Well-posedness and qualitative properties of a dynamical model for the ideal free distribution

Understanding the spatial distribution of populations in heterogeneous environments is an important problem in ecology. In the case of a population of organisms that can sense the quality of their environment and move to increase their fitness, one theoretical description of the expected distribution of the population is the ideal free distribution, where individuals locate themselves to optimize fitness. A model for a dynamical process that allows a population to achieve an ideal free distribution was proposed by the Cosner (Theor Popul Biol 67:101–108, 2005). The model is based on a reaction–diffusion–advection equation with nonlinear diffusion which is similar to a porous medium equation with additional advection and population growth terms. We establish that the model is well-posed, show that solutions stabilize, determine the stationary states, discuss their stability, and describe the biological interpretation of the results.

The research of CC was partially supported by NSF Grant 1118623.

Authors and Affiliations

1. Department of Mathematics, University of Miami Coral Gables, Florida, USA

   Chris Cosner

2. Institut für Mathematik, Universität Paderborn, 33098 , Paderborn, Germany

   Michael Winkler

Corresponding author

Correspondence to Chris Cosner.

Appendix A: Well-posedness. Proof of Theorem 2.1

In view of well-known results on the lack of regularity in degenerate diffusion problems of type (1.3) (cf. e.g. Aronson 1986), it seems adequate to study (1.3) in the framework of weak solutions. For definiteness we explicitly formulate the solution concept pursued below as follows. Here and in the following section, there is actually no need to require \(u_0\) to be continuous, whence we shall suppose here that \(u_0\) belongs to \(L^\infty (\Omega )\) only. Further relaxations are of course possible, including solutions possibly unbounded near \(t=0\); but since our focus below will rather be on the large time behavior of solutions, we shall not pursue the question of optimal initial regularity any further.

Definition 4.1

Let \(T>0\), and suppose that \(u_0 \in L^\infty (\Omega )\) is nonnegative. Then by a weak solution of (1.3) in \(\Omega \times (0,T)\) we mean a nonnegative function

with the property that for each \(t\in (0,T]\), the identity

holds for any nonnegative \(\varphi \in C^1(\bar{\Omega }\times [0,t])\).

A weak subsolution (resp. supersolution) of (1.3) in \(\Omega \times (0,T)\) is a function from the above class which satisfies (4.2) with ‘\(=\)’ replaced by ‘\(\le \)’ (resp. ‘\(\ge \)’). Finally, a measurable function \(u:\Omega \times (0,\infty ) \rightarrow \mathbb {R}\) is said to be a global weak (sub-/super-) solution of (1.3) if \(u\) is a weak (sub-/super-)solution of (1.3) in \(\Omega \times (0,T)\) for any \(T>0\).

1.1 A comparison principle. Uniqueness

We first make sure that in the framework of weak (sub- and super-)solutions, (1.3) allows for a comparison principle. In the verification thereof we follow the lines of a celebrated approach introduced in Aronson et al. (1982) for equations of the form \(u_t=\Delta \phi (u) + f(u)\), which involves solutions of a problem dual to (1.3) in an appropriate sense. Though the basic idea in Lemma 4.2 will thus be straightforward, it turns out that the presence of the convective term in (1.3) requires some additional efforts; to mention one particular technical detail in this respect, we note that unlike in the original argument, it seems that the auxiliary function \(\psi _\delta \) introduced below (cf. 4.5) fails to possess an appropriate second-order regularity property, e.g. boundedness of some weighted space–time \(L^2\) norm of \(\Delta \psi _\delta \), guaranteeing suitable decay of error terms stemming from regularization of the degeneracy (cf. the last term in 4.10).

The comparison principle provided by the following lemma will be essential to most of our results below. Since we could not find a version thereof in the literature which is sufficient for our purpose, we therefore include a full proof here.

Lemma 4.2

Let \(T>0\), and suppose that \(u\) resp. \(v\) are a weak sub- resp. supersolution of (1.3) in \(\Omega \times (0,T)\), corresponding to the nonnegative initial data \(u_0 \in L^\infty (\Omega )\) resp. \(v_0 \in L^\infty (\Omega )\). Furthermore, assume that both \(u\) and \(v\) belong to \(C^{\theta ,\frac{\theta }{2}}(\bar{\Omega }\times (0,T])\) for some \(\theta \in (0,1)\), and that

for some \(p\in (0,1)\). Then

with \(M_0=\Vert m_+\Vert _{L^\infty (\Omega )}\) as in Theorem 2.1. In particular, if \(u_0 \le v_0\) in \(\Omega \) then \(u \le v\) in \(\Omega \times (0,T)\).

Proof

Since (4.1) requires \(u\) and \(v\) to be continuous in \([0,T]\) as \(L^1(\Omega )\)-valued functions, replacing \(t\) by \(t+\tau \) and taking \(\tau \searrow 0\) if necessary we may restrict ourselves to proving (4.4) under the stronger assumption that \(u\) and \(v\) belong to \(C^{\theta ,\frac{\theta }{2}}(\bar{\Omega }\times [0,T])\).

Then performing a variant of a well-established procedure presented in Aronson et al. (1982), given \(t_0 \in (0,T), \delta \in (0,1)\) and \(\chi \in S:=\{\tilde{\chi }\in C_0^\infty (\Omega ) \ | \ 0 \le \tilde{\chi }\le 1 \}\), we let \(\psi _\delta \) denote the solution of the problem

Since \(u+v\) is nonnegative and Hölder continuous in \(\bar{\Omega }\times [0,T]\), standard theory for linear non-degenerate parabolic equations (Ladyzenskaja et al. 1968) asserts that (4.5) indeed possesses a classical solution \(\psi _\delta \in C^{2+\theta ,1+\frac{\theta }{2}}(\bar{\Omega }\times [0,t_0])\) which, according to the classical comparison principle, satisfies

Let us first make sure that there exists \(c_1>0\) such that with \(p\in (0,1)\) as in (4.3) we have

To see this, we multiply (4.5) by \(\psi _\delta \) and integrate by parts over \(\Omega \) to obtain

Clearly, \(I_1 \le \frac{|\Omega |}{2}\), \(I_6 \le 0\) and

and another integration by parts combined with (4.6) shows that

where \(M_2:=\Vert (-\Delta m)_+\Vert _{L^\infty (\Omega )}, M_1:=\Vert \nabla m\Vert _{L^\infty (\Omega )}\) and \(|\partial \Omega |_{n-1}\) denotes the \((n-1)\)-dimensional Lebesgue measure of \(\partial \Omega \).

As for \(I_3\), we invoke Young’s inequality to estimate

where according to (4.6),

Here we apply the elementary inequality

valid for all \(\xi \ge 0\) and \(\eta >0\) with \(c(p):=(1-p)^\frac{1-p}{p}-(1-p)^\frac{1}{p}>0\), to \(\xi :=u\) and \(\eta :=(\frac{c(p)}{\delta })^p\) to see that

Therefore, since \(\nabla u=0\) a.e. in \(\{u=0\}\) (Simader 1972),

and proceeding similarly for \(v\) we conclude using (4.3) that (4.8) implies

with some \(c_4\,{>}\,0\) independent of \(\delta \,{\in }\, (0,1)\), \(t_0\,{\in }\, (0,T)\) and \(\chi \,{\in }\, S\). Since \(u+v+\delta \ge \delta \), this yields (4.7).

We now choose \(\varphi _\delta (x,t):=\psi _\delta (x,t_0-t)\), \((x,t)\in \bar{\Omega }\times [0,t_0]\), as a test function in (4.2).

This first gives

where (4.6) implies that

Since \(\frac{\partial \varphi _\delta }{\partial \nu }=0\) on \(\partial \Omega \), an integration by parts shows that

whence from (4.9) we obtain

According to (4.5), we thus infer that

where we once again integrate by parts and estimate using the Cauchy-Schwarz inequality to see that

Thanks to (4.7), however, we know that

whence from (4.10) we conclude that

Since \(\chi \in S\) and \(t_0 \in (0,T)\) were arbitrary, this yields (4.4). \(\square \)

Fortunately, the integrability condition (4.3) is automatically fulfilled by any weak solution of (1.3).

Lemma 4.3

Suppose that \(u_0 \in L^\infty (\Omega )\) is nonnegative, and that \(u\) is a weak solution of (1.3) in \(\Omega \times (0,T)\) for some \(T>0\). Then for all \(p\in (0,1)\) there exists \(C(p)>0\) such that

Proof

We first note that \(u\) is continuous in \(\bar{\Omega }\times (0,T]\) according to (Porzio and Vespri (2003), Theorem 1.3), and hence the corresponding positivity set \(\mathcal{P} := \{(x,t) \in \bar{\Omega }\times (0,T] \ | \ u(x,t)>0\}\) is relatively open in \(\bar{\Omega }\times (0,T]\). Therefore, standard parabolic regularity results (Ladyzenskaja et al. 1968) assert that \(u\) even belongs to \(C^{2,1}(\mathcal{P})\) and satisfies the PDE and the boundary conditions in (1.3) classically in \(\mathcal{P}\), because \(m \in C^{2+\kappa }(\bar{\Omega })\).

Based on this information, we plan to study the evolution of a regularized version of the functional \(\int _\Omega u^p(\cdot ,t)\) by performing a suitable truncation near \(u=0\). Due to the lack of both convexity and differentiability near zero of \(0 < \xi \mapsto \xi ^p\), however, appropriate modification of straightforward cut-off procedures appears to be in order.

To achieve this, for \(\delta \in (0,1)\) we let \(\chi _\delta \in C^\infty ([0,\infty )\) be nondecreasing and such that \(\chi _\delta \equiv 0\) on \([0,\frac{\delta }{2}]\) and \(\chi _\delta \equiv 1\) on \([\delta ,\infty )\), and such that \(\chi _\delta \nearrow 1\) on \((0,\infty )\) as \(\delta \searrow 0\). We next let

for \(\delta \in (0,1)\), and observe that \(\psi _\delta \) is nonnegative and smooth on \([0,\infty )\) with

thanks to the above properties of \(\chi _\delta \) and the fact that \(p<1\). The primitive

clearly inherits nonnegativity from \(\psi _\delta \), and moreover we have the upper estimate

Now since \(u\in C^{2,1}(\mathcal{P})\), the smoothness of \(\Psi _\delta \) implies that \(\Psi _\delta (u)\) belongs to \(C^{2,1}(\bar{\Omega }\times (0,T])\). We thus may differentiate and integrate by parts to obtain for fixed \(\tau \in (0,T)\) the identity

from which we infer that

for any \(\delta \in (0,1)\) and \(\tau \in (0,T)\). Here, by nonnegativity of \(\Psi _\delta \) we have

whereas (4.13) entails that

Moreover, using that \(s(1-\chi _\delta (s)) \le \delta \) and hence

we can estimate

Likewise, we obtain

As for \(I_3\), we once more integrate by parts to find that

holds with

Since by (4.13) and (4.17) we have

again using that \(m\in C^2(\bar{\Omega })\) we conclude that

Collecting (4.15), (4.16), (4.18), (4.19) and (4.20), from (4.14) we conclude that

for all \(\delta \in (0,1)\) and \(\tau \in (0,T)\), where we observe that \(c_6\) is finite due to the evident fact that \(c_1,c_3,c_4\) and \(c_5\) are all increasing in \(\delta \). In light of (4.12), this implies that

for all \(\delta \in (0,1)\) and \(\tau \in (0,T)\), so that twice applying the monotone convergence theorem in taking \(\delta \searrow 0\) and then \(\eta \searrow 0\) yields the claim. \(\square \)

As a consequence, we may apply Lemma 4.2 in a standard way so as to obtain first that the nonlinear semigroup generated by (1.3) is quasi-contractive in \(L^1(\Omega )\).

Corollary 4.4

Suppose that \(u_0\in L^\infty (\Omega )\) and \(v_0\in L^\infty (\Omega )\) are nonnegative, and that \(u\) and \(v\) are weak solutions of (1.3) in \(\Omega \times (0,T)\) for some \(T>0\) with initial data \(u_0\) and \(v_0\), respectively. Then with \(M_0\) as in Theorem 2.1, we have

Proof

If \(u\) and \(v\) are two weak solutions, both \(u\) and \(v\) are Hölder continuous in \(\bar{\Omega }\times (0,T]\) by Porzio and Vespri (2003). In view of Lemma 4.3, \(u\) and \(v\) satisfy the regularity requirements from Lemma 4.2, whence twice applying the latter we easily obtain (4.21). \(\square \)

In particular, this immediately implies that weak solutions are unique.

Corollary 4.5

Let \(u_0\in L^\infty (\Omega )\) be nonnegative and \(T>0\). Then the problem (1.3) possesses at most one weak solution in \(\Omega \times (0,T)\).

1.2 Global existence and approximation of solutions

Our next goal is to construct a solution of (1.3) by performing a suitable approximation procedure. To this end, we consider the regularized versions of (1.3) as given by

for \(\varepsilon \in (0,1)\). According to Cantrell et al. (2008), for each \(\varepsilon \in (0,1)\) this problem possesses a global classical solution \(u_\varepsilon \in C^{2,1}(\bar{\Omega }\times (0,\infty )) \cap C^0(\bar{\Omega }\times [0,\infty ))\) which is nonnegative in \(\bar{\Omega }\times [0,\infty )\) and thus, as a consequence of the strong maximum principle, even strictly positive in \(\bar{\Omega }\times (0,\infty )\).

We shall next derive \(\varepsilon \)-independent estimates for the solutions of (4.22). We first invoke an argument based on the classical maximum principle to establish a pointwise upper bound for \(u_\varepsilon \).

Lemma 4.6

There exists \(C>0\) such that for all \(\varepsilon \in (0,1)\) we have

Proof

We let

Then \(\rho _\varepsilon '>0\) on \((0,\infty )\) and \(\rho _\varepsilon ((0,\infty ))=\mathbb {R}\). The inverse function \(\rho _\varepsilon ^{-1}:\mathbb {R}\rightarrow (0,\infty )\) is thus increasing on \(\mathbb {R}\) and satisfies \(\rho _\varepsilon ^{-1}(1)=1\) and

because since \(\ln s \le s\) for \(s>0\) we have

Likewise, the inequality \(\rho _\varepsilon (s) \ge s\) for \(s>1\) entails that

In view of (4.24), it is possible to fix \(A>1\) large enough such that

With this value of \(A\) being fixed henceforth, given \(\varepsilon \in (0,1)\) we now define

for \(x\in \bar{\Omega }\) and \(t\ge 0\). Then by (4.22) and the fact that \(u_\varepsilon \) is positive,

Moreover,

and

according to (4.24).

On the other hand, \(\overline{v}\) satisfies \(\frac{\partial \overline{v}}{\partial \nu }=0\) on \(\partial \Omega \) and

thanks to (4.26), and furthermore we have

Here we use (4.26) along with the monotonicity of \(\rho _\varepsilon ^{-1}\) and the nonnegativity of \(m\) to estimate

to obtain that

Therefore, the comparison principle ensures that \(v\le \overline{v}\) in \(\Omega \times (0,\infty )\) and thus, again since \(\rho _\varepsilon ^{-1}\) increases, that

In light of (4.25) and the fact that \(A>1\), this proves that (4.23) is valid with \(C:=A+\Vert m\Vert _{L^\infty (\Omega )}\). \(\square \)

We next use the uniformity of the upper bound in Lemma 4.6 along with well-established regularity theory for degenerate parabolic equations to conclude the following.

Lemma 4.7

There exists \(\theta >0\) with the property that for all \(\tau \in (0,1)\) one can find \(C(\tau )>0\) such that

In particular,

and

If moreover \(u_0 \in C^1(\bar{\Omega })\) then \(C(\tau )\) can be chosen to be independent of \(\tau \in (0,1)\), and then \((u_\varepsilon )_{\varepsilon \in (0,1)}\) is relatively compact even in \(C^0_{loc}(\bar{\Omega }\times [0,\infty ))\).

Proof

For \(\varepsilon \in (0,1)\), the function \(v_\varepsilon :=u_\varepsilon -\varepsilon \) is a classical solution of the problem

with porous medium-type diffusive term. Since \((v_\varepsilon )_{\varepsilon \in (0,1)}\) is bounded in \(L^\infty (\Omega \times (0,\infty ))\) by Lemma 4.6, the estimate (4.27) therefore directly follows from the interior Hölder regularity result in (Porzio and Vespri (2003), Theorem 1.3). Whereas (4.28) and (4.29) are immediate consequences of (4.27) and the Arzelà–Ascoli theorem, the claimed independence of \(C(\tau )\) of \(\tau \) is asserted by the corresponding boundary regularity statement ((Porzio and Vespri (2003), Remark 1.4)). \(\square \)

In the limit procedure \(\varepsilon \searrow 0\) to be carried out in Lemma 4.9 below, we shall moreover need a convenient compactness property of \((\nabla u_\varepsilon )_{\varepsilon \in (0,1)}\).

Lemma 4.8

There exists \(C>0\) such that for all \(\varepsilon \in (0,1)\) we have

Proof

Recalling that \(u_\varepsilon \) is positive, we may multiply (4.22) by \(\ln u_\varepsilon \) and integrate by parts over \(\Omega \) to obtain

for all \(t>0\), where clearly

and, by Young’s inequality,

Since moreover Lemma 4.6 ensures that for some \(c_1>0\) and \(c_2>0\) we have

(4.31) entails that

for any \(t\ge 0\) and \(\varepsilon \in (0,1)\). \(\square \)

We can now use the boundedness and compactness properties collected above in a standard way to make sure that indeed (1.3) is globally weakly solvable.

Lemma 4.9

There exists a nonnegative function

with the property that for each \(T>0\),

as \(\varepsilon \searrow 0\). Moreover, \(u\) is a global weak solution of (1.3) in the sense of Definition 4.1.

Proof

In view of Lemmas 4.7 and 4.8 it is evident upon standard extraction arguments that by passing to an appropriate sequence \((\varepsilon _k)_{k\in \mathbb {N}}\) such that \(\varepsilon _k\searrow 0\) as \(k\rightarrow \infty \), we can achieve that for each \(T>0\) and \(\tau \in (0,T)\) we have

with some \(u \in C^0(\bar{\Omega }\times (0,\infty )) \cap L^2_{loc}([0,\infty );W^{1,2}(\Omega ))\). Clearly, \(u\) is nonnegative, and moreover Lemma 4.6 warrants that \(u\) is bounded. It remains to verify that along some subsequence of \((\varepsilon _k)_{k\in \mathbb {N}}\) this convergence in fact is uniform in \(\Omega \times (0,1)\). Indeed, once this has been shown, (4.32) and hence the regularity requirements in (4.1) are satisfied, and the validity of (4.2) can readily be checked on testing (4.22) by \(\varphi \in C^1(\bar{\Omega }\times [0,t])\) and using (4.33) in a straightforward manner; finally, it will then follow from the uniqueness of weak solutions and a standard argument that the convergence statements in (4.33) actually hold along the entire net \(\varepsilon \searrow 0\).

To assert the claimed convergence property, given \(\eta >0\) we fix \(\overline{u}_0 \in C^1(\bar{\Omega })\) such that \(u_0 \le \overline{u}_0 \le u_0+\frac{\eta }{4}\). From the classical comparison principle we then obtain that the corresponding solutions of (4.22) satisfy \(\overline{u}_\varepsilon \ge u_\varepsilon \) in \(\Omega \times (0,\infty )\). Moreover, another application of Lemma 4.7 shows that since \(\overline{u}_0 \in C^1(\bar{\Omega })\), along a subsequence – still denoted by \((\varepsilon _k)_{k\in \mathbb {N}}\) – we have \(\overline{u}_\varepsilon \rightarrow \overline{u}\) in \(C^0_{loc}(\bar{\Omega }\times [0,\infty ))\) with a certain limit function \(\overline{u}\). Now since \(\overline{u}\) is continuous at \(t=0\), there exists \(\tau _\eta \in (0,1)\) such that \(\overline{u}(\cdot ,t) \le \overline{u}_0+\frac{\eta }{8}\) for all \(t\in (0,\tau _\eta )\), and since \(\overline{u}_\varepsilon \rightarrow \overline{u}\) uniformly in \(\Omega \times (0,\tau _\eta )\) as \(\varepsilon =\varepsilon _k\searrow 0\), this implies that \(\overline{u}_\varepsilon (\cdot ,t) \le \overline{u}_0+\frac{\eta }{4}\) for all \(t\in (0,\tau _\eta )\), provided that \(\varepsilon \in (\varepsilon _k)_{k\in \mathbb {N}}\) is suitably small. Thus, for any such \(\varepsilon \) we find that

and by a similar approximation from below we obtain that on passing to a further subsequence and diminishing \(\tau _\eta \) if necessary, also

holds for all sufficiently small \(\varepsilon \in (\varepsilon _k)_{k\in \mathbb {N}}\). Combining (4.35) and (4.36) shows that

for all small \(\varepsilon ,\varepsilon ' \in (\varepsilon _k)_{k\in \mathbb {N}}\). Together with (4.34), this proves that \((u_{\varepsilon _k})_{k\in \mathbb {N}}\) forms a Cauchy sequence in \(C^0(\bar{\Omega }\times [0,1])\), as desired. \(\square \)

Proof of Theorem 2.1

The existence statement is part of the result provided by Lemma 4.9. Uniqueness and the estimate (2.1) are precisely asserted by Corollaries 4.5 and 4.4. \(\square \)

Appendix B: Weak steady states. Proof of Proposition 2.2

The natural counterpart of Definition 4.1 is the following.

Definition 5.1

A stationary weak (sub-/super-)solution of (1.3) is a nonnegative function \(w \in L^\infty (\Omega ) \cap W^{1,2}(\Omega )\) satisfying

for all nonnegative \(\psi \in C^1(\bar{\Omega })\).

We proceed to identify a family of stationary weak sub- and supersolutions of (1.3).

Lemma 5.2

Suppose that \(\Omega _0 \subset \Omega \) is open, and that

with some \(a\in \mathbb {R}\). Then

defines a stationary weak subsolution of (1.3) if \(a \le 0\), and a stationary weak supersolution of (1.3) if \(a\ge 0\).

Proof

We first note that (5.2) asserts that \(w\) is nonnegative and continuous in \(\bar{\Omega }\) and hence also belongs to \(W^{1,2}(\Omega )\), because \(m\in C^1(\bar{\Omega })\). We thus only need to check the respective integral inequalities in (5.1).

In fact, if \(a\ge 0\) then for any nonnegative \(\psi \in C^1(\bar{\Omega })\) we have

because \(a=const.\), and

for \(m+a \ge 0\) in \(\Omega _0\) by (5.2). This shows that \(w\) is a stationary weak supersolution in this case, whereas the subsolution property in the case \(a\le 0\) can be seen similarly. \(\square \)

This already proves one part of Proposition 2.2:

Corollary 5.3

Let \(\sigma =(\sigma _i)_{i\in \mathbb {N}} \subset \{0,1\}\). Then the function \(m_\sigma \) defined in (1) is a stationary weak solution of (1.3).

Proof

We only need to apply Lemma 5.2 to the set

which clearly satisfies (5.2) with \(a=0\). \(\square \)

In fact, the above corollary already reveals all conceivable steady states. In order to verify this, and thus to complete the proof of Proposition 2.2, we provide an elementary preparation.

Lemma 5.4

Suppose that \(w\in C^0(\bar{\Omega })\) is nonnegative and such that

and let \(G\subset \{m>0\}\) be open and connected. Then either \(w\equiv 0\) in \(G\) or \(w\equiv m\) in \(G\).

Proof

By continuity of \(w\) and \(m\), the sets

are both open, and we first claim that \(G_1 \cap G_2 = \emptyset \). Indeed, this results from the fact that if \(x\in G_1\) then \(w(x)>0\), and hence (5.4) enforces that \(w(x)=m(x)\), that is, \(x\not \in G_2\).

Moreover, (5.4) along with the positivity of \(m\) inside \(G\) implies that \(G=G_1 \cup G_2\): To see this, we pick any \(x\in G\) and then obtain from (5.4) that either \(w(x)=0\) or \(w(x)=m(x)\). In the former case, clearly \(w(x)\ne m(x)\) and thus \(x\in G_2\), whereas in the latter we have \(w(x)>0\) and therefore \(x\in G_1\).

Now since \(G\) is connected, the above properties imply that either \(G_1=G\) or \(G_2=G\), again in view of (5.4) meaning that either \(w\equiv m\) in \(G\) or \(w\equiv 0\) in \(G\). \(\square \)

We are now in the position to fully characterize the set of all weak equilibria of (1.3).

Proof

of Proposition 2.2.    In view of Corollary 5.3, we only need to make sure that any stationary weak solution \(w\) of (1.3) is of the form in (2.2) with some \((\sigma _i)_{i\in \mathbb {N}} \subset \{0,1\}\). To see this, we first observe that since \(\Omega \times (0,\infty ) \ni (x,t) \mapsto w(x)\) is a global weak solution of (1.3), it follows from (Porzio and Vespri (2003), Theorem 1.3) that \(w\) is continuous in \(\bar{\Omega }\). Since (5.1) clearly extends so as to remain valid for each \(\psi \in W^{1,2}(\Omega )\), we may choose \(\psi :=w-m\) as a test function there to obtain

Since \(w\) is nonnegative, the claim therefore easily results from Lemma 5.4. \(\square \)

Appendix C: Large time behavior. Proof of Theorem 2.3

We next examine the large time behavior of the solution \(u\) gained above. Our analysis in this direction will be based on the integral inequality (6.2) satisfied by the approximate solutions \(u_\varepsilon \). This inequality can be viewed as an approximate version of the energy identity

formally associated with (1.3).

Lemma 6.1

Suppose that \(u_0\in C^0(\bar{\Omega }\) be nonnegative, and let \(\varepsilon \in (0,1)\). Then given any \(\eta \in (0,1)\), the solution \(u_\varepsilon \) of (4.22) satisfies the inequality

for all \(t>0\).

Proof

We test (4.22) against \(u_\varepsilon +\varepsilon \ln u_\varepsilon - m\) to obtain

for \(t>0\). On simple rearrangements, we see that

and

where by Young’s inequality

so that (6.2) is a consequence of (6.3). \(\square \)

We now take \(\varepsilon \searrow 0\) here to conclude that \(u\) at least satisfies a weakened version of (6.1).

Lemma 6.2

Assume that \(u_0 \in C^0(\bar{\Omega })\) is nonnegative, and let \(t_0\ge 0\). Then for the solution \(u\) of (1.3) we have

In particular,

and

Proof

For fixed \(t_0>0\) and \(t>t_0\) and arbitrary \(\eta \in (0,1)\) we may integrate (6.2) over the time interval \((t_0,t)\) to see upon dropping a nonnegative term on the left that

holds for all \(\varepsilon \in (0,1)\). Now according to Lemma 4.6 and the nonnegativity of \(u_\varepsilon \) we know that there exist \(c_1>0\) and \(c_2>0\) such that

for all \(\varepsilon \in (0,1)\), whence it follows that

Moreover, using the uniform convergence \(u_{\varepsilon } \rightarrow u\) in \(\bar{\Omega }\times [t_0,t]\) asserted by Lemma 4.9 we find that

and

as \(\varepsilon \searrow 0\). In view of (6.8), we thus conclude from (6.7) in the limit \(\varepsilon =\searrow 0\) that

for each \(\eta \in (0,1)\), which clearly implies (6.4). \(\square \)

Remark

In fact, in accordance with (6.2) it is possible to include a nonnegative space–time integral involving \(\nabla u\) on the left-hand side of (6.4), and thus obtain an energy inequality more closely related to (6.1). The additional information thereby obtained with regard to the total energy dissipation is not needed in the sequel, however.

As a first – though yet rather weak – information on the asymptotics of \(u\), the inequality (6.6) says that there exists some \((t_k)_{k\in \mathbb {N}} \subset (0,\infty )\) such that \(t_k \rightarrow \infty \) and

as \(k\rightarrow \infty \). This will be sharpened in the following lemma so as to cover convergence along the entire net \(t\rightarrow \infty \).

Lemma 6.3

For any choice of \(0 \le u_0 \in C^0(\bar{\Omega })\), the solution \(u\) of (1.3) satisfies

Proof

Let us set

Then assuming that the claim be false, we would obtain \((t_k)_{k\in \mathbb {N}} \subset (1,\infty )\) and \(\delta >0\) such that \(t_k\rightarrow \infty \) as \(k\rightarrow \infty \) and

where we may assume without loss of generality that \(t_{k+1} \ge t_k+1\) for all \(k\in \mathbb {N}\). Now since \(u\) is uniformly continuous in \(\bar{\Omega }\times [1,\infty )\) according to Lemma 4.7, it follows that \(I\) is uniformly continuous in \([1,\infty )\). It is therefore possible to find \(\tau \in (0,1)\) such that

which implies that

Consequently, an integration in time shows that

and thereby contradicts Lemma 6.2. \(\square \)

Now thanks to the fact that \((u(\cdot ,t))_{t>1}\) is equicontinuous by Lemma 4.7, one particular outcome of Lemma 6.3 is that for each individual \(x\in \bar{\Omega }\) we either have \(u(x,t)\rightarrow 0\) or \(u(x,t)\rightarrow m(x)\) as \(t\rightarrow \infty \). The main concern of the following lemma is to make sure that within this alternative the selection is in fact independent of \(x \in G\) for each component \(G\) of \(\{m>0\}\).

Lemma 6.4

Assume that \(u_0 \in C^0(\bar{\Omega })\) is nonnegative, and let \(u\) denote the solution of (1.3). Moreover, suppose \((t_k)_{k\in \mathbb {N}} \subset (1,\infty )\) is such that \(t_k\rightarrow \infty \) and

as \(k\rightarrow \infty \) with some \(u_\infty \in C^0(\bar{\Omega })\). Then for each component \(G\) of \(\{m>0\}\),

Proof

According to Lemma 6.3, \(u_\infty \) satisfies the identity \(u_\infty \cdot (u_\infty -m)^2 \equiv 0\) in \(\Omega \). Therefore, Lemma 5.4 states that either \(u_\infty \equiv 0\) in \(G\) or \(u_\infty \equiv m\) in \(G\). \(\square \)

As a crucial step towards the proof of Theorem 2.3, we next make sure that for all sufficiently small nontrivial initial data which vanish outside some component \(G\) of \(\{m>0\}\), always the latter alternative in (6.12) occurs. The proof of this will essentially rely on the monotonicity property in Lemma 6.2, which rules out the possibility that such a solution, being nontrivial and lying below \(m\) in \(G\), decays to zero in the large time limit. Solutions satisfying such special initial conditions will play an important role as comparison functions in the proof of Lemma 6.6.

Lemma 6.5

Let \(G\) be a component of \(\{m>0\}\), and let \(u_0 \in C^0(\bar{\Omega })\) be nonnegative and such that \(u_0 \not \equiv 0\) and

Then the solution \(u\) of (1.3) satisfies

Proof

Since \(m \cdot \chi _G\) is a stationary weak solution of (1.3) by Proposition 2.2, (6.13) together with the comparison principle in Lemma 4.2 implies that

Now suppose that (6.14) be false. Since \((u(\cdot ,t))_{t>1}\) is relatively compact in \(C^0(\bar{\Omega })\) by Lemma 4.7, we can then find a sequence \((t_k)_{k\in \mathbb {N}} \subset (1,\infty )\) fulfilling \(t_k\rightarrow \infty \) and \(u(\cdot ,t_k) \rightarrow u_\infty \) in \(C^0(\bar{\Omega })\) as \(k\rightarrow \infty \) with some nonnegative \(u_\infty \in C^0(\bar{\Omega })\) such that \(u_\infty \not \equiv m \cdot \chi _G\). But from (6.15) we know that \(u_\infty \le m\cdot \chi _G\) in \(\Omega \), so that in light of Lemma 6.4 we must have \(u_\infty \equiv 0\) in \(\Omega \). This means that \(u(\cdot ,t_k) \rightarrow 0\) in \(C^0(\bar{\Omega })\), whence in particular

However, when combined with the downward monotonicity of \(t\mapsto \int _\Omega (u(\cdot ,t)-m)^2\) as asserted by Lemma 6.2, (6.16) says that we should have

This implies that

and hence

because \(u_0 \le m \cdot \chi _G\). Since \(u_0 \cdot m\) is nontrivial and nonnegative according to our assumption, this is impossible, so that actually (6.14) must be valid. \(\square \)

Based on the above lemma, we can proceed to prove the main part of Theorem 2.3 by means of a comparison argument.

Lemma 6.6

Let \(u_0\in C^0(\bar{\Omega })\) be nonnegative. Then for each component \(G\) of \(\{m>0\}\), the solution \(u\) of (1.3) has the property that

Proof

Suppose that \(u\not \equiv 0\) in \(G \times [0,\infty )\). Then there exists \(t_0 \ge 0\) such that \(u(\cdot ,t_0) \not \equiv 0\) in \(G\). Since \(u(\cdot ,t_0)\) is continuous, we can therefore pick some nontrivial nonnegative \(\psi \in C_0^\infty (\Omega )\) such that \(\mathrm{supp} \, \psi \subset G\) and \(\psi \le u(\cdot ,t_0)\) in \(\Omega \) , where we can also achieve that \(\psi \le m\) in \(G\), because \(m\) is positive in \(G\).

We now let \(\underline{u}\) denote the weak solution of

which by uniqueness can be obtained by letting \(\underline{u}(x,t):=\tilde{u}(x,t-t_0)\), \((x,t) \in \bar{\Omega }\times [t_0,\infty )\), where \(\tilde{u}\) solves (1.3) with initial data \(\psi \).

According to Lemma 6.5, since \(\psi \le m \cdot \chi _G\) in \(\Omega \) and \(\psi \not \equiv 0\), we have

On the other hand, a comparison argument involving Lemma 4.2 asserts that \(u\ge \underline{u}\) in \(\Omega \times (t_0,\infty )\). In conjunction with (6.17) this shows that

so that an application of Lemma 6.4 warrants that indeed \(u(x,t) \rightarrow m(x)\) as \(t\rightarrow \infty \), uniformly with respect to \(x\in G\). \(\square \)

Now our main result on the asymptotics in (1.3) actually reduces to a corollary:

Proof of Theorem 2.3

The claimed solution behavior in \(\{m>0\}\) is precisely described by Lemma 6.6. Outside \(\{m>0\}\), however, Lemma 6.3 and Lemma 4.7 easily imply that \(u(\cdot ,t) \rightarrow 0\) in \(C^0(\{m \le 0\})\) as \(t\rightarrow \infty \). \(\square \)

Appendix D: Decay to zero. Proof of Proposition 2.4

Clearly, Theorem 2.3 implies that if \(m\le 0\) throughout \(\Omega \) then any solution \(u\) of (1.3) will satisfy \(u(\cdot ,t) \rightarrow 0\) in \(C^0(\bar{\Omega })\) as \(t\rightarrow \infty \). In the case when \(\{m>0\}\) is not empty, however, the question whether such nontrivial but decaying solutions exist appears to be nontrivial. The following lemma provides our main step in identifying a class of situation where such phenomena occur.

Lemma 7.1

Suppose that \(\Omega _1 \subset \Omega \) is open and such that

Then there exists a nontrivial nonnegative \(w \in C^0(\bar{\Omega })\) with \(w \equiv 0\) in \(\Omega \setminus \Omega _1\) such that whenever \(u_0 \in C^0(\bar{\Omega })\) is nonnegative fulfilling \(u_0 \le w\) in \(\Omega \), the solution \(u\) of (1.3) satisfies

Proof

Fixing \(x_1 \in \bar{\Omega }_1\) such that \(m(x_1)=M:=\sup _{x\in \Omega _1} m(x)\), we know from (7.1) that \(x_1\) is an interior point of \(\Omega _1\), and that moreover for some nonnegative \(a\) with \(-a<M\) we have

We now let \(\Omega _0\) denote the component of \(\{m>-a\}\) which contains \(x_1\), and then evidently have \(m+a=0\) throughout \(\partial \Omega _0 \cap \Omega \). Since \(a\ge 0\), Lemma 5.2 says that the nonnegative function \(w\) defined in (5.3) is a stationary weak supersolution of (1.3), and the fact that \(-a<M\) asserts that \(w(x_1)\ne 0\), hence \(w\not \equiv 0\). As \(w\equiv 0\) in \(\Omega {\setminus } \Omega _0 \supset \Omega {\setminus } \Omega _1\), the claim thus results upon an application of the comparison principle in Lemma 4.2. \(\square \)

Now combined with Theorem 2.3, the above indeed yields a family of such decaying solutions under the assumption that \(m\) has the property (2.7).

Proof of Proposition 2.4

Observing that (2.7) is stronger than (7.1), we can apply Lemma 7.1 to fix \(w \in C^0(\bar{\Omega })\) with the properties listed there. we then take any nonnegative \(u_0 \in C^0(\bar{\Omega })\) such that \(u_0 \le w\) in \(\Omega \) and let \(u\) denote the solution of (1.3). Then (7.2) says that we trivially have \(u(\cdot ,t) \rightarrow 0\) in \(C^0(\bar{\Omega }{\setminus } \Omega _1)\) as \(t\rightarrow \infty \). Inside \(\Omega _1\), however, we know that \(m\le 0\) by (2.7), and hence Theorem 2.3 guarantees that \(u(\cdot ,t) \rightarrow 0\) also in \(C^0(\bar{\Omega }_1)\) as \(t\rightarrow \infty \). \(\square \)

Appendix E: Stability and attractivity properties of equilibria

1.1 Stability

The proof of Proposition 2.5 will be an immediate consequence of the following more general result.

Proposition 8.1

Let \(w\) be a stationary weak solution of (1.3). Then \(w\) is stable from below in both \(X_1\) and \(X_\infty \). Moreover, \(m_+\) is stable from above in \(X_1\), while if \(w\ne m_+\) then \(w\) is unstable from above in both \(X_1\) and \(X_\infty \).

Proof

We first note that any weak solution \(u\) of (1.3) satisfies

which can easily be seen on choosing \(\varphi \equiv 1\) as a test function in (4.2).

Now let \(w\) be a weak steady state of (1.3), so that \(w\equiv m\) in \(\{w>0\}\) according to Proposition 2.2. Moreover, let \(u_0 \in C^0(\bar{\Omega })\) be such that \(0\le u_0 \le w\) in \(\Omega \). Then thanks to the comparison principle in Lemma 4.2 we know that \(u(\cdot ,t)\le w\) in \(\Omega \), whence in particular \(u(\cdot ,t) \le m\) in \(\{w>0\}\) and \(u(\cdot ,t)\equiv 0\) in \(\Omega {\setminus } \{w>0\}\) for all \(t>0\). Therefore (8.1) implies

Accordingly,

which entails that \(w\) is stable from below in \(X_1\).

Next, given \(u_0\in C^0(\bar{\Omega })\) such that \(0 \le u_0 \le w\) in \(\Omega \), we let \(\delta :=\Vert w-u_0\Vert _{L^\infty (\Omega )}\), so that \(u_0 \ge (w-\delta )_+\) in \(\Omega \). Since \((w-\delta )_+\) is a stationary weak subsolution of (1.3) by Lemma 5.2, the comparison principle in Lemma 4.2 says that \(w\ge u(\cdot ,t) \ge (w-\delta )_+\) in \(\Omega \) and thus also

implying stability from below in \(X_\infty \).

The stability of \(w=m_+\) from above in \(X_1\) can be seen similarly: Namely, if \(u_0 \in C^0(\bar{\Omega })\) is such that \(u_0 \ge m_+\) in \(\Omega \) then \(u(\cdot ,t)\ge m_+ \ge m\) by comparison and hence

implying that

Finally, to see that any weak steady state \(w\ne m_+\) is unstable from above in both \(X_1\) and \(X_\infty \), it is evidently sufficient to observe that from each \(\delta >0\), the solution \(u\) of (1.3) emanating from \(u_0:=w+\delta \) satisfies \(u(\cdot ,t)\rightarrow m_+ \not \equiv w\) in \(C^0(\bar{\Omega })\) as \(t\rightarrow \infty \). \(\square \)

Our proof of the fact that \(m_+\) is stable from above also with respect to the norm in \(L^\infty (\Omega )\) is more involved, and it will be postponed to the separate Lemma 8.4 below. Before, we provide two preparatory lemmata which address approximate versions of the steady-state problem associated with (1.3).

Lemma 8.2

There exists a sequence \((w_k)_{k\in \mathbb {N}} \subset C^2(\bar{\Omega })\) of positive classical solutions to the boundary-value problems

which satisfy

whenever \(k\ge 2\).

Proof

Given \(k\in \mathbb {N}\), we let

and can then fix a positive function \(\phi _k\in C^\infty (\mathbb {R})\) such that \(\phi _k(s)=s\) for all \(s\ge \eta _k\) and \(\phi _k'>0\) on \(\mathbb {R}\). Therefore, the boundary-value problem

is uniformly elliptic, and we claim that for each \(k\in \mathbb {N}\) it possesses a classical solution \(w_k\in C^2(\bar{\Omega })\) such that

with a suitably function \(w_0 \in C^2(\bar{\Omega })\). To construct the latter, we first fix any \(\psi \in C^2(\bar{\Omega })\) such that \(\frac{\partial \psi }{\partial \nu } \le -1\) on \(\partial \Omega \), and then choose a positive constant \(A\) large enough fulfilling

and

as well as

which is clearly possible because \(m\in C^1(\bar{\Omega })\) and \(\psi \in C^2(\bar{\Omega })\).

Upon these choices, we let \(w_0\in C^2(\bar{\Omega })\) be defined by

and then obtain from (8.9) that

In particular, \(w_0\ge 1\) and hence \(\phi _1(w_0)\equiv w_0\), so that

according to (8.10). Moreover, (8.11) implies that

because \(\frac{\partial \psi }{\partial \nu }\le -1\) on \(\partial \Omega \).

We next make sure that for all \(k\in \mathbb {N}\), the functions defined by

both are subsolutions with regard to (8.7). Indeed, using (8.6) we compute

and

Similarly,

by (8.6) and, obviously,

It thus follows that \(\underline{w}_k:=\min \{\hat{w}_k,\tilde{w}_k\} \equiv m_+ +\eta _k\) is a subsolution of (8.7) in the Nagumo sense, which satisfies \(\underline{w}_k \le w_0\) in \(\Omega \) thanks to (8.12). Consequently, standard monotonicity methods for elliptic boundary-value problems (Tomi 1969) can be applied to first assert the existence of a classical solution \(w_1 \in C^2(\bar{\Omega })\) of (8.7) complying with (8.8).

Next, in order to recursively construct \(w_l\) for \(k\ge 2\), we suppose that for some \(k\ge 2\) we already have found \(w_{k-1}\in C^2(\bar{\Omega })\) fulfilling \(\mathcal{E}_{k-1} w_{k-1}=0\) in \(\Omega \), \(\mathcal{B}_{k-1} w_{k-1}=0\) on \(\partial \Omega \) and \(w_{k-1} \ge m_+ +\eta _{k-1}\) in \(\Omega \). Then since \(\eta _{k-1} \ge \eta _k\) by (8.6), this implies that \(w_{k-1} \ge m_+ + \eta _k=\underline{w}_k\), and moreover we have

as well as

Therefore, \(\underline{w}_k\) and \(w_{k-1}\) form a properly ordered pair of sub- and supersolutions for (8.7) on the sense of Nagumo, whence again we infer the existence of a solution lying in between, that is, of a classical solution \(w_k\in C^2(\bar{\Omega })\) fulfilling (8.8). \(\square \)

Lemma 8.3

The solutions \(w_k\) of (8.4) constructed in Lemma 8.2 satisfy

Proof

Since \(w_k>0\) in \(\bar{\Omega }\) by (8.5), we may test (8.4) by \(\ln w_k\) to find that

Now from (8.5) we obtain that there exists \(c_1>0\), independent of \(k\in \mathbb {N}\), such that \(w_k \le c_1\) in \(\Omega \). Since \(\int _\Omega \nabla w_k \cdot \nabla m \le \frac{1}{2} \int _\Omega |\nabla w_k|^2 + \frac{1}{2} \int _\Omega |\nabla m|^2\) by Young’s inequality, it thus follows from (8.14) that \((w_k)_{k\in \mathbb {N}}\) is bounded in \(W^{1,2}(\Omega )\). In light of the ordering in (8.5), this means that there exists \(w\in W^{1,2}(\Omega ) \cap L^\infty (\Omega )\) such that \(w_k\searrow w\) in \(\bar{\Omega }\) and \(w_k \rightharpoonup w\) in \(W^{1,2}(\Omega )\) as \(k\rightarrow \infty \), where the former combined with Beppo Levi’s theorem implies that also \(w_k \rightarrow w\) in \(L^p(\Omega )\) as \(k\rightarrow \infty \) for any \(p\in (1,\infty )\). Given any \(\psi \in C^1(\bar{\Omega })\), in the identity

gained from (8.4) on testing by \(\psi \), we may thus let \(k\rightarrow \infty \) in each integral separately to conclude that \(w\) is a weak solution of the limit problem (1.3) in the sense of Definition 5.1. However, since \(w\ge m_+\) in \(\Omega \) by (8.5), in view of Proposition 2.2 this already means that \(w\equiv m_+\) in \(\Omega \). In particular, \(w\) is continuous in \(\bar{\Omega }\), and hence Dini’s theorem asserts that the monotone convergence \(w_k\rightarrow w\) is even uniform in \(\bar{\Omega }\). \(\square \)

Lemma 8.4

The stationary weak solution \(m_+\) of (1.3) is stable from above in \(X_\infty \).

Proof

Given \(\delta >0\), from Lemma 8.3 we obtain some large \(k_\delta \in \mathbb {N}\) such that the solution \(w_{k_\delta }\) of (8.4) constructed in Lemma 8.2 there satisfies

Since \(w_{k_\delta }>m_+\) in \(\Omega \) by (8.5), the number

is positive. Now let \(u_0 \in C^0(\bar{\Omega })\) be nonnegative and such that \(u_0 \le m_+ + \eta _\delta \) in \(\Omega \). Then \(u_0 \le w_{k_\delta }\) in \(\Omega \) by (8.16), and since \(w_{k_\delta }\) clearly is a stationary weak supersolution of (1.3), by comparison we conclude that the solution \(u\) of (1.3) satisfies \(u(\cdot ,t)\le w_{k_\delta }\) in \(\Omega \) and hence, according to (8.15), \(u(\cdot ,t) \le m_+ + \delta \) in \(\Omega \) for all \(t>0\). \(\square \)

Summarizing, we obtain the announced result on stability properties of all weak steady states of (1.3).

Proof of Proposition 2.5

The claim is part of the statements of Proposition 8.1 and Lemma 8.4. \(\square \)

1.2 Domains of attraction

We now investigate the attractivity properties of the weak steady states of (1.3) in more detail.

Proof of Proposition 2.6

Let \(\sigma =(\sigma _i)_{i\in \mathbb {N}} \subset \{0,1\}\), and suppose that \(u_0 \in C^0(\bar{\Omega })\) is nonnegative and such that \(u_0 \le m_\sigma \) and \(u_0 \not \equiv 0\) in each component \(G_i\) of \(\{m>0\}\) with \(\sigma _i=1\). Then according to the comparison principle in Lemma 4.2 we have \(u(\cdot ,t) \le m_\sigma \) in \(\Omega \) for all \(t>0\). Thus it follows that we have \(u\not \equiv 0\) in \(G_i \times [0,\infty )\) for some \(i\in \mathbb {N}\) if and only if \(\sigma _i=1\). Therefore Theorem 2.3 says that \(u(\cdot ,t) \rightarrow m_\sigma \) in \(X\) as \(t\rightarrow \infty \). \(\square \)

The proof of Proposition 2.7 is also immediate.

Proof of Proposition 2.7

Since \(X_+:=\{\psi \in X \ | \ \psi >0 \text{ in } \bar{\Omega }\}\) is dense in \(X\), (2.8) implies that \(\mathcal{D}(m_+)\) contains the dense set \(X_+\) and thus itself is dense in \(X\). On the other hand, if \(w\ne m_+\) then again using (2.8) it can easily be checked that any \(u_0\in X_+\) does not belong to the closure of \(\mathcal{D}(w)\). \(\square \)

Some more efforts are necessary in the study of openness.

Proof of Proposition 2.8

For definiteness, in giving details we fix \(X=X_\infty \); the case \(X=X_1\) can be addressed using minor modifications.

First, since for each \(\delta >0\) the function \(w+\delta \) belongs to \(\mathcal{D}(m_+)\) by (2.8), it follows that \(\mathcal{D}(w)\) is not open in \(X\) whenever \(w\ne m_+\).

Next, assume that \(w=m_+\) and that \(\{m>0\}\) has infinitely many components, whence in the notation of Proposition 2.2 we have \(G_0 \ne \emptyset \) for infinitely many \(i\in \mathbb {N}\). Then if \(\mathcal{D}(m_+)\) were open in \(X\), there would exist \(\delta >0\) such that

Now it is easy to check that by continuity of \(m\) in \(\bar{\Omega }\) we must necessarily have \(\Vert m\Vert _{L^\infty (G_i)} \rightarrow 0\) as \(i\rightarrow \infty \). It is therefore possible to fix some large \(i\in \mathbb {N}\) such that \(G_i \ne \emptyset \) and \(\Vert m\Vert _{L^\infty (G_i)} \le \frac{\delta }{2}\). Then

clearly coincides with one of the weak steady states identified in Proposition 2.2, and hence the corresponding solution \(u\) of (1.3) satisfies \(u(\cdot ,t)\equiv u_0\) for all \(t>0\). Since \(G_i \ne \emptyset \), however, we have \(u_0 \not \equiv m_+\) in \(\Omega \), so that \(u_0\) does not belong to \(\mathcal{D}(m_+)\). This contradiction to (8.17) shows that \(\mathcal{D}(m_+=\) is not open in \(X\) in this case.

It remains to be verified that \(\mathcal{D}(m_+)\) is open in \(X\) if \(\{m>0\}\) consists of at most finitely many components only, where in the case \(\{m>0\}=\emptyset \) this is obvious, because then Theorem 2.3 says that \(\mathcal{D}(m_+)=X\). Thus, still referring to the notation from Proposition 2.2, we may assume that for some \(n\in \mathbb {N}, \{m>0\}\) precisely consists of the nonempty and mutually disjoint components \(G_1,\ldots ,G_n\). We then fix and arbitrary \(u_0\in \mathcal{D}(m_+)\) and let \(u\) denote the solution of (1.3) evolving from \(u_0\). In order to construct a suitable neighborhood of \(u_0\) which is fully contained in \(\mathcal{D}(m_+)\), we introduce the positive numbers

Then since \(u_0 \in \mathcal{D}(m_+)\) implies that \(u(\cdot ,t) \rightarrow m_+\) with respect to the norm in \(L^1(\Omega )\), we can find \(t_0>0\) such that

In particular, this means that

Now let

with \(M_0:=\Vert m_+\Vert _{L^\infty (\Omega )}\) as in Lemma 4.2. Then for any \(\tilde{u}_0 \in X\) fulfilling \(\Vert \tilde{u}_0-u_0\Vert _{L^\infty (\Omega )} < \delta \), according to Corollary 4.4 the corresponding solution \(\tilde{u}\) of (1.3) satisfies

Therefore, using (8.18) we find that

which entails that \(\tilde{u}(\cdot ,t_0) \not \equiv 0\) in \(G_i\) for each \(i\in \{1,\ldots ,n\}\). In light of Theorem 2.3, this means that as \(t\rightarrow \infty \) we have \(\tilde{u}(\cdot ,t)\rightarrow m\) in \(C^0(\bar{G}_i)\) for all \(i\in \{1,\ldots ,n\}\) and thus implies that \(\tilde{u}(\cdot ,t) \rightarrow m_+\) in \(C^0(\bar{\Omega })\), because \(\{m>0\}=\bigcup _{i=1}^n G_i\). Accordingly, we conclude that indeed \(\tilde{u}_0 \in \mathcal{D}(m_+)\) for any such \(\tilde{u}_0\), and that hence \(\mathcal{D}(m_+)\) is open in \(X\). \(\square \)

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