Asymptotic behavior of solutions for nonlinear parabolic operators with natural growth term and measure data

Abstract

We are interested in the asymptotic behavior, as t tends to $+\infty$, of finite energy solutions and entropy solutions $u_{n}$ of nonlinear parabolic problems whose model is

$$\begin{aligned} {\left\{ \begin{array}{ll} u_{t}-\Delta _{p}u+g(u)|\nabla u|^{p}=\mu &{}\text { in }(0,T)\times \Omega ,\\ u(0,x)=u_{0}(x)&{}\text { in }\Omega ,\\ u(t,x)=0&{}\text { on }(0,T)\times \partial \Omega \end{array}\right. } \end{aligned}$$

(0.1)

where $\Omega \subseteq \mathbb {R}^{N}$ is a bounded open set, $N\ge 3$, $u_{0}\in L^{1}(\Omega )$ is a nonnegative initial data, while $g:\mathbb {R}\mapsto \mathbb {R}$ is a real function in $C^{1}(\mathbb {R})$ which satisfies sign condition with positive derivative and $\mu$ is a nonnegative measure independent on time which does not charge sets of null p-capacity.

Résumé

Comportement asymptotique des solutions pour des opérateurs paraboliques non linéaire avec un terme de croissance naturelle et une donnée mesure. Nous somme interessés au comportement asymptotique, quant t tend vers $+\infty$, des solutions énergétiques finies et des solutions entropiques $u_{n}$ des problèmes paraboliques non linéaires dont le modèle est (0.1) où $\Omega \subseteq \mathbb {R}^{N}$ est un ouvert borné, $N\ge 3$, $u_{0}\in L^{1}(\Omega )$ est une donnée initiale non négative, tandis que $g:\mathbb {R}\mapsto \mathbb {R}$ est une fonction réelle de classe $C^{1}(\mathbb {R})$ qui satisfait la condition du signe avec une dérivée positive et $\mu$ est une mesure non négative indépendante du temps qui ne prend pas en charge les parties de p-capacité nulle.

Appendix (In connection with G-convergence)

In this section, we shall study, following an idea of [81], the connection between the G-convergence of a sequence $(u_{\tau })_{\tau >0}$ and the asymptotic behaviour of the corresponding solutions $u_{\tau }(t,x)$ relative to the operators $A:u\mapsto \text {div}(a(x,u_{\tau })_{\tau >0})$ when $u_{\tau }(0,x)=u_{0}^{\tau }(x)=0$ in $\Omega$. We prove a convergence result for entropy solutions of a nonlinear parabolic problem with nonnegative measure $\mu \in {\mathcal {M}}_{0}(\Omega )$ with $\mu \ne 0$ using the theory of G-convergence [6, 19, 42, 72, 76, 81]. To this aim, let us consider the following initial boundary value problem

$$\begin{aligned} {\left\{ \begin{array}{ll} (u_{\tau })_{t}-\text {div}(a(x,u_{\tau }))+g(u_{\tau })|u_{\tau }|^{p}=\mu _{\tau }\text { in }Q=(0,T)\times \Omega ,\\ u_{\tau }(0,x)=0\text { in }\Omega ,\quad u_{\tau }(t,x)=0\text { on }(0,T)\times \partial \Omega \end{array}\right. } \end{aligned}$$

(5.1)

where $T>0$ is any positive constant and $\mu \in {\mathcal {M}}_{0}(\Omega )$ ($\mu \ne 0$) is a Radon measure with bounded variation which does not charge the sets of zero p-capacity and which does not depend on the time variable t (in accordance with the definition given in Theorem 2.3).

Theorem 5.1

Let $\mu _{\tau }\in {\mathcal {M}}_{0}(\Omega )$ be a measure such that $\mu _{\tau }\ne 0$. Let $u_{\tau }(t,x)$ ($\tau >0$) be the entropy solution of parabolic problem (5.1) corresponding to $\mu _{\tau }$ and $v_{\tau }(x)$ the entropy solution of the following corresponding elliptic problem

$$\begin{aligned} {\left\{ \begin{array}{ll} -{\text {div}}(a(x,v_{\tau }))+g(v_{\tau })|v_{\tau }|^{p}=\mu _{\tau }&{}\text { in }\Omega ,\\ u_{\tau }=0&{}\text { on }\partial \Omega . \end{array}\right. } \end{aligned}$$

(5.2)

Then, we have

$$\begin{aligned} \underset{\tau \rightarrow \infty }{\text {lim }}\underset{T\rightarrow \infty }{\text {lim }}u_{\tau }(t,x)=\underset{\tau \rightarrow \infty }{\text {lim }}v_{\tau }(x)=+\infty \text { a.e. in }\Omega . \end{aligned}$$

Proof

First, note that $u_{\tau }(0,x)=u_{0}^{\tau }(x)=0$ (this condition is essential in order to deal with some difficulties) and suppose that $\mu \in W^{-1,p'}(\Omega )$ (independent of time), we have

$$\begin{aligned} \mu \in W^{-1,p'}(\Omega )\quad \text { if and only if }\mu =-\text {div}(G)\text { with }G\in L^{p'}(\Omega )^{N}. \end{aligned}$$

(5.3)

Then for $\mu =\mu _{\tau }$ where

$$\begin{aligned} \mu _{\tau }={\left\{ \begin{array}{ll} \tau \mu &{}\text { if }f=0,\\ \tau f-\text {div}(G)&{}\text { if }f=0 \end{array}\right. } \end{aligned}$$

with $f\ge 0\in L^{1}(\Omega )$ and $G\in L^{p'}(\Omega )^{N}$. We have

$$\begin{aligned}&\int _{0}^{T}\langle (u_{\tau })_{t},\varphi \rangle dt+\int _{Q}a(x,\nabla u_{\tau })\cdot \nabla \varphi dx dt+\int _{Q}g(u_{\tau })|\nabla u_{\tau }|^{p}\varphi dx dt\nonumber \\&\quad =\tau \int _{Q}G\cdot \nabla \varphi dx dt \end{aligned}$$

(5.4)

for every $\varphi \in L^{p}(0,T;W^{1,p}_{0}(\Omega ))\cap L^{\infty }(Q)$ such that $\varphi _{t}\in L^{p'}(0,T;W^{-1,p'}(\Omega ))$ and $\varphi (T,x)=0$. Hence, for $\varphi =u_{\tau }$, (5.4) becomes

$$\begin{aligned}&\int _{0}^{T}\langle (u_{\tau })_{t},u_{\tau }\rangle dt+\int _{Q}a(x,\nabla u_{\tau })\cdot \nabla u_{\tau }dx dt+\int _{Q}g(u_{\tau })|\nabla u_{\tau }|^{p}u_{\tau }dx dt\nonumber \\&\quad =\tau \int _{Q}G\cdot \nabla u_{\tau }dx dt. \end{aligned}$$

(5.5)

Moreover, by (2.6) we have

$$\begin{aligned} \begin{aligned}&\int _{0}^{T}\langle (u_{\tau })_{t},u_{\tau }\rangle dt+\alpha \int _{Q}|\nabla u_{\tau }|^{p}dx dt+\int _{Q}g(u_{\tau })|\nabla u_{\tau }|^{p}u_{\tau }dx dt\\&\quad \le \tau \int _{Q}G\cdot \nabla u_{\tau }dx dt\le \tau \Vert G\Vert _{L^{p'}(\Omega )^{N}}\Vert \nabla u_{\tau }\Vert _{L^{p}(Q)^{N}} \end{aligned} \end{aligned}$$

(5.6)

and then, using the integration by parts and assumptions (2.14) (recall that $u_{0}^{\tau }(0)=0$) we get

$$\begin{aligned} \int _{\Omega }\frac{[u_{\tau }(T)]^{2}}{2}dx+\alpha \int _{Q}|\nabla u_{\tau }|^{p}dx dt\le \tau \Vert G\Vert _{L^{p'}(\Omega )^{N}}\Vert \nabla u_{\tau }\Vert _{L^{p}(Q)^{N}}. \end{aligned}$$

(5.7)

Now, since $u_{\tau }\ne 0$ and the fact that the first term is nonnegative, we can divide the above expression by $\tau \Vert \nabla u\Vert _{L^{p}(Q)^{N}}$, getting

$$\begin{aligned} \frac{1}{\tau }\left( \int _{Q}|\nabla u_{\tau }|^{p}dxdt\right) ^{\frac{p-1}{p}}=\left( \int _{Q}\left| \nabla \left( \frac{u_{\tau }}{\tau ^{\frac{1}{p-1}}}\right) \right| ^{p}dx dt\right) ^{\frac{p-1}{p}}\le \Vert G\Vert _{L^{p'}(\Omega )^{N}}.\nonumber \\ \end{aligned}$$

(5.8)

Therefore, we have that $\frac{u_{\tau }}{\tau ^{\frac{1}{p-1}}}$ is bounded in $L^{p}(0,T;W^{1,p}_{0}(\Omega ))$, and so there exist a function $u\in L^{p}(0,T;W^{1,p}_{0}(\Omega ))$ and a subsequence, such that, up to subsequences, $\frac{u_{\tau }}{\tau ^{\frac{1}{p-1}}}$ weakly converges in $L^{p}(0,T;W^{1,p}_{0}(\Omega ))$ (and then a.e.) to u as $\tau$ tends to infinity. So, it is enough to prove that $u>0$ almost everywhere on Q to conclude the proof. To this aim, for every $\tau >0$, let us define

$$\begin{aligned} a_{\tau }(x,\zeta )=\frac{1}{\tau }a\left( x,\tau ^{\frac{1}{p-1}}\zeta \right) \text { (for the }p\text {--Laplacian, we have }a_{\tau }\equiv a) \end{aligned}$$

(5.9)

and then we can easily check that such an operator satisfies assumptions (2.6)–(2.8) (with the same constants $\alpha$ and $\beta$). Now, $\frac{u_{\tau }}{\tau ^{\frac{1}{p-1}}}$ satisfies the parabolic problem

$$\begin{aligned} {\left\{ \begin{array}{ll} \tau ^{\frac{1}{p-1}}\left( \frac{u_{\tau }}{\tau ^{\frac{1}{p-1}}}\right) _{t}-\text {div}\left( a_{\tau }\left( x,\nabla (\frac{u_{\tau }}{\tau ^{\frac{1}{p-1}}})\right) \right) \\ +g\left( \tau ^{\frac{1}{p-1}}\left( \frac{u_{\tau }}{\tau ^{\frac{1}{p-1}}}\right) \right) \left| \tau ^{\frac{1}{p-1}}\nabla \left( \frac{u_{\tau }}{\tau ^{\frac{1}{p-1}}}\right) \right| ^{p}=\mu &{}\text { in }(0,T)\times \Omega ,\\ \frac{u_{\tau }}{\tau ^{\frac{1}{p-1}}}(0,x)=0&{}\text { in }\Omega ,\\ \frac{u_{\tau }}{\tau ^{\frac{1}{p-1}}}(t,x)=0&{}\text { on }(0,T)\times \partial \Omega \end{array}\right. } \end{aligned}$$

(5.10)

in a variational sense. Indeed, for every $\varphi \in L^{p}(0,T;W^{1,p}_{0}(\Omega ))\cap L^{\infty }(Q)$ such that $\varphi _{t}\in L^{p'}(0,T;W^{-1,p'}(\Omega ))$ and $\varphi (T,x)=0$, we have

$$\begin{aligned}&\int _{0}^{T}\left\langle (u_{\tau })_{t},\varphi \right\rangle dt+\int _{Q}a_{\tau }\left( x,\nabla \left( \frac{u_{\tau }}{\tau ^{\frac{1}{p-1}}}\right) \right) \cdot \nabla \varphi dx dt+\int _{Q}g(u_{\tau })|\nabla u_{\tau }|^{p}\varphi dx dt\nonumber \\&\quad =\int _{Q}G\cdot \nabla \varphi dx dt. \end{aligned}$$

(5.11)

Moreover, thanks to [6, Theorem 3.1], we have that the family of operators $(a_{\tau })$G-converges in the class of Leray–Lions operators, that is, there exists a Carathéodory function ${\overline{a}}$ satisfying assumptions (2.6)–(2.8), and a sequence of indices $\tau _{k}=\tau (k)$ (called $\tau$ again) such that

$$\begin{aligned} a_{\tau }(x,\nabla u_{\tau })\underset{\text { G-converges }}{\longrightarrow }{\overline{a}}(x,\nabla u_{\tau }). \end{aligned}$$

(5.12)

So, because of that, being u the weak limit of $\frac{u_{\tau }}{\tau ^{\frac{1}{p-1}}}$ in $L^{p}(0,T;W^{1,p}_{0}(\Omega ))$, we get that

$$\begin{aligned} a\left( x,\nabla \left( \frac{u_{\tau }}{\tau ^{\frac{1}{p-1}}}\right) \right) \underset{\tau _{k}\rightarrow \infty }{\rightharpoonup }{\overline{a}}(x,\nabla u)\text { weakly in }L^{p'}(Q)^{N}. \end{aligned}$$

(5.13)

Therefore, using this result in (5.10), we have

$$\begin{aligned} \int _{0}^{T}\langle u_{t},\varphi \rangle dt+\int _{Q}{\overline{a}}(x,\nabla u)\cdot \nabla \varphi dx dt+\int _{Q}g(u)|\nabla u|^{p}\varphi dx dt=\int _{Q}G\cdot \nabla \varphi dx dt\nonumber \\ \end{aligned}$$

(5.14)

for every $\varphi \in L^{p}(0,T;W^{1,p}_{0}(\Omega ))\cap L^{\infty }(Q)$ such that $\varphi _{t}\in L^{p'}(0,T;W^{-1,p'}(\Omega ))$ and $\varphi (T,x)=0$; and so, u is a variational solution of problem

$$\begin{aligned} {\left\{ \begin{array}{ll} u_{t}-\text {div}({\overline{a}}(x,\nabla u))+g(u)|\nabla u|^{p}=\mu &{}\text { in }(0,T)\times \Omega ,\\ u(0,x)=0&{}\text { in }\Omega ,\\ u(t,x)=0&{}\text { on }(0,T)\times \partial \Omega . \end{array}\right. } \end{aligned}$$

(5.15)

Then, recalling that $\mu \ne 0$ and using a suitable Harnack type result adapted to parabolic inequalities [99], we deduce that $u(t,x)>0$ a.e. on Q. Now, if $\mu \in {\mathcal {M}}_{0}(\Omega )$, we have

$$\begin{aligned} \mu _{\tau }={\left\{ \begin{array}{ll} \tau \mu &{}\text { if }f=0,\\ \tau f-\text {div}(G)&{}\text { if }f\ne 0 \end{array}\right. } \end{aligned}$$

where $f\in L^{1}(\Omega )$ a nonnegative function and $G\in L^{p'}(\Omega )^{N}$ (see [26]), we can suppose, without loss of generality, that $u_{\tau }=\tau \chi _{E}-\text {div}(G)$ for a suitable set $E\subseteq \Omega$ of positive measure; indeed, f, being nonidentically zero, it turns out to be strictly bounded away from zero on a suitable $E\subseteq \Omega$, and so there exists a constant C such that $f\ge C\chi _{E}$, and then, once we proved our result for such a $\mu _{\tau }$, we can easily prove the statement by applying again a comparison argument. Now, reasoning analogously as above we deduce that $\frac{u_{\tau }}{\tau ^{\frac{1}{p-1}}}$ solves the parabolic problem

$$\begin{aligned} {\left\{ \begin{array}{ll} \tau ^{\frac{1}{p-1}}\left( \frac{u_{\tau }}{\tau ^{\frac{1}{p-1}}}\right) _{t}-\text {div}\left( a_{\tau }\left( x,\nabla (\frac{u_{\tau }}{\tau ^{\frac{1}{p-1}}})\right) \right) \\ +g\left( \tau ^{\frac{1}{p-1}}\left( \frac{u_{\tau }}{\tau ^{\frac{1}{p-1}}}\right) \right) \left| \tau ^{\frac{1}{p-1}}\nabla \left( \frac{u_{\tau }}{\tau ^{\frac{1}{p-1}}}\right) \right| ^{p}=\chi _{B}-\frac{1}{\tau }\text {div}(G)\text { in }Q,\\ \frac{u_{\tau }}{\tau ^{\frac{1}{p-1}}}(0,x)=0\text { in }\Omega ,\\ \frac{u_{\tau }}{\tau ^{\frac{1}{p-1}}}(t,x)=0\text { on }(0,T)\times \partial \Omega \end{array}\right. } \end{aligned}$$

(5.16)

Moreover

$$\begin{aligned} \chi _{B}-\frac{1}{\tau }\text {div}(G)\longrightarrow \chi _{B} \text { stronly in }W^{-1,p'}(\Omega )\text { as }\tau \rightarrow \infty . \end{aligned}$$

(5.17)

Therefore, since the G-convergence is stable under such type of convergence of data, we have, that the weak limit u of $\frac{u_{\tau }}{\tau ^{\frac{1}{p-1}}}$ in $L^{p}(0,T;W^{1,p}_{0}(\Omega ))$ solves

$$\begin{aligned} {\left\{ \begin{array}{ll} u_{t}-{\text {div}}({\overline{a}}(x,\nabla u))+g(u)|\nabla u|^{p}=\chi _{B}&{}\text { in }Q=(0,T)\times \Omega ,\\ u(0,x)=0&{}\text { in }\Omega ,\\ u(t,x)=0&{}\text { on }(0,T)\times \partial \Omega , \end{array}\right. } \end{aligned}$$

(5.18)

and so we deduce, as above, that $u(t,x)>0$ a.e. on $Q=(0,T)\times \Omega$, which implies that $u_{\tau }$ goes to infinity as $\tau$ and T tends to infinity and then we conclude the result of Theorem 5.1. $\square$