\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2015 (2015), No. 184, pp. 1--9.\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-2015/184\hfil $L^p$-continuity of solutions] {$L^p$-continuity of solutions to parabolic free boundary problems} \author[A. Lyaghfouri, E. Zaouche \hfil EJDE-2015/184\hfilneg] {Abdeslem Lyaghfouri, El Mehdi Zaouche} \address{Abdeslem Lyaghfouri \newline American University of Ras Al Khaimah, Department of Mathematics and Natural Sciences, Ras Al Khaimah, UAE} \email{abdeslem.lyaghfouri@aurak.ac.ae} \address{El Mehdi Zaouche \newline Ecole Normale Sup\'erieure, D\'epartement de Math\'ematiques, 16050 Kouba, Alger, Alg\'erie. \newline Universit\'e d'EL Oued, D\'epartement de Math\'ematiques, 39000 EL Oued, Alg\'erie} \email{elmehdi-zaouche@univ-eloued.dz} \thanks{Submitted May 25, 2015. Published July 7, 2015.} \subjclass[2010]{35B65, 35D30, 35R35} \keywords{Parabolic equation; free boundary; monotonicity; $L^p$-continuity} \begin{abstract} In this article, we consider a class of parabolic free boundary problems. We establish some properties of the solutions, including $L^\infty$-regularity in time and a monotonicity property, from which we deduce strong $L^p$-continuity in time. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \allowdisplaybreaks \section{Introduction}\label{s1} In this work, we study the following weak formulation which describes a class of nonstationary free boundary problems: \noindent\textbf{Problem (p).} Find $(u, \chi) \in L^2(0,T;H^1(\Omega))\times L^\infty (Q)$ such that \begin{itemize} \item[(i)] $u \geq 0$, $0\leq \chi\leq 1$, $u(1-\chi) = 0$ a.e. in $Q$; \item[(ii)] $u=\phi$ on $\Sigma_2$; \item[(iii)] \[ \int_Q \big[\big( a(x)\nabla u +\chi H(x)\big) \cdot\nabla\xi-(\alpha u+\chi)\xi_t\big] dx\,dt\\ \leq \int_\Omega (\chi_0(x)+\alpha u_0(x))\xi(x,0)\,dx \] for all $\xi \in H^1(Q)$, $\xi=0$ on $\Sigma_3, \xi \geq 0$ on $\Sigma_4$, $\xi(x,T)=0$ for a.e. $x\in \Omega$, \end{itemize} where $\alpha, T$ are positive numbers, $\Omega$ is a bounded domain in $\mathbb{R}^n (n\geq2)$ with Lipschitz boundary $\partial\Omega=\Gamma_1\cup \Gamma_2$, $Q=\Omega\times (0,T)$, $\Sigma_1=\Gamma_1\times(0,T)$, $\Sigma_2=\Gamma_2\times(0,T)$, $\Sigma_3=\Sigma_2\cap \{\phi>0\}$ and $\Sigma_4=\Sigma_2\cap \{\phi=0\}$, with $\phi$ a nonnegative Lipschitz continuous function defined in $\overline{Q}$. For a.e. $x\in\Omega$, $ a(x)=(a_{ij}(x))_{ij}$ is an $n\times n$ matrix, $H:\Omega \to \mathbb{R}^n$ is a vector function satisfying for some positive constants $\lambda, \Lambda$ and $\overline{H}$: \begin{gather}\label{e1.1} \forall \xi\in \mathbb{R}^n,\quad\text{a.e. } x\in\Omega \quad \lambda |\xi|^2\leq a(x) \xi\cdot \xi, \\ \forall \xi\in \mathbb{R}^n,\quad\text{a.e. } x\in\Omega \quad |a(x)\xi|\leq \Lambda|\xi|, \label{e1.2}\\ |H(x)|\leq \overline{H} \quad \text{a.e. } x\in\Omega. \label{e1.3} \end{gather} Moreover, we assume that \begin{equation} \label{e1.4} \operatorname{div}(H(x)) \in L^{2}(\Omega), \end{equation} and the functions $u_0, \chi_0: \Omega \to \mathbb{R}$ satisfying \begin{gather}\label{e1.5} u_0, \chi_0 \in L^\infty(\Omega),\\ u_0(x)\geq 0\quad\text{for a.e. } x\in\Omega, \label{e1.6}\\ 0\leq \chi_0(x)\leq 1\quad\text{for a.e. } x\in\Omega. \label{e1.7} \end{gather} Note that problem (p) describes in particular the weak formulation of the non-steady state dam problem \cite{C1,C2,CG,DF,G}. For the heterogeneous stationary dam problem, we refer for example to \cite{ChL1,Ly1}. Another free boundary problem described by the above formulation is the one-phase Stefan problem (see for example \cite{Rod,[Rou]}). Under assumptions \eqref{e1.1}-\eqref{e1.7}, existence of a solution is proved in \cite{Za}. The proof is based on the Tychonoff fixed theorem and combines technics from \cite{C1,G}, where existence was established for the unsteady filtration problem in a homogeneous porous medium respectively in the incompressible and compressible cases. Another approach with quasi-variational inequalities was adopted in \cite{To} for rectangular domains. Uniqueness of the solution was proved for dams with general geometry and rectangular dams respectively in \cite{C2} and \cite{DF} with different methods. Extensions to a quasilinear operator modeling incompressible fluid flow governed by a generalized nonlinear Darcy's law with Dirichlet, Neuman, or generalized boundary conditions were considered in \cite{CaL,Ly2,Ly3,Ly4}. In this article, we are concerned with the $L^p(\Omega)$-continuity in time of the functions $u$ and $\chi$. We recall that regularity of the solution was investigated in \cite{CG,C2}, when $a(x)=I_n$ and $H(x)=e=(0,\dots,0,1)\in \mathbb{R}^n$, where it was proved that $\chi \in C^0([0,T],L^p(\Omega))$ for all $p\geq 1$ in both incompressible and compressible cases, and that $u \in C^0([0,T],L^p(\Omega))$ for all $ 1\leq p\leq 2$, in the compressible case. Extensions to the quasilinear case were obtained in \cite{Ly2,Ly3,Ly4} in both homogeneous and nonhomogeneous frameworks. \section{Properties}\label{s2} We shall denote by $(u,\chi)$ a solution of the problem (p). \begin{proposition}\label{prop2.1} We have $$ \alpha u+\chi\in C^0([0,T];V'),\quad \text{where }V=\{v\in H^1(\Omega): v=0\text{ on }\Gamma_2\}. $$ \end{proposition} For a proof of the above proposition see \cite{Za}. \begin{proposition}\label{prop2.2} If $\alpha>0$, then we have \begin{equation}\label{e2.1} u \in L^{\infty}(0,T;L^2(\Omega)). \end{equation} \end{proposition} \begin{proof} Let $\zeta$ be a smooth function such that $d(\operatorname{supp}(\zeta),\Sigma_{2})> 0$ and $\operatorname{supp}(\zeta)\subset \mathbb{R}^n\times (0,T)$. Then there exists $0<\tau_00$, we conclude that $ u \in L^{\infty}(0,T;L^{2}(\Omega))$, which is \eqref{e2.1}. \end{proof} The following result will be used to establish a monotonicity property of $\chi$ which is the key point to prove the main result of the paper. \begin{proposition}\label{prop2.3} We have \begin{equation} \label{e2.7} \operatorname{div}(\chi H(x))-\chi_{\{u>0\}}\operatorname{div}(H(x)) -\chi_t\leq 0\quad \text{in }\mathcal{D}'(Q). \end{equation} \end{proposition} \begin{proof} Arguing as in the beginning of the proof of Proposition \ref{prop2.2}, we have for any $\zeta \in L^{2}(0,T;H^{1}(\Omega))$ such that $\zeta=0$ on $\Sigma_{2}$ and $\zeta=0$ on $\Omega\times ((0,\tau_0)\cup (T-\tau_0,T))$, with $\tau_0>0$ \begin{equation} \label{e2.8} \begin{aligned} &\int_{Q}(a(x)\nabla u(x,t+\tau)+\chi(x,t+\tau) H(x)).\nabla\zeta(x,t)\,dx\,dt \\ &=-\frac{\partial}{\partial\tau}\Big(\int_{Q}(\alpha u(x,t+\tau)+\chi(x,t+\tau))\zeta(x,t)\,dx\,dt\Big) \quad \forall \tau\in(-\tau_0,\tau_0). \end{aligned} \end{equation} Now, let us consider $\epsilon>0$, $\xi\in \mathcal{D}(\Omega\times (\tau_0,T-\tau_0))$ such that $\xi\geq 0$, and choose $\zeta(x,t)=\min\big(\frac{u(x,t+\tau)}{\epsilon},1\big)\xi$ in \eqref{e2.8}. We obtain \begin{equation} \label{e2.9} \begin{aligned} &\int_{Q}(a(x)\nabla u(x,t+\tau)+\chi(x,t+\tau) H(x)).\nabla\Big(\min\Big(\frac{u(x,t+\tau)}{\epsilon},1\Big)\xi(x,t)\Big) \,dx\,dt \\ &=-\frac{\partial}{\partial\tau}\Big(\int_{Q}(\alpha u(x,t+\tau)+\chi(x,t+\tau))\min\Big(\frac{u(x,t+\tau)}{\epsilon},1\Big) \xi(x,t)\,dx\,dt\Big) \end{aligned} \end{equation} for all $\tau\in (-\tau_0,\tau_0)$. Obviously the integral at the left hand side of \eqref{e2.9} is continuous in $(-\tau_0,\tau_0)$. Consequently the function $$ G(\tau)=\int_{Q}(\alpha u(x,t+\tau)+\chi(x,t+\tau))\min\Big(\frac{u(x,t+\tau)}{\epsilon},1\Big)\xi(x,t) \,dx\,dt $$ is a $C^1$ function in $(-\tau_0,\tau_0)$. For $\tau=0$, we obtain \begin{equation}\label{e2.10} \int_{Q}(a(x)\nabla u+\chi H(x)).\nabla\Big(\min\Big(\frac{u}{\epsilon},1\Big) \xi\Big) \,dx\,dt= -G'(0). \end{equation} Since \begin{align*} G(\tau)&= \int_Q (\alpha u(x,t)+\chi(x,t))\min\Big(\frac{u(x,t)}{\epsilon},1\Big) \xi(x,t-\tau)\,dx\,dt\\ &= \int_Q (\alpha u(x,t)+1)\min\Big(\frac{u(x,t)}{\epsilon},1\Big)\xi(x,t-\tau) \,dx\,dt, \end{align*} we obtain \begin{equation} \label{e2.11} G'(0)= -\int_Q (\alpha u+1)\min\Big(\frac{u}{\epsilon},1\Big)\xi_t\,dx\,dt. \end{equation} Hence from \eqref{e2.10} and \eqref{e2.11} we obtain \begin{align*} &\int_{Q}(a(x)\nabla u+\chi H(x))\cdot \nabla\Big(\min\Big(\frac{u}{\epsilon},1\Big)\xi\Big) \,dx\,dt \\ &=\int_Q (\alpha u+1)\min\Big(\frac{u}{\epsilon},1\Big)\xi_t \,dx\,dt \end{align*} which leads to \begin{align*} &\int_{Q}a(x)\nabla u\cdot \nabla\Big(\min\Big(\frac{u}{\epsilon},1\Big)\xi\Big) -\alpha u\min\Big(\frac{u}{\epsilon},1\Big)\xi_{t}\,dx\,dt \\ &=-\int_{Q}\chi H(x).\nabla\Big(\min\Big(\frac{u}{\epsilon},1\Big)\xi\Big) \,dx\,dt+\alpha\int_{Q}u \min\Big(\frac{u}{\epsilon},1\Big)\xi_{t}\,dx\,dt \\ &=-\int_{Q} H(x).\nabla\Big(\min\Big(\frac{u}{\epsilon},1\Big)\xi\Big)\,dx\,dt +\alpha\int_{Q}u \min\Big(\frac{u}{\epsilon},1\Big)\xi_{t}\,dx\,dt. \\ &=\int_{Q}\operatorname{div}( H(x)).\min\Big(\frac{u}{\epsilon},1\Big)\xi \,dx\,dt+\alpha\int_{Q}u \min\Big(\frac{u}{\epsilon},1\Big)\xi_{t}\,dx\,dt \end{align*} or \begin{equation} \label{e2.12} \begin{aligned} &\int_{Q}\min\Big(\frac{u}{\epsilon},1\Big)a(x)\nabla u.\nabla \xi-\alpha u\min\Big(\frac{u}{\epsilon},1\Big)\xi_{t}\,dx\,dt \\ &=\int_{Q}\operatorname{div}( H(x)).\min\Big(\frac{u}{\epsilon},1\Big)\xi \,dx\,dt+\alpha\int_{Q}u \min\Big(\frac{u}{\epsilon},1\Big)\xi_{t}\,dx\,dt \\ &\quad -\int_{Q\cap \{u<\epsilon\}}\xi a(x)\nabla u.\nabla u\,dx\,dt \\ &\leq\int_{Q}\operatorname{div}( H(x)).\min\Big(\frac{u}{\epsilon},1\Big)\xi \,dx\,dt+\alpha\int_{Q}u \min\Big(\frac{u}{\epsilon},1\Big)\xi_{t}\,dx\,dt. \end{aligned} \end{equation} Letting $\epsilon\to 0$ in \eqref{e2.12}, we obtain \[ \int_{Q}a(x)\nabla u.\nabla\xi-\alpha u\xi_{t}\,dx\,dt\leq \int_{Q}\chi_{\{u>0\}}\operatorname{div}(H(x))\xi \,dx\,dt \] or \begin{equation} \label{e2.13} \operatorname{div}(a(x)\nabla u)+\chi_{\{u>0\}}\operatorname{div}(H(x))-\alpha u_{t}\geq 0~~\text{in }\mathcal{D}'(Q). \end{equation} Now using $\pm\xi$ as a test function in (p), we obtain \begin{equation} \label{e2.14} \operatorname{div}(a(x)\nabla u+ \chi H(x))-\alpha u_{t}- \chi_{t}=0 \quad \text{in } \mathcal{D}'(Q). \end{equation} Taking into account \eqref{e2.13} and \eqref{e2.14}, we obtain \begin{align*} &\operatorname{div}(\chi H(x))-\chi_{\{u>0\}}\operatorname{div}(H(x))-\chi_{t} \\ &=-\operatorname{div}(a(x)\nabla u)-\chi_{\{u>0\}} \operatorname{div}(H(x))+\alpha u_{t}\leq 0 \quad \text{in } \mathcal{D}'(Q), \end{align*} which is \eqref{e2.7}. \end{proof} \section{Monotonicity property}\label{s3} In all what follows, we shall assume that \begin{gather}\label{e3.1} H(x)=(h_{1}(x),\dots ,h_{n}(x))\in C^{0,1}(\overline{\Omega},\mathbb{R}^n) \\ \operatorname{div}(H(x))\geq0 \quad \text{a.e. } x \in \Omega \end{gather} and for two positive constants $\underline{h}$ and $\overline{h}$, \begin{equation}\label{e3.2} 0<\underline{h}\leq h_{n}(x)\leq \overline{h}, \quad |h_{i}(x)|\leq \overline{h} \quad \forall x\in \overline{\Omega}, \; i=1,\dots ,n-1. \end{equation} Since $H\in C^{0,1}(\overline{\Omega})$, there exists by Kirszbraun's theorem (see \cite[Theorem 2.10.43 p. 210]{F}) an extension ${\widetilde{H}}\in C^{0,1}(\mathbb{R}^n)$ of $H$ with the same Lipschitz constant. Then the function $\overline{H}=(\overline{H}_1,\dots ,\overline{H}_{n-1},\overline{H}_n)$ defined by \begin{gather*} \overline{H}_i=\min(\bar{h},\max(\widetilde{H}_i,-\bar{h}))\quad i=1,\dots ,n-1\\ \overline{H}_n=\min(\bar{h},\max(\widetilde{H}_n,\underline{h})) \end{gather*} satisfies $\overline{H}\in C^{0,1}(\mathbb{R}^n)$, $\overline{H}_{/\overline{\Omega}}=H$, and \[ 0<\underline{h}\leq \overline{H}_n (x)\leq \overline{h}, \quad |\overline{H}_i(x)|\leq \overline{h} \quad \forall x\in \mathbb{R}^n, \; i=1,\dots ,n-1. \] For simplicity, we will denote $\overline{H}$ by $H$. Let $h_0\in {\mathbb R}$ such that $\Omega$ is located strictly above the hyperplane $x_n=h_0$. We consider for each $\omega\in {\mathbb R}^{n-1}$ the differential equation \begin{equation} \label{Eomega} %(E_\omega) \begin{gathered} X'(s,\omega)=H(X(s,\omega))\\ X(0,\omega)=(\omega,h_0). \end{gathered} \end{equation} Then we have the following proposition. \begin{proposition} \label{prop3.1} There exists a unique maximal solution $x(\cdot,\omega)$ of \eqref{Eomega} defined on $(-\infty,\infty)$. Moreover $x$ is of class $C^{0,1}$ with respect to $\omega$, $C^{1,1}$ with respect to $s$, and we have \begin{equation} \label{e3.4} \lim_{s\to \pm\infty}x_n(s,\omega)=\pm\infty. \end{equation} \end{proposition} \begin{proof} By the classical theory of ordinary differential equations there exists a unique maximal solution $x(\cdot,\omega)$ of \eqref{Eomega} defined on $(\alpha_{-}(\omega),\alpha_{+} (\omega))$. Moreover since $H$ is of class $C^{0,1}$, $x$ is of class $C^{0,1}$ with respect to $\omega$, $C^{1,1}$ with respect to $s$. For \eqref{e3.4}, we refer to the proof of \eqref{e2.4} in \cite{Ly4}. \end{proof} \begin{theorem}\label{thm3.1} The mappings ${\mathcal T} : {\mathbb R}^n \to {\mathbb R}^n$ defined by ${\mathcal T}(s,\omega)=x(s,\omega)$ is a $C^{0,1}$-homeomorphism from ${\mathbb R}^n$ to ${\mathbb R}^n$. Moreover \[ Y(s,\omega)={\mathcal J}{\mathcal T}(s,\omega) = (-1)^{n+1}h_n(\omega,h_0) \exp\Bigl(\int_0^s (div H) (x(\sigma,\omega)) \, d\sigma \Bigl)\neq 0, \] where ${\mathcal J}$ denotes the Jacobian. \end{theorem} \begin{proof} We refer to the proof of \cite[Theorem 2.2]{ChL2} and to the proof of \cite[Theorem 2.1]{Ly4}. \end{proof} \begin{remark}\label{rmk3.1}\rm Let $\mathcal{O}={\mathcal T}^{-1}(\Omega)$. Then $\mathcal{O}$ is a domain of ${\mathbb R}^n$ and ${\mathcal T}\,:\mathcal{O}\to\,\Omega$ is a $C^{0,1}$-homeomorphism. \end{remark} Let $f(s,\omega,t)=\chi( T(s,\omega),t)$. In the following theorem we show that $f$ satisfies a monotonicity result similar to the one in \cite[Theorem 2.1]{ChL2} for the stationary case and to \cite[Theorem 2.2]{Ly4} for the nonstationary case. This extends the well known monotonicity in the homogeneous case i.e. $\chi_n-\chi_t\geq0$ in $\mathcal{D}'(Q)$ when $a(x)=I_n$ (see \cite{C2,CG}). This result will be the key point for the proof of the $L^p$-continuity of $\chi$ and $u$. \begin{theorem}\label{thm3.2} Let $(u, \chi)$ be a solution of {\rm (p)}. We have \begin{equation}\label{e3.5} % \label{e3.12} (\frac{\partial}{\partial s}-\frac{\partial}{\partial t})f\leq0 \quad \text{ in } \mathcal{D}'(\mathcal{O}\times (0,T)). \end{equation} \end{theorem} \begin{proof} Let $\phi\in \mathcal{D}(\mathcal{O}\times (0,T))$, $\phi\geq0$. Since $\mathcal{T}^{-1}\in C^{0,1}(\Omega)$, by approximation we can use $\phi\circ \mathcal{T}^{-1}$ as a test function in \eqref{e2.7}. So we have \begin{equation}\label{e3.6} \begin{aligned} &\int_{\mathcal{T}(\mathcal{O})\times (0,T)}\Big\{\chi H(x).\nabla(\phi\circ \mathcal{T}^{-1})+\chi_{\{u>0\}}\operatorname{div}(H(x)).\phi\circ \mathcal{T}^{-1} \\ &\quad-\chi(\phi\circ \mathcal{T}^{-1})_{t}\Big\}\,dx\,dt\geq0. \end{aligned} \end{equation} Since ${\mathcal T}$ is a $C^{0,1}$-homeomorphism from $\mathcal{O}$ to $\Omega$, we can use the change of variables formula \cite[p. 52]{Zi} to obtain, from \eqref{e3.6}, \[ \int_{\mathcal{O}\times (0,T)}\Big(\chi\circ \mathcal{T}.\frac{\partial \phi}{\partial s}+\chi_{\{u\circ \mathcal{T}>0\}}(\operatorname{div}(H))\circ \mathcal{T}.\phi-\chi\circ \mathcal{T}.\frac{\partial\phi}{\partial t} \Big)|Y|\,ds\,d\omega\,dt\geq0 \] which, given that $\frac{\partial |Y|}{\partial s}=|Y|.(\operatorname{div}(H))\circ \mathcal{T}$, leads to \begin{equation}\label{e3.7} \begin{aligned} &\int_{\mathcal{O}\times (0,T)}\Big(\chi\circ \mathcal{T}.\frac{\partial (|Y|.\phi)}{\partial s}-\chi\circ \mathcal{T}.\frac{\partial (|Y|.\phi)}{\partial t}\Big)\,ds\,d\omega\,dt \\ &= \int_{\mathcal{O}\times (0,T)}\Big(\chi\circ \mathcal{T}.\frac{\partial \phi}{\partial s}|Y|+\chi\circ \mathcal{T}.(\operatorname{div}(H))\circ \mathcal{T}.\phi|Y|-\chi\circ \mathcal{T}.\frac{\partial \phi}{\partial t}|Y|\Big)\,ds\,d\omega\,dt \\ &\geq\int_{\mathcal{O}\times (0,T)}\Big(\chi\circ \mathcal{T}.\frac{\partial \phi}{\partial s}+\chi_{\{u\circ \mathcal{T}>0\}}.(\operatorname{div}(H))\circ \mathcal{T}.\phi-\chi\circ \mathcal{T}.\frac{\partial \phi}{\partial t}\Big)|Y|\,ds\,d\omega\,dt \\ &\geq 0. \end{aligned} \end{equation} By approximation, \eqref{e3.7} holds for any nonnegative function $\phi$ with compact support such that $\phi_{s},\phi_{t}\in L^1(\mathcal{O}\times (0,T)).$ Since $Y, Y_{s}\in L^\infty(\mathcal{O}\times (0,T))$, one can choose $\phi=\frac{\psi}{|Y|}$, with $\psi \in \mathcal{D}(\mathcal{O}\times (0,T))$ and $\psi\geq0.$ Thus we get the result. \end{proof} \section{Continuity of $\chi$ and $\alpha u$}\label{s4} The main result of the this article is the following theorem. \begin{theorem}\label{thm4.1} Let $(u, \chi)$ be a solution of problem {\rm (p)}. Then we have \begin{gather}\label{e4.1} \chi\in C^{0}([0,T];L^{p}(\Omega)) \quad \forall p \in [1,\infty),\\ \text{If }\alpha>0,\text{ then } u\in C^{0}([0,T];L^p(\Omega)) \quad \forall p\in[1,2]. \label{e4.2} \end{gather} \end{theorem} \begin{proof} Let $v=uo\mathcal{T}^{-1}$. Since $\mathcal{T}$ is a $C^{0,1}$-homeomorphism, we get from Propositions \ref{prop2.1} and \ref{prop2.2} \begin{gather}\label{e4.3} f+\alpha v\in C^{0}([0,T];H^{-1}(\mathcal{O})),\\ v\in L^\infty([0,T];L^2(\mathcal{O})). \label{e4.4} \end{gather} Taking into account \eqref{e4.3}-\eqref{e4.4}, the monotonicity of $f$ in \eqref{e3.5}, and arguing as in the proof \cite[Theorem 2.4]{C2}, we obtain \[ f\in C^{0}([0,T];L^{p}(\mathcal{O})) \quad \forall p \in [1,\infty), \] which by using the change of variables $\mathcal{T}$ leads to \begin{equation}\label{e4.5} \chi\in C^{0}([0,T];L^p(\mathcal{T}(\mathcal{O}))) =C^0\big([0,T],L^p(\Omega)\big) \quad \forall p \in [1,\infty). \end{equation} Assume that $\alpha>0$. Since $\chi,\phi \in C^0([0,T],L^2(\Omega))$, we deduce from the last part of the proof of Proposition \ref{prop2.2} that $u\in C^0(0,T;L^2(\Omega))$, and since $\Omega$ is bounded \eqref{e4.2} follows. \end{proof} \begin{remark}\label{rmk4.1} \rm If $\alpha>0$ and $u\in L^{\infty}(0,T;L^p(\Omega))$ for some $p>2$, we have, $u \in C^0([0,T];L^p(\Omega))$. In particular, if $u \in L^{\infty}(Q)$, we have \begin{align*} u\in C^{0}([0,T];L^p(\Omega)) \quad \forall p\geq 1. \end{align*} If $\alpha=0$, in general, $u \notin C^{0}([0,T];L^{p}(\Omega))$ (see \cite[Remark 3.9]{CG}). \end{remark} \subsection*{Acknowledgments} The second author is grateful to Prof. J. F. Rodrigues for kindly inviting him to the CMAF where he enjoyed excellent research conditions during the preparation of his Ph.D. Thesis. \begin{thebibliography}{99} \bibitem{C1} J. 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