\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2015 (2015), No. 38, pp. 1--7.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2015 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2015/38\hfil Nonuniqueness for Parabolic $p$-Laplacian] {Nonuniqueness of solutions of initial-value problems for parabolic $p$-Laplacian} \author[J. Benedikt, V. E. Bobkov, P. Girg, L. Kotrla, P. Tak\'a\v{c} \hfil EJDE-2015/38\hfilneg] {Ji\v{r}\'i Benedikt, Vladimir E. Bobkov, Petr Girg,\\ Luk\'{a}\v{s} Kotrla, Peter Tak\'a\v{c}} \address{ Ji\v{r}\'i Benedikt \newline Department of Mathematics and NTIS, Faculty of Applied Scences, University of West Bohemia, Univerzitn\'{\i} 22, CZ-306\,14~Plze\v{n}, Czech Republic} \email{benedikt@kma.zcu.cz} \address{Vladimir E. Bobkov \newline Fachbereich Mathematik, Universit\"at Rostock, Germany} \email{vladimir.bobkov@uni-rostock.de} \address{Petr Girg \newline Department of Mathematics and NTIS, Faculty of Applied Scences, University of West Bohemia, Univerzitn\'{\i} 22, CZ-306\,14~Plze\v{n}, Czech Republic} \email{pgirg@kma.zcu.cz} \address{Luk\'a\v{s} Kotrla \newline Department of Mathematics and NTIS, Faculty of Applied Scences, University of West Bohemia, Univerzitn\'{\i} 22, CZ-306\,14~Plze\v{n}, Czech Republic} \email{kotrla@ntis.zcu.cz} \address{Peter Tak\'a\v{c} \newline Fachbereich Mathematik, Universit\"at Rostock, Germany} \email{peter.takac@uni-rostock.de} \thanks{Submitted January 29, 2015. Published February 10, 2015.} \subjclass[2000]{35B05, 35B30, 35K15, 35K55, 35K65} \keywords{Quasilinear parabolic equations with $p$-Laplacian; nonuniqueness for initial-boundary value problem; sub- and supersolutions; comparison principle} \begin{abstract} We construct a positive solution to a quasilinear parabolic problem in a bounded spatial domain with the $p$-Laplacian and a nonsmooth reaction function. We obtain nonuniqueness for zero initial data. Our method is based on sub- and supersolutions and the weak comparison principle. Using the method of sub- and supersolutions we construct a positive solution to a quasilinear parabolic problem with the $p$-Laplacian and a reaction function that is non-Lipschitz on a part of the spatial domain. Thereby we obtain nonuniqueness for zero initial data. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \allowdisplaybreaks \section{Introduction} The problem of \emph{uniqueness} and \emph{nonuniqueness} of solutions to various types of initial (and boundary) value problems for quasilinear parabolic equations has been an interesting research topic for several decades (see, e.g., Fujita and Watanabe \cite{fujitawatanabe} and the references therein, Guedda \cite{Guedda}, Ladyzhenskaya and Ural'tseva \cite{ladyzural1962}, and Oleinik and Kruzhkov \cite{oleinikkruzhkov}). In this work we focus on the following problem with the $p$-Laplacian and a (partly) nonsmooth reaction function: \begin{equation} \label{Eq_P} \begin{gathered} \frac{\partial u}{\partial t} - \Delta_p u = q(x) |u|^{\alpha-1} u \quad\text{for }(x,t)\in\Omega\times(0,T)\,; \\ u(x,t)=0 \quad\text{for }(x,t)\in\partial\Omega\times(0,T)\,, \\ u(x,0)= 0 \quad\text{for }x\in\Omega\,. \end{gathered} \end{equation} Here, $\Delta_p u :=\operatorname{div} \big(|\nabla u|^{p-2}\nabla u\big)$ denotes the $p$-Laplacian for $10$ for some $x_0\in\Omega$. \end{itemize} We assume that $\Omega\subset\mathbb{R}^{N}$ is a bounded domain with a $C^{1+\mu}$-boundary $\partial\Omega$ where $\mu\in (0,1)$. In particular, we deal with degenerate (singular) diffusion if $20$ then the function $u\mapsto f(x_0, u)$ satisfies neither a local Lipschitz nor an Osgood (see \cite{Osgood}) condition near $u=0$ provided $\alpha\in (0,1)$. The case $p=2$ (the Laplace operator) was treated in Fujita and Watanabe \cite{fujitawatanabe} by entirely different methods based on the Green's function for the heat equation. An important special case, $N = 1$, $1 < p < \infty$, and $q(x)\equiv \lambda > 0$ (a constant), was treated in Guedda \cite{Guedda} also by different methods. The main purpose of the present article is to fill in the gap left open for $1 < p < \infty$, $p\neq 2$, and $q\in C(\overline{\Omega})$, $q\geq 0$, where $q$ is not necessarily positive everywhere in~$\Omega$. Because of this possibly non\-uniform positivity of $q$ over~$\Omega$, the method used in \cite{Guedda} cannot be applied here. We use a different approach based on sub- and supersolutions and the weak comparison principle. As a trivial consequence of the fact that problem \eqref{Eq_P} possesses a nontrivial non\-negative solution (see our main result, Theorem~1), we conlude that the weak comparison principle does not hold for problem \eqref{Eq_P} considered with nontrivial initial conditions, say, in $W_0^{1,p}(\Omega)$. Observe that our assumption (Q) implies that there exists $R>0$ such that $q(x)\geq q_0\equiv \mathrm{const} > 0$ for all $x\in B_R(x_0)$ where $$ B_R(x_0) :=\{ x \in \mathbb{R}^N: |x - x_0| < R \} \subset \Omega. $$ Let $(\lambda_1, \varphi_{1,R})$ denote the first eigenpair for the operator $-\Delta_p\colon W_0^{1,p}(B_R(x_0)) \to W^{-1, p'}(B_R(x_0))$; that is, \begin{equation}\label{Eq:eigenvalue:problem} \begin{gathered} - \Delta_p\varphi_{1,R} = \lambda_{1,R} \, \varphi_{1,R}^{p-1} \quad\text{in }B_R(x_0) \,; \\ \varphi_{1,R} = 0 \quad\text{on }\partial{B_R(x_0)} \,, \end{gathered} \end{equation} and $\varphi_{1,R}\in W_0^{1,p}(B_R(x_0))$ is normalized by $\varphi_{1,R}(x_0)=1$. Note that this normalization yields $0<\varphi_{1,R}(x)\leq 1$ for all $x\in B_R(x_0)$. Moreover, we denote by \begin{equation} \label{def:tilde:phi:1} \widetilde{\varphi}_{1,R}(x) :=\begin{cases} \varphi_{1,R}(x) & \text{for }x\in B_R(x_0)\,; \\ 0 & \text{for }x\in \overline{\Omega}\setminus B_R(x_0)\,, \end{cases} \end{equation} the natural zero extension of $\varphi_{1,R}$ from $B_R(x_0)$ to the whole of $\overline{\Omega}$. Our main theorem is the following nonuniqueness result. \begin{theorem} \label{thm:1} Assume that $0<\alpha<\min\{1, p-1\}$ and {\rm (Q)} are satisfied. Then there exists $T>0$ small enough, such that problem \eqref{Eq_P} possesses (besides the trivial solution $u\equiv 0$) a nontrivial, nonnegative weak solution \[ u\in C\big([0, T]\to L^2(\Omega)\big) \cap L^p\big((0,T)\to W^{1,p}(\Omega)\big) \] \hfil\break which is bounded below by a subsolution $\underline{u}:\Omega\times(0,T)\to\mathbb{R}_{+}$ of type $$ \underline{u}(x,t) =\theta(t) \widetilde{\varphi}_{1,R}(x)^{\beta}\geq 0 \quad\text{in }\Omega\times (0,T)\,, $$ where $\theta\colon [0,T]\to\mathbb{R}_{+}$ is a strictly increasing, continuously differentiable function with $\theta(0)=0$, and $\beta\in (1, \infty)$ is a suitable number. \end{theorem} In contrast with this nonuniqueness result, several uniqueness results have been established in \cite{BobkovTakac}. \begin{remark} \label{rmk2} \rm Assume that $q\in L^{\infty}(\Omega)$ satisfies $0\leq q(x)\leq \lambda_1$ a.e. in $\Omega$, where $\lambda_1$ stands for the principal eigenvalue of $-\Delta_p$ with zero Dirichlet boundary conditions on $\Omega$. Then the condition $\alpha 0$. This means that it suffices to treat the constant case $q(x) \equiv q_0 = \mathrm{const} > 0$ and then use the resulting solution as a subsolution for the general case $q(x) \geq q_0 = \mathrm{const} > 0$. In contrast, our Theorem \ref{thm:1} above does not assume $q_0 > 0$; we assume only $q \geq 0$ and $q \not\equiv 0$ in $\Omega$. Nevertheless, our proof of this result, especially our construction of a nonzero subsolution, is simpler than in \cite{merchanmontoroperal}. \section{Proof of Theorem \ref{thm:1}} Note that $\widetilde{\varphi}_{1,R}$ defined in \eqref{def:tilde:phi:1} is continuous on $\overline \Omega$ and $\widetilde{\varphi}_{1,R}^{\beta}$ is continuously differentiable for any constant $\beta>1$. We need to establish a few additional properties of $\varphi_{1,R}(x)\equiv \varphi_{1,R}(|x-x_0|) = \varphi_{1,R}(r)$, with $r=|x-x_0|$ and the usual harmless abuse of notation. \begin{lemma} \label{lem:varphi_1} If $\beta\in (0,\infty)$ then \begin{equation} \label{eq:plapl:phibeta} -\Delta_p \Big(\varphi_{1,R}^{\beta}\Big) = \beta^{p-1} \varphi_{1,R}^{(p-1)(\beta-1)-1} \big[\lambda_{1,R} \varphi_{1,R}^p - (p - 1)(\beta - 1)|\nabla\varphi_{1,R}|^p \big] \end{equation} holds pointwise a.e. in $B_R(x_0)$. In particular, for $\beta\geq 1$ we have \begin{equation} \label{Eq:estimation:1} \frac{- \Delta_p( \varphi_{1,R}^{\beta}) }{\varphi_{1,R}^{\beta}}\leq C \equiv \mathrm{const} < \infty \quad\text{pointwise a.e. in } B_R(x_0) \,. \end{equation} \end{lemma} \begin{proof} Any function $u\colon B_R(x_0)\to \mathbb{R}$ that is radially symmetric around $x_0$ can be written as $u(x)=u(r)$ where $r=|x-x_0|$. Using this notation we obtain, by formal differentiation, \begin{equation} \label{e5} \begin{aligned} \Delta_p u(|x-x_0|) &=\operatorname{div} \Big(|u'(r)|^{p-2}u'(r) \frac{x-x_0}{r}\Big)\\ &=\Big( |u'(r)|^{p-2} u'(r)\Big)'+ \frac{N-1}{r} |u'(r)|^{p-2} u'(r)\,. \end{aligned} \end{equation} It is well-known that the first eigenfunction $\varphi_{1,R}$ is radially symmetric around $x_0$, positive, and $C^2$ in $\overline{B_R}(x_0)\setminus\{ x_0\}$, see e.g. \cite{Tilak}. Therefore, we get a.e. in $B_R(x_0)$, \begin{align*} &\Delta_p \Big(\varphi_{1,R}^{\beta}(r)\Big)\\ &=\Big(\beta^{p-1}\varphi_{1,R}^{(p-1)(\beta-1)} |\varphi_{1,R}'|^{p-2}\varphi_{1,R}'\Big)'\\ &\quad + \frac{N-1}{r} \beta^{p-1}\varphi_{1,R}^{(p-1)(\beta-1)} |\varphi_{1,R}'|^{p-2}\varphi_{1,R}'\\ & =\beta^{p-1} \Big\{ (p-1)(\beta-1)\varphi_{1,R}^{ (p-1)(\beta-1)-1} |\varphi_{1,R}'|^p \\ &\quad + \varphi_{1,R}^{(p-1)(\beta-1)} \Big(|\varphi_{1,R}'|^{p-2}\varphi_{1,R}'\Big)' +\frac{N-1}{r} \varphi_{1,R}^{(p-1)(\beta-1)} |\varphi_{1,R}'|^{p-2}\varphi_{1,R}'\Big\}\\ & = \beta^{p-1} \varphi_{1,R}^{(p-1)(\beta-1)-1} \Big\{(p-1)(\beta-1) |\varphi_{1,R}'|^p -\lambda_{1,R} \varphi_{1,R}^p\Big\} \\ & =\beta^{p-1} \varphi_{1,R}^{(p-1)\beta} \Big\{(p-1)(\beta-1)\,\frac{|\varphi_{1,R}'|^p}{\varphi_{1,R}^p} - \lambda_{1,R} \Big\}\,. \end{align*} Hence, $$ -\Delta_p \big( \varphi_{1,R}^{\beta}\big) \leq \beta^{p-1}\lambda_{1,R} \varphi_{1,R}^{(p-1)\beta} $$ for $\beta\geq 1$. For $p \geq 2$ this yields $$ \frac{-\Delta_p\big(\varphi_{1,R}^{\beta}\big)}{\varphi_{1,R}^{\beta}} \leq \beta^{p-1} \lambda_{1,R} \varphi_{1,R}^{(p-2)\beta} \leq \beta^{p-1} \lambda_{1,R}\,, $$ thanks to our normalization $0<\varphi_{1,R}\leq 1$. On the other hand, for $1 0$ such that $\varphi_{1,R}'(r) < \varphi_{1,R}'(R)/2 < 0$ for all $r\in (R-\varepsilon, R)$. Hence, \eqref{Eq:estimation:calc} implies \eqref{Eq:estimation:1} for $R-\varepsilon\leq r < R$ provided $\varepsilon>0$ is small enough, such that $$ \lambda_{1,R} - (p-1)(\beta-1)\varphi_{1,R}^{-p} |\varphi_{1,R}'|^{p} \leq 0 \quad \text{for } R-\varepsilon\leq r < R\,. $$ At the same time, the ratio $-\Delta_p\big(\varphi_{1,R}^{\beta}\big) /{\varphi_{1,R}^{\beta}}$ is bounded for $0 < r\leq R-\varepsilon$. Thus, estimate \eqref{Eq:estimation:1} holds a.e. in $B_R(x_0)$. \end{proof} \begin{proposition} \label{prop4} Assume that $0 < \alpha < \min\{1, p-1 \}$ and {\rm (Q)} are satisfied. Given any fixed number $S\in (0,\infty)$, we define \begin{equation*} \underline{u}(x,t) := \theta(t) \widetilde{\varphi}_{1,R}(x)^\beta \quad\text{ for }\, (x,t)\in \Omega\times [0,S] \,, \end{equation*} where $\beta > 1$, $\widetilde{\varphi}_{1,R}$ is given by \eqref{def:tilde:phi:1}, and $\theta: [0,S] \to \mathbb{R}_+$ is the positive solution of the Cauchy problem \begin{equation} \label{def:theta} \frac{\mathrm{d} \theta}{\mathrm{d} t}(t) = \frac{q_0}{2} \theta^{\alpha}(t) \quad\text{for } t\in (0,S) \,; \quad \theta(0) = 0 \,, \end{equation} such that $0 < \theta(t) < \infty$ for every $t\in (0,S)$. Then $\underline{u}: \Omega \times (0,S) \to \mathbb{R}_+$ is a subsolution of problem \eqref{Eq_P} in a smaller domain $\Omega\times (0,\underline{\sigma})$, i.e., for $t\in (0,\underline{\sigma})$ only, where $\underline{\sigma}\in (0,S)$ is small enough. \end{proposition} \begin{proof} We will show that the following inequality holds $$ \frac{\partial \underline{u}}{\partial t} - \Delta_p \underline{u} \leq q(x) |\underline{u}|^{\alpha-1} \underline{u}. $$ Using $0 < \alpha < \min\{1, p-1\}$, equation \eqref{def:theta}, and the continuity of $\theta\colon [0,S)\to \mathbb{R}_{+}$, we get \begin{equation} \label{ineq:theta} \frac{\mathrm{d} \theta}{\mathrm{d} t} \leq {}-C \theta(t)^{p-1} + q_0 \theta(t)^{\alpha} \quad\text{for all } t\in [0, \underline{\sigma}]\,, \end{equation} where $\underline{\sigma}\in (0,S)$ is small enough, such that $\theta(t)^{p-1-\alpha}\leq {q_0}/{(2 C)}$ holds for all $t\in [0, \underline{\sigma}]$. Inserting the inequality \[ \varphi_{1,R}^{-\beta}\Delta_p(\varphi_{1,R}^{\beta}) \geq -C\equiv\mathrm{const} \] in $\Omega$ from Lemma \ref{lem:varphi_1}, inequality \eqref{Eq:estimation:1}, into \eqref{ineq:theta}, we obtain \begin{align*} \frac{\mathrm{d} \theta}{\mathrm{d} t} & \leq \varphi_{1,R}^{-\beta} \Delta_p(\varphi_{1,R}^{\beta}) \theta(t)^{p-1} + q_0 \theta(t)^{\alpha}\\ &\leq \varphi_{1,R}^{-\beta} \Delta_p(\varphi_{1,R}^{\beta}) \theta(t)^{p-1} + q_0 \varphi_{1,R}^{(\alpha-1) \beta} \theta(t)^{\alpha}, \end{align*} thanks to the normalization $0<\varphi_{1,R}\leq 1$ in $B_R(x_0)$ combined with $(\alpha-1)\beta < 0$. Finally, multiplying by $\varphi_{1,R}^{\beta}$, we arrive at \begin{align*} \frac{\mathrm{d} \theta}{\mathrm{d} t} \varphi_{1,R}^{\beta} & \leq \Delta_p(\varphi_{1,R}^{\beta}) \theta(t)^{p-1} + q_0 \theta(t)^{\alpha} \varphi_{1,R}^\alpha \\ & \leq \Delta_p(\varphi_{1,R}^{\beta}) \theta(t)^{p-1} + q(x) \theta(t)^{\alpha} \varphi_{1,R}^\alpha\,. \end{align*} This inequality, combined with our definition of the function $\widetilde{\varphi}_{1,R}$, guarantees that $\underline{u}(x,t)= \theta(t) \widetilde{\varphi}_{1,R}(x)$ is a subsolution to problem \eqref{Eq_P}. \end{proof} \begin{proof}[Proof of Theorem \ref{thm:1}] First, let us observe that $\overline{u}(x,t) = \|q\|_{\infty}^{\frac{1}{1-\alpha}} t$ is a supersolution of \eqref{Eq_P} for $0 < t\leq 1$. Indeed, a straightforward calculation shows that $$ \frac{\partial \overline{u}}{\partial t} - \Delta_p \overline{u} = \|q\|_{\infty}^{\frac{1}{1-\alpha}} \geq q(x) \Big(\|q\|_{\infty}^{\frac{1}{1-\alpha}} t\Big)^\alpha = q(x) |\overline{u}|^{\alpha - 1} \overline{u} $$ holds for $0 < t \leq 1$, since $q\in C(\overline\Omega)$, $q\geq 0$, and $\|q\|_{\infty}=\sup_{x\in\Omega} q(x)$. Second, we show now that $\underline{u} \leq \overline{u}$ for all $x\in \Omega$ and all $t > 0$ sufficiently small, say, $0 < t\leq \overline{\sigma}$. Evidently, $$ \underline{u}(x,t) = \theta(t) \widetilde{\varphi}_1(x)^\beta = c_1 t^{\frac{1}{1-\alpha}} \widetilde{\varphi}_1(x)^\beta \leq c_1 t^{\frac{1}{1-\alpha}} \leq \overline{u}(x,t) = \|q\|_{\infty}^{\frac{1}{1-\alpha}} t $$ for $00. $$ Let us define a sequence of functions $u_n\colon \Omega\times (0,T)\to \mathbb{R}$ recursively for $n=1,2,3,\dots$, such that $u_n$ is the unique weak solution of \begin{equation} \label{eq:P:mon:it} \begin{gathered} \frac{\partial u_n}{\partial t} - \Delta_p u_n = q(x)|u_{n-1}|^{\alpha-1} u_{n-1}, \quad (x,t) \in \Omega \times (0, T), \\ u_n(x, 0) = 0, \quad x \in \Omega, \\[0.4em] u_n(x,t) = 0, \quad (x,t) \in \partial\Omega \times (0,T), \end{gathered} \end{equation} with $u_0 = \underline{u}$. By a weak solution of \eqref{eq:P:mon:it}, we mean a Lebesgue\--measurable function $u_n\colon \Omega \times (0,T) \to \mathbb{R}$ that satisfies $$ u_n \in C ( [0,T] \to L^2(\Omega) ) \cap L^p \big( (0,T) \to W_0^{1,p}(\Omega) \big) $$ and the equation \begin{equation} \label{weakn} \begin{aligned} &\int_\Omega u_n(x,t) \phi(x,t)\,\mathrm{d}x - \int_0^t \int_{\Omega}u_n(x,s) \frac{\partial \phi}{\partial t}(x,s)\, \mathrm{d}x\,\mathrm{d}s\\ &+\int_0^t \int_{\Omega} |\nabla u_n(x,s)|^{p-2} \langle \nabla u_n(x,s),\nabla \phi(x,s)\rangle \, \mathrm{d}x\,\mathrm{d}s \\ & = \int_{0}^{t} \int_{\Omega} q(x) |u_{n-1}(x,s)|^{\alpha-1} u_{n-1}(x,s) \phi(x,s) \,\mathrm{d}x\,\mathrm{d}s \end{aligned} \end{equation} for every $t\in (0,T)$ and every test function $$ \phi\in C \left([0, T]\to L^2(\Omega)\right) \cap L^p\left((0, T)\to W_0^{1,p}(\Omega)\right) \cap W^{1,p'}\left((0,T)\to W^{-1,p'}(\Omega)\right)\,. $$ The questions of existence and uniqueness of weak solutions of problems of type \eqref{eq:P:mon:it} obtained by monotone iterations have been discussed in \cite[Appendix A, \S A.1]{takac2010}. Let us deduce from the fact that $u_0 = \underline{u}$ is a subsolution of \eqref{Eq_P} the inequalities $u_{n-1} \leq u_n$ in $\Omega \times (0, T)$ for every $n= 1,2,3,\dots$. The proof is by induction on $n$. The first inequality, $u_0 \leq u_1$ in $\Omega\times (0,T)$, holds by the Weak Comparison Principle (see \cite[Lemma 4.9,~p.~618]{takac2010}) and the fact that $u_0=\underline{u}$ is a subsolution of \eqref{Eq_P}. Now assume that $u_{n-1} \leq u_n$ in $\Omega\times (0,T)$ for some $n \in \mathbb{N}$. Then we have $$ \frac{\partial u_n}{\partial t} - \Delta_p u_n = |u_{n-1}|^{\alpha-1} u_{n-1} \leq |u_{n}|^{\alpha-1} u_{n} = \frac{\partial u_{n+1}}{\partial t} - \Delta_p u_{n+1} $$ in $\Omega\times (0,T)$ and consequently $u_{n} \leq u_{n+1}$ in $\Omega\times (0,T)$ again, by \cite[Lemma 4.9,~p.~618]{takac2010}. Therefore, monotonicity holds: $\underline{u}=u_0\leq u_1\leq u_2 \leq \dots\leq \overline{u}$ in $\Omega\times (0,T)$. The comparison with the supersolution $\overline{u}$ is deduced again from the Weak Comparison Principle. Hence, $u_n$ is uniformly bounded in $\Omega\times (0,T)$ by $\underline{u}\leq u\leq \overline{u}$. A global regularity result from \cite[Theorem 0.1, p. 552]{lieberman1993} (cf. \cite[Lemma 4.6, p. 617]{takac2010}) guarantees $u_n \in C^{1+\gamma, \frac{1+\gamma}{2}}(\overline{\Omega}\times[0, T])$ uniformly for $n \in \mathbb{N}$, where $\gamma \in (0,1)$ is independent of $n$. We follow the notations and definitions of H\"{o}lder spaces of functions on $\Omega\times [0,T]$ from \cite[Chpt.~1, p.~7]{ladyzsolural}. Thus, by the Arzel\`{a}-Ascoli theorem, $\{ u_n \}$ is relatively compact in $C^{1, 0}(\overline{\Omega}\times[0, T])$. Hence, the sequence $\{u_n\}$ possesses a subsequence which converges to $u \in C^{1, 0}(\overline{\Omega}\times[0, T])$. Therefore, in the weak formulation of \eqref{weakn} we may pass to the limit as $n \to \infty$, thus verifying that the limit function $u$ is a weak solution of \eqref{Eq_P} in $\Omega \times (0,T)$, such that $\underline{u} \leq u \leq \overline{u}$. \end{proof} \subsection*{Acknowledgments} All authors were partially supported by a joint exchange program between the Czech Republic and Germany: By the Ministry of Education, Youth, and Sports of the Czech Republic under the grant No. 7AMB14DE005 (exchange program ``MOBILITY'') and by the Federal Ministry of Education and Research of Germany under grant No. 57063847 (D.A.A.D.\ Program ``PPP''). The research of Peter Tak\'{a}\v{c} was partially supported also by the German Research Society (D.F.G.), grant No. TA 213 / 15-1 and the research of Vladimir E. 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