JOURNAL

OF XIATHEMATICAL

ANALYSIS

AND

APPLICATIONS

35, 361-373 (1971)

Remarks on the One-Phase Stefan Problem for the

Heat Equation with the
Flux Prescribed on the Fixed Boundary*
J. R. CANNON
Mathematics

Department,

The University

of Texas, Austin,

Texas 78712

AND
MARIO
Universitd

PRIMICERIO

di Firenze, Istituto Matematico Ulisse Dini,

67/A, 50134, Firenze, Italy

Viale Morgagni,

Submitted by Richard Bellman

1. INTRODUCTION
This paper is concerned with the following one-phase Stefan problem:
Lu = 24,. - Ut = 0,

in

0 < x < s(t), 0 < t d T,

(1.1)

u.40, t) = fqt),

O<t<TT;

(1.2)

4% 0) = @p(x),

O<x<b,

(1.3)

0 < t < T, s(O) = b > 0,

(1.4)

0 < t -< T,

(1.5)

UW), 4 = 0,

i(t) = - %(S@>,
4,

where T is an arbitrarily fixed positive number.

As is well known, the problem (l.l)-(1.5) is a mathematical description for
the unidimensional heat conduction in a plane infinite slab of homogeneous
thermally isotropic material with a phase occurring at one limiting plane and
the thermal flux prescribed on the other.
For sake of simplicity, in writing down (1. l)-( 1.5) we choose a system of
variables such that the thermal coefficients (conductivity, heat capacity,
density, latent heat) disappear.
* This research was supported in part by the National
G.P. 15724 and the NATO Senior Fellowship program.

361

Science Foundation

Contract

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CANNON

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PRIMICERIO

Problems of this type have been considered by various authors [l, 6, 7, 9,

10, 11, 12, 131.Our discussion of this problem applies the maximum principle
as the major tool in a constructive existence proof via the method of retarding
the argument in the free boundary condition (1.5). Hence, existence is
obtained under minimal smoothness assumptions upon the data. In the following sections we shall discuss existence (global), uniqueness, stability, monotone dependence, and the asymptotic behavior of the solution of (l.l)-(1.5).
The techniques used and the results obtained are similar to those in [2, 3,
4, 51.
The assumptions we shall require on the Stefan data are as follows:
(A)

H(t) is a bounded piecewise continuous nonpositive function;

(B)

Q(x) is a piecewise continuous function

0 < CD(X)<L(b

such that:
- x).

Obviously, the assumption (B) on the Lipschitz continuity of a(x) near

x = b has a significance only in the case b > 0.
The assumption on the sign of H(t) means that heat is entering the region,
so that, for each t there is only one phase, say the liquid one (of course, the
same reasoning holds if a(x) < 0 and H(t) 3 0).
We make the following definitions:
DEFINITION

the auxiliary

1. We say that a real-valued function u(x, t) is a solution of

problem (1. I)-( 1.4) f or a g iven real-valued function s(t) (s(t) > 0),

if:

(4

u,, and ut E C, u,, = ut , 0 < x < s(t), 0 < t < T;

(b) u E C in 0 < x < s(t), 0 <

of Q(x);
(c)

t<

T except at points of discontinuity

u(x, 0) = Q(x) at points of continuity

of D(x) and

0 < lim u(x, t) < iii% u(x, t) < co

t+0

t-10

at points of discontinuity;
(4

lim,,

u2(x, t) = H(t) at points of continuity of H and

at points of discontinuity.

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ONE-PHASE STEFAN PROBLEM

DEFINITION2. By a solution of the given Stefan Problem (l.l)-(1.5) we

mean a pair of real-valued functions (s(t), u(x, t)), such that s(O) = b, and for
O<t<T:
(ii)

s(t) E Cl, s(t) > 0; t > 0;

U(X, t) is the solution of the corresponding

(iii)

uJs(t), t) exists and is continuous;

(iv)

(1.5) is satisfied.

(i)

auxiliary

problem;

It is well known that (see [8, 2]), if s(t) is Holder continuous with Holder
coefficient > 1, if (A) and (B) are satisfied and if b 3 0, the auxiliary problem
has a unique solution.
We state a useful result concerning the reformulation of the free boundary
condition (1.5).
LEMMA 1. Under assumptions (A) and (B), if s(t) is a Lipschitz
function for 0 < t < T, then condition (1.5) is equivalent to:

s(t) = b + 1; CD(X)dx -

1; H(T) dT -

continuous

,I u(x, t) dx.

U-6)

Proof. Suppose (u, s) is a solution of (l.l)-(1.5)

and integrate (1.1) over
its domain of validity; using (1.2)-( 1.5), (1.6) follows directly.
Suppose conversely that (u, s) satisfies (1 .l)-(1.4) and (1.6); by differentiating (1.6), (1.5) follows directly if u%(s(t), t) exists and is continuous for
0 < t < T. But this last assumption is guaranteed by Lemma 1 of [2]; consequently Lemma 1 is proved.

2. EXISTENCE (b > 0)
First, we need:

LEMMA2. Under the assumptions (A) and (B), let s(t) be a Lipschitx
tinuous, monotonic nondecreasing function, and define
A = max&a~,

I Wt)l,

Ll.

Then the solution u(x, t) of the corresponding auxiliary

- A < z&(s(t), t) < 0.

con-

(2.1)
problem is such that

(2.2)

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Proof. Let us prove (2.2) for each fixed instant t,, . First note that, from
the maximum principle, it follows that 11(x,t) is non-negative and since
u(s(t), t) = 0 the second inequality in (2.2) is proved. For each to E [0, T]
consider the function
v(x, t) = A(&)

- x) - u(x, t)

for 0 < t < to , 0 < x < s(t). Clearly,

Lv 3 v,, - vt = 0,
~(0, t) = - A - H(t) < 0
v(s(t>, t) = A(+,)
v(x, 0) = A(&)

by (2.1) and (A),

by the monotonicity of s(t),

- s(t)) B 0
- x) - CD(x)3 0

by (2.1) and (B).

Hence, by the maximum principle, v(x, t) > 0 in its domain of definition.

Since v(s(t,) to) = 0, it follows that v,(s(t,) to) < 0 which implies (2.2) for
each to and completes the proof of the Lemma.
Next, we will apply the retarded argument technique to construct the
solution of the given problem. For each ti E (0, b), let us define

CD(x)
1z y

for
for

O<x<b--8
b - 0 < x < b,

and find the solution 18(x, t) of the auxiliary problem (l.l)-(1.4), where @P(X)
is replaced by P(X), s(t) is replaced by s(t) = b, and T by 8. In this case it is
easy to show that ~,~(b, t) = u&s(t), t) exists and is continuous in [0, 01
and that by Lemma 2 we have - A < uze(se(t),t) < 0. In the second timeinterval t9 < t < 28 let us define:

t
se(t)= b - I ~,ep(~
e

e),T - e)dT

(2.3)

and solve the auxiliary problem for this choice of s(t); these are the first steps
of an inductive process that we can perform for each 0,O < 8 < b. We prove
the following result.
LEMMA
3. For each 6 E (0, b), there exists a sol&ion (se,ue) of (l.l)-(1.4)
where Q(x) = Qe(x). The function se(t) is equal to b in t E [0, 01, satisfies (2.3)
in t E [O, T] and is Cl in [0, T]. Moreover,

0 < Se(t) < A.

(2.4)

Proof. Suppose that by the method given above we have constructed a

pair (se,ue) for 0 < t < no. Assume that s@(t)E c1 and Se> 0. By Lemma 1

ONE-PHASE

STEFAN

365

PROBLEM

ofPI, he(@),
q exists, is continuous

and satisfies (2.2). Suppose finally that

(9, u) satisfy (2.3) for 0 < t < no. In the next step no < t < (n + 1) 0, let
us define seby (2.3) and solve the auxiliary problem until t = (rz + 1) 8. By
the hypothesis on uo(Se(t),t) in [O, n6], se(t) is Cl in [no, (n + 1) 01and satisfies
(2.4). Hence, uZe(se(t),t) is continuous in the same time interval and, by
Lemma 2, satisfies (2.2).
We can now prove the main result of this section.
THEOREM
1. Under the assumptions (A) and (B), there exists a solution
(s, u) for the Stefun probEem (1 .l)-( 1S) when b > 0. The free boundary is Cl in
(0, T], is monotonically nondecreasing and satisjies:

o < t(t) < A

for 0 < t < T,

(2-5)

where A is defined by (2.1).

Proof,
By (2.4), the functions s(t) form an equicontinuous, uniformly
bounded family. Hence Ascoli-Arzelas theorem holds and we can select a
subsequence se(t) that converges uniformly to a monotonic Lipschitz continuous function s(t) as 0 tends to zero. Let U(X, t) be the unique solution of the
auxiliary problem with that choice of s. It is easy to show that, given any E > 0,
it is possible to find a 8* such that, for all 0 < 0* / z/(x, t) - u(x, t)l <; E.
Indeed consider the difference ue - u = w in the region

0 < x < max(se(t), s(t)),

O<t<T,

(where the two functions are extended by setting them identically zero outside
their domain of definition). Since w&O, t) = 0, we can reflect the domain of
definition about the line x = 0. Then, the maximum principle gives:

I US@,
t) - u(x, t>l < maxhp I Qe- @I , II~~(4~1,41t , IIu(se(4 TM
where for any function f = f (t)

By the continuity of Us and u the right side of the inequality can be made
less than E,provided 0* is chosen such that 1se(t) - s(t)1 is sufficiently small.
So we have shown that the subsequence us, corresponding to the setending to
s, converges uniformly to u. In order to prove that (s, u) is a solution of the
Stefan problem, we must prove (i), (iii), (iv) of the Definition 2; (iii) follows
directly by the Lipschitz continuity of s(t). By Lemma 1 in order to prove (iv)
409/35/=9

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CANNON AND PRIMICERIO

it suffices to prove that (s, U) satisfies (1.6). Integrating (1.1) over its domain
of definition, it is easy to see
s@(t+ 0) = b + j, Q(x) dx -

j: H(T) d7 -

jr

uB(x, t) dx.

Taking the limit as 0 tends to zero it follows from the uniform convergence
of seto s, of @sto @ and of ue to u that (s, u) satisfies (1.6). Consequently (iv)
is demonstrated and (i) follows directly by (1.5).

3. STABILITY

AND UNIQUENESS

(b 2

0)

THEOREM 2. If (si , u.,) is a solution of (1. I)-(1.5) for data H,(t), Qi(x),
sa tfy
is ing assumptions (A) and (B) b, 3 6, 3 0, then there exists a
bi(i=1,2)
constant C = C(A, T) such that:

I sl(t>- s&)1 < C 1I b, - b, I + jb I @l(x)- @z(x)ldx +

j::

I W4 dx
(3.1)

Proof.

The proof is given in [l] and will not be repeated here.

THEOREM 3. Under the assumptions (A) and (B), the solution to the Stefan
problem (1 .l)-( 1.5) with b > 0 is unique.

Proof.

Theorem 3 is an immediate corollary of Theorem 2.

4. EXISTENCE (b = 0)
THEOREM 4. Under assumptions (A) and (B), there exists a solution (s, u)
to the Stefun problem (l.l)-( I .5) when b = 0. The free boundary is Cl in (0, T],
is monotonically nondecreasing and satisJies (2.5).

Proof.
For each 0 < b < b, let (sb,u) be the unique solution of the Stefan
Problem (1.1)-( 1.5) with a(x) = 0. Note that the proof of Lemma 2 and
the constant A are independent on b. Hence, we have: 0 < Sb(t) < A for
0 < t < T and b E (0, b,). Consequently, the functions sb(t) form an equicontinuous, uniformly bounded family. Choose a sequence of bs tending
to zero and apply the Ascoli-Arzela Theorem to obtain an subsequence sb
converging uniformly to s. Let u(x, t) be the unique solution of the auxiliary

ONE-PHASE STEFAN PROBLElLl

367

problem (l.l)-(1.4) f or such a choice of s. With the same argument used in

Theorem 1, one can show that z@(x,t) converges uniformly to U(X, t). Moreover, from (1.6) for each b we have

9(t)
=b- ss(t)

ub(x, t) dx -

t H(T) d7,
s0

and from the uniform convergence of {s(t)> and (~~(2, t)}, we obtain that
(s, U) satisfies (1.6). Since s(t) is a Lipschitz continuous function it follows
that u=(s(~),t) exists and is continuous for t > 0. Consequently condition (iii)
of Definition 2 is fulfilled. Since Lemma 1 applies, (iv) and (i) are satisfied.
Since (ii) follows from the definition of U(X, t), the existence of a solution
is proved.
5. MONOTONE DEPENDENCE (b >, 0)
Consider two sets {H,(t), Qi(x), bi} i = 1, 2 of Stefan data satisfying
assumptions (A) and (B). Theorem I, 3 and 4 state the existence of an unique
solution (si , ui) to each one of the two problems. We shall prove the following
result.
THEOREM 5.

Under the above assumptions, if

0 < b, < b, ,

@I G @z 7

f4 2 4

then
s1(t) < s&>.

(5.1)

Proof.
Consider first the case 0 < b, < b, . We shall show that sl(t) < s2(t).
If not, then there exists a first time to such that

s&o) = S&o)

and

Go) 2 ~dto).

(5.2)

Consider the difference u,(x, t) - u2(x, t) in 0 < x < sl(t), 0 < t < to . By
the maximum principle we have u,(x, t) - u~(x, t) < 0. Since
~l(~l(to)~ to> - %WO)> to) = 0,
%Mto),

to) - %?rMto), to) > 0,

or S,(t,) < S,(t,). For the case b, = b, > 0, we define b,* = b, + 6 = 6, + 6

and construct the solution (sa*, ~a) to the Stefan problem with data H, ,
bS6and Ibzswhere:
0 < x < b, = b, ;
b, < x < 6, + 6.

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CANNON AND PRIMICERIO

By the previous argument, ss6> s, , but, the stability theorem implies that
sss(t) converges uniformly to ss(t) as 6 tends to zero. Hence, (5.1) is proved.

6. ASYMPTOTIC BEHAVIOR
Throughout this section we shall be concerned with the asymptotic
behavior of the free boundary x = s(t) of the Stefan problem (l.l)-(1.5) as
t -+ + cc. Under the assumptions (A) and (B), we have the existence and
uniqueness of the solution in either case b > 0 or b = 0.
THEOREM 6. If

tiir 1 H(T) d7 = -co,

0

then
liIiIs(t)

= +co.

If
lim t H(T) d7 = -F,
t-x s o

O<F<+co,

then
li+is(t)

=b++SP@(x)dx+F=lo

(6.1)

Proof. Consider first the case of H(t) with compact support, i.e. suppose
H(t) = 0 for each t > 2.
From (1.6) we have
s(t) = to - I: u(x, t) dx

(6.2)

for t 2 Z.
By the maximum principle, u(x, t) is dominated by yr(x, t) + ys(x, t),
where yr and ys solve the heat equation in the half-space x > 0 with the
following conditions:
Y&9 0) = l;(xJv

O<x<b,
b<xxq

Yl&

and
Y&9 0) = 0,

ysx(O, t> = H(t);

4 = 0;

(6.3)

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ONE-PHASE STEFAN PROBLEM

respectively. But for t > 2,

yl(x, t) == c Sr J$$

(x - ?Y
4t
] d[ < cbt-r/a max @

exp [ -

(6.3)

and
y2(x, t) =~==
c 1: -$&

exp [-

&]

dT < c 11H jlz{Z/t-

vtj.

(6.4)
From (6.4) and (6.3) we find that lim,,,
the case of H(t) with compact support.
For general H(t), set

U(X, t) = 0 and (6.1) is proved in

and define the corresponding Ldn) and sfn). Now lim,, dn)(t) = tin). Since
Ii, > H, it follows from Theorem 5 that W(t) < s(t). But, from (1.6)
s(t) < to . Hence,
P

= lim s()(t) < lim s(t) < iii5 s(t) < 8, .

t+m
t--x
t-=c

Now, let n--f CO. Since &jnj --f to, the proof of Theorem 6 is complete.
Next, we shall perform a deeper analysis of the behavior of s(t) in the case
s(t) = co. First we prove the following result.
lb+,
THEOREM 7. Assume lim,,, $ H(T) dr = - CO and consider the solution
(s, u) of the given problem and the solution (u, v) of the Stefan problem
Lv = 0,
v&t
44,

0 < x < a(t),

u(t,) = 0,

t) = H(t),
t) = 0,
b(t) = -%c(u(t),

t),

to < t < co,

to < t

(6.5)

to < t.

Then, as t -+ CO,

$$=l.,($)),
which implies in particular

that s(t) N a(t).

(6.6)

CANNON

370

AND

PRIMICERIO

Proof. From the monotone dependence it follows a(t) < s(t); and from
the maximum principle we have V(X, t) < U(X, t). Now,
u(t) < s(t) = s(to) + j,

u(x, to) dx -

j;, H(T) dr -

r: u(x, t) dx

< o(t) + s(t,) + ,: u(x, to) dx.

Hence,

or, i.e.,
$j=1+0(-&).
Next we demonstrate the following result.
THEOREM 8. If (s, u) is the solution of the given Stefan problem and if
,&ix - f H(T) dr = co

(C)

and

then
s(t) -

t
s0

H(T)

dT

(6.7)

as t+cO.
Proof.

Consider o(t) as defined by (6.5) with

-

s H(T) dT 0

yz(x,

to

= 0. We have

t) dx < u(t) < -1:

H(T)

dT,

(6.8)

where y&x, t) is defined by (6.4) and (6.4). F rom the first inequality in (6.8)
we get

(6.9)

ONE-PHASE

STEFAN

371

PROBLEM

where c is a positive constant. Hence, the result follows immediately from (C)
and (D).
Remark. The condition (C) and (D) are not contradictory. They express
the delicate area where the total energy input (C) is infinite and yet the
boundary temperature tends to zero as t + co. As an example, consider

-1,
l-9

H(t) =

-1
t

0<t<l,
1<t<m.

An elementary quadrature yields

tIff(~)l
dT
sodt--d;

-J- log 1

Other examples are

o<t<1,

-1,
H(t) =

;<,<1.

1<t<oo,

i-- ty

The details are left to the reader.

Next, we study some particular casesin which

THEOREM

9. Let (s, u) be the solution of the given Stefan problem. If

H(t) N - c exp[c2t],

c > 0,

(6.10)

then
s(t) -ct.

(6.11)

And if,
H(t) -

--nil

O +-lr(2na
- cy+ 1) ttnavn
F(2n) F(2ncr - 01- n + 1)

(6.12)

then
s(t)

CP.

01 >

g.

(6.13)

372

CANNON AND PRIMICEFUO

Proof.

It is known that the formula [4, p. 4341

(6.14)

provides, whenever it makes sense, an explicit solution to the inverse Stefan

problem. Consequently the asymptotic behavior of H(t) corresponding to
(6.11) can easily be found:
w,(O, t) = -

f
C2n-1 (2n - 1)(2n - 2) .-* (n) * P-l
n=l (2n - l)!

= c exp[&]

Conversely, it is easy to prove that an asymptotic behavior like (6.10) generates

a solution *hose boundary satisfies (6.11). The application of the same
method provides the proof of the second statement of the Theorem. Details
are omitted, as well as special cases of (6.12), as for example 01= Q in which

i.e. the boundary temperature is asymptotically finite (but not equal to zero)
and the s(t) goes to infinity approaching a parabola.
*

7. REGULARITY

OF THE BOUNDARY

For the case that b > 0 and Q(x) 3 0, we can state the following result.
THEOREM

qx>

10.

The free boundary s is injinitely

difSerentiable if b > 0 and

= 0.

Proof. The techniques of [5] can easily be applied to yield the results of
the theorem.

REPERPNCES
1. J. R. CANNON AND JIM DOUGLAS, JR., The stability of the boundary in a Stefan
problem, Ann. Scuola. Norm. Sup. Piss XXI (1967), 83-91.
2. J. R. CANNON AND C. D. HILL, Existence, uniqueness, stability, and monotone
dependence in a Stefan problem for the heat equation, J. Mat/z. Me&. 17 (1967),
l-20.
3. J. R. CANNON, JIM DOUGLAS, JR., AND C. D. HILL, A multi-boundary
Stefan
problem and the disappearance of phases, J. Math. Me&. 17 (1967), 21-34.

ONE-PHASE

STEFAN

PROBLEM

373

4. J. R. CANNON AND C. D. HILL, Remarks on a Stefan problem, 1. Math. Me&.

17 (1967), 433-442.
5. J. R. CrlNNON AND C. D. HILL, On the infinite differentiability
of the free boundary
in a Stefan problem, /. Math. Anal. Appl. 22 (1968), 385-397.
6. J~nr DOIJGLAS, JR., A uniqueness theorem for the solution of a Stefan problem,
Proc. Amer. Math. Sot. 8 (1957), 402-408.
7. ABNER FRIEDMAN, Remarks on Stefan-type free boundary problems for parabolic
equations, J. Math. Me&. 9 (1960), 885-903.
8. ;II. c;EVREY, Sur les Cquations aux d&i&es
partielles du type parabolique,
J.
Moth. Ser. 6 9 (1913), 305-471.
9. I. I. KOLODNER,
Free boundary problem for the heat equation with applications
to problems of change of phase, Comm. Pure Appl. Math. 9 (1956), 1-31.
IO. W. I?. KYNER, An existence and uniqueness theorem for non-linear
Stefan
problem, /. Math. Mech. 8 (1959), 483-498.
Il. W. I. ~~IRANKER, A free boundary value problem for the heat equation, Quart.
Appl. Math. 16 (1958), 121-130.
12. 1%. PRIMICERIO, Stefan like problems with space dependent latent heat, to appear
in Meccanica J. of the Italian Assoc. of Theoretical
and Appl. Mech.
13. L. I. RUBINSTEIN, On the determination
of the portion of the free boundary
which separates two phases in the one dimensional
problem of Stefan, Dokl.
Skad. Nauk SSSR 68 (1947), 217-220.