (u~ A~) (u P- 2AI) u(x, r)A", (x) dxdr
" Kp
~ 'Y(t - "l)!-,Alllull!;'~.
As for J~I,l), since ID(I ~ 2/ p, again by (5.5) and (5.3)
J~l,l) ~ 'Y(t - "l)!-,AI~ull!;'~.

326 XI. Non-negative solutions in Er. The case 1'>2
Combining these estimates in (5.6).
(5.7)
where we have changed pinto 2p. Next, for all ,., ~ t ~ T - E
6. Uniqueness for data in Lloc(RN)
THEOREM 6.1. Letu and v be two non-negative local weak solutionsof(l.5) in

ET. satisfying
Then u==v in ET.

PROOF: Fix r>O and some EE (0. T) and set
Ilullr.T-~ + Ilvllr.T~ == A.
Then and u and v satisfy all the estimates of Theorem 2.1 with the quantities
llu. vllr.T~ replaced by A within the strip ETa' where
0< To = min{T;-y.A-(p-2)}.
and -y. is the constant appearing in (2.5). It will suffice to prove uniqueness within
the strip ETo' The difference w=u-v satisfies
6. Uniqueness for data in Lloc(RN ) 327
(6.1)
where
ai,j(z, t) = (iID(S' + (1 - S)V)Ip-2ds) 6,;
+ (p - 2) j ID(su + (1 - s)v)IP-4
o
X (su + (1 - s)v)x. (su + (1 - s)v)xjds.
The matrix (ai,j) is positive semi-definite and for all eE RN and (x, t) E ETo
ao(x, t)lel 2 S ai,j (x, t)eiej S (p - l)a o(x, t)leI 2 ,
{ l
(6.2)
ao(x, t) = flD(su + (1 - s)v)IP- 2ds, (x, t) E ETo.
o
Let Ao(x) be the weight introduced in (5.1) with Q satisfying (5.2). In the argu-
ments below, 'Y denotes a positive constant that can be determined a priori only in
terms of N, p, (T and A.
6-0). Auxiliary lemmas

LEMMA 6.1. There exists a constant 'Y='Y(N,p, (T, A) such that ifw(·, t) -+0 in
Lloc(RN) as f\,O, then
j1w(x, t)IAa(x) dx S 'Ytl/", 0< t < To.

RN
PROOF: The functions w± are both weak subsolutions of (6.1), i.e.,

wf - (ai'i(x, t)w~)x., SO weakly in ETo·
By working separately with w+ and w- we may assume that w is a non-negative
subsolution of (6.1). In the weak formulation of (6.1) take the test function x-+
Ao(x)«x), where (is the usual cutoff function in Kp. Using the assumptions of
the lemma we deduce
t
j1w(x, t)IAo(x)( dx S 'Y j jODvl + IDvD p - 1 IDAo(1 dxdr

Kp OKp
t
S 'Y j jODvl + IDvD p - 1 AoID(1 dxdr

OKp
t
+'Y j jODvl + IDvDP-IIDAoldxdr.

OKp
328 XI. Non-negative solutions in ET. The case p>2
In the last integral.IDAal $'YAa+l/p and in the first integral. since IDcl =0 on
Kp/2 we have AalDCI $'YAa+l/p for p > 1. Therefore letting p-+oo
t
jlw(x.t)IAa(X)dX $ 'Y jjODvl + IDvI)P-l Aa+l/pdxdr,
RN ORN
and the conclusion follows from Theorem 5.1.
PROOF: Let fiE (0, *,) be fixed. Then 'v'tE (0, To)
j1w{X, t) I1+'1 A a +fJ /(p-2) {x)dx

RN
$ jIW{X, t)l fJ A fJ /(p-2) {x)lw(x, t)IAa{x)dx.

RN
By (5.5).lw(x, t)lfJA fJ /(p_2) (x) $ 'Yt-If 'I. so that by Lemma 6.1.
jIW(x, t) I1+'1 A a+fJ /(p-2) {x)dx $ 'Yc IYf j1w(X, t)IAa{x) dx

RN RN
$ 'Yt! (l-NfJ).
6-0;). Proof of Theorem 6.1

In (6.1) we may assume. by working separately with w+ and W-. that w ~ O.
In its weak: formulation we take the testing functions
Integrating over K p x (c, t). 0 < C< t $ To. we obtain

6. Uniqueness for data in L:""(RN ) 329
(6.3) 1 ~ 11 !(W + 6)1+'7AQ(2dx

Kp
t
+11 If
6Kp
IDwI2 (1)2
ao(x,r)(W+6)1-'7 A dxdr Q (
~ 1 ~ 11 !
Kpx{6}
(w + 6)1+'7AQ(2dx
+'1 !!ao(X,r)(W~~~ (w+6)l:f1

6Kp
X (A!() ID (A!() Idxdr,

where ao(x, t) has been defined in (6.2). By the Schwartz inequality the last inte-
gral is majorized by
t
!
+ '1(11) !ao(x, r)(w + 6)1+'7 (AQID(1 2 + IDA! 12) dxdr.
6K p
We absorb the integral involving IDwl2 on the left-hand side of (6.3) and discard
the resulting non-negative term. Finally, we observe that by the definition of AQ
and the structure of ( we have
AQID(1 2 + IDA! 12 ~ 'YAQ(X) Alp (x).

Carrying these remarks in (6.3) gives
(6.4) !(w + 6)1+'7AQ(2dx ~ !(w + 6)1+'7AQ(x) dx

Kpx{t} Kpx{t}
t
+'Y!!ao(X,r)A;(x)(w + 6)1+'7AQ(x)dxdr.
6Kp
Next by (6.2) and (2.4)
a (x r)A.a(x) < '1 Ixl2 Ai (p-2)r-(Nr> (p-2).

0, p - (1 + Ixl p)2/p
Substitute this last estimate in (6.4) and let 6 - 0 for p 2: 1 fixed so that by Lemma
6.2
330 XI. Non-negative solutions in ET. The case p>2
j (w + 6)1+f/Ao(x)dx --+ °as 6 - 0.

Kpx{6}
Then we let p - 00. The net result is
j1w(x, t) 11+'1 Ao(x) dx

RN
t
:5 'Y j.,.- (Ntl) (p-2) j1w(x, "')11+'1 Ao(x) dxd.,..
o RN
Since.,.- (Nti) (p-2) E L1 (0, t), this implies
t - j1w(x,tW+f/Ao(X)dx == 0,
RN
by Gronwall's lemma, provided
t - /lw(x,tW+f/Ao(X)dX E VlO(O,To)'
RN
Now the parameter Q in the calculations above is arbitrary and only restricted by
(5.2). If Q is replaced by Q+,,/(p-2). then Lemma 6.2 and its proof ensure the
Loo(O, To) requirement and the theorem follows.
Remark 6.1. For non-negative solutions u and v of (1.5) in ET. the quantities
(6.5) Ilullr,T-E, IIIvllr,T_ are fmite.

The proof of Theorem 6.1 uses only this information. Indeed by Remark 2.2 such
a growth condition implies all the estimates of Theorem 2.1. We conclude that
the uniqueness theorem for initial data taken in the sense of Lloc(RN) holds for
solutions of variable sign provided (6.5) holds.
7. Solving the Cauchy problem

Consider the Cauchy problem
u EC (0, T; Lloc(RN»nLfoc (0, T; W,!;:(RN») , p>2,

(7.1)
{
Ut - div (lDulp-2Du) = ° in ET, for some T>O
u(·,O) = Uo E Lloc(RN).
As indicated in the firstof(7.1) the initial datum is taken in the sense of Lloc(RN).
By Theorem 6.1 and Remark 6.1 there is at most one solution to (7.1) within the
class of functions u satisfying
7. Solving the Cauchy problem 331
(7.2) Ilullr,T-£ < 00 for some EE (0, T).

Existence of a solution satisfying (7.2) can be established if the initial datum U o
satisfies the growth condition
(7.3)
- f
Iluolir = :~~
luo(x)1
P"/(p-2) dx<oo, for some r > O.
Kp
Since U o ELloc(RN) if Iluollir is finite for some r >0, it is finite for all r>O.
THEOREM 7.1. Let U o satisfy (7.3) for some r > O. There exists a constant 'Y. =
'Y.(N,p) such that defining
(7.4)
there exists a unique solution u to (7.1) in ET. Moreover u satisfies (7.2) for all
eE(O, T) and the estimates (2.2)-(2.4) of Theorem 2.1.
Remark 7.1. This is an existence theorem local in time and the largest existence
time is estimated by (7.4). The functional dependence in (7.4) is optimal as shown
by the following explicit solution.
1'(x,t)= { A ( -T- )~ + (p---2) ..\ _~ ( Ixl )p!r}~

p=-r --
P
,
T-t P T-t
where A and T are two positive parameters. By direct calculation we have

~
~111>(.,O)llr = ~ (P;2)P- ("\T)-~,
where W N is the area of the unit sphere in R N. Therefore 1'( x, t) exists up to the
blow-up time
-<P-2)
T = 'Y. { ~ 1111>(·,OHlr } ,
where
'Y. = ..\-~ (~r-2 (p; 2)P-l
For n= I, 2, ... ,consider the sequence of truncated initial data
( ) = {max{-n;
uo,n x - 0,
min{uo(x);n}}, for Ixl < n
for Ixl ~ n.
It is apparent that for all n= 1, 2, ... ,
(7.5)
Consider also the family of approximating problems

Un,t - div IDUnl p - 2 Dun = 0, in RN xR+

{
un(·,O) = uo,n·
Since uo,n are compactly supported in RN, (7.1)n can be uniquely solved as indi-
cated in §12 of Chap. VI. By the maximum principle the solutions Un are bounded
by n. Therefore the quantities
III unlIII r,t -= sup sup

!un(X, 'T)
P>-/( -2) dx
O<.,.<tp>r P
- Kp
are finite for all r, t > o. It follows that the sequence {Un} satisfies (2.2)-(2.4) of
1beorem 2.1. We will tum such n-dependent information into a quantitative sup-
estimate of {un} independent of n. Let x --+ '(x) be the standard cutoff function
in K 2p • Then (7.l)n implies
We divide by p>'/ (p- 2) and take the supremum over all p> r. Taking into account
(7.5) and (2.3) this gives
for two constants 'Yi ='Yi(N), i=O, 1. Let tn be defined by
P-2) 1/>. 1
'Yl ( tn IIlunllr,t = 2·
Then from (7.6) for all t E (0, t n )
IIlunllr,t ~ 2'Yo Iluolir.

This implies that tn ~ Tr for all n = I, 2, ... " where Tr is defined by
We summarise:
LEMMA 7.1. Let {Un} be the sequence of the approximating solutions (7.1)n.
There exists a constants 'Y = 'Y(N,p) and 'Y. ='Y.(N,p) independent ofn. such
that
(7.7)
where
(7.8)
Given such an estimate, the Cauchy problem (7.1) can be solved by a standard
limiting process. Indeed by Theorem 2.1 the sequences
{ -f)
Un } , i = 1,2, ... ,N,
f)xi nEN
are locally equibounded and equi-HOlder continuous in RN x (O, Tr). This gives
the existence of a unique solution in ETr • The largest time of existence can be
calculated from (7.8) by letting r -+ 00. In particular the solution to (7.1) is global
in time if
.
lim sup
p>r
f'-tOO
j uo{x)
>./( -2) dx = O.
P P
Kp
Theorem 2.1 is taken from [41]. A weaker version of (2.2) in I-space dimension
is due to Kalashnikov [58]. It is remarkable that in (2.4) one can also control the
behaviour of the space-gradient IDul as Ixl-+ 00. Since IDul 2 is a non-negative
subsolution of a porous medium-type equation (see (1.8) of Chap. IX) the same
techniques yield a version of (2.2) for such degenerate p.d.e. The analog of (2.2)
for the porous medium equation is due to Benilan-Crandall-Pierre [10] in the
context of an existence theorem. A rather general version is in [4]. Perhaps the
most relevant estimate of Theorem 2.1 is the integral gradient bound (2.3) proved
in [41]. A version of such a local bound, for the porous medium equation is in [4]
and reads
jlDuml dxdr ~ -yt /"p1+w!=r "lu"I!;.:~l,

1 K. = N{m - 1) + 2,
Kp
where -y=-y{N, m) and
IIIulllr,T-E
- sup sup j
= u(x,t)
,./(m-l)dx.
O<tST~p>r p
Kp
The estimate holds for small time intervals and for general non-linearities. We
refer to [4] for details. There is no analog of (2.4) for the porous medium equation.
Theorems 4.1 is taken from [41]. The analog for the porous medium equations is
in [6] and for general non-linearities [4]. It would be desirable to have a version
of the uniqueness Theorem 6.1 for initial data measures. This would parallel the
analogous theory for the heat equation.
XII
Non-negative solutions in E T .
The case 1<p<2
1. Introduction
We will investigate the structure of non-negative solutions in the strip ET of the

singular p.d.e.
(1.1) Ut - div IDulp-2 Du = 0, I<p<2.
A striking feature of these singular equations is that, unlike the degenerate case
p>2, non-negative solutions of (1.1) are not restricted by any 'growth condition'
as Ixl- 00. Nevertheless they have initial traces that are Radon measures. More-
over they are unique whenever the initial traces are in Lloc{RN ). Accordingly, the
Cauchy problem for (1.1) associated with an initial datum-
(1.2) U o ~O,
is uniquely solvable, regardless of the behaviour of x-uo(x) as Ixl-oo.

The case 1 2, both in terms
of results and techniques. The main difference stems from the fact that, unlike the
degenerate case, solutions of (1.1) are not, in general, locally bounded. In a precise
way, if
2N
(1.3) and P>-N
+r '
1. Inttoduction 335
then the solution '1£ of (1.1)-(1.2) belongs to Lroc{ST) , "It> O. This is the content
of Theorem 5.1 of Chap. V. In §13 we will give a counterexample that shows that
if '1£0 violates (1.3), then '1£ ¢ L~c{ET). The basic formal energy estimate for (1.1)
is
VO<8<t~T, VKp
t
(1.4) sup ju 2 {x, r) dx + f flDulPdxdr
s<r<t
- - Kp
11
sKp
Thus ifu e L~oc{ET), the left hand side of (1.4) is finite and IDul e Lfoc{ET)'
However if '1£0 e Ltoc{RN), there is no a priori information to guarantee that
(1.5)
We have spoken oholutions of (1.1); however if (1.5) fails, one of the main prob-
lems is to make precise what it is meant by solution. Thus the starting point of the
theory is to give a precise meaning to Du to make sense out of (1.1). The previous
remarks suggest that IDul might fail to be in Lfoc{ET ), roughly speaking at those
points where '1£ is unbounded. Motivated by these remarks, we have given a novel
formulation of non-negative weak solutions. Such solutions are 'regular' in the
sense that the truncations
(1.6) Vk > 0, Uk = min{u, k},
satisfy
(1.7)
Then (1.1) can be interpreted weakly against testing functions that vanish 'when-
ever '1£ is large'. A suitable choice of such testing functions is
(~- '1£)+ == max{(~ - u);O}, ~ e C~{ET); ET'
The notion is introduced and discussed §§2 and 3. We prove that these solutions
coincide with the distributional ones if (1.5) holds and that the truncations Uk are
distributional super-SOlutions of (1.1) Vk > O. We derive a spectrum of properties
of such local weak solutions, regardless of their initial datum. In particular we
investigate the behaviour of DUk as k -+ 00. A relevant fact is the estimate
(1.8) frlDulP-l dxdr

11
~ "Y s<r<t
sup fu{x, r) dx + "Y (t -"s) ~ ,
P
BKp - - K2P
VO<8<t~T, VK2p,
336 XII. Non-negative solutions in E r . The case I <p<2
where A=N(p- 2) + P and 'Y='Y(N,p). We remark that in Chap. XI an estimate

of the local integral nonn of IDuI P- 1 was crucial to establish the existence of
initial traces. In the singular case 1 < p < 2 it is precisely (1.8) that pennits one to
prove an integral Harnack-type inequality, which in turns implies the existence of
initial traces. The estimate (1.8) is essential also for the solvability of the Cauchy
problem. A solution to (l.l )-(1.2) is constructed by using the increasing sequence
{ u o,n} of approximating initial data
(1.9) Uo,n = min{u o ; n}, n = 1,2, ... ,

and solving the approximating problems
Un E C (0, T; L~oc(RN»nLP (O,T; WI~:(RN») ,

(1.10) {
Un,t - div IDu n IP- 2 DUn = 0, in ET,
un(·,O) = uo.n , in the sense of Lloc(RN).
The comparison principle and (1.8) yield the Lloc(ET) convergence of the approx-
imating solutions {un}. A one-sided bound on uo,n and hence on Un is crucial to
this process in view of the regularising effect of Proposition 6.1 of Chap. VI.
In §5 we show uniqueness of weak solutions if they take their initial datum
in the sense of Lloc(RN). Namely, if U and v solve (1.1) weakly and if
then the difference w = U - v satisfies
for a constant 'Y = 'Y( N, p, q). The theorem follows by letting p -+ 00 after we
choose q so large that N(p - 2) + pq>O.
If, in (1.3), r = 1 and p > J~ l' the existence and uniqueness theory remains
valid if U o E Lloc (RN) with no sign restriction. Indeed in such a case the sequences
are locally equibounded and equi-HOlder continuous in ET.

If 1 < p < J~2' the singular equation (1.1) is not fully understood. For ex-
ample it would be of interest to investigate questions of existence and uniqueness
for the Cauchy problem (1.1 )-( 1.2) if the initial datum is a measure JL. Finally, we
notice that all the results of this chapter hold true for equations of the type
N
Ut - L(lux;IP-2 ux;)x, =0 in ET.
i=l
2. Weak solutions 337
2. Weak solutions
A measurable function U : ET -+ R + is a local weak solution of (1.1) in ET if
(2.1) uEC(O,T:Lloc(RN )), !DUk!ELfoc(ET), :tUkELloc(ET)
for all k>O and'v'ep E C:'(E T ),
(2.2) !!{Ut(ep - u)+ + IDuIP- 2 DuD(ep - u)+}dxdr = o.

ET
Introduce the spaces
(2.3)
(2.4) XloC (ET) == {ep E Xloc(ET) !ep(x, t) = 0, 'v'lxl > p} .

'v't E (0, T), for some p >0
o
By density, (2.2) holds for all ep E X loc (ET). We denote with S the set of all
non-negative local weak solutions of (1.1) in ET.
LEMMA 2.1. LetuES. Then 'v'1/JEXloc(ET) and'v'l1EC:'(ET).
(2.5) !!{Ut(1/J - U)+l1 + IDul p - 2 DuD[(1/J - U)+l1)} dxdr = o.

ET
PROOF: Let IC c IC' be compact subsets of ET such that dist (8IC, 8IC') = d > 0
and let (E C:'(IC') be such that 0 ~ (:51 and (== Ion IC. Choose 1/J E Xloc(ET)
and in (2.2) take
where
(2.6) 11 E C;:"(IC) and k= 111/Jlloo,K:/.

We have a.e. in IC'\IC
(ep - u)+ = ((1/J - U)+l1 + Uk( - u)+

= (Uk(-U)+ =0.
Moreover
(ep - u)+ = ((1/J - U)+l1 + Uk - u)+, a.e. in IC.
This vanishes unless U< 1/J. In such a case, Uk = U and
a.e. in IC.
338 XII. Non-negative solutions in ET. The case I <p<2
We conclude that this holds a.e. in I:T and (2.5) follows.

Let (f E (0, 1) and let x - «x) denote the standard cutoff function in K p that
equals one on K tTp , (f E (0, 1). By density. (2.5) implies
(2.7) 'Vt/J E X'oc(I: T ), 'VO<s<t~T,

t
j j {Ut(t/J - u)+(P + IDul p- 2DuD[(t/J - u)+(PJ} dxdT = O.

BRN
Conversely. if t/J E C~ (I:T ). we may write (2.7) for s < t such that supp{ t/J} C
RN x (s,t). By taking (so that p>2diam(supp{t/J}). we obtain (2.2). We con-
clude that the fonnulations (2.2), (2.5) and (2.7) are equivalent.
LEMMA 2.2. Let UES satisfy
Then
Ut - div IDul p- 2Du = 0 in 1>'(I:T).
PROOF: In (2.5) take t/J=un + 1 E X'oc(I:T). nEN. We obtain 'VTJE C~(I:T)
j j{UtTJ + IDulp-2 DuDTJ}(un - u + l)+dxdT = j j IDuI P'1 dxdT .
I:T I:Tn[n<u<n+1!
Since IDul E Lfoc(I:T ). the right-hand side tends to zero as n- 00. The left-hand
side converges to
j j {UtTJ + IDuI P- 2 DuDTJ} dxdT = O.

I:T
LEMMA 2.3. Let U E S. Then/or all k > 0, Uk is a distributional super-solution

0/{1.1) in I: T .
PROOF: Fix k>O and Q,eE(O, I), and in (2.5) take
t/J = Uk + [(k - u)+ + el o E X'oc(I:T)

to obtain 'VTJEC~(I:T). TJ~O
jj{UtTJ + IDuI P - 2 DuDTJ}(t/J - u)+dxdT = jjlDulP'1 dXdT

I:T I:Tn(k<UO/I]
+ Q j jIDUkIP[(k - u)+ + elo-l'1dxdT ~ O.

I:T
First we let e-O as Q E (0,1) remains fixed. Since

2. Weak solutions 339
(1/J - u)+ - (k - u)+

we deduce
j j{u'TI + IDul,,-2 DuD.,.,}(k - u)+dxdr ~ 0, Voe(o, 1).

I:T
Now letting 0 - 0 gives for every non-negative TI e Or:'(I:T )
(2.8) jj {! UkTl + IDUkl,,-2DUk.DTI} dxdr ~ O.

I:T
The next proposition pennits a large class of testing functions in (2.5). H ko > 0,
let F(ko ) denote the set of all the Lipschitz-continuous functions f : R+ - R
such that f(k)=O, Vk>ko, and set
PROPOSITION 2.1. Let u e S. Then 'If e F and VTI e Or:' (I:T ).
j j{Utf(u)TI + IDul,,-2 Du·D(f(u)TI)}dxdr = o.

I:T
PROOF: Assume first that f e 0 2 (0,00). Write (2.5) for 1/J = k, multiply it by
- f" (k) and integrate in dk over (0, 00). By interchanging the order of integration
with the aid of Fubini's theorem we obtain
00
j j {U'Tlj!"(k)(k - u)dk
I:T U
+ IDuIP-'D,..D [~ P"(k)(k - U)dk] }dzdT = o.
Since
00
j!"(k)(k - u)dk = f(u),

U
the assertion follows for f e 0 2(0,00). TIle general case is proved by approxima-
tion.
340 XU. Non-negative solutions in Er. The case 1<p<2
3. Estimating IDul
LEMMA 3.1. There exists a constant 'Y = 'Y(N, p) such that
Yk> 0, Yp > 0, YO<s<t::5T, Yu E S
t; s) .
t
j jlDUkl PdxdT ::5 'Y kPlKpl (k2-P +

BRN
PROOF: Let ( be the standard cutoff function in K2p. Then from (2.7) with 1/J =k
t t
j jIDUk/P(PdxdT::5 p jj/DUk/P-1(P-l(k - u)+ID(ldxdT

BRN SRN
t
+~ JJ:T (k - u)!(PdxdT
BRN
p; J
t
::5 1 jIDUkIP(PdXdT
SRN
J
t
+ pp-l fik - u)~ID(IPdxdT

SRN
+ ~ fik - u)!(Pdx.
RNX{t}
For all 0< s < t::5T and all p>O set
(3.1) MB,t(P) = sup fU(X,T)dx.

1'e(B,t)
Kp
LEMMA 3.2. LetueS.ThenYo:e(O,p-l)
I
IDuP-~-Q e Lroc(I:T ),
tmd there exists a constant 'Y='Y(N,p) such thatYO< s<t::5T and/or all p>O
PROOF: Fix k > 0 and e e (0,1). and in (2.8) take '1 = (P1/J-Q. where ( is the
standard cutoff function in K 2p and
3. Estimating IDul 341
u>e
u ~e.
We obtain
t
o /JIDuIPu-Q-l(Px[e<u<kj dxdr
BRN
t
~ P //IDuIP-lu-Q(P-IID(lx[e<u<kjdXdT
sRN
(3.3)
t
+ p / /IDu~IP-le-Q(P-IID(1 dxdr
sRN
By Young's inequality, the first integral on the right-hand side is majorised by

t
i //IDuI Pu-Q-1(Px[e < u < kjdxdr
sRN
By virtue of Lemma 3.1 the second integral tends to zero as e -+ 0 at the rate of
eP - 1 - Q • Combining these calculations we deduce
t
o j/IDuIPu-Q-lx[e<u<kl dxdr
BKp
~ O(eP-1-Q) + oP
"(_1 { (sup
'TE(s,t)
ju(x, T)dx) l-Q (2p)QN
K2p
(3.4)
+ (~) (sup
pi' 'TE(s,t)
ju(x,r)dx)P-I-Q (2P)N(2-P+Q)}
K2p
~ 0:-1{[M ,t(2P)j1-Q + (t ; s) [MB,t(2P)jP-I-Q} pN

s
+ O(eP - 1- Q ).
342 XII. Non-negative solutions in ET. The case I <p<2
If
c S)
~ ~ [Ms ,t(2p)]2-",
the quantity in braces on the rightmost side of (3.4) is majorised by [Ms ,t(2p)j1-0<.
Otherwise it is majorised by
[Ms ,t(2p)]1-0< + ( -;:;-

t-S)~ .
In either case
t
(3.5) JJIDul"u-(O<+l)x.[e<u<k]dxdT
sKp
,; O(e,-(.+1») + :, pH { M •.• (2p) + (t;.) ~ }

1-. ,
and the lemma follows by letting first e -+ 0 and then k -+ 00.

Estimate (3.2) deteriorates as Q -+ O. TIle next lemma gives some information
for the case Q =O.
LEMMA 3.3. Let uES. There exists 'Y='Y(N,p) such that
\t'O<s<t~T, \t'p > 0, \t'n ~ 1
t
J
sKp
f IDu¥ I"x.[n < u< n + 1] dxdT
( + n1) [M ,t(2p) + (t-S)~l

~ 'Yin 1 s -;:;- .
PROOF: In (2.8) we take ,,= (1/J, where ( is standard cutoff function

the in K 2p
and 1/J=ln+ (~).Here
= {n, if 0 < n
u(n) u ~
u, >ifu n.
We get
t
(3.6) JJIDul"u-1x.[n<u<n + 1] dxdT

B Kp
~ jJ:T un+lln+ (:~)1) ("dxdT

SK2p
+ ~ JJI Dul,,-lln+ (:~)1) dxdT = I~l) + ~ I~2).

SK2p
3. Estimating IDul 343
Setting, for simplicity of notation,

A = K 2p x (s,t),
we have
I~2) ~ln (1 +~) ff IDuIP-1u-(Ot:+l) (P;ll u(Ot:+1) (P;1) dxdT
An[u<n+l]
$ ~ In(,+ ;) ([.rID.."¥' I'dxdT) · Vi

.P=l
U<Q +l)(p-l) dxdT)'

.1
If Q E (0, P - 1) is so small that (Q + 1) (p - 1) ~ I, both integrals in parentheses

are finite. Taking Lemma 3.2 into account in estimating the first integral we have
~ I~2) ~ 1pN In (1 + ~) ,
.1
[Ms,t(2 P) + (t ~ 8) ,.!;; ] (l-Ot:)¥ (~ j / U(Ot:+1HP_l)dxdT) P
SK2p
The last integral above is estimated by
(~ ft
/U(Ot:+lHP-l) (x, T) dxdT )

.1
SK2p
Therefore
~ t.') $ ~pNIn (I+~) [M••• (2P) + (' ;;;:t].

As for I~l) we write
I~l) = ff In (1 +~) +ff In+ (n: 1)

An[u<n]
Ut (PdxdT
An[u>n]
Ut (PdxdT
= ff! In (1 +~)Un +ff In+ (n: 1)

(PdxdT :T u(n) (Pdxdr
A A
~1pNln(1+~)Ms,t(2P)+ ff :T (jln+ (n;')d{\ ("dxdT.

An) +
344 XII. Non-negative solutions in ~T' The case 1<p<2
COROLLARY 3.1. Let u E S and define

u{x,t)
(x,t) -> z(x,t) = / (~lnl+E~)-;dx, E E (O,p - 1).

e
Then IDzl E Lfoc(ET) and there exists "Y="Y(N,p) such that'v'O < s< t ~ T and
'v'p>O,
PROOF: Divide both sides of the inequality of Lemma 3.3 by lnl+E n, and add
over all n=2,3, ...
The estimate (3.7) deteriorates as E-O. The following corollary gives some
information in the case E = O.
COROLLARY 3.2. Let uES. Then 'v'O<s<t$T. andforall C> 1.
t
lim
k .....
ffIDuIP-I-l-
oo}} U nu
X[k<u<Ckjdxdr = O.
s Kp
PROOF: Without loss of generality we may assume that k and Ck are positive
integers. Divide both sides of the inequality of Lemma 3.3 by In n and add for
n=k, k + 1, ... ,Ck. This gives
t
//IDunUln u)-lX[k<u<Ckj dxdr

s Kp
~"Y{lnlnCk-Inlnk) [ M s ,t(2p)+ ( t - Ii'S)~l
= "YIn ( 1 + InC)
Ink [ M s •t (2p) + (tIi'S) ~l .
4. The weak Harnack inequality and initial traces

In the definition of local weak solutions of (2.1) in ET, no reference has been
made to initial data. We will show that each u E S has a unique non-negative
u-finite Borel measure J.I. as the initial trace. The existence of such a trace will be
a consequence of the following weak Harnack-type estimate.
4. The weak Harnack inequality and initial traces 345
THEOREM 4.1. Let U E S. There exists "I = "I(N,p), such that "10 < s < t $. T
andVp>O
(4.1) sup
TE(s,t)
!
Kp
u(x, r)dx $. "I
K2p
j u(x, t)dx + "I ( t-S)~
P
-.>.- ,
A = N(p - 2) + p.
The uniqueness of the initial trace J.I. relies on the next gradient estimates.
LEMMA 4.1. Let uES. There exists a constant "I="I(N,p) such that
VO<s<t$.T, Vp> 0, "10' E (0,1), "Iv> 0,
(4.2) h
1 IDulp-1dxdr
p}} $. "I (t-S)~
7-
8 Kp
:!iE..=..ll
+ "I (t -/) *{
P
sup ju(x, r)
S<T<t
dx} P
K2p
Moreover
(4.3) -
p
Ih'lt IDulp-1dxdr $. "I sup
S<T<t
j u(x, r) dx + "I (t-S)~
P
-.>.- .
S Kp K2p
PROOF: The proof is the same as that of Propositions 4.1 and 4.2 of Chap. VII.
The only difference is that instead of working with the solution u we work with
the truncations Uk and use the fact that these are supersolutions. In (2.8) we take
the testing functions
t/J = (t -r)*(uk + v)l-: E X'oc(E T ),

where v> 0 is arbitrary. We proceed as in Chap. VII and then let k ~ 00.
THEOREM 4.2. Every u E S has a unique Radon measure J.I. as initial trace at
t=O.
PROOF: From Theorem 4.1 it follows that V'1EC~(RN), the net
{ jU(r)'1dx}
RN TE(O,t)
is equibounded, with bound depending only upon I '1 II oo,RN • A subnet indexed
with {r'} converges to a Radon measure 1', in the sense of the measures, i.e.,
346 XII. Non-negative solutions in ET. The case 1<p<2
Suppose now that there exist another subnet, indexed with {r"} and a Radon mea-
sure jJ., such that
We will prove that J.I. == jJ.. Let u E (0,1) and write (2.8) with 1/J == 1 and ( the
standard cutoff function in K(l+u)p. Letting k -+ 00, standard calculations give
VO<s<tST
t
(4.4) jU(S)dXS j u(t)dx+ :pjjIDuIP-1dxdr.

Kp K(1+")p BK2p
We estimate the last tenn by using (4.2) and let s '\. 0 along r' while t> 0 remains
fixed. Then we let t '\. 0 along the net r" to get
Since uE (0;1) is arbitrary, interchanging the role of J.I. and jJ. proves the theorem.
5. The uniqueness theorem

Let S· denote the subclass of S of those non-negative local weak solutions of (1.1)
in ET, satisfying
(5.1)
for some "'( = ",((N,p, t), Vk E R+,
(5.2) lim frJf

k-oo
IDuI P.!. dxdr = 0,
u
K:n[k<u<Ck]
for every compact subset K:. C ET and for all C> 1. In section §§8-12 we will
construct solutions of the Cauchy problem (1.1)-(1.2) that satisfy both (5.1) and
(5.2); therefore S· is not empty. Corollary 3.2 suggests that (5.2) is almost satisfied
s. The uniqueness theorem 347
by all solutions in S. It would be of interest to know whether the inclusion S* c S

is strict. .
THEOREM 5.1. Let Ul. U2 E S* satisfy
5-(i). Preliminaries
LEMMA 5.1. LetuEs*. Then/oraIlO<s<t'5,T, Vp>O, VC> 1,
t
lim ff1ut/x[k<u<CkJ dxdr

1c-oojj
= O.
BKp
PROOF: Consider (2.8) written for Uk replaced by UCk. against testing functions
7J = ( In (k/2w k,C)
where x-(x) is the standard cutoff function in K2p. and
!k, o '5, u '5, !k

Wk,C == { u, !k < u < Ck
Ck, u~Ck.
It follows from these definitions that 7J '5, 0 a.e. in ET and 7J = 0 a.e. on the set
[O<u'5, !kJ. By calculation from (2.8) we obtain
t t
(5.3) I I ! uC1c7Jdxdr '5, IIIDU1P;X(k/2<u<CkJdxdr

SK2p SK2p
+In2C IIIDUCkIJl-IX(U > k/211D(ldxdr.

BK2p
The fll'St integral on the right hand side of (5.3) tends to zero as k - 00 by virtue
of (5.2). We estimate the second integral. formally. by
348 XII. Non-negative solutions in ~T. The case I <p<2
=In-2C
p
- ljlD
t
UCk
,
IP-l U- (<>+I)(p-l)
P U(<>+I)(p-l)
P [
XU> k/2]dxdr
SK2p
cl
~ In;c (p _~ _0) p-l (if,Ducr 1PdXdr) P
SK2p
X (if u(O+!)(P-l)X[U > k/2]dxdr) P

SK2p
If we choose 0 E (O,p - 1) so small that (0 + l)(p - 1) :5 1, the estimate is

rigorous and the last tenn in the right hand side of (5.3) tends to zero as k --+ 00,
since UE LJoc(ET)' These remarks in (5.3) give
t
ffUt In (2W:,c) 'X[Ut < 0] X [(k/2) <u<Ck] dxdr

SK2p
:5 ff Ut lin 2W:,c I'X[Ut ~ O]X[u > k/2] dxdr + 0 (~) .

SK2p
In view of the definition of Wk,C this gives in turn

t t
ff1utlX[k<U<Ck]dxdr ~ 'YffUtX[Ut > O]X[u > k/2]dxdr+O (~).

SK2p SK2p
1be last integral is estimated by means of (5.1) and the lemma follows.
Remark 5.1. The assertion of the lemma is trivial if Ut E LJoc{ET)'
We give next a weak fonnulation for the difference of two solutions U1, U2.
First we recall that, by Lemma 2.3, the truncated function
ifO<U2<k
if u2~k
is a distributional supersolution of (1.1), Vk > O. We write (2.8) for U2,k against

the testing functions
S. The uniqueness rheorem 349
where ( is a non-negative piecewise smooth cutoff function in K(1+a)p, 0' E (0, 1),
such that
(5.4) ( == 1 on Kp and ID(I:5 1/O'p.
In view of the definition of X 10c (ET) and the regularity properties (2.1) of Ui, i =
1,2, such a choice of testing function is admissible, modulo a density argument.
On the other hand the weak formulation (2.7) of Ul holds against the same testing
functions. Therefore setting
kER+,
we obtain by difference the weak formulation

t
(5.5) / / {!W(k)(1/J - ulh(P + J kD(1/J - Uil+(p} dxdr

SK(l+,,)p
:5 -p / /Jk(1/J - ud+(p-l D(dxdr \:/1/J E Xloc(E T ),

SK(l+O')P
where
Jk == IDuIlp-2Dul -IDu2,klp-2Du2,k
1 .
= / ~ {ID (~Ul + (1 - ~)U2,kW-2 D (~Ul + (1 - ~)U2,k) } d{

o
~ (iID({Ul + (1 - O....)IP-'d{) Ow(')
+ (p - 2) ( /ID(~Ul + (1 - ~)U2,k)IP-4
o
XD(~Ul + (1 - ~)U2,k)(~Ul + (1- ~)U2,k)zjd{ )W(k),Zj'
Set also
1
Ao == /ID(~Ul + (1 - ~)U2,k)IP-2d{.
o
LEMMA 5.2. Ao:5P~1IDw(k)IP-2.
PROOF: If IDu2,kl ~ IDw(k)l, we have
350 xu. Non-negative solutions in ET. The case I <p<2
ID(eUI + (1 - e)U2,1c)1 = IDu2,1c + eDW(1c) I
~ IIDu2,1cI- eI DW(1c)11
~ (1 - e)IDw(1c) I·
1berefore
A. $ (/<1 -(~'d{) IDw(.r'

= ~IIDW(1c)IP-2.
p-
where eoE(O, 1) is defmed by
eo =_ ID
IDu2,1c I ( )
W(1c)
leo, 1 .
From the definitions set forth and Lemma 5.2 we have
JlcDw(lc) ~ (p - I)AoIDw(Ic)12,
(5.6) {
IJlcl S AoIDw(1c)1 s p~IIDw(1c)IP-I.
In what follows we will use these inequalities without specific mention.
6. An auxiliary proposition
PROPOSITION 6.1. Let Ui E S· ,i = 1, 2, satisfy
wet) == (UI - U2)(t) - 0 in Lloc(RN) as t - O.

6. An auxiliary proposition 3S I
Then W E Loo (0, Tj Lfoc(RN ») , Vq E [1,00). Moreover Vq ~ 1 there exists a

constant ",(=",(N,p, q), such that
t
(6.1) jlw(tWdX ~ (0';)" j jlwl9+(,,-2)dxdT,

Kp OK(1+O')p
for all p>Oandforall O'E (0,1).

The proof is based on an iteration procedure and uses recursive inequalities
obtained from (5.5) with suitable choices of testing functions 1/1.
6-(;). Testingfunctions in (5.5)

For h>O, set
~O
{:/i)
ifW(k)
(6.2) W/i).h" (UI - ", •• )t - ifw(k) < h
ifw(k) ~ h
and in (5.5) consider the testing function
(6.3) 1/1 == Ul,l/r: + ~ (w(t),n + a(w(t),m +

e) e) b E X'oc(E T ) ,
where
eE(O,I), a,b>O, n,mENj n>m+1.
We obtain
(6.4) j W(k) (1/1 - ul)+("dx - j W(k)(1/1 - ud+("dx

RNX{t} RNX{B}
t t
- jjW(k)!(1/1-ud+("dxdT+ jjJkD(1/1- u d+("dXdT

BRN BRN
t
::; -p j jJ k (1/1 - Ud+(,,-l D(dxdT.
BRN
In using 1/1 as a testing function in (6.4) we keep in mind that the truncated functions
Ui,h, i = 1,2, Vh > 0, are regular in the sense of (2.1). In particular the first two
integrals on the left hand side of (6:4) are well defined V0 < 8 < t ~ T. We will
eliminate the parameters e,k,8,n,m by letting e-O, k-oo, 8-0, n,m-oo
in the indicated order.
352 XII. Non-negative solutions in I:r. The case I <p<2
6-(U). The limit as E-+O

We multiply both sides of (6.4) by E and let E -+ 0, while k, 8, n, m remain
fixed. From the definition (6.3) of t/J it follows that \;IrE (0, T] the net [W(k)(Et/J-
Eud+](" r), is equiboundt:d in Lloc(RN). Moreover it converges to
[W(k) (w~).nf (w~).m) b] (., r) a.e. K 2p ,
and it is majorised a.e. in RN by
W(k) (w~).n + If (w~).m + I)b (·,r) ELtoc(RN).

1berefore for all 0 < r ~ T, as E -+ 0
(6.5) f
RNx{r}
W(k)(t/J - ud+(?dx -+ f W(k)
RNx{r}
(w~).nf (w~).m)b (Pdx.
This determines the limit for the first two terms on the left hand side of (6.4). To
examine the remaining terms we let 'iii, i = 1, 2, be arbitrarily selected but fixed
representatives out of the equivalence classes Ui, define iii, iii(k) accordingly, and
let
Next
t
ff
LE == -E
sRN
W(k) :r (t/J - ud+(Pdxdr
= ff w~).n (w~).n +Er- 1(W~).m

-a w~).n(PX(gE)dxdr +E)b!
SRN
t
ff w~).m(w~).n +Er(W~).m +E)b-l ! w~).m(PX(gE)dxdr

-b
sRN
t
-ff BRN
w(k)(1 - EUlhx(FE)(Pdxdr
== L~l) + L~2) + L~3) .

We claim that L~3) - 0 as E -+ O. Indeed
t
IL~3)1::; JJeIW(k)II!UIIX(Fe)dxdr.
aK2p
On the set Fe we have
1
-
e
::; Ul ::; -
1
e
1
+ -(n
e
+ 1)a+b == -,
'Y
e
eIW(k)/ ::; 'Y a.e. Fe·
Therefore
(6.6)
and the assertion follows from Lemma 5.1.

Since k, n, m are fixed, the integrands in L~i) •i = 1, 2, are in Lloc(ET) uni-
fonnly in e. Moreover they have a.e. limits that are in Lloc(ET) and their abso-
lute value is majorised almost everywhere in ET, unifonnly in e, by functions in
Ltoc(ET). Therefore as e-+O
(6.7)
:: -a: 1 jJ! (W~),n)a+1 (W~),m)b (,Pdxdr

aRN
t
-b! 1 !
JJ (w~),n) a (wtk),m) b+l (?dxdr.
aRN
Since n>m + 1, .
+ )a Or{) (+
(w(k),n w(k),m )b+l = (wtk),m )a Or
{) (wtk),m )b+l
-- a +b+b +1 1 Or
{) (
w+(k),m )a+b+l ' a.e.ET.
We obtain from (6.7)
(6.7') c:: - a: 1 J (wtk),n) a+l (wtk),m) b(Pdx

RNX{t}
b
- (a + l)(a + b + 1)
J( w~),m )a+b+ 1 (Pdx
RNX{t} .
+ a: 1 J (wtk),n) a+1 (w~),m) b(Pdx

RNx{a}
+ (a + 1)(: + b + 1) J (wtk),mf+b+ 1 (Pdx.

RNx{a}
354 XII. Non-negative solutions in ET. The case 1<p<2
We combine this with (6.5) and conclude that the sum of the first three tenns on
the left hand side of (6.4) has a limit. as e - 0, that is minorised by
(6.8) 1
a + b+ 1
I(w~).m )4+/1+1
("dx
aNx{t}
- a + b+ 1
1 I (w~) w~).n
)4+/1
("dx.
aNx{s}
We tum to estimate below the lim-inf as e - 0 of the last integral on the left hand
side of (6.4):
t
e IIJIcD(1/J - ud+("dxdr
saN
t
= a I I JIcDw~).n ( w~).n + e) 4-1

saN
x(w~).m + e) /I ("x(g,Jdxdr
+b IIJkDW~).m (w~).n +e)4

saN
/1-1
(6.9)
X ( w~).m + e) ("X(g,;}dxdr
t
+ IIJ kD(1- eud("X(F,;}dxdr

saN
t
~ a(p - 1) I I AoIDw~).nI2 (w~).n + e) 4-1

saN
- p ~ 1 I I AoIDw(k)IIDull("x(F~)dxdr
saN
== H~I) + H~2) .
By weak lower semicontinuity
(6.10) lim in! H~I)
~-o
t
~ a(p-1) I IAoIDw~).nI2 (w~).nr-l (w~).mt ("dxdr.

saN
We claim that H~2) - 0 as €- O. Using Lemma 5.2 we have
t
IH~2)1 ::; €C(N,p) jfiDUl - DU2,IcIP-1IDullx(FE)dxdr
aK2p
t t
::; C€ j jIDU1IPx(FE)dXdr + C€ j jIDu2, Ic IPx(FE) dxdr
aK2p SK2p
=
-
H(2) +H(2)
E,l E,2·
Since IDu2,1c1 E LfoAET) the second tenD tends to zero as € - o. As for H!~l
write
where "1= 1 + (n + l)G(m + l)b. This implies. since Ul ES·
t
H!~l::;c(P,n,m)jjIDu~IPx(~ ::;Ul::; ~)dxdr-o as €-o.
aK2p .
We fmally estimate above the lim-sup as € - 0 of the integral on the right-hand

side of (6.4). Using the definition (6.3) of 1/J and (5.6)
(6.11) Ip jjJIc(1/J-Ut}+(P- 1D( dxdr l

saN
t
j AoIDw(lc) I (w~),n +
::; "1 j €) a
aaN
t
+ 'Y j j AoIDw(lc) IWIX(FE)(P-l ID(ldxdr.
saN
The last integral tends to zero as € - O. Indeed it can be majorised by

356 XII. Non-negative solutions in Er. The case 1<p<2
t
(6.12) 'Y !fiDW(k)IP-IX(:Fe)dxdr
SK2p
t
:$ 'Y !!IDUIIP-1x[Ul ~ :ldxdr

t
+ 'Y ! !I DU2,kI P- 1x[Ul ~ :ldxdr,

sK2p
for a constant 'Y = 'Y(P, n, m, a, b). The second integral on the right hand side of
(6.12) tends to zero as e ..... O. since Ul E Lloc(ET). As for the frrst integral. let
oE (O,p - 1) be so small that (0 + 1)(p - 1) < 1. Then
'Y //IDUI!P-IX[UI > :ldxdr

SK2p
t
{{ _ (a+1)(e-l) (a+l)(p-1)
= 'Y JJ IDuI!p-l ul P u1 P X[UI > : ldxdr
sK2p
t
_ {{ p-1-a 1 (a+1)(p-1)
=7 JJ IDu l P IP- U1 P X[UI > : Jdxdr

sK2p
~
<~IIDu P-~-Q II P- 1
- 1 p,K2px(s,t)
(rJJ(u(O:+l)(P-l)X[Ul >
1 •
)P
1 1dxdr
SK2p
---+ 0 as e ..... O.
We examine the lim-sup as e ..... 0 of the first integral on the right-hand side of
(6.11). The numbers k E R + , n E N being fixed. if e is small enough. we have the
inclusion
We write
t
ffAoIDW(k)1 (w~),n +ef (w~),m +e)b (P-1ID(IX(Qe)dxdT

BRN
=ffAoIDW~),nl (w~),n +ef (w~),m +e)b (P-1ID(ldxdT

aRN
(6.13)
t
+f f AoIDw~)I(n + e)a(m + e)b(P-IID(lx[w~) > nlx(Qe)dxdT

BRN
+f f AOIDw~)lea+b(p-IID(IX(Qe)dxdT = K~l) + K~2) + K~3).

BRN
As for K~l) the integrand tends to

AoIDw~),nl(w~),n)a( w~),m)b(P-IID(1 a.e. K2p x (s, t),
in a decreasing way. Therefore
t
K~l) -+ f f AoIDw~),nl(w~),nt(w~),m)b(P-lID(ldxdT.
BRN
The last integral tends to zero as e -+ O. Indeed
The operation Dw~) coincides with the weak derivative of w~) only on those sets
Ai where w~) is bounded by a positive constant i, i.e.,
Dw~)x (At) == Dw~),t.
Since Dw~) is not well defined a.e. in the whole strip ET we estimate K~2) as
follows:
t
K~2) -:; 'Y (m ~ l)b ffiDUl - DU2,kIP-lu~X[Ul > n + U2,k]X(Qe) dxdT

aK2p
t
-:; 'Y (m ~ l)b ffIDUIIP-IU~X[Ul > n]x(Qe) dxdT

aK2p
b t
+ 'Y
(m+l)
Up Jff a
J IDU2,kI P- 1 UIX[UI > n + U2,k]X(Qe) dxdT.
aK2p
358 XII. Non-negative solutions in Er. The case I<p<2
If 0 E (O,p - 1). write

t
JJIDUIIP-IU~X[UI > njx(ge)dxdr
aK2p
t
=JJIDUIIP-IU~ (Q+l~p-l) u~Q+1)('P-ll+"P X[UI > njx(ge)dxdr
aK2p
c.! !
~ 'Y (jjIDu;- :-° 1, /hd) •rjj .\0+1 )(,-1 )+«, xl'l > n]dzdT Y
BK2p ') ~BK2P ')
Choose 0 and a> 0 so small that

(6.14) (0 + l)(P - 1) + ap ~ 1.
Then u~Q+l)(p-l)+ap E Ltoc(ET) and \fEE (0,1)
t
JJIDUIIP-IU~X[UI > njx(ge)dxdr ~ 0 (;).

BK2p
Analogously
t
JfiDU2'kIP-IU~X[Ul > n + U2,k]X(ge)dxdr

aK2p
c.!
x (jj,:'u!,o+I)(,-I)xl'l
BK2p
> n + ....]dzdT I·
)
~ (jfiDu~IPdxdr) ~
'Y
BK2p
We conclude that
provided 0: and a> 0 are chosen so that (6.14) holds. Combining these estimates
and limiting processes as parts of (6.4) we obtain
a+ b1 + 1 J(w(k),m+ )a+b+l
(P dx
RNx{t}
t
+ a(p - 1) JJ AoIDw~),~J (w~),n) a-I (w~),m)" (Pdxdr
BRN
(6.16) $ a+ ~ + 1 Jwt (w~),nr+" (Pdx

RNx{a}
t
+ ~JJAoIDW~),nl (w~),nr (w~),m)" (P-1dxdr

BRN
+-y(m+ 1)"0 (~).
6-(iii). The limits as k--+oo and 8--+0

If n E N and k > 0 are fixed, we let iii{k),n and Dw~),n be arbitrarily selected
but fixed representatives out of the equivalence classes w~),n and Dw~),n and
introduce the sets
where C is the constant appearing in the last integral on the right hand side of
(6.16). This integral is estimated as follows:
360 XII. Non-negative solutions in ~T. The case I <p<2
t
~ I IAoIDw~),nl (w~),nr (w~),m)b (P- 1dxdr
sRN
t
$ ~/IAOIDw~),nl (w~),nr (w~),m)b (P-l x(tddxdr

sRN
t
+ (p ~~)crp I fiDW~),nIP-l (w~),n) (w~),m) b(p-1 X(t2 )dxdr

4
SRN
$
a(p-l)
2
It/ AoIDw~),nl
2(w~),n )4-1 (w~),m )b (Pdxdr
sRN
+ 4PC P
aP-1(p _ l)p(crp)p
1/( t
w+
(k),n
)P-l+4 (w+ (k),m
)b dxdr
•
SRN
We carry this estimate in (6.16), move the integral involving IDw~),nI2 on the
left hand side and discard the resulting non-negative tenn to obtain
(6.17)
-y(p)
+ (crp)P It I( w~),n )P-l+4 (+ w(k),m
)b dxdr
s K(l+")p
+ -y(m + l)bO (~).

We let now k - 00 while B > 0, n, mEN remain fixed. Since w~) - w+ in a
decreasing way we may pass to the limit under the integrals in (6.17) and obtain
the same integral inequality written for w+. In particular the first integral on the
right hand side takes the fonn
(6.18) 1 jW+(W+)4+b(Pdx.
a+b+l n
RNx{s}
Now letting 8-0, the integral in (6.18) tends to zero since it can be majorised by
as B - 0.
These limiting processes yield
j(W:at+b+l (Pdx ~ 'Y(P)(~;)! + I)! t
j(W:;y-1+ a (W:a)b dxdr

(6.19) RNX{t} OK(1+")p
+'Y(a+b+ 1)(m+ l)bO(~).
6-(ivJ. Proof of Proposition 6.1

We let n -+ 00 in (6.19), while mEN remains fixed. The integrand in the
last integral tends to (w+)p-1+a(w~)b a.e. in K(1+CT)p x (0, t) in an increasing
fashion. Moreover if a is so small that
(6.20) p - 1 + a E (0,1),
it is dominated, uniformly in n, by the function
The limit process gives

t
(6.21) j(W~t+b+1 (Pdx ~ 'Y(P)(~;)! + 1) j J(w+)p-l+a(w~)bdXdT.

RNX{t} OK(l+ .. )p
This inequality holds true "1m E N, Vb ~ 0, "10- E (0,1), Vp > O. The positive
number a is fixed, satisfying the restrictions (6.14) and (6.20). The sequence {w~}
increases to w+ a.e. in ET . Therefore as m -+ 00, we may pass to the limit under
the integrals in (6.21) for those b ~ 0 for which
(w+)p-1+ a+b E LtoAET).
If bi ~0 is one such b, letting m -+ 00 we find that
which implies that
(w+)p-1+a+bi+l E LloAE T ), bi +l=bi +2-p>bi .
Let bo ~ 0 be defined by p - 1 + a + bo = 1. Then the previous remarks show that
(w+)p-l+a+b o+i(2- p) == (W+)1+i(2- p) E Ltoc(ET), i = 0,1,2, ....

Interchanging the role of UI and U2 proves the Proposition.
362 xu. Non-negative solutions in ET. The case I <p<2
7. Proof of the uniqueness theorem
From (6.1) by HOlder's inequality, since pE (1, 2)
(7.1)
Let p > 0 be fixed and for n = 1, 2, ... defme
Pn = (~
~
2-i) K= K
p, n P.. , q
n -- 2-(n+1) ,
,=0
Rewrite (7.1) over Kn and Kn+l to obtain
(7.2)
By the interpolation Lemma 4.3 of Chap. I we conclude that for every q E [1,00)
there exists a constant -y=-y(N,p, q), independent of p, such that for all tE (0, T)
(7.3)
To prove the theorem we choose q so large that
N(P - 2) + pq > 0
q
and then, such a q being fixed. we let p-oo in (7.3).
8. Solving the Cauchy problem
We will establish the existence of a unique non-negative solution to the Cauchy

problem (1.1)-(1.2) where "0
is non-negative and merely in L}oc(RN). For n =
1, 2, ... consider the sequence of approximating problems
9. Compacbless in the space variables 363
(8.1)n
I Un
Un
E C (O,Tj
( )_
x,O -
L10c(RN»)nV
/:rUn - div IDun l
U on=
'0
p - 2 DUn
_ {min{uojn}
(0, Tj W,!;:(RN»)
= 0, in ET
for Ixl < n
II
for x ~n.
The initial data are bounded and compactly supported in RN. Therefore the unique
solvability of (8.1)n can be established as indicated in §12 of Chap. VI. Since the
initial data {uo,n}f1S\l form an increasing sequence of functions in Lloc<RN) we
have
(8.2) 't/p>O.
The solution of (1.1 )-( 1.2) will be constructed as the limit of the sequence {u n }f1S\l
in a suitable topology. For this we establish flJ'St some basic compactness of
{un}f1S\l.
LEMMA 8.1. There exists a constant "'( = "'( N, p) independent of n such that for
all t,p>O
MoreoverforallaE(O,p-l),
PROOF: The Lloc-estimate follows from (4.1) with s = 0, and the gradient es-
timate (8.4) is a consequence of (4.2) with s = O. Finally (8.5) is the content of
Lemma 3.2.
9. Compactness in the space variables

LEMMA 9.1. Let a E (0, p - 1) be so small that (a + 1)(P - 1) < 1. There exists
a constant ",(='Y(N,p, a) such that
364 XII. Non-negative solutions in I:T. The case 1<p<2
'VO < t ~ T, 'Vk, 'VC > 1, 'Vn = 1,2, ... ,

t
(9.1) / /IDunIPu;;tx.lk<un <Ck] dxdT

OKp
< 'Vk
_ I
_(l-<O+lHP-l»)
p P
No2=.!
p
(t )*{/
-
pP u 0 dx +
(- )~}P_Q~P_l)
t
p>'
K3p
+ InC / uoX[Uo > k] dx.

K2P
The constant 'Y( 0) /00 as either 0'" 0 or 0 / p - l.

PROOF: We drop the subscript n for simplicity of notation. If C > 1 is fixed, let
u~2 == {:
ifO<u~k
if k < u < Ck
Ck if u? Ck
and in the weak formulation of (8.1)n, take the testing function
(k»)
( In U~k ((x),
where x -+ (( x) is the standard cutoff function in K 2p that equals one on K p. We

obtain
t t
/ f!DUIP~X[k<U<Ck]dXdT:5 2; / !IDuIP-1X[U > kJdxdT

OK p OK3p
-j/ (i
OKb
:T
k
In min{~jCk} de)
+
((x)dxdT == G~l) + G~2).
Let 0 be any positive number satisfying
·OE(O,p-l) and (0+ l)(P-l)<1.
Then by virtue of (8.5)

9. Compactness in the space variables 36S
G{l)
k ~-
2')' jjlD IP-l -
t
U U
(o+l)(p-l)
pup
(o+l)(p-l)
xu>kdxdT
[ j
p
OK2p
')'(a,p) jilDu
=--
t
PUP I
.-1-0 P- 1 (0+1)(,-1) [
XU> kjdxdT
p
OK2p
(0+1)(,-1)
X{ sup
O<T'<t
jU(X, T)dx} P
K2p
1-(0+1)(,-1)
X { sup
O<T'<t
K2p
J X[U > kj dx} P
~')'(a,p)(~);{Juodx+(:~)~} P
K4p
1-(o+l)(p-l)
SUP
{ O<T'<t
jX[U> kj dx} P
K2p
366 XII. Non-negative solutions in E1. The case 1<p<2
The last step follows by use of (8.3). Using it again we obtain
1berefore
As for G~2) it is estimated above by
10. Compactness in the t variable
LEMMA 10.1. Let a E (O,p - 1). There exists a constant "{ = ,,{(N,p, a) such
that
VO<s<t~T, VB ~ a + 1, Vn = 1,2, ... ,
,,{p-aN
+ -s -
{f ( )~}l-a
uodx+ ~
P
t
K2p
The constant ,,{(N,p,a) /00 as either a'\.O or a/(p - 1).

PROOF: Let 0 < s < t ~ T and p> 0 be fixed. Consider the cylinders
QoEKpx(s,t), Ql EBtpX(i,t),
and let (x,

r) -(x, r) be a non-negative pieceWise smooth cutoff function in Ql
which equals one on Qo and such that ID(I ~ 2/ p and (t ~ 2/ s. At first we will
proceed formally. The calculations below will be made rigorous later. In the weak
formulation of (8.1 )n, take the testing function
10. Compactness in the t variable 367
!.&n,t (un + 1)-8(2,

and integrate by parts over Ql. Dropping the subscript n, we obtain
II{u+ 1)-8u~(2dxdr = - IIIDuI P- 2DUD{Ut (U+ 1)-8(2)dxdT

Ql Ql
= -t I I !IDuIP(u + 1)-8(2dxdr
Ql
f
+ (J IIDuIP(u + 1)-8-1Ut(2dxdT
Ql
- 2 IIIDu IP- 2Duut{u + 1)-8(D( dxdr

Ql
~ ~ (p - 1) IIIDuIP{u + 1)-8-1Ut(2dxdr
Ql
+ ~ IIIDuIP(U + 1)-8((Tdxdr
Ql
+ ~ IIIDuIP-1(u + 1)-t ({u + 1)-8u~(2) I dxdr

Ql
= n(l) + nP) + n(3).
In estimating 'R,( 1) we use the regularising effect of Proposition 6.1 of Chap. VI,
1 u
(10.2) Ut < - - - .
- 2-p t
Then,
By Young's inequality
'R,(3) ~ ~ II{u + 1)-8u~(2dxdT + ; I I IDuI 2(p-l){U + 1)-8dxdr.

Qo Qo
Since 1 < p < 2, this last integral is majorised by
Combining these estimates we find that

368 XU. Non-negative solutions in I:T. The case 1<p<2
By (8.5). if aE (O,p - 1). this is estimated by
~ ffiDuIPu-(O+1)(u + 1)-[6-(o+1)ldxdr
Ql
~ "Y(a,~)pON JJIDUp-~-a IPdxdr

Ql
and the lemma follows by fonnal calculations. The calculations are fonnal since
Un,t (un + 1) -6 (2. need not be an admissible testing functions in (8.1)n. The ar-
guments would be rigorous if
(10.3)
Indeed. if so. we may take in the weak fonnulation (8.1)n the testing function
Un(t + h) - un(t) ( h (0 T _ )
h un + 1)-6/"2
.. , E, Is, Is~t<T-h.
The limit as h - t ° is justified and we may proceed as before.
lO-(i). Approximating estimates

Therefore to prove the lemma it suffices to establish (10.1) for a sequence of
approximating solutions satisfying (10.3). The unique solution of (8.l)n. can be
approximated by the solutions of
Vn,j E C (0, T; L2 (Bj)}nLP (0, T; W:,p (Bj ») ,

{ j = n + 1, n + 2, ... ,
(10.4)
/rvn,j - div (lDv n,jlp- 2 Dvn,;) = °in B; x (O,T),
vn,;(·,t) 11%1=;= 0, vn,;("O) = uo,n,;,
where B j is the ball of radius j about the origin and {uo,n,; };:n+1' is a sequence
offunctions in C':' (B n +1). such that .
Uo,n,; --+ uo,n in L~oc (Bn+t> ,
and
10. Compactness in the t ¥ariable 369
/ uo,n,;dx ::; 2/uodx Vp > O.

Kp Kp
As indicated in §12 of Chap. VI.

{) {)
Vn,;, -{) Vn,; -- Un, -{) Un, in C1!c (ET)
Xl Xl
Vl= 1,2, ... ,N,
for some Q E (0,1). The unique solvability of (10.4) can be established by a
Galerlcin procedure. Such a method also yields
To establish (10.1) for Vn,j is suffices to show that
(10.6) ID! vn,;1 E L?oc (B;).
In the remarks below we drop the subscript n, j and write v = vn ,;. We write
(10.4) for the time levels t+h and t and set
w = vet + h~ - vet) , hE (0, T - !s), !s::; t < T - h.
By difference
(10.7) Wt-h-ldivJh=O inBjx(O,T-h),
where
Jh =IDv(t + hW- Dv(t + h) -IDv(t)IP- Dv(t).
2 2
In the weak formulation of (10.7) take the testing function W (t- V+ which van-
ishes on Ixl =j and for t::; ~. This gives
T-h T-h t+h
(10.8) / /(t -
! Bj
i) + Ao,;IDwI dxdr ::; 'Y /
2 /1 f
0 Bj
:r
t
vex, r)drr dxdr,
where
1
Ao,; = /ID(svn,;(t + h) + (1 - s)vn,;(t))IP- 2 ds.

o
If /C is a compact subset of B; x (s, T). we have
A o,; ~ 'YIIDvn ,; lI~i .

It follows from (10.8) that
370 XU. Non-negative solutions in 1::1. The case I <p<2
The last integral is finite by virtue of (10.5) and the lemma follows.
11. More on the time-compactness
We record a simple consequence of Lemma 10.1. If x - ((x) is the usual cutoff

function in K 2p that equals one on Kp. we find from the weak fonnulation (8.i)n.
VO<s<t$T.
t t t
ffiUt)-(dxdT - ffiUt)+(dXdT =- ffUt(dxdT

aK2p aK2p aK2p
t
= f f1Du1JI-2 DuD( dxdT.

aK2p
1berefore
The fust integral on the right hand side is estimated by (10.2). i.e .•
t-s{f
s-
$ 'Y- uodx + (t),!p}
p>' •
K4p
Estimating the second integral by Proposition 8.1 gives

LEMMA 11.1. There exists a constant 'Y ='Y( N, p), such that
(l1.1) VO<s<t$T, Vn = 1,2, ... ,

if t-s{f
!{~un.tldXdT $ 'Y -s- K4puodx + (t)¢P}
p>' •
12. The limiting process 371
12. The limiting process

By construction Un /' U a.e. in ET and by (8.3)
for all O<t~T. Moreover by (11.1)
(12.2)
p-l-..
By Lemma 8.1 the sequence {'Un J> } is equibounded in
V (0, Ti W1,P(Kp ») , 'rip> 0, provided 0 E (O,P -1).
Since the whole sequence {un },1E:N -+u in Lloc(RN).
'Un
p-l-..
J> -+ U
p-l-..
J> weakly in LP(O, Ti W1,P(K p », 'rip> 0.
This implies that the sequences
Un,A: = Un 1\ k = min{un,k}
are equibounded in £P (0, Ti W1,P(Kp )) , 'rip> 0. and
(12.3) Un,k -+ U1\ k weakly in LP (0, T; W1,P(Kp ») , 'rip> 0, 'rIk > O.
LEMMA 12.1. DUn,A: -+ DUA: strongly in Lfoc(ET).

PROOF: In the weak fonnulation of (8.I)n. take the testing function
to obtain
(12.4) jjIDUn,A:IPl()dxdT = jjIDUnIP-2DUn.DUkl()dxdr

ET ET
+ jjlDunlP-2 D(u + v)(UA: - Un,k)Dl()dxdr
ET
+j j Un,t (Uk - Un,k)tpdxdr == 10 + It + 12.
ET
We ftrst estimate the integrals Ii, i = 1, 2. Let 0 E (O,p - 1) be so small that
(0 + l)(p - 1) ~ 1. Then by Lemma 8.1
372 XU. Non-negative solutions in !:T. The case I <p<2
1_ ~ riD
II I < fJ Un 11'-1 Un-(01+1)7 Un(a+1)7(Uk - Un,k )cp .l-dT «=
ET
1
I' ~ IIDu:-~-"IIP-1 p,supp{ <P }

(I
ET
1u~+1)(p-1)(Uk - un'k)Pcp"dXdT) ;
(flu
.1
"~(Q,P' u.,'I') + l)(u, - Un"l"'I"'dzdT) ,
--+ 0 as n-oo.
We estimate 1121 by making use of the regularising inequality (10.2).
1121 ~ ;~~ Ifun (Un,k - Uk) cpdxdT

ET
2-p
--+ 0 as n--oo.
p
We return to (12.4) and estimate
10 = IIIDun,kIP-1IDUkl cp dxdT
ET
~ p; 1 IllDun,klPCPdxdT + ~ IIIDUkIPcpdxdT.
ET ET
Combining these calculations in (12.4) gives
IllDun,klPcpdxdT ~ IllDUklPcpdXdT + o(~) .

ET ET
From this, by lower semicontinuity
IflDUklPcpdxdT ~ l~~~ IIIDUn,kIPcpdxdT

ET ET
~ IIIDukIPcpdxdT.
ET
This proves the lemma.
Next, by Lemma 10.1 the sequence
2-.} nEN
{ ata (Un + 1)2"
is equibounded in L~oAET) for all (J~ 0: + 1 and for all o:E (O,p - 1). Therefore
a (Un + 1)2"
at 2-' -+
a (u + 1)"'-
at 2-' weakly in L
2{ s, t; L 2(Kp) ) ,
for all 0 < s < t :5 T and all p> O. This implies that
{:t Uk,n} E L~ocET

unifonnly in n and
(12.5)
o
Choose 1/J EXloc (ET) and in (S.I)n consider the testing function r.p= (1/J - u)+.
Fix O<s<t:5T and let
Then
(1/J - u)+ = (1/J - U" k)+ o
EXloc (ET),
so that r.p is an admissible testing function. It gives
t
(12.6) !!{ !Un(1/J - u)+ + IDunIP-2Dun ·D(1/J - U)+} dxdr = O.

SRN
Since Un :5 u, Vn EN, we have
Therefore in view of (12.5)

t t
n~!! :r un (1/J - u)+dxdr = ! j<u" kh(1/J - u" k)+dxdr

SRN sRN
t
== ! !ut (1/J - u)+dxdr.

SRN
Analogously,
374 XII. Non-negative solutions in l::r. The case 1<p<2
IDunlp- 2Dun D('I/1 - u)+

= ID(Un " k)IP-2 D(un "k)D('I/1 - U" k)+
= (IDun,kIP- 2DUn,k -IDuklp-2 DUk) .D('I/1 - Uk)
+ IDuIP- 2 Du·D('I/1 - u)+.
By a calculation similar to that in Lemma 5.2 and leading to (5.6) we have
Therefore taking into account Lemma 12.1 and letting n-+oo in (12.6) gives
j j {Ut('I/1 - u)+ + IDulp-2 Du·D ('1/1 - u)+} dxdr = 0,

·a N
o
for all '1/1 EXloc (ET)' It remains to prove that u takes the initial datum U o in the
sense of Lloc(RN) and that ueS*.
12-(i). Continuity in Lloc(RN) at t=O
Fix p > 0 and let uo,e be a net of functions satisfying
for Ixl > 4p

{ uo,e == 0,
uo,e --+ u o, in L1 (K2p )'
Such a family can be constructed by first defining a function that coincides with
U o in K 3p and zero otherwise and then by mollifying the function so obtained.
Let also Ue be the unique solution of (1.1) with initial datum uo,e' We take the
difference of (8.1)n and the equation satisfied by Ue. In the p.d.e. so obtained take
the testing function
tp = [(un - u e)+ + 6]",(
where (1,6 e (0,1) and x-((x) is the usual cutoff function in K 2p that equals one
on Kp. We perfonn an integration by parts and let 6-+0, 8-0, (1-0. to obtain
j(Un(t) - ue(t»+dx ~ j(Uo,n - uo,e)+dx

Kp K2p
t
+ 2; j j (IDunIP-1 + IDueIP-1) dxdr.
OK2p
We use (8.4), interchange the role of Un and Ue and, for t > 0 fixed, let n -+ 00.
This gives
jIU(t} - UE(t}ldx ~ jluo - uo,EI dx

Kp K2p
From this
jlu(t} - uoldx ~ 2 j luo - uo,Eldx + jluE(t} - uo,Eldx + O(t;) .

Kp K2p Kp
Letting t '\. 0
lim-suPt'\,o jlu(t} - uo}ldx ~2j luo - uo,Eldx, 'VeE(O, I}.

Kp K2p
12-(ii). uES·
By (10.2). 'Vn E N and for all k > 0
-
o(U I\k } < -1- -Un.
at n - 2-p t
As n-+oo
1 u
(12.7) (u 1\ k}t ~ -- - a.e. in ET.
2-pt
The limit is flISt taken in 1)'(0, T) and then (12.7) holds almost everywhere in ET
in view of (12.5). Next from Lemma 9.1 it follows that 'VC> 1
t
jjlDunlP ~ X[k < unlx[u < CkldxdT = O(~).

sKp
Here we have used the fact that Un / u implies [un < Ckl ~ Iu < Ckl. Letting
n -+ 00 for k > 0 and C> 1 fixed yields by lower semicontinuity
t
jjIDuIP~X[k<U<CkldxdT = 0 (~).
sKp
We conclude by remarking that the requirement u E S· is necessary and sufficient

for uniqueness. Indeed. if solutions in S are unique. they can be constructed start-
ing from their traces on t = T E (0, T) to yield u E S·. Vice versa solutions in S·
are unique.
376 XII. Non-negative solutions in Er. The case I <p<2
13. Bounded solutions. A counterexample

Let r 2: 1 satisfy Ar ::= N(p - 2) + rp > O. If U o E L'oc(RN ), then by energy
estimates, the sequence of approximating solutions of (8.1)n satisfies
{un} E L'oA~T) uniformly in n.

Therefore by Theorem 5.1 of Chap. V, {un} E L~(ET) uniformly in n. It follows
from the regularity results of Chaps. IV and IX that
{un}, {un.x;} E C;:'c(~T)' j=l, 2, ... ,N, uniformly in n,

for some a E (0, 1) depending only upon N and p. This gives a regular solutions
to the Cauchy problem (1.1)-(1.2). A similar analysis can be carried if the initial
datum is a measure JI. and Al > 0, i.e.,
2N
(13.1)
p> N+l'
We show next that if(l3.1) is violated, then initial data in Ltoc(RN) might produce
unbounded solutions.
13-(i). A counterexample
Let a E (0, 1) be a positive constant and let Be denote the ball of radius a in
RN centered at the origin. Consider the functions
(a 2-lxI2)2
(13.2) z= + and v = (1 - ht)+ z,
Ixl N lin Ix121f3
where {j, h > 1 are to be chosen. One verifies that
Consider also the Cauchy problem

Ut - div IDulp-2 Du = 0, in E1 ::=RN x (0,1),
(13.3) {
u(·,O)=z.
The p.d.e. is meant in the sense of (2.1 )-(2.2) and the initial datum is taken in the
sense of Ltoc(RN).
LEMMA 13.1. Assume that N(p - 2) + p = O. The constants a E (0,1) and
(j, h > 1 can be determined a priori so that v is a non-negative, weak subsolution
0/(13.3) in E 1 •
PROOF: By calculation on the set 0 < Ixl < a,
z .
Dz = -lxl 2 Fx,
13. Bounded solutions. A counterexample 377
where
2/3 41X12}
F = { N + In Ixl2 + a2 _ Ix l2 .
We choose a =e -k and k> 1 so large that F> O. Compute .
_ zp-IFp-1
IDzlP 2Dz =- x
. Ixl p
zp-2 FP-l zp-l FP-l
div(lDzlp-2 Dz) = -(p - 1) Ixl p Dz . x + P Ixl1>+1 Dlxl· X
Zp-l FP-l zp-l FP-2
-N Ixl p -(P-I) Ixl p DF·x.
Using the fonnulae
Dz . x = -zF, Dlxl· x = Ixl

DF . x = -4/3 + 81xl 2 + 81xl 4
In21xl 2 (a 2 -lxI 2) (a 2 - Ix12)2'
we obtain
We calculate the expression in braces on the right-hand side using the definition
of F and the fact that N(p - 2) + p=O, to obtain
Consider the sets
Cil) == {~e-2k ~ Ixl 2< e- 2k } , ci2) == {lxl2 < ~e-2k }, k > 1.

One verifies that on Cil) we have
1t >
-
8(2 _!!.)
k
_Nf3.
2k
378 XII. Non-negative solutions in I:T. The case I <p<2
Therefore 'H. ~O on £1 if k is sufficiently large. On£~2) we have F~ (N - f3/k) >

1. Therefore
div(IDzIP-2 Dz) > zp-1 'Y(N,p).
- Ixl P In Ixl 2
Finally we compute in {O< Ixl <a}
.c(v) == Vt - div(IDvI P- 2Dv)
= -hz - (1 - ht)~-l div(IDzIP-2 Dz).
On £(1)
k'
.c(v) -< 0 and on £(2)
k
zp-2 'Y(N,P)]
.c(v) ~ z [-h - Ixl p In Ixl2 .
By calculation on £~2) •
zp-2 'Y(N,p) (a 2 _lxI2)2(p-2)

- Ixl p In Ixl 2 ~ 'Y(N,p) lin IxI21~(p-2)+l '
where we have used the fact that Al == N (p - 2) + p = O. We select f3 > 1 so that
f3(P- 2) +1 > O. This gives
zp-2 'Y(N,p) •
-lxlP Inlxl2 ~'Y (N,p,k).
Therefore
.c(w) ~ z( -h + 'Y·(k».
Cltoosing h ='Y. ( k) proves that
(13.4) .c(V) ~ 0 on {O<lxl<a}x(O,I).
To prove that indeed v is a weak subsolution in the whole E .. multiply (13.4) by
a non-negative function x - cp(x) E C~ (E1 ). and integrate over the cylindrical
domain with annular cross section Q~ =={e< Ixl <a-e}X (0,1). We obtain
II{
1:1
vtCP+IDvI P- 2Dv.Dcp} dxdr
= lim
~\,O
jrJ{{vtcp+IDvlp-2 Dv.Dcp} dxdr
Q.
I
~ ~~ II
o{lzl=G-~}
IDvl p- 2Dv· 1:1 tpdtrdr
I
- lim
~ .... "'O
J"J{IDvIP-2 Dv . -ixix tpdtrdr,
I
O{lzl=~}
where du denotes the surface measure on {Ixl =e} and on {Ixl =e}. The limits
on the right hand side are zero. In particular we have
Vt/J E x'oc (Ed, V( E C~(RN), ( ~ 0,

(13.5) jj{vt(t/J - v)++IDvI P- 2Dv·D(t/J - v)+} dxdr ~ O.
1:1
One also verifies by direct calculation that v satisfies (5.1) and (5.2) and therefore
is a subsolution of (13.3) in the class S·.
Next we return to (13.3). This problem has a unique solution U E S·, by
the construction of §§8-12 and the uniqueness theorem 7.1. By the comparison
principle U ~ v and therefore U is not bounded. The comparison principle here
is applied as follows. By the definition of weak solution the truncated functions
Uk == mint Ui k} are, for all k > 0 distributional subsolutions of (13.3). Setting
w == v - U and W(k) == v - Uk
and using (13.5) we find
VO<s<t~T, Vt/JeX,oc (Ed, V(EC~(RN), ( ~ 0
+ [lDvlp-2 Dv -IDuklp-2 DUk]·D «t/J - v)+() }dxdr ~ O.

Observe that w(t)-O as t'\,O in Lloc(RN). Therefore we may proceed as in the
proof of the uniqueness theorem and establish an analog of Proposition 6.1, i.e.
VO<t~T, Vq>l, Vp>O, Vue (0, 1)
j (w+(t»)q dx ~ _'Y_
(up)p
11
t
(w+)P-2+ q dxdr.
Kp OK(1+a)p
Proceeding as in the proof of Theorem 7.1 we find w+ = O.

Remark 13.1. If N(p-2)+p>0 then v satisfies (13.4) but it is not a subsolution
of (13.3) in the whole E 1 • In particular it does not satisfy the requirement (5.2) of
the class S·. If N(P-2)+p<0 then v does not satisfy (13.4).

Equations of the type of (1.1) arise in modelling of non-newtonian fluids (see
Kalashnikov [57], Martinson-Paplov [74,75], Antonsev [5] and Joseph-Nield-
Papanicolau [56]). Questions of solvability, even though in a different context,
380 XU. Non-negative solutions in ET. The case 1<p<2
were fmt investigated by Btizis and Friedman [18J. The notion of weak solution
introduced in §2 is taken from [42J. B6nilan has infonned us of a more general no-
tion of solution. introduced in [II J. that would include solutions of variable sign.
The remainder of the chapter is essentially taken from [42J. It would be of interest
to investigate questions of existence/Uniqueness for (1.1) in ET when the initial
datum is of variable sign or is a measure. Singular equations are little understood.
mostly if p violates (13.1). Preliminary investigations seem to indicate questions
of limiting Sobolev exponent (see [19]) and differential geometry.
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@ 1993 Springer-Verlag New York, Inc.

ISBN 0-387-9402C).() Springer-Verlag New York Berlin Heidelberg

1. Elliptic equations: Harnack estimates and HOlder

Here 0 < A ~ A are two given constants and cp E Ll::(n) is non-negative. As a

(1.5) div IDuIP-2 Du = 0, in n, p> 1.

2. Parabolic equations: Harnack estimates and Holder

(2.1) Qp == Bp(xo) x (to - p2, tol, Bp(xo) == {Ix - xol < p} .

(2.2) u(xo, to) ~ "y sup u(x, to _ p2) .

The proof is based on local representations by means of heat potentials. A striking

u E V 1 ,2(nT) == Loo (O,T;L2(n»)nL2 (o,T;Wl,2(n» ,

u E V~,p(nT) == Loo (0, T; L2(n»nV (O,~; Wl,p(n»),

(2.5) Ut - div IDu IP-2 Du = 0,

3. Parabolic equations and systems

This is a parabolic version of a similar finding observed in [99,I00J for elliptic

and its quasilinear versions. Such an equation is degenerate at those points of OT

(4.1) Ar == N(p - 2) + rp > 0

In Chaps. VI and VII we present an intrinsic version of the Harnack estimate

(4.3) u~? E C/:'c(ilT), i = 1, 2, ... ,n, j = 1, 2, ... , N,

A similar spectrum of results could be developed for equations of the type

1. HOlder continuity and boundedness of solutions (Chapters I-V)

I. Notation and function spaces

II. Weak solutions and local energy estimates

III. HOlder continuity of solutions of degenerate

IV. HOlder continuity of solutions of singular

v. Boundedness of weak solutions

§7. Local iterative inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

VI. Harnack estimates: the case p>2

VII. Harnack estimates and extinction profile for

VIII. Degenerate and singular parabolic systems

IX. Parabolic p-systems: HOlder continuity of Du

X. Parabolic p-systems: boundary regularity

XI. Non-negative solutions in ET • The case p> 2

§5. Estimating IDulp - 1 in ET ............................ 323

XII. Non-negative solutions in E T . The case 1 <p<2

Bibliography ...................................... 381

8T == afl x [0, TJ, r == 8 T u(flx {O})

ISliBp(xo)I ::; (1 - o:*)IBp(xo)l,

(1.3) [fjJh IC == sup IfjJ(x)1 + sup IfjJ(x) - fjJ~Y)I.

11/11 ••.,,,, '" (l ([ IfI'dz) dT) f 1 < 00.

Also 1 E Lf::C(llT), iffor every compact subset IC of II and every subinterval

Whenever q = r we set Lq,r(llT) == Lq(llT), Lf::C(llT) == Lfoc(llT) and

W l,p (ll) is the completion of Coo (ll) under the norm

Equivalently WI,P(ll) is the Banach space of functions v E LP(ll) whose gener-

We let WI,oo(n) denote the space of functions v E LOO(n) whose disttibu-

2. Basic facts about W 1,P(il) and W~'P(il)

where Q E [0, 1], p, q ~ 1, are linked by

and their admissible range is:

qE [s, :!p] if s~ :!p'

PROOF: We may take Q = 1 and S = 1 in (2.2-ii).

THEOREM 2.2. Assume that an is piecewise smooth. There exists a constant C

1 (N - I)P] if < p < N,

If an is piecewise smooth. the space W;,"(n) can be defined equivalently

(2.5) (v - k)+ == max{(v - k) ; O}, (v - k)_ == max{ -(v - k) ; O}.

LEMMA 2.1. LetvEW1,"(n). Thenforall kER, (v=Fk)± E Wl'''(n). Assume

Ilvlloo,an :5 ko, for some ko > o.

w == min {Vl,V2, ... ,vn } E W1,"(n).