\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2017 (2017), No. 32, pp. 1--32.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu} \thanks{\copyright 2017 Texas State University.} \vspace{8mm}} \begin{document} \title[\hfilneg EJDE-2017/32\hfil Nonlinear parabolic problems] {Nonlinear parabolic problems with variable exponent and $L^1$-data} \author[S. Ouaro, A. Ou\'edraogo \hfil EJDE-2017/32\hfilneg] {Stanislas Ouaro, Arouna Ou\'edraogo} \address{Stanislas Ouaro \newline Universit\'e Ouaga I-Professeur Joseph KI ZERBO, Unit\'e de Formation et de Recherche en Sciences Exactes et Appliqu\'ees, D\'epartement de Math\'ematiques, 03 BP.7021 Ouagadougou 03, Burkina Faso} \email{souaro@univ-ouaga.bf, ouaro@yahoo.fr} \address{Arouna Ou\'edraogo \newline Universit\'e de Koudoguou, Unit\'e de formation et de recherche en sciences et technologie, D\'epartement de math\'ematiques, BP 376 Koudogou, Burkina Faso} \email{arounaoued2002@yahoo.fr} \dedicatory{Communicated by Jesus Ildefonso Diaz} \thanks{Submitted October 9, 2013. Published January 29, 2017.} \subjclass[2010]{35K55, 35D05} \keywords{Parabolic equation; variable exponent; entropy solution; $L^1$-data} \begin{abstract} In this article, we prove the existence and uniqueness of entropy solutions to nonlinear parabolic equation with variable exponent and $L^1$-data. The functional setting involves Lebesgue and Sobolev spaces with variable exponent. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \newtheorem{definition}[theorem]{Definition} \allowdisplaybreaks \section{Introduction} The purpose of this article is to study the existence and uniqueness of entropy solutions to the nonlinear parabolic problem involving the $p(x)$-Laplacian type operator \begin{equation}\label{pbmere} \begin{gathered} u_{t} - \operatorname{div}a(x,\nabla u) = f \quad \text{in } Q = (0,T)\times \Omega \\ u = 0 \quad \text{on }\Sigma_{T} = (0,T)\times \partial \Omega \\ u(0,\cdot) = u_0(\cdot) \quad \text{in }\Omega, \end{gathered} \end{equation} where $\Omega \subset \mathbb{R}^N (N \geq 3)$ is a bounded open domain with smooth boundary and $T$ is a positive fixed final time. The study of various mathematical problems with variable exponent has received considerable attention in recent years. These problems concern applications (see \cite{Anton, Chen,Dien2, Rajagopal, Ruzicka}) and raise many difficult mathematical problems. The operator $- \operatorname{div}a(x, \nabla u)$ is called $p(x)$-Laplacian type operator and is a generalization of the $p(x)$-Laplace operator $-\Delta_{p(x)}(u):= - \operatorname{div}(|\nabla u|^{p(x)-2}\nabla u)$ and the generalized mean curvature operator $- \operatorname{div}\big(\big(1 + |\nabla u|^2\big)^{(p(x)-2)/2}\nabla u\big)$. Therefore, the problem \eqref{pbmere} can be viewed as a generalization of the $p(x)$-Laplace problem \begin{equation}\label{pb BWZ} \begin{gathered} u_{t} - \operatorname{div}(|\nabla u|^{p(x)-2}\nabla u) = f \quad \text{in } Q = (0,T)\times \Omega \\ u = 0 \quad \text{on } \Sigma_{T} = (0,T)\times \partial \Omega \\ u(0,\cdot) = u_0(\cdot) \quad \text{in }\Omega. \end{gathered} \end{equation} and the generalized mean curvature problem \begin{equation}\label{pb curv} \begin{gathered} u_{t} - \operatorname{div}\Big(\big(1 + |\nabla u|^2\big)^{(p(x)-2)/2} \nabla u\Big) = f \quad \text{in } Q = (0,T)\times \Omega \\ u = 0 \quad \text{on }\Sigma_{T} = (0,T)\times \partial \Omega \\ u(0,\cdot) = u_0(\cdot) \quad \text{in }\Omega. \end{gathered} \end{equation} The existence and uniqueness of renormalized solutions to problems \eqref{pb BWZ} and \eqref{pb curv} are nowadays well-known (see \cite{Benwittzim, ZZ}). We recall that the notion of renormalized solutions was introduced for the first time by Diperna and Lions \cite{Dip} in their study of the Boltzmann equation. An equivalent notion of solutions, called entropy solutions, was introduced independently by B\'enilan and al.\ in \cite{Benboc}. Following \cite{Benboc} and using the same notion of solution, Ouaro and Traor\'e (see \cite{Ouarotraore1}) studied the problem \begin{equation}\label{pb OT} \begin{gathered} u - \operatorname{div}a(x,\nabla u) = f \quad \text{in }\Omega \\ u = 0 \quad \text{on }\partial \Omega, \end{gathered} \end{equation} where they proved the existence and uniqueness of entropy solution for a data $f \in L^1(\Omega)$. Relying on these results and applying nonlinear semigroup theory, it is easy to deduce the existence of a unique mild solution for the abstract Cauchy problem corresponding to \eqref{pbmere} and arbitrary $L^1-$data (cf.\ section 4). In this paper, we use the abstract semigroup theory to prove the existence and uniqueness of entropy solution to \eqref{pbmere} for arbitrary $L^1-$data. We recall that Wittbold and Zimmermann in \cite{WZ} studied and proved the existence and uniqueness of a renormalized solution to the stationary problem \begin{equation}\label{pb WZ} \begin{gathered} \beta(u) - \operatorname{div}a(x,Du) - \operatorname{div}F(u) \ni f \quad \text{in }\Omega, \\ u = 0 \quad \text{on }\partial \Omega, \\ \end{gathered} \end{equation} where $f \in L^1(\Omega), \Omega$ a bounded domain of $\mathbb{R}^N (N \geq 1)$ with Lipschitz boundary $\partial \Omega$ (if $N \geq 2$), $F: \mathbb{R} \to \mathbb{R}^N$ locally Lipschitz continuous, $\beta: \mathbb{R} \to 2^{\mathbb{R}}$ a set valued, maximal monotone mapping such that $0 \in \beta(0)$, $a: \Omega \times \mathbb{R}^N \to \mathbb{R}^N$ a Carath\'eodory function and $p(\cdot): \overline{\Omega} \to (1,\infty)$ a continuous variable exponent such that $1 < \min_{x \in \overline{\Omega}}{p(x)} < N$. Relying on these above results and applying nonlinear semigroup theory (see \cite{Bencrand}), Bendahmane, Wittbold and Zimmermann proved (see \cite{Benwittzim}) the existence and uniqueness of a renormalized solution to the problem \eqref{pb BWZ}. Apart from the work by Bendahmane and al \cite{Benwittzim}, Zhang and Zhou \cite{ZZ} studied the problem \eqref{pb BWZ} by using other methods, where they proved the existence and uniqueness of entropy solutions. They also proved the equivalence between entropy and renormalized solutions of \eqref{pb BWZ}. The method used in \cite{ZZ} was the following: They employed first the difference and variation methods to prove the existence and uniqueness of a weak solution for the approximate problem of \eqref{pb BWZ} under appropriate assumptions. Then they constructed an approximate solution sequence and established some \textit{ a priori} estimates. Next, they drew a subsequence to obtain a limit function and proved that this function is a renormalized solution. Based on the strong convergence on the truncations of approximate solutions, they obtained that the renormalized solution to problem \eqref{pb BWZ} is also an entropy solution, which leads to an equality in the entropy formulation. Finally, by choosing suitable test functions, they proved the uniqueness of renormalized solutions and entropy solutions and thus, the equivalence of renormalized solutions and entropy solutions. The main operator in problem \eqref{pbmere} is more general than the $p(\cdot)$-Laplace operator of \eqref{pb BWZ} as we will see later. The aim of our paper is to extend the results in \cite{Ouarotraore1}, to the case of parabolic equations. Inspired by \cite{Benwittzim} and \cite{ZZ}, we first define two notions of solutions of problem \eqref{pbmere}: The notion of entropy solution and the notion of renormalized solution. Next, we show that the two notions are equivalent which will permit us to use both notion when convenient. After that, according to the results in \cite{Ouarotraore1}, we prove some properties of the entropy solutions of problem \eqref{pbmere}, by using nonlinear semigroup theory. Next, we prove the existence and uniqueness of entropy solutions to problem \eqref{pbmere}. This article is organized as follows: In section 2 we recall some results of \cite{Ouarotraore1}, the assumptions of problem \eqref{pbmere} and some basic notations and properties of Lebesgue and Sobolev spaces with variable exponents. In section 3, we give the definition of entropy and renormalized solutions to problem \eqref{pbmere} and prove that the two notions are equivalent. In section 4, using the results of \cite{Ouarotraore1}, we prove some properties of entropy solutions to problem \eqref{pbmere}. Finally, in section 5 we prove the existence and uniqueness of entropy solutions of \eqref{pbmere}. \section{Preliminaries} In this article, we study problem \eqref{pbmere} with the following assumptions on the data: \begin{equation}\label{p(x)} p(\cdot): \Omega \to \mathbb{R}\text{ is a measurable function such that } 1 < p_{-} \leq p_{+} < + \infty, \end{equation} where $p_{-} := \operatorname{ess\,inf}_{x \in \Omega}{p(x)}$ and $p_{+} := \operatorname{ess\,sup}_{x \in \Omega}{p(x)}$. For the vector field $a(\cdot,\cdot)$, we assume that $a(x,\xi): \Omega \times \mathbb{R}^N \to \mathbb{R}^N$ is Carath\'eodory and is the continuous derivative with respect to $\xi$ of the mapping $ A: \Omega \times \mathbb{R}^N \to \mathbb{R}$, i.e. $a(x,\xi) = \nabla_{\xi}A(x,\xi)$ such that: \begin{equation}\label{A} A(x,0) = 0\quad\text{for almost every }x \in \Omega\,. \end{equation} There exists a positive constant $C_1$ such that \begin{equation}\label{C1} |a(x,\xi)| \leq C_1(j(x) + |\xi|^{p(x)-1}) \end{equation} for almost every $x \in \Omega$ and for every $\xi \in \mathbb{R}^N$ where $j$ is a nonnegative function in $ L^{p'(\cdot)}(\Omega)$, with $ \frac{1}{p(x)} + \frac{1}{p'(x)} = 1$. The following inequalities hold \begin{equation}\label{coerc} (a(x,\xi) - a(x,\eta))\cdot(\xi-\eta) > 0, \end{equation} for almost every $x \in \Omega$ and for every $\xi, \eta \in \mathbb{R}^N$, with $\xi \neq \eta$ and \begin{equation}\label{encadr} \frac{1}{C}|\xi|^{p(x)} \leq a(x,\xi).\xi \leq Cp(x)A(x,\xi) \end{equation} for almost every $x \in \Omega$, $C > 0$ and for every $\xi \in \mathbb{R}^N$. Assumption \eqref{coerc} is imposed to obtain uniqueness of the solution to problem \eqref{pbmere}. \begin{remark}\label{remarque a(x,0)} \rm (1) Strict monotonicity (see assumption \eqref{coerc}) of the vector field is certainly not needed to prove uniqueness of the entropy solution. It was assumed it here only just for simplicity. (2) $a(x,0)=0$ for a.e. $x\in\Omega$. Indeed for a.e. $x\in\Omega$ fixed, denote $z=a(x,0)\in\mathbb{R}^N$. By the continuity of $a(x,\cdot)$, we have $ \lim_{\xi\to 0} a(x,\xi)=z$. Suppose now that $z\neq 0$ (if $z=0$, there is no need to make a proof; this is the case for example when $a(x,\xi)=|\xi|^{p-2}\xi$) and choose $\xi_0=-sz$ with $s>0$ used to tend toward 0; then $a(x,\xi_0)\cdot\xi_0=-s(z+\epsilon(s))\cdot z = -s|z|^2-sz \epsilon(s) \leq -s|z|^2+s|z\|\epsilon(s)|$, where $ \lim_{s\to 0}|\epsilon (s)|=0$. Therefore, for $s$ sufficiently small, $-s|z|^2+s|z\|\epsilon(s)|<0$, which is a contradiction by assumption \eqref{encadr}. Thus, $z=0$. (3) As examples of models with respect to assumptions \eqref{A}-\eqref{encadr} for problem \eqref{pbmere}, we can give the following.\\ (i) Set $A(x,\xi) = (1/ p(x))|\xi|^{p(x)}, a(x,\xi) = |\xi|^{p(x)-2}\xi$, where $p(x) \geq 2$. Then we obtain the $p(x)$-Laplace operator $$ \operatorname{div}(|\nabla u|^{p(x)-2}\nabla u). $$ (ii) Set $A(x,\xi) = (1/ p(x))[\Big(1 + |\xi|^2\Big)^{p(x)/2} - 1]$, $a(x,\xi) = \big(1 + |\xi|^2\big)^{(p(x)-2)/2}\xi$, where $p(x) \geq 2$. Then we obtain the generalized mean curvature operator $$ \operatorname{div}\Big(\big(1 + |\nabla u|^2\big)^{(p(x)-2)/2}\nabla u\Big). $$ \end{remark} As the exponent $p(x)$ appearing in \eqref{C1} and \eqref{encadr} depends on the variable $x$, we must work with Lebesgue and Sobolev spaces with variable exponents. We define the Lebesgue space with variable exponent $L^{p(\cdot)}(\Omega)$ as the set of all measurable functions $u: \Omega \to \mathbb{R}$ for which the convex modular \begin{equation*} \rho_{p(\cdot)}(u):= \int_{\Omega} |u|^{p(x)} \,dx \end{equation*} is finite. If the exponent is bounded, i.e., if $p_{+} < + \infty$, then the expression \begin{equation*} |u|_{p(\cdot)}:= \inf{\{\lambda > 0: \rho_{p(\cdot)}(u/\lambda) \leq 1\}} \end{equation*} defines a norm in $L^{p(\cdot)}(\Omega)$, called the Luxembourg norm. The space $ (L^{p(\cdot)}(\Omega),|.|_{p(\cdot)})$ is a separable Banach space. Moreover, if $1 < p_{-} \leq p_{+} < + \infty$, then $L^{p(\cdot)}(\Omega)$ is uniformly convex, hence reflexive, and its dual space is isomorphic to $L^{p'(\cdot)}(\Omega)$, where $ \frac{1}{p(x)} + \frac{1}{p'(x)} = 1$. Finally, we have the H\"{o}lder type inequality. \begin{equation}\label{ineholder} \big|\int_{\Omega} uv \, dx\big| \leq \Big(\frac{1}{p_{-}} + \frac{1}{p_{+}}\Big)|u|_{p(\cdot)}|v|_{p'(\cdot)}, \end{equation} for all $u \in L^{p(\cdot)}(\Omega)$ and $v \in L^{p'(\cdot)}(\Omega)$. Now, let \begin{equation*} W^{1,p(\cdot)}(\Omega):=\Big\{u \in L^{p(\cdot)}(\Omega): |\nabla u| \in L^{p(\cdot)}(\Omega)\Big\}, \end{equation*} which is a Banach space equipped with the norm \begin{equation*} \|u\|_{1,p(\cdot)} = |u|_{p(\cdot)} + |(|\nabla u|)|_{p(\cdot)}. \end{equation*} The space $ \big(W^{1,p(\cdot)}(\Omega),\|u\|_{1,p(\cdot)}\big)$ is a separable and reflexive Banach space. Next, we define $ W_0^{1,p(\cdot)}(\Omega)$ as the closure of $\mathcal{C}_0^{\infty}(\Omega)$ in $ W^{1,p(\cdot)}(\Omega)$ under the norm $$ \|u\|:= |(|\nabla u|)|_{p(\cdot)}. $$ The space $ \big(W_0^{1,p(\cdot)}(\Omega),\|u\|\big)$ is a separable and reflexive Banach space. For the interested reader, more details about Lebesgue and Sobolev spaces with variable exponent can be found in \cite{DHHR} (see also \cite{Kovarak}). An important role in manipulating the generalized Lebesgue and Sobolev spaces is played by the modular $\rho_{p(\cdot)}$ of the space $L^{p(\cdot)}(\Omega)$. We have the following result (cf. \cite{Fanzhao}). \begin{lemma}\label{lemmaineq} If $u_n, u \in L^{p(\cdot)}$ and $p_{+} < +\infty$, then the following properties hold: \begin{enumerate} \item $|u|_{p(\cdot)} > 1 \Rightarrow |u|_{p(\cdot)}^{p_{-}} \leq \rho_{p(\cdot)}(u) \leq |u|_{p(\cdot)}^{p_{+}}$; \item $|u|_{p(\cdot)} < 1 \Rightarrow |u|_{p(\cdot)}^{p_{+}} \leq \rho_{p(\cdot)}(u) \leq |u|_{p(\cdot)}^{p_{-}}$; \item $|u|_{p(\cdot)} < 1$ (respectively $= 1; > 1) \Longleftrightarrow \rho_{p(\cdot)}(u) < 1$ (respectively $= 1; > 1$); \item $|u_n|_{p(\cdot)} \to 0$ (respectively $\to +\infty) \Longleftrightarrow \rho_{p(\cdot)}(u_n) \to 0$ (respectively $\to +\infty$); \item $\rho_{p(\cdot)}\big(u/|u|_{p(\cdot)}\big) = 1$. \end{enumerate} \end{lemma} Following \cite{Benwittzim}, we extend a variable exponent $p: \overline{\Omega} \to [1, +\infty)$ to $\overline{Q} = [0,T] \times \overline{\Omega}$ by setting $p(t,x):= p(x)$ for all $(t,x) \in \overline{Q}$. We also consider the generalized Lebesgue space \begin{equation*} L^{p(\cdot)}(Q) = \big\{u: Q\to \mathbb{R}\text{ measurable such that } \iint{Q} |u(t,x)|^{p(x)}\, d(x,t) < \infty\Big\}, \end{equation*} endowed with the norm \begin{equation*} \|u\|_{L^{p(\cdot)}(Q)}:= \inf\Big\{\lambda > 0: \iint_{Q} \big|\frac{u(t,x)}{\lambda}\big|^{p(x)}\, d(x,t) < 1\Big\}, \end{equation*} which shares the same properties as $L^{p(\cdot)}(\Omega)$. We now recall the main result of \cite{Ouarotraore1} for the study of \eqref{pb OT}. We first recall the definition of the weak and entropy solutions of \eqref{pb OT}. \begin{definition} \label{def2.3} \rm A weak solution of \eqref{pb OT} is a function $u \in W_0^{1,1}(\Omega)$ such that $a(\cdot,\nabla u) \in \big(L^1_{\rm loc}(\Omega)\big)^N$ and \begin{equation}\label{weaksolOT} \int_{\Omega} a(x, \nabla u)\cdot\nabla \varphi \,dx + \int_{\Omega} u \varphi \, dx = \int_{\Omega} f(x) \varphi \, dx, \end{equation} for all $\varphi \in C_0^{\infty}(\Omega)$. A weak energy solution is a weak solution such that $u \in W_0^{1,p(\cdot)}(\Omega)$. \end{definition} \begin{definition} \label{def.24} \rm A measurable function $u$ is an entropy solution to problem \eqref{pb OT} if, for every $t > 0$, $T_{t}(u) \in W_0^{1,p(\cdot)}(\Omega)$ and \begin{equation}\label{entrsolOT} \int_{\Omega} uT_{t}(u - \varphi) \, dx + \int_{\Omega} a(x, \nabla u)\cdot\nabla T_{t}(u - \varphi) \,dx \leq \int_{\Omega} f(x) T_{t}(u - \varphi) \, dx, \end{equation} for all $\varphi \in W_0^{1,p(\cdot)}(\Omega) \cap L^{\infty}(\Omega)$. \end{definition} Now, we recall the two main results in \cite{Ouarotraore1}. \begin{theorem}[{\cite[Theorem 3.2]{Ouarotraore1}}] \label{theoOT1} Assume that \eqref{p(x)}-\eqref{encadr} hold and $f \in L^{\infty}(\Omega)$. Then there exists a unique weak energy solution of \eqref{pb OT}. \end{theorem} \begin{theorem}[{\cite[Theorem 4.3]{Ouarotraore1}}] \label{theoOT2} Assume that \eqref{p(x)}-\eqref{encadr} hold and $f \in L^1(\Omega)$. Then there exists a unique entropy solution to problem \eqref{pb OT}. \end{theorem} \begin{remark} \label{rmk2.7} \rm Theorems \ref{theoOT1} and \ref{theoOT2} were generalized by Bonzi and Ouaro (see \cite[Theorem 3.2 and 4.3]{BO}). According to \cite[Theorem 3.2]{BO}, \cite[Theorem 3.2 ]{Ouarotraore1} hold for $f \in L^{(p_{-})'}(\Omega)$. \end{remark} \section{Equivalence between entropy and renormalized solutions} Let $T_{k}$ denote the truncation function at height $k$, that is \[ T_{k}(s) = \begin{cases} s& \text{if }|s| \leq k,\\ k\operatorname{sign}(s) &\text{if }|s| > k . \end{cases} \] For the notion of entropy solution to problem \eqref{pbmere}, we will use the primitive of the truncation function at height $k \geq 0$ denoted by $\Theta_{k}: \mathbb{R} \to \mathbb{R}^{+}$ such that \begin{equation*} \Theta_{k}(r) = \int_0^{r} T_{k}(s)\,ds = \begin{cases} r^2/2 & \text{if }|r| \leq k, \\ k|r| - \frac{k^2}{2} & \text{if } |r| \geq k. \end{cases} \end{equation*} It is obvious that $\Theta_{k}(r) \geq 0$ and $\Theta_{k}(r) \leq k|r|$. We denote \begin{align*} \mathcal{T}_0^{1,p(\cdot)}(Q) = \Big\{&u: \Omega \times (0,T] \to \mathbb{R}\text{ measurable }; T_{k}(u) \in L^{p_{-}} (0,T; W_0^{1,p(\cdot)}(\Omega)),\\ & \text{with } \nabla T_{k}(u) \in \big(L^{p(\cdot)}(Q)\big)^N, \text{ for every } k > 0\Big\}. \end{align*} Next, we define the weak gradient of a measurable function $u \in \mathcal{T}_0^{1,p(\cdot)}(Q)$. The proof follows from \cite[Lemma 2.1]{Benboc} due to the fact that $ W_0^{1,p(\cdot)}(\Omega) \subset W_0^{1,p_{-}}(\Omega)$. \begin{proposition} \label{prop3.1} For every measurable function $u \in \mathcal{T}_0^{1,p(\cdot)}(Q)$, there exists a unique measurable function $\nu: Q \to \mathbb{R}^N$, which we call the weak gradient of $u$ and denote $\nu = \nabla u$, such that \begin{equation*} \nabla T_{k}(u) = \nu \chi_{\{|u|0$}, \end{equation*} where $\chi_{E}$ denotes the characteristic function of a measurable set $E$. Moreover, if $u$ belongs to $ L^1(0,T; W_0^{1,1}(\Omega))$, then $\nu$ coincides with the gradient of $u$. \end{proposition} The notion of the weak gradient allows us to give the following definitions of entropy and renormalized solutions to problem \eqref{pbmere}. We define the spaces: \begin{gather*} V=\big\{f\in L^{p_{-}}(0,T;W^{1,p(\cdot)}_0(\Omega)):|\nabla f| \in L^{p(\cdot)}(Q)\big\}, \\ E=\big\{\varphi \in V \cap L^{\infty}(Q):\varphi_{t}\in V^{*} + L^1(Q)\big\}. \end{gather*} According to \cite{Porreta}, we have $E \subset \mathcal{C}([0,T]; L^1(\Omega))$. \begin{definition} \label{def3.2} \rm An entropy solution to problem \eqref{pbmere} is a function $u \in \mathcal{T}_0^{1,p(\cdot)}(Q) \cap L^{\infty}(Q)$ such that the mapping $$ [0,T] \ni t \mapsto \int_{\Omega} \Theta_{k}(u-\phi)(t,x)\, dx $$ is a.e. equal to a continuous function for all $k > 0$ and all $\phi \in E$, and \begin{equation}\label{entrsol} \begin{aligned} &\int_{\Omega} \Theta_{k}(u - \phi)(T) \, dx - \int_{\Omega} \Theta_{k}(u_0 - \phi(0))\,dx \\ &+ \int_{Q} \phi_{t} T_{k}(u - \phi)\,dx\,dt + \int_{Q} a(x, \nabla u)\cdot\nabla T_{k}(u - \phi) \,dx\,dt \\ &= \int_{Q} fT_{k}(u - \phi) \, dt\, dx, \end{aligned} \end{equation} for all $k>0$ and $\phi \in E$. \end{definition} \begin{definition}\label{def3.3} \rm A function $u \in \mathcal{T}_0^{1,p(\cdot)}(Q) \cap L^{\infty}(Q)$ is a renormalized solution to problem \eqref{pbmere} if the following conditions are satisfied: (i) \begin{equation}\label{intprop} \lim_{n \to +\infty} \int_{\{(t,x) \in Q: n \leq |u(t,x)| \leq n+1\}} |\nabla u|^{p(x)}\, dt\, dx = 0; \end{equation} (ii) for all $S$ in $W^{2,\infty}(\mathbb{R})$ such that $S'$ has a compact support, \begin{gather}\label{renormsol} \frac{\partial}{\partial t} S(u) - \operatorname{div}\big[S'(u) a(x, \nabla u) + S"(u)a(x,\nabla u)\cdot\nabla u \big] = f S'(u) \quad\text{in }\mathcal{D}'(Q), \\ \label{initcond} S(u)(0) = S(u_0)\quad \text{in }L^1(\Omega)\,. \end{gather} \end{definition} \begin{remark} \label{rmk3.4} \rm Using the fact that for any function $\varphi\in V\cap L^{\infty}(Q)$, there exists functions $\varphi_n\in\mathcal{D}(Q)$ that converge strongly to $\varphi$ in $V$ and weak-* in $L^{\infty}(Q)$, we see that in \eqref{entrsol} and \eqref{renormsol} we cannot only use the test functions in $\mathcal{D}(Q)$, but also functions in $V\cap L^{\infty}(Q)$. In fact, we can replace \eqref{renormsol} by \begin{equation}\label{renormsol2} \begin{aligned} &\big\langle \frac{\partial S(u)}{\partial t},\varphi\big\rangle + \int_0^{T}\int_{\Omega} \big[S'(u) a(x, \nabla u)\cdot\nabla \varphi + S"(u)a(x,\nabla u)\cdot\nabla (u) \varphi\big]\, dx\,dt \\ &= \int_0^{T}\int_{\Omega} f S'(u) \varphi\,dx\,dt, \end{aligned} \end{equation} where $\langle\cdot,\cdot\rangle$ denotes the duality pairing between $V^{*} + L^1(Q)$ and $V \cap L^{\infty}(Q)$. \end{remark} To find more estimates for entropy solutions and also to get useful a priori estimates of approximate solutions to the equation \eqref{solfaiblmere2} below, the following integration by parts formula plays a crucial role. \begin{lemma}\label{integration by parts} Let $f: \mathbb{R} \to \mathbb{R}$ be a continuous piecewise $C^1$ function such that $f(0)=0$ and $f'$ is zero outside a compact set of $\mathbb{R}$. Let us denote $F(s) = \int_0^{s} f(r)\, dr$. If $u \in V$ is such that $u_{t} \in V^{*} + L^1(Q)$ and if $\psi \in \mathcal{C}^{\infty}(\overline{Q})$, then we have $$ \langle u_{t}, f(u)\psi\rangle = \int_{\Omega} F(u(T))\psi(T)\, dx - \int_{\Omega} F(u(0))\psi(0)\, dx - \int_{Q} \psi_{t}F(u)\,dx\,dt, $$ where $\langle \cdot,\cdot \rangle$ denotes the duality pairing between $V^{*}+L^1(Q)$ and $V \cap L^{\infty}(Q)$. \end{lemma} The proof of the above lemma follows the same lines as the proof of \cite[Lemma 7.1]{DP}; we omit it. Next, we have a result showing the equivalence between entropy and renormalized solutions of \eqref{pbmere}. \begin{theorem}\label{theorem equiv} A function $u$ is an entropy solution of \eqref{pbmere} if and only if it is a renormalized solution. \end{theorem} The proof of the above theorem is the same as in constant exponent case; see \cite{DP}. \section{Properties of entropy solutions} In this section, we prove the existence of mild solutions of \eqref{pbmere} satisfying an $L^1$-comparison principle. A classical method to prove that consists in approximating \eqref{pbmere} for $\epsilon > 0$, by an implicit time-discretization \begin{equation}\label{pb discret} \begin{gathered} \frac{u_i^{\epsilon}-u_{i-1}^{\epsilon}}{t_i^{\epsilon}-t_{i-1}^{\epsilon}} = \operatorname{div}a(x,\nabla u_i^{\epsilon}) + f_i^{\epsilon }\quad \text{in $D'(\Omega)$, for $i=1,\dots,n$}, \\ u_i^{\epsilon} \in W_0^{1,p(\cdot)}(\Omega) \cap L^{\infty}(\Omega), \end{gathered} \end{equation} where $n \in \mathbb{N}$, $0 = t_0^{\epsilon} < t_1^{\epsilon} < \dots < t_n^{\epsilon} \leq T$ and $f_i^{\epsilon} \in L^{\infty}(\Omega)$, $i = 1,\dots, n$ such that \begin{gather*} \sum_{i=1}^{n}\int_{t_{i-1}^{\epsilon}}^{t_i^{\epsilon}} \|f(t) - f_i^{\epsilon}\|_{L^1(\Omega)} dt\to 0,\quad \max_{i=1,\dots ,n}{(t_i^{\epsilon} - t_{i-1}^{\epsilon})}\to 0,\\ T - t_n^{\epsilon }\to 0, \quad \|u_0 - u_0^{\epsilon}\|_{L^1(\Omega)} \to 0\quad \text{as } \epsilon \to 0. \end{gather*} The function $u_{\epsilon}$ is piecewise constant, defined by \begin{equation*} u_{\epsilon} = u_i^{\epsilon }\quad \text{on } (t_{i-1}^{\epsilon},t_i^{\epsilon}], \; i = 1,\dots , n;\quad u_{\epsilon}(0) = u_0^{\epsilon}. \end{equation*} This method is actually the method of nonlinear semigroup theory. Naturally, we are led to give the following concept. \begin{definition} \label{def4.1} \rm A mild solution of \eqref{pbmere} is a function $u\in C([0,T]; L^1(\Omega))$ which is the uniform limit of the piecewise constant function $ u_{\epsilon}$. \end{definition} The main result of this section is the following. \begin{theorem} \label{thm4.2} For any $(u_0,f) \in L^1(\Omega) \times L^1(Q)$, there exists a unique mild solution $u$ of \eqref{pbmere}. Moreover, the following contraction principle holds: for any $0 \leq t \leq T$, if $u$ (resp. $\hat{u}$) is a mild solution of \eqref{pbmere} with respect to $(u_0,f) \in L^1(\Omega) \times L^1(Q)$ (resp. $(\hat{u}_0,\hat{f}) \in L^1(\Omega) \times L^1(Q)$), then \begin{equation*} \|u(t) - \hat{u}(t)\|_{L^1(\Omega)} \leq \|u_0 - \hat{u}_0\|_{L^1(\Omega)} + \int_0^{t} \big |\big|f(s) - \hat{f}(s)\big |\big|_{L^1(\Omega)}\,ds. \end{equation*} \end{theorem} According to the nonlinear semigroups theory (see \cite{Bencrand}), the preceding result is, essentially, a consequence of the result of Proposition \ref{propacr} below. Before stating the proposition, we need to recall some definitions. Let $A$ be a (possibly) multivalued nonlinear operator in $L^1(\Omega)$ that is $A: L^1(\Omega) \to \mathcal{P}(L^1(\Omega))$; as usual, $A$ is identified with its graph $\{(u,v) \in L^1(\Omega) \times L^1(\Omega); v \in Au\}$. The operator $A$ is called accretive if \begin{equation*} \|u - \hat{u}\|_1 \leq \|u - \hat{u} + \sigma (v - \hat{v})\|_1, \quad \text{for any }(u,v), (\hat{u},\hat{v}) \in A, \sigma > 0; \end{equation*} i.e., for any $\sigma > 0$, the resolvent of $A, (I + \sigma A)^{-1}$, is a single-valued operator and a contraction in the $L^1-$norm. The operator $A$ is called $T$-accretive if $ \|(u - \hat{u})^{+}\|_1 \leq \|(u - \hat{u})^{+} + \sigma (v - \hat{v})^{+}\|_1$, for any $(u,v), (\hat{u},\hat{v}) \in A$, $\sigma > 0$; equivalently, if \[ \int_{\{u > \hat{u}\}} (v - \hat{v})^{+} + \int_{\{u = \hat{u}\}} (v - \hat{v})^{+} \geq 0, \] for any $(u,v), (\hat{u},\hat{v}) \in A$. Finally, the operator $A$ is called $m$-accretive (resp. $m-T$-accretive) if $A$ is accretive (resp. $T$-accretive) and, moreover, $R(I + \sigma A) = L^1(\Omega)$, for any $\sigma > 0$ (cf. \cite{Bencrand} ). \begin{proposition}\label{propacr} There exists an operator \begin{equation*} A = \{(u,f) \in L^1(\Omega) \times L^1(\Omega); u \text{ is an entropy solution of }\eqref{pb OT}\} \end{equation*} such that \begin{itemize} \item[(i)] $A$ is $T$-accretive (and even completely accretive, cf. \cite{BC}); \item[(ii)] $R(I + \sigma A) = L^1(\Omega)$, for any $\sigma > 0$; \item[(iii)] $\overline{D(A)} = L^1(\Omega)$. \end{itemize} \end{proposition} \begin{proof} (i) Let $u$ (resp. $\hat{u}$) be a weak solution of \eqref{pb OT} for $f$ (resp. $\hat{f}) \in L^{\infty}(\Omega)$. We use $ \frac{1}{k}T_{k}(u - \hat{u})^{+}$ as test function in \eqref{weaksolOT} for $k > 0$ to get upon addition \begin{align*} &\int_{\Omega} (u - \hat{u})\frac{1}{k}T_{k}(u - \hat{u})^{+} \, dx + \int_{\{|u-\hat{u}| 0$, and $f \in L^{\infty}(\Omega)$. We denote $ u_{\alpha}:= (I + \alpha A)^{-1}f$. Then $ (u_{\alpha}, \frac{1}{\alpha}(f - u_{\alpha})) \in A$. As $f \in L^{\infty}(\Omega)$ then, according to Theorem \ref{theoOT1}, $u_{\alpha}$ is a weak energy solution of \eqref{pb OT}. Let's take $\phi \in D(\Omega)$ as a test function in \eqref{weaksolOT} to obtain \begin{equation}\label{Testphi} \alpha \int_{\Omega} a(x,\nabla u_{\alpha})\cdot\nabla \phi \, dx + \int_{\Omega} u_{\alpha} \phi \, dx = \int_{\Omega} f(x) \phi \, dx. \end{equation} The following Lemma provides $L^{\infty}$-a priori estimates of a solution $u$ and is crucial for the next of the proof. \begin{lemma}\label{lemmanorm u} Let $u$ be a weak energy solution of \eqref{pb OT}, then \begin{equation*} \|u\|_s \leq C \|f\|_s, \quad \text{for } 1 \leq s \leq + \infty. \end{equation*} \end{lemma} \begin{proof} The proof is rather classical (see. \cite{Ouarotraore1}). For the sake of completeness, let us recall the arguments. For $p \in P_0 = \big\{ p \in C^{\infty}(\mathbb{R}); 0 \leq p' \leq 1$, $\operatorname{supp}p'$ is compact, $0 \notin \operatorname{supp} p\big\}$, we take $p(u_{\alpha})$ as a test function in \eqref{Testphi} to obtain \begin{equation}\label{Testpualfa} \begin{aligned} &\int_{\Omega} p(u_{\alpha})f(x) \,dx \\ & = \alpha \int_{\Omega} a(x,\nabla u_{\alpha})\cdot\nabla p(u_{\alpha}) \, dx + \int_{\Omega} p(u_{\alpha}) u_{\alpha} \, dx \\ & = \alpha \int_{\Omega} \big[a(x,\nabla u_{\alpha}) - a(x,0)\big] \cdot\nabla u_{\alpha} p'(u_{\alpha}) \, dx + \alpha \int_{\Omega} a(x,0)\cdot\nabla u_{\alpha}p'(u_{\alpha}) \, dx \\ &\quad + \int_{\Omega} p(u_{\alpha}) u_{\alpha} \, dx \\ & \geq \alpha \int_{\Omega} a(x,0)\cdot\nabla u_{\alpha}p'(u_{\alpha}) \, dx + \int_{\Omega} p(u_{\alpha}) u_{\alpha} \, dx \\ & = \int_{\Omega} p(u_{\alpha}) u_{\alpha} \, dx \quad (\text{by the divergence formula}). \end{aligned} \end{equation} Next, we choose $p$ such that $ p(k) = |k|^{s-2}k$ for $1 \leq s < + \infty$ in \eqref{Testpualfa} to obtain \begin{equation}\label{Inegualpha} \int_{\Omega} |u_{\alpha}|^{s-2}u_{\alpha}f \, dx \geq \int_{\Omega} |u_{\alpha}|^{s} \, dx. \end{equation} By H\"{o}lder inequality, from \eqref{Inegualpha} we obtain \begin{equation*} \int_{\Omega} |u_{\alpha}|^{s} \, dx \leq \|f\|_s\Big(\int_{\Omega} \big(|u_{\alpha}|^{s-1}\big)^{s'} \, dx \Big)^{1/s'}, \end{equation*} which gives \begin{equation}\label{Normualpha} \|u_{\alpha}\|_s \leq \|f\|_s. \end{equation} As $f \in L^{\infty}(\Omega)$, then \eqref{Normualpha} implies $\|u_{\alpha}\|_{\infty} \leq \|f\|_{\infty}$. \end{proof} Now, let us come back to the proof of Proposition \ref{propacr}. We take $u_{\alpha}$ as a test function in \eqref{Testphi} to obtain \begin{equation}\label{Testualpha} \begin{aligned} \alpha \int_{\Omega} a(x,\nabla u_{\alpha})\cdot\nabla u_{\alpha} \, dx & = - \int_{\Omega} u_{\alpha}^2 \, dx + \int_{\Omega} f(x) u_{\alpha} \, dx \\ & \leq \|u_{\alpha}\|_{q}\|f\|_{p}. \end{aligned} \end{equation} Then, by Lemma \ref{lemmanorm u} and \eqref{encadr}, from \eqref{Testualpha} we deduce that \begin{equation}\label{Inenabla} \alpha \int_{\Omega} |\nabla u_{\alpha}|^{p(x)} dx \leq C \|f\|_{p}\|f\|_{q} < \infty \quad \text{(because $ f \in L^{\infty}(\Omega)$)}. \end{equation} Now, by using the H\"{o}lder type inequality, for all $\phi \in D(\Omega)$, we have \begin{equation}\label{Inegtypeholder} \begin{aligned} &\big| \alpha \int_{\Omega} a(x,\nabla u_{\alpha})\cdot\nabla \phi dx \big|\\ & \leq C_1 \alpha \int_{\Omega} \big(j(x) + |\nabla u_{\alpha}|^{p(x)-1}\big)|\nabla \phi| dx \\ & \leq C' \alpha \|j\|_{p'(\cdot)}\|\nabla \phi\|_{p(\cdot)} + C' \alpha \big|\big(|\nabla u_{\alpha}|^{p(x)-1}\big)\big|_{p'(\cdot)}\|\nabla \phi\|_{p(\cdot)}\\ & \leq C \max \Big(\alpha^{1-\frac{1}{p_{-}}}(\alpha \rho_{p(\cdot)} (\nabla u_{\alpha}))^{\frac{1}{p_{-}}} , \alpha^{1-\frac{1}{p_{+}}} (\alpha \rho_{p(\cdot)}(\nabla u_{\alpha}))^{\frac{1}{p_{+}}}\Big)+C\alpha. \end{aligned} \end{equation} According to \eqref{Inenabla}, from \eqref{Inegtypeholder} we deduce that \begin{equation}\label{Limalphaa} \big| \alpha \int_{\Omega} a(x,\nabla u_{\alpha})\cdot\nabla \phi dx \big| \to 0 \quad \text{as } \alpha \to 0. \end{equation} From \eqref{Testphi} by using \eqref{Limalphaa} we obtain \begin{equation}\label{Convfaiblualph} u_{\alpha} \to f \quad \text{as $\alpha \to 0$, in }\mathcal{D}'(\Omega). \end{equation} Note also that $ (u_{\alpha})_{\alpha>0}$ is uniformly bounded by Lemma \ref{lemmanorm u}, then up to a subsequence, $u_{\alpha} \to f$ in $ L^{p}(\Omega)$, for all $1 < p < +\infty$ and a.e. in $\Omega$. Now, $ \|u_{\alpha}\|_{p} \leq \|f\|_{p}$ for all $1 < p < +\infty$ by Lemma \ref{lemmanorm u}, then by the Lebesgue dominated convergence theorem, we deduce that \begin{equation}\label{ConvLPualph} u_{\alpha} \to f \quad \text{as $\alpha \to 0$, in }L^{p}(\Omega), \; \forall 1 < p < +\infty. \end{equation} As $\Omega$ is bounded, \eqref{ConvLPualph} implies \begin{equation}\label{ConvL1ualph} u_{\alpha} \to f \quad \text{in $L^1(\Omega)$ as } \alpha \to 0. \end{equation} Therefore, by \eqref{ConvL1ualph}, we deduce that $ \overline{D(A)} = L^1(\Omega)$. \end{proof} By Proposition \ref{propacr}, the nonlinear operator $A$ is $m$-accretive in $L^1(\Omega)$. Then, by the general theory of nonlinear semigroups (see \cite{Bencrand}) we conclude that the abstract evolution problem corresponding to \eqref{pbmere} admits a unique mild solution $ u \in C([0,T];L^1(\Omega))$ for any initial datum $ u_0 \in \overline{D(A)}^{\|.\|_{L^1(\Omega)}}$ and any right-hand side $ f \in L^1(0,T;L^1(\Omega))$. \begin{lemma}\label{lemma bornugradu} Let $u$ be an entropy solution to problem \eqref{pbmere}, then \begin{gather}\label{bornu} \|u\|_{L^{\infty}(0,T; L^1(\Omega))} \leq \|f\|_{L^1(Q)} + \|u_0\|_{L^1(\Omega)},\\ \label{borngradu} \begin{aligned} &\|\nabla T_{k}(u)\|_{L^{p(\cdot)}(Q)} \\ &\leq k \max{\Big\{ \big(\|f\|_{L^1(Q)} + \|u_0\|_{L^1(\Omega)}\big)^{1/p_{-}}, \big(\|f\|_{L^1(Q)} + \|u_0\|_{L^1(\Omega)}\big)^{1/p_{+}}\Big\}}. \end{aligned} \end{gather} \end{lemma} \begin{proof} \textbf{Step 1:} Proof of \eqref{borngradu}. Taking $\phi = 0$ as a test function in \eqref{entrsol}, we obtain \begin{equation}\label{testentrosol} \begin{aligned} &\int_{\Omega} \Theta_{k}(u)(T) \, dx - \int_{\Omega} \Theta_{k}(u_0)\,dx + \int_{Q} a(x, \nabla u)\cdot\nabla T_{k}(u) \,dx\,dt \\ &= \int_{Q} fT_{k}(u) \,dx\,dt. \end{aligned} \end{equation} By the definition of $\Theta_{k}$, we have $\Theta_{k}(u)\geq 0$. Using hypothesis \eqref{encadr}, inequality \eqref{testentrosol} becomes \begin{align*} \frac{1}{C}\int_{Q} |\nabla T_{k}(u)|^{p(x)} \,dx\,dt & \leq \int_{\Omega} \Theta_{k}(u_0)\,dx + \int_{Q} fT_{k}(u) \,dx\,dt \\ & \leq \int_{\Omega} k|u_0| dx + \int_{Q} fT_{k}(u) \,dx\,dt \\ & \leq k\Big( \int_{\Omega} |u_0| dx + \int_{Q} |f| \,dx\,dt\Big), \end{align*} then, according to Lemma \ref{lemmaineq}, we deduce that \begin{align*} &\|\nabla T_{k}(u)\|_{L^{p(\cdot)}(Q)} \\ &\leq k \max{\Big\{ \big(\|f\|_{L^1(Q)} + \|u_0\|_{L^1(\Omega)}\big)^{1/p_{-}}, \big(\|f\|_{L^1(Q)} + \|u_0\|_{L^1(\Omega)}\big)^{1/p_{+}}\Big\}}. \end{align*} \noindent\textbf{Step 2:} Proof of \eqref{bornu}. In the following, we use the function $ S_n: \mathbb{R} \to \mathbb{R}$ defined by: \begin{equation}\label{def Sn} S_n(s) = \int_0^{s} h_n(r) dr, \quad\text{where } h_n(s) = 1 - |T_1(s - T_n(s))|. \end{equation} Note that $S_n$ satisfies \begin{equation}\label{Sn propr} \begin{gathered} S_n(r) = S_n(T_{n+1}(r)), \|S'_n\|_{L^{\infty}(\mathbb{R})} \leq 1,\\ \operatorname{supp}S'_n \subset [-(n+1) , n+1], S''_n = 1_{[-n-1, -n]} - 1_{[n, n + 1]}. \end{gathered} \end{equation} Let $ t_1 \in (0,T)$ and $ \theta_{\epsilon}(t) = \big(1 - \frac{(t-t_1)^{+}}{\epsilon}\big)^{+}$. Then $\theta_{\epsilon}$ is continuous on $[0,+\infty)$, $\theta_{\epsilon} = 1$ on $[0,t_1], \theta_{\epsilon} = 0$ on $[t_1+\epsilon, +\infty)$ and $\theta_{\epsilon}$ is linear on $[t_1,t_1+\epsilon]$. Using $\varphi = \frac{1}{k}T_{k}(u) \theta_{\epsilon}$ as a test function in \eqref{renormsol}(since entropy and renormalized solutions are equivalent) and taking $S = S_n$, we obtain \begin{equation}\label{testvarphi2} \begin{aligned} &\frac{1}{k} \int_0^{T}\int_{\Omega} \theta_{\epsilon}(S_n(u))_{t}T_{k}(u)\, dx\,dt\\ &+ \frac{1}{k} \int_0^{T}\int_{\Omega} S'_n(u) a(x, \nabla u) \cdot\nabla(T_{k}(u)\theta_{\epsilon}) dx\,dt \\ &= \frac{1}{k}\int_0^{T}\int_{\Omega} f S'_n(u)T_{k}(u)\theta_{\epsilon} \,dx\,dt\\ &\quad - \frac{1}{k}\int_0^{T}\int_{\Omega} S''_n(u) a(x, \nabla u) \cdot\nabla u T_{k}(u)\theta_{\epsilon} \,dx\,dt. \end{aligned} \end{equation} Since $S''_n(s) = 0$ for $|s| \notin [n, n+1]$, we can write \begin{equation*} S''_n(u) a(x, \nabla u)\cdot\nabla u T_{k}(u) = S''_n(u) a(x, \nabla T_{n+1}(u))\cdot\nabla (T_{n+1}(u))T_{k}(u) \in L^1(Q). \end{equation*} Since $ \theta_{\epsilon} \to 1_{[0,t_1]}$ and is bounded by $1$ as $\epsilon \to 0$, using Lebesgue dominated convergence theorem in equality \eqref{testvarphi2}, we obtain \begin{equation}\label{eqintpart2} \begin{aligned} &\frac{1}{k}\int_0^{t_1}\int_{\Omega} (S_n(u))_{t} T_{k}(u) \,dx\,dt + \frac{1}{k}\int_0^{t_1}\int_{\Omega}S'_n(u) a(x, \nabla u)\cdot\nabla(T_{k}(u)) \,dx\,dt \\ &+ \frac{1}{k}\int_0^{t_1}\int_{\Omega} S''_n(u) a(x, \nabla u)\cdot\nabla u T_{k}(u) \,dx\,dt \\ &= \frac{1}{k}\int_0^{t_1}\int_{\Omega} f S'_n(u)T_{k}(u) \,dx\,dt. \end{aligned} \end{equation} Let $n \geq M$. We have $ T_{k}(u) = T_{k}(S_n(u))$ (since $S_n(s) = s$ on $[-M,M], |S_n(s)| \geq M$ and sign($S_n(s)$) = sign($s$) outside $[-M,M]$), $ (S_n(u))(t_1) \to u(t_1,\cdot)$ in $L^1(\Omega)$, $ S_n(u_0) \to u_0$ in $L^1(\Omega)$ and $S'_n(u) \to 1$ a.e. in $Q$ as $n \to +\infty$. Since $|S''_n(s)| \leq 1$ and $S''_n(s) \neq 0$ only if $|s| \in [n,n+1]$, using \eqref{C1} we can write \begin{align*} & \big| \int_0^{t_1}\int_{\Omega} S''_n(u) a(x, \nabla u)\cdot\nabla u T_{k}(u) \,dx\,dt\big| \\ & \leq k\int_{\{n \leq |u| \leq n+1\}} \Big|a(x, \nabla u)\cdot\nabla u \Big| \,dx\,dt \\ & \leq k \int_{\{n \leq |u| \leq n+1\}} C_1\Big(j(x) + |\nabla u|^{p(x)-1}\Big) |\nabla u| \,dx\,dt \\ & \leq k \int_{\Omega} C_1\Big(j(x) + |\nabla u|^{p(x)-1}\Big) |\nabla u| 1_{\{n \leq |u| \leq n+1\}} \,dx \to 0 \quad \text{as } n \to +\infty. \end{align*} Passing to the limit in \eqref{eqintpart2} as $n \to +\infty$, we obtain \begin{equation}\label{limeqintpart2} \begin{aligned} &\frac{1}{k}\int_0^{t_1}\int_{\Omega} u_{t} T_{k}(u) \,dx\,dt + \frac{1}{k}\int_0^{t_1}\int_{\Omega} a(x, \nabla T_{k}(u)) \cdot\nabla(T_{k}(u)) \,dx\,dt \\ &= \frac{1}{k}\int_0^{t_1}\int_{\Omega} f T_{k}(u) \,dx\,dt, \end{aligned} \end{equation} for all $t_1 \in (0,T)$. By \eqref{encadr}, from \eqref{limeqintpart2}, we obtain \begin{equation*} \frac{1}{k}\int_0^{t_1}\int_{\Omega} u_{t} T_{k}(u) \,dx\,dt \leq \frac{1}{k} \int_0^{t_1}\int_{\Omega} fT_{k}(u) \,dx\,dt. \end{equation*} Letting $k\to 0$ in the inequality above, we obtain \begin{equation*} \int_0^{t_1}\int_{\Omega} u_{t} \operatorname{sign}_0(u)\,dx\,dt \leq \int_0^{t_1}\int_{\Omega} f \operatorname{sign}_0(u) \,dx\,dt. \end{equation*} which implies \begin{equation*} \|u(t_1,\cdot)\|_{L^1(\Omega)} \leq \|f\|_{L^1(Q)} + \|u_0\|_{L^1(\Omega)}, \quad \text{for all } t_1 \in (0,T) \end{equation*} i.e. \begin{equation*} \|u\|_{L^{\infty}(0,T; L^1(\Omega))} \leq \|f\|_{L^1(Q)} + \|u_0\|_{L^1(\Omega)}\,. \end{equation*} This completes the proof. \end{proof} Now, for any continuous and monotonic function $\psi$, we define the proper lower semi-continuous and convex or upper semi-continuous and concave function \begin{equation*} B_{\psi}(s) = \int_0^{s} \psi(r)\, dr. \end{equation*} To prove the existence of weak solutions, we need an energy estimate similar to the one given in \cite[Lemma 1.5]{Alt}. \begin{lemma} \label{lemmaBpsi} Let $\psi \in C^{0,1}(\mathbb{R})$ be monotone, let $u$ be a measurable function such that $u \in L^{p_{-}}(0,T; W_0^{1,p(\cdot)}(\Omega))$. Then $ B_{\psi}(u) \in L^{\infty}(0,T; L^1(\Omega))$ and, for almost every $t \in [0,T]$, \begin{equation}\label{prepalemma} \begin{aligned} &\int_{\Omega} B_{\psi}(u(t))\xi(t)\, dx - \int_{\Omega} B_{\psi}(u_0)\xi(0)\, dx \\ &= \int_0^{t} \int_{\Omega} u_{t} \psi(u) \xi \, dx\,dt + \int_0^{t}\int_{\Omega} B_{\psi}(u)\xi_{t}\, dx\,dt \end{aligned} \end{equation} for any $\xi \in C^{0,1}(\overline{Q})$ such that $\psi(u)\xi \in L^2(0,T; W_0^{1,1}(\Omega))$. \end{lemma} For the proof of the above lemma, see the proof of \cite[Lemma 4]{carillo}. By a weak solution of \eqref{pbmere} we understand a solution in the sense of distributions that belongs to the energy space, i.e., \begin{equation} \label{weaksolu} \begin{gathered} u\in V = \big\{f\in L^{p_{-}}(0,T;W^{1,p(\cdot)}_0(\Omega)); |\nabla f|\in L^{p(\cdot)}(Q)\big\}, \\ \frac{\partial u}{\partial t}-\operatorname{div}a(x,\nabla u) = f\text{ in }\mathcal{D}'(Q), \quad u(0,\cdot) = u_0\,. \end{gathered} \end{equation} To complete this section we prove the following proposition. \begin{proposition}\label{prop weaksol} Assume that \eqref{p(x)}-\eqref{encadr} hold, $ u_0 \in L^{\infty}(\Omega)$, $ f \in L^{\infty}(Q)$ and $u$ is the unique mild solution of \eqref{pbmere}. Then $u$ is a weak solution of \eqref{pbmere}. \end{proposition} \begin{proof} For $i = 0, 1,\dots , n$, let $ u_i^{\epsilon}$ be the unique weak energy solution of \begin{equation*} \epsilon f_i^{\epsilon} + u_{i-1}^{\epsilon} \in (I + \epsilon A)u_i^{\epsilon}. \end{equation*} We have \begin{equation}\label{weaksoldiscr} \int_{\Omega} a(x, \nabla u_i^{\epsilon})\cdot\nabla \varphi dx + \int_{\Omega} \frac{u_i^{\epsilon} - u_{i-1}^{\epsilon}}{\epsilon} \varphi dx = \int_{\Omega} f_i^{\epsilon} \varphi dx, \end{equation} for all $\varphi \in W_0^{1,p(\cdot)}(\Omega)$. Taking $\varphi = u_i^{\epsilon}$ as test function in \eqref{weaksoldiscr}, integrating over $ (t_{i-1}^{\epsilon}, t_i^{\epsilon})$ and summing up the inequalities over $i = 1,\dots ,n$, we obtain \begin{equation}\label{testueps} \begin{aligned} &\sum_{i=1}^{n}\int_{t_{i-1}^{\epsilon}}^{t_i^{\epsilon}} \int_{\Omega} \frac{u_i^{\epsilon} - u_{i-1}^{\epsilon}}{\epsilon} u_i^{\epsilon}\, dx\,dt + \sum_{i=1}^{n}\int_{t_{i-1}^{\epsilon}}^{t_i^{\epsilon}} \int_{\Omega} a(x, \nabla u_i^{\epsilon})\cdot\nabla u_i^{\epsilon}\, dx\,dt \\ &= \sum_{i=1}^{n}\int_{t_{i-1}^{\epsilon}}^{t_i^{\epsilon}} \int_{\Omega} f_i^{\epsilon} u_i^{\epsilon}\, dx\,dt. \end{aligned} \end{equation} By \eqref{encadr} and as $B_{Id}$ is convex, from \eqref{testueps} we deduce that \begin{align*} & \sum_{i=1}^{n}\int_{t_{i-1}^{\epsilon}}^{t_i^{\epsilon}} \int_{\Omega} \frac{B_{Id}(u_i^{\epsilon}) - B_{Id}( u_{i-1}^{\epsilon})}{\epsilon} \, dx\,dt + \sum_{i=1}^{n}\int_{t_{i-1}^{\epsilon}}^{t_i^{\epsilon}} \frac{1}{C}\int_{\Omega} |\nabla u_i^{\epsilon}|^{p(x)} \, dx\,dt \\ &\leq \sum_{i=1}^{n}\int_{t_{i-1}^{\epsilon}}^{t_i^{\epsilon}} \int_{\Omega} f_i^{\epsilon} u_i^{\epsilon}\, dx\,dt. \end{align*} Consequently, if we set $\epsilon = t_i^{\epsilon}- t_{i-1}^{\epsilon}$, then $f_{\epsilon}(t) = f_i^{\epsilon}$ and $ u_{\epsilon}(t) = u_i^{\epsilon}$ for $t \in ( t_{i-1}^{\epsilon}, t_i^{\epsilon}]$, $i = 1,\dots ,n$; $u_{\epsilon}(0) = u_0^{\epsilon}$. It follows that \begin{align*} & \int_{\Omega} \Big[B_{Id}(u_{\epsilon}(T)) - B_{Id}( u_{\epsilon}(0))\Big] \,dx + \frac{1}{C} \int_0^{T}\int_{\Omega} |\nabla u_{\epsilon}|^{p(x)} \, dx\,dt \\ &\leq \int_0^{T}\int_{\Omega} f_{\epsilon} u_{\epsilon}\, dx\,dt. \end{align*} As $ B_{Id}(u_{\epsilon}(T)) - B_{Id}( u_{\epsilon}(0)), u_{\epsilon}, f_{\epsilon} \in L^{\infty}(\Omega)$, we obtain \begin{equation}\label{intgradueps} \int_0^{T}\int_{\Omega} |\nabla u_{\epsilon}|^{p(x)} \, dx\,dt \leq C \Rightarrow \int_0^{T}\int_{\Omega} |\nabla u_{\epsilon}|^{p_{-}} \, dx\,dt \leq C . \end{equation} Using the Poincar\'e inequality with constant exponent, we deduce that $ (u_{\epsilon})_{\epsilon > 0}$ is uniformly bounded in $ L^{p_{-}}(0,T;W_0^{1,p(\cdot)}(\Omega))$. So, there exists a subsequence still denoted $ (u_{\epsilon})_{\epsilon > 0}$, such that \begin{gather}\label{weakconvueps} u_{\epsilon} \rightharpoonup u \quad\text{in } L^{p_{-}}(0,T;W_0^{1,p(\cdot)}(\Omega)) \text{ as } \epsilon \to 0, \\ \label{weakconvgradueps} \nabla u_{\epsilon} \rightharpoonup \nabla u \quad \text{in } \big(L^{p(\cdot)}(Q)\big)^N \text{ as } \epsilon \to 0. \end{gather} Since $(\nabla u_{\epsilon})_{\epsilon>0}$ is uniformly bounded in $ \big(L^{p(\cdot)}(Q)\big)^N$, by \eqref{C1} we deduce that $ \big(a(x, \nabla u_{\epsilon})\big)_{\epsilon > 0}$ is uniformly bounded in $ \big(L^{p'(\cdot)}(Q)\big)^N$ and then we can assume that \begin{equation}\label{weakconvaeps} a(x, \nabla u_{\epsilon}) \rightharpoonup \Phi \quad \text{in } \big(L^{p'(\cdot)}(Q)\big)^N \text{ as } \epsilon \to 0. \end{equation} From \eqref{weaksoldiscr}, we have \begin{equation}\label{weaksolueps} \int_{\Omega} a(x, \nabla (u_{\epsilon}))\cdot\nabla \varphi\, dx + \int_{\Omega} \frac{u_{\epsilon}(t) - u_{\epsilon}(t-\epsilon)}{\epsilon} \varphi \,dx = \int_{\Omega} f_{\epsilon}(t) \varphi dx, \end{equation} for all $ \varphi \in W_0^{1,p(\cdot)}(\Omega)$. Then, taking $\psi \in W^{1,1}(0, T; W^{1,1}(\Omega) \cap L^{\infty}(\Omega)) \cap E$, $\psi(T) = 0$ as a test function in \eqref{weaksolueps}, we obtain \begin{equation}\label{weaksolueps2} \begin{aligned} &\int_0^{T}\int_{\Omega} a(x, \nabla u_{\epsilon}(t))\cdot\nabla \psi(t) \,dx\,dt + \int_0^{T} \int_{\Omega} \frac{u_{\epsilon}(t) - u_{\epsilon} (t-\epsilon)}{\epsilon} \psi(t) \,dx\,dt \\ &= \int_0^{T}\int_{\Omega} f_{\epsilon}(t) \psi(t) \,dx\,dt. \end{aligned} \end{equation} We have \begin{align*} & \int_0^{T} \int_{\Omega} \frac{u_{\epsilon}(t) - u_{\epsilon} (t-\epsilon)}{\epsilon} \psi(t)\, dx\,dt \\ &= \int_0^{T} \int_{\Omega} \frac{u_{\epsilon}(t)}{\epsilon} \psi(t)\,dx\,dt - \int_0^{T} \int_{\Omega} \frac{u_{\epsilon}(t-\epsilon)}{\epsilon} \psi(t) \, dx\,dt \\ &= \int_0^{T} \int_{\Omega} \frac{u_{\epsilon}(t)}{\epsilon} \psi(t)\,dx\,dt - \int_{-\epsilon}^{T-\epsilon} \int_{\Omega} \frac{u_{\epsilon}(s)}{\epsilon} \psi(s+\epsilon)\, dx\,ds \quad (\text{where } s = t - \epsilon) \\ &= \int_0^{T-\epsilon} \int_{\Omega} \frac{u_{\epsilon}(t)}{\epsilon} \psi(t)\,dx\,dt + \int_{T-\epsilon}^{T} \int_{\Omega} \frac{u_{\epsilon}(t)}{\epsilon} \psi(t)\, dx\,dt \\ &\quad - \int_{-\epsilon}^{0} \int_{\Omega} \frac{u_{\epsilon}(s)}{\epsilon} \psi(s+\epsilon)\, dx\, ds - \int_0^{T-\epsilon} \int_{\Omega} \frac{u_{\epsilon}(s)}{\epsilon} \psi(s+\epsilon)\, dx\,ds \\ &= - \int_0^{T-\epsilon} \int_{\Omega} u_{\epsilon}(t) \frac{\psi(t+\epsilon) - \psi(t)}{\epsilon}\, dx\,dt + \int_{T-\epsilon}^{T} \int_{\Omega} \frac{u_{\epsilon}(t) \psi(t)}{\epsilon} dx\,dt \\ &\quad - \int_0^{\epsilon} \int_{\Omega} \frac{u_{0, \epsilon}(t) \psi(t)}{\epsilon}\, dx\,dt, \\ &\to - \int_0^{T} \int_{\Omega} u(t) \psi_{t}\, dx\,dt - \int_{\Omega} u_0 \psi(0)\, dx\,dt \quad \text{as } \epsilon \to 0, \end{align*} where $u_{\epsilon}(t) = u_0$ for $t \leq 0$. Therefore, taking limit in \eqref{weaksolueps2} as $\epsilon \to 0$, we obtain \begin{equation}\label{limweaksolueps2} \begin{aligned} &\int_0^{T}\int_{\Omega} \Phi\cdot\nabla \psi\, dx\,dt - \int_0^{T} \int_{\Omega} u \psi_{t}\, dx\,dt - \int_{\Omega} u_0 \psi(0)\,dx\,dt \\ &= \int_0^{T}\int_{\Omega} f(t) \psi \,dx\,dt. \end{aligned} \end{equation} Thus, to complete the proof of Proposition \ref{prop weaksol}, we only need to show that $\Phi = a(x, \nabla u)$. To do so, we apply the Minty-Browder's method. Firstly, we prove that \begin{equation}\label{limsupepsa} \limsup_{\epsilon \to 0} \iint{Q} a(x, \nabla u_{\epsilon}). \nabla u_{\epsilon}\, dx\,dt \leq \int_{Q} \Phi\cdot\nabla u \, dx\,dt. \end{equation} Using \eqref{testueps}, we have \begin{equation}\label{testueps3} \begin{aligned} &\int_0^{T} \int_{\Omega} a(x, \nabla u_{\epsilon})\cdot\nabla u_{\epsilon} \, dx\,dt \\ &\leq - \int_{\Omega} \Big[B_{Id}(u_{\epsilon}(T)) - B_{Id}( u_0)\Big] \, dx + \int_0^{T}\int_{\Omega} f_{\epsilon} u_{\epsilon}\, dx\,dt. \end{aligned} \end{equation} Since $ B_{Id}(u_{\epsilon}(T)) \geq 0$, then by Fatou's lemma, we have \begin{equation}\label{ineqBid} \int_{\Omega} \liminf_{\epsilon \to 0} B_{Id}(u_{\epsilon}(T))\,dx \leq \liminf_{\epsilon \to 0} \int_{\Omega} B_{Id}(u_{\epsilon}(T))\,dx. \end{equation} Because of the lower semi-continuity of $ B_{Id}$, we have \begin{equation}\label{ineqBid 2} \int_{\Omega} B_{Id}(u(T))\,dx \leq \int_{\Omega} \liminf_{\epsilon \to 0} B_{Id}(u_{\epsilon}(T))\,dx . \end{equation} Inequalities \eqref{ineqBid} and \eqref{ineqBid 2} imply \begin{equation*} \int_{\Omega} B_{Id}(u(T))\,dx \leq \liminf_{\epsilon \to 0} \int_{\Omega} B_{Id}(u_{\epsilon}(T))\,dx, \end{equation*} i.e. \begin{equation*} - \liminf_{\epsilon \to 0} \int_{\Omega} B_{Id}(u_{\epsilon}(T)) \, dx \leq - \int_{\Omega} B_{Id}(u(T)) \, dx . \end{equation*} Then, passing to the limit in \eqref{testueps3} as $\epsilon \to 0$ and according to Lemma \ref{lemmaBpsi} we have \begin{equation}\label{limsup a} \begin{aligned} &\limsup_{\epsilon \to 0} \int_0^{T} \int_{\Omega} a(x, \nabla u_{\epsilon})\cdot\nabla u_{\epsilon}\, dx\,dt \\ &\leq - \int_{\Omega} \Big[B_{Id}(u(T)) - B_{Id}( u(0))\Big] \, dx + \int_0^{T}\int_{\Omega} f u \, dx\,dt \\ &= \langle f -u_{t}, u\rangle. \end{aligned} \end{equation} Now, we prove that \begin{equation}\label{div a} \iint{Q} a(x, \nabla u). \nabla \xi \,dx\,dt = \iint{Q} \Phi\cdot\nabla \xi \,dx\,dt, \end{equation} for any $\xi \in L^{p_{-}}(0,T; W_0^{1,p(\cdot)}(\Omega))$.\\ By the monotonicity of $a$, for any $\rho \in L^{p_{-}}(0,T; W_0^{1,p(\cdot)}(\Omega))$, \begin{equation}\label{inerho} \iint{Q} a(x, \nabla \rho). \nabla(u_{\epsilon}-\rho)\,dx\,dt \leq \iint{Q}a(x, \nabla u_{\epsilon}). \nabla(u_{\epsilon}-\rho) \,dx\,dt. \end{equation} Since $ u_{\epsilon}$ is a weak energy solution of $ \epsilon f_i^{\epsilon} + u_{i-1}^{\epsilon} \in (I + \epsilon A) u_i^{\epsilon}$ then, by \cite[Proposition 4.11]{Ouarotraore1}, $ \nabla u_{\epsilon}$ converges in measure to $\nabla u$. We can then extract a subsequence such that $ \nabla u_{\epsilon} \to \nabla u$ a.e. in $Q$. Then according to \eqref{C1}, we may apply Lebesgue dominated convergence theorem and pass to the limit in \eqref{inerho} as $\epsilon \to 0$ to obtain \begin{equation}\label{lim inerho} \liminf_{\epsilon \to 0} \int \int_{Q}a(x, \nabla u_{\epsilon} ). \nabla(u_{\epsilon}-\rho) \,dx\,dt \geq \int \int_{Q} a(x, \nabla \rho). \nabla(u - \rho)\,dx\,dt. \end{equation} Combining \eqref{limsup a} and \eqref{lim inerho}, we have \begin{equation*} \langle f - u_{t}, u - \rho \rangle \geq \int \int_{Q} a(x, \nabla \rho). \nabla(u - \rho)\,dx\,dt, \end{equation*} for all $\rho \in L^{p_{-}}(0,T; W_0^{1,p(\cdot)}(\Omega))$. Choosing $\rho = u + \sigma \xi, \sigma \in \mathbb{R}, \xi \in L^{p_{-}}(0,T; W_0^{1,p(\cdot)}(\Omega))$, we obtain \begin{equation}\label{inesigma} \langle f - u_{t}, \sigma \xi \rangle \leq \sigma \int \int_{Q} a(x, \nabla ( u + \sigma \xi)). \nabla \xi \,dx\,dt. \end{equation} Dividing inequality \eqref{inesigma} by $\sigma > 0$, resp. $\sigma < 0$ and passing to the limit with $\sigma \downarrow 0$, resp. $\sigma \uparrow 0$, we obtain \begin{equation}\label{liminesigma} \langle f - u_{t}, \xi \rangle = \int \int_{Q} a(x, \nabla u ). \nabla \xi \,dx\,dt, \end{equation} for any $\xi \in L^{p_{-}}(0,T; W_0^{1,p(\cdot)}(\Omega))$. By \eqref{limweaksolueps2}, we have \begin{equation}\label{eg Phi} \begin{aligned} &\int_0^{T}\int_{\Omega} \Phi\cdot\nabla \psi\, dx\,dt \\ & = \int_0^{T} \int_{\Omega} u \psi_{t}\, dx\,dt + \int_{\Omega} u_0 \psi(0)\,dx + \int_0^{T}\int_{\Omega} f \psi \, dx\,dt \\ & = \langle f-u_{t}, \psi \rangle. \end{aligned} \end{equation} Combining \eqref{liminesigma} and \eqref{eg Phi} yields \eqref{div a}. To conclude, we pass to the limit in \eqref{weaksolueps2} as $\epsilon \to 0$ to obtain \begin{equation}\label{solfaiblmere} \begin{aligned} \int_0^{T}\int_{\Omega} f \phi \,dx\,dt &= - \int_0^{T}\int_{\Omega} u \phi_{t} \,dx\,dt - \int_{\Omega} (u \phi)(0) dx \\ &\quad + \int_0^{T}\int_{\Omega} a(x, \nabla u)\cdot\nabla \phi \,dx\,dt, \end{aligned} \end{equation} for all $ \phi \in E \cap L^{\infty}(Q)$. Hence $u$ is a weak solution of \eqref{pbmere}. \end{proof} Our aim is to prove that this weak solution is also an entropy solution of \eqref{pbmere}. The proof of this result consists of two main steps. Firstly, we prove $\epsilon-$uniform a-priori-estimates in certain Bochner spaces as well as in appropriate variable exponent Lebesgue spaces for $u_{\epsilon}$ and $\nabla u_{\epsilon}$. Secondly, we pass to the limit in the entropy relation as $\epsilon \to 0$. \section{Existence and uniqueness of an entropy solution} \begin{theorem}\label{theorementr} Let \eqref{p(x)}-\eqref{encadr} hold. Let $ u_0 \in L^1(\Omega)$, $ f \in L^1(Q)$. There exists a unique entropy solution for \eqref{pbmere}. \end{theorem} The proof of the above theorem is done in several steps. \subsection{A priori estimates} As $ u_0 \in L^1(\Omega)$, $ f \in L^1(Q)$ and $ L^{\infty}$ is dense in $ L^1$, then we can find two sequences of functions $ \big(f_{\epsilon}\big)_{\epsilon > 0} \subset L^{\infty}(Q)$ and $ \big(u_{0,\epsilon}\big)_{\epsilon > 0} \subset L^{\infty}(\Omega)$ strongly converging respectively to $f$ and $ u_0$ such that \begin{equation}\label{normfu0} \|f_{\epsilon}\|_{L^1(Q)} \leq \|f\|_{L^1(Q)},\quad \|u_{0,\epsilon}\|_{L^1(\Omega)} \leq \|u_0\|_{L^1(\Omega)}. \end{equation} Now, let $ u_{\epsilon}$ be a weak solution to problem \eqref{pbmere} with $ f_{\epsilon}$ and $ u_{0, \epsilon}$ as data, i.e. \begin{equation}\label{solfaiblmere2} \begin{aligned} \int_0^{T}\int_{\Omega} f_{\epsilon} \phi \,dx\,dt &= - \int_0^{T}\int_{\Omega} u_{\epsilon} \phi_{t} \,dx\,dt - \int_{\Omega} u_{0,\epsilon} \phi(0,\cdot) dx \\ &\quad + \int_0^{T}\int_{\Omega} a(x, \nabla u_{\epsilon})\cdot\nabla \phi \,dx\,dt, \end{aligned} \end{equation} for all $ \phi \in E \cap L^{\infty}(Q)$. \begin{lemma}\label{lemma bornueps} The estimates in Lemma \ref{lemma bornugradu} hold with $u$ replaced by $u_{\epsilon}$, and all the constants are independent of $\epsilon$, i.e. \begin{gather}\label{bornueps} \|u_{\epsilon}\|_{L^{\infty}(0,T; L^1(\Omega))} \leq \|f\|_{L^1(Q)} + \|u_0\|_{L^1(\Omega)}, \\ \label{borngradueps} \begin{aligned} &\|\nabla T_{k}(u_{\epsilon})\|_{L^{p(\cdot)}(Q)} \\ &\leq k \max{\Big\{ \big(\|f\|_{L^1(Q)} + \|u_0\|_{L^1(\Omega)}\big)^{1/p_{-}}, \big(\|f\|_{L^1(Q)} + \|u_0\|_{L^1(\Omega)}\big)^{1/p_{+}}\Big\}}. \end{aligned} \end{gather} \end{lemma} The proof of the above lemma is similar to that of Lemma \ref{lemma bornugradu}. \subsection{Basic convergence results} The a priori estimates in lemmas \ref{lemma bornugradu} and \ref{lemma bornueps}, together with the $ C([0,T]; L^1(\Omega))$-convergence guaranteed by nonlinear semigroup theory, imply the following basic convergence results. \begin{lemma}\label{lemmaconvfaible} For a subsequence $ \big(u_{\epsilon}\big)_{\epsilon > 0}$ as $\epsilon \to 0$: \begin{gather}\label{convueps p.p.} u_{\epsilon} \to u \quad \text{a.e. in }Q, \\ \label{convgradfaib} \nabla T_{k}(u_{\epsilon}) \rightharpoonup \nabla T_{k}(u) \quad \text{in }(L^{p(\cdot)}(Q))^N, \\ \label{convtroncfaib} T_{k}(u_{\epsilon}) \to T_{k}(u) \text{in }L ^{p_{-}}(0,T; W_0^{1,p(\cdot)}(\Omega)) \end{gather} for all $k > 0$. \end{lemma} \begin{proof} Proof of \eqref{convueps p.p.}. Let $ u_{\epsilon_1}$ and $ u_{\epsilon_2}$ be two weak solutions of problem \eqref{pbmere}. Choosing $ \theta_{\epsilon}T_1(u_{\epsilon_1} - u_{\epsilon_2})$ as a test function corresponding to $ u_{\epsilon_1}$ and $ \theta_{\epsilon}T_1(u_{\epsilon_2} - u_{\epsilon_1})$ as a test function corresponding to $ u_{\epsilon_2}$, we obtain \begin{equation}\label{solfaiblmereueps1} \begin{aligned} &\int_0^{T}\int_{\Omega} \theta_{\epsilon}(u_{\epsilon_1})_{t} T_1(u_{\epsilon_1} - u_{\epsilon_2})\,dx\,dt \\ &= \int_0^{T}\int_{\Omega}\theta_{\epsilon} a(x, \nabla u_{\epsilon_1}) \cdot\nabla T_1(u_{\epsilon_1} - u_{\epsilon_2})\,dx\,dt \\ &\quad + \int_0^{T}\int_{\Omega} \theta_{\epsilon} f_{\epsilon_1} T_1(u_{\epsilon_1} - u_{\epsilon_2})\, dx\,dt \end{aligned} \end{equation} and \begin{equation}\label{solfaiblmereueps2} \begin{aligned} &\int_0^{T}\int_{\Omega} \theta_{\epsilon}(u_{\epsilon_2})_{t} T_1(u_{\epsilon_2} - u_{\epsilon_1})\,dx\,dt \\ & = \int_0^{T}\int_{\Omega}\theta_{\epsilon} a(x, \nabla u_{\epsilon_2}) \cdot\nabla T_1(u_{\epsilon_2} - u_{\epsilon_1})\,dx\,dt \\ &\quad + \int_0^{T}\int_{\Omega} \theta_{\epsilon}f_{\epsilon_2} T_1(u_{\epsilon_2} - u_{\epsilon_1})\, dx\,dt. \end{aligned} \end{equation} Adding \eqref{solfaiblmereueps1} and \eqref{solfaiblmereueps2}, then by using \eqref{coerc} and letting $\epsilon$ approach zero we have \begin{equation}\label{solfaiblmereueps12} \begin{aligned} &\int_0^{t_1}\int_{\Omega} (u_{\epsilon_1} - u_{\epsilon_2})_{t} T_1(u_{\epsilon_1} - u_{\epsilon_2})\,dx\,dt \\ &= \int_0^{t_1}\int_{\Omega} \big(a(x, \nabla u_{\epsilon_1}) - a(x, \nabla u_{\epsilon_2})\big)\cdot\nabla T_1(u_{\epsilon_1} - u_{\epsilon_2})\,dx\,dt \\ &\quad + \int_0^{t_1}\int_{\Omega} \big(f_{\epsilon_2} - f_{\epsilon_1}\big) T_1(u_{\epsilon_1} - u_{\epsilon_2})\, dx\,dt \\ &\leq \int_0^{t_1}\int_{\Omega} \big(f_{\epsilon_2} - f_{\epsilon_1}\big) T_1(u_{\epsilon_1} - u_{\epsilon_2})\, dx\,dt. \end{aligned} \end{equation} From \eqref{solfaiblmereueps12} we deduce that \begin{equation}\label{ineqTheta} \begin{aligned} &\int_{\Omega} \Theta_1(u_{\epsilon_1} - u_{\epsilon_2})(t_1)\,dx \\ & \leq \int_{\Omega} \Theta_1(u_{0,\epsilon_1} - u_{0,\epsilon_2})\,dx + \|f_{\epsilon_2} - f_{\epsilon_1}\|_{L^1(Q)} \\ & \leq \|u_{0,\epsilon_1} - u_{0,\epsilon_2}\|_{L^1(\Omega)} + \|f_{\epsilon_2} - f_{\epsilon_1}\|_{L^1(Q)} := a_{\epsilon_1\epsilon_2}. \end{aligned} \end{equation} By the definition of $ \Theta_1$, we have \begin{equation*} \Theta_1(u_{\epsilon_1} - u_{\epsilon_2})(t_1) = \begin{cases} \frac{[(u_{\epsilon_1} - u_{\epsilon_2})(t_1)]^2}{2} & \text{if } |u_{\epsilon_1}(t_1) - u_{\epsilon_2}(t_1)| < 1 \\[4pt] |(u_{\epsilon_1} - u_{\epsilon_2})(t_1)| & \text{if } |u_{\epsilon_1}(t_1) - u_{\epsilon_2}(t_1)| \geq 1. \end{cases} \end{equation*} On the set $\{|u_{\epsilon_1} - u_{\epsilon_2}| \geq 1\}$, we have $ \frac{|(u_{\epsilon_1} - u_{\epsilon_2})(t_1)|}{2} \leq |u_{\epsilon_1}(t_1) - u_{\epsilon_2}(t_1)|$. Then, from \eqref{ineqTheta} we deduce \begin{align*} &\int_{\{|u_{\epsilon_1} - u_{\epsilon_2}| < 1\}} \frac{(u_{\epsilon_1} - u_{\epsilon_2})^2(t_1)}{2}\, dx + \int_{\{|u_{\epsilon_1} - u_{\epsilon_2}| \geq 1\}} \frac{|u_{\epsilon_1}(t_1) - u_{\epsilon_2}(t_1)|}{2}\, dx \\ &\leq \int_{\Omega} \Theta_1(u_{\epsilon_1} - u_{\epsilon_2})(t_1)\,dx \leq a_{\epsilon_1\epsilon_2}. \end{align*} Using H\"{o}lder inequality, \begin{equation}\label{bornuesp12} \begin{aligned} &\int_{\Omega} |u_{\epsilon_1}(t_1) - u_{\epsilon_2}(t_1)|\, dx \\ & = \int_{\{|u_{\epsilon_1} - u_{\epsilon_2}| < 1\}} |u_{\epsilon_1}(t_1) - u_{\epsilon_2}(t_1)|\, dx + \int_{\{|u_{\epsilon_1} - u_{\epsilon_2}| \geq 1\}} |u_{\epsilon_1}(t_1) - u_{\epsilon_2}(t_1)|\, dx \\ & \leq \Big( \int_{\{|u_{\epsilon_1} - u_{\epsilon_2}| < 1\}} |u_{\epsilon_1}(t_1) - u_{\epsilon_2}(t_1)|^2\, dx\Big)^{1/2} \operatorname{meas}(\Omega)^{1/2} + 2 a_{\epsilon_1\epsilon_2} \\ & \leq (2\operatorname{meas}(\Omega))^{1/2} a_{\epsilon_1\epsilon_2}^{1/2} + 2 a_{\epsilon_1\epsilon_2}. \end{aligned} \end{equation} Since $ \big(f_{\epsilon}\big)_{\epsilon > 0}$ and $ \big(u_{0,\epsilon}\big)_{\epsilon > 0}$ are convergent respectively in $L^1(Q)$ and $L^1(\Omega)$, we have $ a_{\epsilon_1\epsilon_2} \to 0$ for $ \epsilon_1, \epsilon_2 \to 0$. Thus from \eqref{bornuesp12} we deduce that $ \big(u_{\epsilon}\big)_{\epsilon > 0}$ is a Cauchy sequence in $C([0,T]; L^1(\Omega))$ and $ u_{\epsilon}$ converges to $u$ in $C([0,T]; L^1(\Omega))$. Then we find an a.e. convergent subsequence (still denoted by $ \big(u_{\epsilon}\big)_{\epsilon > 0}$) in $Q$ such that $ u_{\epsilon} \to u$ a.e. in $Q$. The proof of \eqref{convueps p.p.} is complete. \smallskip Proof of \eqref{convgradfaib} and \eqref{convtroncfaib}. By \eqref{borngradueps}, the sequence $\big(\nabla T_{k}(u_{\epsilon})\big)_{\epsilon > 0}$ is bounded in $ \big(L^{p(\cdot)}(Q)\big)^N$; hence the sequence $\big(T_{k}(u_{\epsilon})\big)_{\epsilon > 0}$ is bounded in $ W_0^{1,p(\cdot)}(Q)$. Then, up to a subsequence we can assume that for any $k > 0$, $\big(T_{k}(u_{\epsilon})\big)_{\epsilon > 0}$ converges weakly to $ \sigma_{k}$ in $ W_0^{1,p(\cdot)}(Q)$ and so $\big(T_{k}(u_{\epsilon})\big)_{\epsilon > 0}$ converges strongly to $ \sigma_{k}$ in $ L^{p_{-}}(Q)$. By \eqref{convueps p.p.}, we have $ u_{\epsilon} \to u$ a.e. in $Q$. As for $k > 0, T_{k}$ is continuous, then $ T_{k}(u_{\epsilon}) \to T_{k}(u)$ a.e. in $Q$ and $ \sigma_{k} = T_{k}(u)$ a.e. in $Q$, which yields \eqref{convtroncfaib}. Using also the boundedness of $\big(\nabla T_{k}(u_{\epsilon})\big)_{\epsilon > 0}$ in $ \big(L^{p(\cdot)}(Q)\big)^N$, we can find a subsequence (still denoted by $ (u_{\epsilon})_{\epsilon > 0}$) from $ (u_{\epsilon})_{\epsilon > 0}$ such that $ \nabla T_{k}(u_{\epsilon})$ converges weakly to $ \nabla T_{k}(u)$ in $ \big(L^{p(\cdot)}(Q)\big)^N$, i.e. \eqref{convgradfaib} holds. \end{proof} \subsection{Strong convergence} We start by recalling a suitable time-regularization procedure, which was first introduced by Landes (see \cite{Landes}) and employed by several authors to solve nonlinear time dependent problems with $L^1$ or measure data (see e.g. \cite{Blanchmurat}). We denote this time regularized function to $ T_n(u)$ by $ (T_n(u))_{\mu}$, with $\mu > 0$. It is defined as the unique solution $ (T_n(u))_{\mu} \in L^{p_{-}}(0,T; W_0^{1,p(\cdot)}(\Omega)) \cap L^{\infty}(Q)$, with $ \nabla(T_n(u))_{\mu} \in \big(L^{p(\cdot)}(Q)\big)^N$, of the equation \begin{equation}\label{eqregul} \partial_{t}(T_n(u))_{\mu} + \mu((T_n(u))_{\mu} - T_n(u)) = 0 \quad \text{in }\mathcal{D}'(Q), \end{equation} with the initial condition \begin{equation}\label{initcond2} (T_n(u))_{\mu}|_{t=0} = w_0^{\mu} \quad \text{in }\Omega, \end{equation} where $ w_0^{\mu}$ is a sequence of functions such that \begin{equation}\label{w0mu} \begin{gathered} w_0^{\mu} \in W_0^{1,p(\cdot)}(\Omega) \cap L^{\infty}(\Omega), \quad \|w_0^{\mu}\|_{L^{\infty}(\Omega)} \leq n \\ w_0^{\mu} \to T_n(u_0) \quad \text{a.e. in $\Omega$ as $\mu \to \infty$}, \\ \frac{1}{\mu}\| w_0^{\mu}\|_{W_0^{1,p(\cdot)}(\Omega)} \to 0 \quad \text{as } \mu \to \infty. \end{gathered} \end{equation} Following \cite{Landes} we can prove that \begin{equation}\label{strongconv} \begin{gathered} \frac{\partial (T_n(u))_{\mu}}{\partial t}\in L^{p_{-}} (0,T;W^{1,p(\cdot)}_0(\Omega))\cap L^{\infty}(Q), \quad \|(T_n(u))_{\mu}\|_{L^{\infty}(Q)}\leq n, \\ (T_n(u))_{\mu} \to T_n(u) \text{ a.e. in $Q$, weak-* in }L^{\infty}(Q)\\ \text{and strongly in } L^{p_{-}}(0,T;W^{1,p(\cdot)}_0(\Omega)). \end{gathered} \end{equation} To continue our proof of Theorem \ref{theorementr}, we need the following result. \begin{proposition}\label{propconv aTk} For all $k > 0$ we have: \begin{itemize} \item[(i)] $ a(x, \nabla T_{k}(u_{\epsilon})) \rightharpoonup a(x, \nabla T_{k}(u))$ in $ \big(L^{p'(\cdot)}(Q)\big)^N$, \item[(ii)] $ \nabla T_{k}(u_{\epsilon}) \to \nabla T_{k}(u)$ a.e. in $Q$, \item[(iii)] $ a(x, \nabla T_{k}(u_{\epsilon}))\cdot\nabla T_{k}(u_{\epsilon}) \to a(x, \nabla T_{k}(u))\cdot\nabla T_{k}(u)$ strongly in $ L^1(Q)$ and a.e. in $Q$, \item[(iv)] $ \nabla T_{k}(u_{\epsilon}) \to \nabla T_{k}(u)$ in $ \big(L^{p(\cdot)}(Q)\big)^N$. \end{itemize} \end{proposition} \begin{proof} (i) The sequence $\big( a(x, \nabla T_{k}(u_{\epsilon}))\big)_{\epsilon > 0}$ is bounded in $ \big( L^{p'(\cdot)}(Q)\big)^N$ according to \eqref{C1}. We can extract a subsequence such that $ a(x, \nabla T_{k}(u_{\epsilon})) \to \zeta_{k}$ in $ \big( L^{p'(\cdot)}(Q)\big)^N$. We have to show that $ \zeta_{k} = a(x, \nabla T_{k}(u))$ a.e. in $Q$. To this end, we take a subsequence $ (u_{\epsilon})_{\epsilon > 0}$ such that $ u_{\epsilon} \to u$ almost everywhere in $Q$. For $h > 2k$, we introduce the function \begin{equation*} w_{\epsilon} = T_{2k}\Big(u_{\epsilon} - T_{h}(u_{\epsilon}) + T_{k}(u_{\epsilon}) - \big(T_{k}(u)\big)_{\mu}\Big), \end{equation*} where $ \big(T_{k}(u)\big)_{\mu}$ is the approximation of $ T_{k}(u)$ defined in \eqref{eqregul}. The use of $ w_{\epsilon}$ as a test function to prove the strong convergence of truncations was first introduced in the stationary case in \cite{Leoporeta}, then adapted to parabolic equations in \cite{Porreta}. The advantage in working with $ w_{\epsilon}$ is that since \begin{equation*} \nabla w_{\epsilon} = \nabla \Big(u_{\epsilon} - T_{h}(u_{\epsilon}) + T_{k}(u_{\epsilon}) - \big(T_{k}(u)\big)_{\mu}\Big)\chi_{E_{\epsilon}}, \end{equation*} with $ E_{\epsilon} = \big\{\big|u_{\epsilon} - T_{h}(u_{\epsilon}) + T_{k}(u_{\epsilon}) - \big(T_{k}(u)\big)_{\mu}\big| \leq 2k\big\}$, in particular we have $ \nabla w_{\epsilon} = 0$ if $ |u_{\epsilon}| > h + 4k$. Thus the estimate on $ T_{k}(u_{\epsilon})$ in $ L^{p_{-}}(0,T; W_0^{1,p(\cdot)}(\Omega))$ appearing in Lemma \ref{lemmaconvfaible} implies that $ w_{\epsilon}$ is bounded in $ L^{p_{-}}(0,T; W_0^{1,p(\cdot)}(\Omega))$. Then by the almost everywhere convergence of $ u_{\epsilon}$ to $u$ as $\epsilon \to 0$, we deduce that \begin{equation}\label{convfaibweps} w_{\epsilon} \rightharpoonup T_{2k}\Big(u - T_{h}(u) + T_{k}(u) - \big(T_{k}(u)\big)_{\mu}\Big) \end{equation} in $L^{p_{-}}(0,T; W_0^{1,p(\cdot)}(\Omega))$ and a.e. in $Q$. In the following, we set $M = h + 4k$, moreover we will denote by $w(\epsilon, \mu, h)$ all quantities (possibly different) such that \begin{equation}\label{limsup w} \lim_{h \to +\infty} \lim_{\mu \to +\infty} \limsup_{\epsilon \to 0} |w(\epsilon, \mu, h)| = 0. \end{equation} Similarly we will write only $w(\epsilon)$ or $w(\epsilon, \mu)$, to mean that the limits are made only on the specified parameters. Choosing $ w_{\epsilon}$ as a test function in \eqref{solfaiblmere2} we have \begin{equation}\label{test weps} \int_0^{T}\int_{\Omega} (u_{\epsilon})_{t} w_{\epsilon} \,dx\,dt + \int_0^{T}\int_{\Omega} a(x, \nabla u_{\epsilon}) \cdot\nabla w_{\epsilon}\, dx\,dt = \int_0^{T}\int_{\Omega} f_{\epsilon} w_{\epsilon}\, dx\,dt. \end{equation} Notice that \begin{align*} &|\int_0^{T}\int_{\Omega} f_{\epsilon} w_{\epsilon}\, dx\,dt| \\ &\leq \int_0^{T}\int_{\Omega} |f_{\epsilon}-f\|T_{2k}(u_{\epsilon} - T_{h}(u_{\epsilon}) + T_{k}(u_{\epsilon}) - (T_{k}(u))_{\mu})|\,dx\,dt \\ &\quad + \int_0^{T}\int_{\Omega} |f T_{2k}(u_{\epsilon} - T_{h}(u_{\epsilon}) + T_{k}(u_{\epsilon}) - (T_{k}(u))_{\mu})|\, dx\,dt \\ &\leq 2k \int_0^{T}\int_{\Omega} |f_{\epsilon}-f|\,dx\,dt + \int_0^{T}\int_{\Omega} |f T_{2k}(u_{\epsilon} - T_{h}(u_{\epsilon}) + T_{k}(u_{\epsilon}) - (T_{k}(u))_{\mu})|\, dx\,dt. \end{align*} Since $ f_{\epsilon}$ is strongly compact in $ L^1(Q)$, using \eqref{convueps p.p.}, the definition of $ \big(T_{k}(u)\big)_{\mu}$ and the Lebesgue dominated convergence theorem, we have \begin{equation*} \lim_{h \to +\infty} \lim_{\mu \to +\infty} \lim_{\epsilon \to 0} \big|\int_0^{T}\int_{\Omega} f_{\epsilon} w_{\epsilon}\, dx\,dt\big| \leq \lim_{h \to +\infty} \int_0^{T}\int_{\Omega} |f T_{2k}(u - T_{h}(u)|\, dx\,dt = 0. \end{equation*} Thus, recalling the notation introduced in \eqref{limsup w}, we have proven that \begin{equation}\label{wfeps} \int_0^{T}\int_{\Omega} f_{\epsilon} w_{\epsilon}\,dx\,dt = w(\epsilon, \mu, h). \end{equation} Let us estimate the second term in \eqref{test weps}. Since $ \nabla w_{\epsilon} = 0$ if $ |u_{\epsilon}| > M = h + 4k$, we have \begin{equation*} \int_0^{T}\int_{\Omega} a(x, \nabla u_{\epsilon})\cdot\nabla w_{\epsilon} \, dx\,dx = \int_0^{T}\int_{\Omega} a(x, \nabla T_{M}(u_{\epsilon}))\cdot\nabla w_{\epsilon} \, dt\,dt. \end{equation*} Next we split the integral in the sets $\{| u_{\epsilon}| \leq k\}$ and $\{| u_{\epsilon}| > k\}$, so that we have, recalling that $h > 2k$, \begin{equation}\label{a decomp} \begin{aligned} &\int_0^{T}\int_{\Omega} a(x, \nabla T_{M}(u_{\epsilon})) \cdot\nabla T_{2k}(u_{\epsilon} - T_{h}(u_{\epsilon}) + T_{k}(u_{\epsilon}) - (T_{k}(u))_{\mu})\, dx\,dt \\ &= \iint_{\{|u_{\epsilon}| \leq k\}} a(x, \nabla u_{\epsilon}) \cdot\nabla (u_{\epsilon} - (T_{k}(u))_{\mu})\,dx\,dt \\ &\quad + \iint_{\{|u_{\epsilon}| > k\}} a(x, \nabla T_{M} (u_{\epsilon}))\cdot\nabla (u_{\epsilon} - T_{h}(u_{\epsilon}))\,dx\,dt \\ &\quad - \iint_{\{|u_{\epsilon}| > k\}} a(x, \nabla T_{M}(u_{\epsilon})) \cdot\nabla (T_{k}(u))_{\mu}\,dx\,dt := I_1 + I_2 + I_3. \end{aligned} \end{equation} Let us estimate $I_2$. Since $ u_{\epsilon} - T_{h}(u_{\epsilon}) = 0$ if $ |u_{\epsilon}| \leq h$, using \eqref{C1}, Remark \ref{remarque a(x,0)} and Young inequality, we obtain \begin{equation}\label{ineq I2} \begin{aligned} &|I_2| \\ & = \Big| \iint_{\{|u_{\epsilon}| > k\}} a(x, \nabla T_{M}(u_{\epsilon}))\cdot\nabla (u_{\epsilon} - T_{h}(u_{\epsilon}))\,dx\,dt\Big| \\ & \leq \iint_{\{h \leq |u_{\epsilon}| \leq M\}} |a(x, \nabla u_{\epsilon})\|\nabla u_{\epsilon}| \,dx\,dt \\ & \leq \iint_{\{h \leq |u_{\epsilon}| \leq M\}} C_1(j(x) + |\nabla u_{\epsilon}|^{p(x)-1})|\nabla u_{\epsilon}| \,dx\,dt \\ & \leq \iint_{\{h \leq |u_{\epsilon}| \leq M\}} C_1j(x)|\nabla u_{\epsilon}| \,dx\,dt + \iint_{\{h \leq |u_{\epsilon}| \leq M\}} C_1|\nabla u_{\epsilon}|^{p(x)} \,dx\,dt \\ & \leq \iint_{\{h \leq |u_{\epsilon}| \leq M\}} \frac{C_1}{p'_{-}}|j(x)|^{p'(x)} \,dx\,dt + \iint_{\{h \leq |u_{\epsilon}| \leq M\}} \frac{C_1}{p_{-}}|\nabla u_{\epsilon}|^{p(x)} \,dx\,dt \\ & \quad + \iint_{\{h \leq |u_{\epsilon}| \leq M\}} C_1|\nabla u_{\epsilon}|^{p(x)} \,dx\,dt \\ & \leq C \iint_{\{h \leq |u_{\epsilon}| \leq M\}} C_1|\nabla u_{\epsilon}|^{p(x)} \,dx\,dt \\ &\quad + C' \iint_{\{h \leq |u_{\epsilon}| \leq M\}} \frac{C_1}{p'_{-}}|j(x)|^{p'(x)} \,dx\,dt. \end{aligned} \end{equation} The functions $j(t,x)$ and $ \big(\nabla u_{\epsilon}\big)_{\epsilon > 0}$ are bounded in $ L^{p'_{-}}(0,T; W_0^{1,p(\cdot)}(\Omega))$ and in $ L^{p_{-}}(0,T; W_0^{1,p(\cdot)}(\Omega))$ respectively, and meas$\{h \leq |u_{\epsilon}| \leq h+4k\}$ converges uniformly to zero as $h$ tends to infinity with respect to $\epsilon$. Then, passing to the limit in \eqref{ineq I2} as $\epsilon \to 0$ and $h \to +\infty$ respectively, and using Lebesgue dominated convergence theorem, we obtain \begin{equation*} I_2 = w(\epsilon, h). \end{equation*} For $ I_3$, let us remark that, since $ \big(\nabla u_{\epsilon}\big)_{\epsilon > 0}$ is bounded in $ L^{p_{-}}(0,T; W_0^{1,p(\cdot)}(\Omega))$, \eqref{C1} implies that $ \big(a(x,\nabla T_{M}(u_{\epsilon}))\big)_{\epsilon > 0}$ is bounded in $ \big( L^{p'(\cdot)}(Q)\big)^N$. The almost everywhere convergence of $ u_{\epsilon}$ to $u$, as $\epsilon \to 0$, implies that $ |\nabla T_{k}(u)|\chi_{\{|u_{\epsilon}| > k\}}$ strongly converges to zero in $ L^{p_{-}}(0,T; W_0^{1,p(\cdot)}(\Omega))$. So that, by the Lebesgue dominated convergence theorem, we have \begin{equation*} \limsup_{\epsilon \to 0} \iint_{\{|u_{\epsilon}| > k\}} a(x, \nabla T_{M}(u_{\epsilon}))\cdot\nabla T_{k}(u)\,dx\,dt = 0. \end{equation*} Thus, we obtain \begin{align*} I_3 & = \iint_{\{|u_{\epsilon}| > k\}} a(x, \nabla T_{M}(u_{\epsilon})) \cdot\nabla (T_{k}(u))_{\mu}\,dx\,dt \\ & = \iint_{\{|u_{\epsilon}| > k\}} a(x, \nabla T_{M}(u_{\epsilon})) \cdot\nabla T_{k}(u)\,dx\,dt \\ & \quad + \iint_{\{|u_{\epsilon}| > k\}} a(x, \nabla T_{M}(u_{\epsilon}))\cdot\nabla ((T_{k}(u))_{\mu} - T_{k}(u))\,dx\,dt \\ & = w(\epsilon) + \iint_{\{|u_{\epsilon}| > k\}} a(x, \nabla T_{M}(u_{\epsilon}))\cdot\nabla ((T_{k}(u))_{\mu} - T_{k}(u))\,dx\,dt. \end{align*} Using the fact that $ \big(a(x,\nabla T_{M}(u_{\epsilon}))\big)_{\epsilon > 0}$ is bounded in $ \big( L^{p'(\cdot)}(Q)\big)^N$ and thanks to \eqref{strongconv}, we can apply the Lebesgue dominated convergence theorem to obtain \begin{equation*} \iint_{\{|u_{\epsilon}| > k\}} a(x, \nabla T_{M}(u_{\epsilon})) \cdot\nabla ((T_{k}(u))_{\mu} - T_{k}(u))\,dx\,dt = w(\epsilon, \mu), \end{equation*} therefore we conclude that $I_3 = w(\epsilon, \mu)$. Then from \eqref{a decomp}, according to the fact that $ I_2$ and $ I_3$ converge to zero, we obtain \begin{equation}\label{test weps2} \begin{aligned} &\int_0^{T}\int_{\Omega} a(x, \nabla u_{\epsilon})\cdot\nabla w_{\epsilon}\,dx\,dt \\ &= \iint_{\{|u_{\epsilon}| \leq k\}} a(x, \nabla u_{\epsilon})\cdot\nabla (u_{\epsilon} - (T_{k}(u))_{\mu})\,dx\,dt + w(\epsilon, \mu, h). \end{aligned} \end{equation} Putting together \eqref{test weps}, \eqref{wfeps} and \eqref{test weps2} we have \begin{equation}\label{test weps3} \begin{aligned} &\int_0^{T}\int_{\Omega} (u_{\epsilon})_{t} w_{\epsilon}\, dx\,dt + \iint_{\{|u_{\epsilon}| \leq k\}} a(x, \nabla u_{\epsilon})\cdot\nabla (u_{\epsilon} - (T_{k}(u))_{\mu})\,dx\,dt \\ &= w(\epsilon, \mu, h). \end{aligned} \end{equation} For the first term of \eqref{test weps3}, we can apply \cite[Lemma 2.1]{Porreta} to obtain \begin{equation*} \int_0^{T}\int_{\Omega} (u_{\epsilon})_{t} w_{\epsilon}\, dx\,dt \geq w(\epsilon, \mu, h). \end{equation*} Hence \eqref{test weps3} becomes \begin{equation}\label{test weps4} \iint_{\{|u_{\epsilon}| \leq k\}} a(x, \nabla u_{\epsilon}) \cdot\nabla (u_{\epsilon} - (T_{k}(u))_{\mu})\,dx\,dt \leq w(\epsilon, \mu, h). \end{equation} Since $ \nabla(T_{k}(u))_{\mu} \to \nabla T_{k}(u)$ strongly in $ \big(L^{p(\cdot)}(Q)\big)^N$ as $\mu \to +\infty$, we deduce from \eqref{test weps4} that \begin{equation}\label{test weps5} \int_0^{T}\int_{\Omega} a(x, \nabla T_{k}(u_{\epsilon})) \cdot\nabla (T_{k}(u_{\epsilon}) - (T_{k}(u)))\, dx\,dt \leq w(\epsilon, \mu, h). \end{equation} Therefore, passing to the limit in \eqref{test weps5} as $\epsilon$ tends to zero, $\mu$ and $h$ tend to infinity respectively, we deduce that \begin{equation}\label{limsup aTk} \limsup_{\epsilon \to 0} \int_0^{T}\int_{\Omega} a(x, \nabla T_{k}(u_{\epsilon}))\cdot\nabla (T_{k}(u_{\epsilon}) - (T_{k}(u)))\, dx\,dt \leq 0. \end{equation} Now, let $\varphi \in \mathcal{D}(Q)$ and $\lambda \in \mathbb{R}^{*}$. Using \eqref{limsup aTk} and \eqref{coerc}, we obtain \begin{equation}\label{limsup aTk2} \begin{aligned} &\lambda \lim_{\epsilon \to 0} \int_0^{T} \int_{\Omega} a(x, \nabla T_{k}(u_{\epsilon}))\cdot\nabla \varphi \,dx \,dt \\ &\geq \limsup_{\epsilon \to 0} \int_0^{T}\int_{\Omega} a(x, \nabla T_{k}(u_{\epsilon}))\cdot\nabla \big[T_{k}(u_{\epsilon}) - T_{k}(u) + \lambda \varphi\big]\, dx\, dt \\ &\geq \limsup_{\epsilon \to 0} \int_0^{T}\int_{\Omega} a(x, \nabla(T_{k}(u) - \lambda \varphi))\cdot\nabla \big[T_{k}(u_{\epsilon}) - T_{k}(u) + \lambda \varphi\big]\, dx \,dt \\ &\geq \lambda \int_0^{T} \int_{\Omega} a(x, \nabla(T_{k}(u) - \lambda \varphi)) \cdot\nabla \varphi\,dx\,dt. \end{aligned} \end{equation} Dividing \eqref{limsup aTk2} by $\lambda > 0$ and by $\lambda < 0$ respectively, passing to the limit with $\lambda \to 0$ it follows that \begin{equation*} \lim_{\epsilon \to 0} \int_0^{T} \int_{\Omega} a(x, \nabla T_{k}(u_{\epsilon}))\cdot\nabla \varphi \, dx\,dt = \int_0^{T} \int_{\Omega} a(x, \nabla T_{k}(u)\cdot\nabla \varphi \, dx\,dt. \end{equation*} This means that for all $k > 0$, \[ \int_0^{T} \int_{\Omega} \zeta_{k} \nabla \varphi\, dx = \int_0^{T} \int_{\Omega} a(x, \nabla T_{k}(u)\cdot\nabla \varphi \, dx\,dt. \] Hence $ \zeta_{k} = a(x, \nabla T_{k}(u))$ a.e. in $Q$ and we have \begin{equation*} a(x, \nabla T_{k}(u_{\epsilon})) \rightharpoonup a(x, \nabla T_{k}(u)) \quad \text{in } \big(L^{p'(\cdot)}(Q)\big)^N. \end{equation*} (ii) From \eqref{test weps5}, we have \begin{equation}\label{test weps6} \begin{aligned} &\int_0^{T}\int_{\Omega} (a(x, \nabla T_{k}(u_{\epsilon})) - a(x, \nabla T_{k}(u)))\cdot\nabla (T_{k}(u_{\epsilon}) - (T_{k}(u)))\, dx\,dt \\ &\leq -\int_0^{T}\int_{\Omega} a(x, \nabla T_{k}(u))\cdot\nabla (T_{k}(u_{\epsilon}) - (T_{k}(u)))\, dx\,dt + w(\epsilon, \mu, h). \end{aligned} \end{equation} The weak convergence of $ \nabla T_{k}(u_{\epsilon})$ to $ \nabla T_{k}(u)$ in $ \big(L^{p(\cdot)}(Q)\big)^N$ allows to conclude that \[ \limsup_{\epsilon \to 0} \int_0^{T}\int_{\Omega} a(x, \nabla T_{k}(u))\cdot\nabla (T_{k}(u_{\epsilon}) - (T_{k}(u)))\, dx\,dt = 0. \] Therefore, passing to the limit in \eqref{test weps6} as $\epsilon$ tends to zero, $\mu$ and $h$ tend to infinity respectively, we deduce that \begin{equation}\label{limsup aTk3} \limsup_{\epsilon \to 0} \int_0^{T}\int_{\Omega} (a(x, \nabla T_{k}(u_{\epsilon})) - a(x, \nabla T_{k}(u))) \cdot\nabla (T_{k}(u_{\epsilon}) - (T_{k}(u)))\, dx\,dt = 0. \end{equation} Now, set \[ g_{\epsilon}(t,x) = \big[a(x, \nabla u_{\epsilon}) - a(x, \nabla u)\big]\cdot\nabla \big[T_{k}(u_{\epsilon}) - T_{k}(u)\big] \geq 0. \] $g_{\epsilon}(t,x) \to 0$ strongly in $ L^1(Q)$ as $\epsilon \to 0$. Up to a subsequence, $ g_{\epsilon}(t,x) \to 0$ a.e. in $Q$, which means that there exists $\omega \subset Q$ such that meas $(\omega) = 0$ and $ g_{\epsilon}(t,x) \to 0$ in $Q \backslash \omega$. Let $(t,x) \in Q \backslash \omega$. Using assumptions \eqref{encadr} and\eqref{C1}, it follows that the sequence $ \big(\nabla T_{k}(u_{\epsilon}(t,x))\big)_{\epsilon > 0}$ is bounded in $ \mathbb{R}\times \mathbb{R}^N$ and so we can extract a subsequence which converges to some $\theta$ in $ \mathbb{R}\times\mathbb{R}^N$. Passing to the limit in the expression of $ g_{\epsilon}(t,x)$, it follows that \begin{equation*} 0 = \big[a(x, \theta) - a(x, \nabla T_{k}(u))\big]\cdot\big[\theta - T_{k}(u)\big] \end{equation*} and it yields $\theta = \nabla T_{k}(u)$ for all $(t,x) \in Q \backslash \omega$. As the limit does not depend on the subsequence, the whole sequence $ \big(\nabla T_{k}(u_{\epsilon}(t,x))\big)_{\epsilon > 0}$ converges to $\theta$ in $ \mathbb{R}\times\mathbb{R}^N$. This means that $\nabla T_{k}(u_{\epsilon}) \to \nabla T_{k}(u)$ a.e. in $Q$. (iii) The continuity of $a(x, \xi)$ with respect to $ \xi \in \mathbb{R}\times\mathbb{R}^N$ gives us \begin{equation*} a(x, \nabla T_{k}(u_{\epsilon})) \to a(x, \nabla T_{k}(u)) \quad\text{a.e. in }Q. \end{equation*} Therefore, \begin{equation*} a(x, \nabla T_{k}(u_{\epsilon}))\cdot\nabla T_{k}(u_{\epsilon}) \to a(x, \nabla T_{k}(u))\cdot\nabla T_{k}(u) \quad \text{a.e. in }Q. \end{equation*} Setting $ z_{\epsilon} = a(x, \nabla T_{k}(u_{\epsilon})) \cdot\nabla T_{k}(u_{\epsilon})$ and $z = a(x, \nabla T_{k}(u))\cdot\nabla T_{k}(u)$, we have \begin{gather*} z_{\epsilon} > 0, \quad z_{\epsilon} \to z \text{ a.e. in }Q,\; z \in L^1(\Omega), \\ \iint_{Q} z_{\epsilon}\,dx\,dt \to \iint_{Q} z\,dx\,dt \end{gather*} and as \[ \iint_{Q} |z_{\epsilon} - z|\,dx\,dt = 2 \iint_{Q} (z - z_{\epsilon})^{+} \,dx\,dt + \iint_{Q} (z_{\epsilon} - z)\,dx\,dt \] and $ (z - z_{\epsilon})^{+} \leq z$, it follows by using the Lebesgue dominated convergence theorem that \begin{equation*} \lim_{\epsilon \to 0} \iint_{Q} |z_{\epsilon} - z|\,dx\,dt = 0, \end{equation*} which implies \begin{equation*} a(x, \nabla T_{k}(u_{\epsilon}))\cdot\nabla T_{k}(u_{\epsilon}) \to a(x, \nabla T_{k}(u))\cdot\nabla T_{k}(u) \quad \text{strongly in } L^1(Q) \text{ and a.e. in }Q. \end{equation*} To prove (iv), we need the following lemmas. \begin{lemma}[\cite{Fanzhao}] \label{lemmaconv} Let $u, u_n \in L^{p(\cdot)}(Q)$, $n = 1, 2, \ldots$. Then the following statements are equivalent to each other: \begin{itemize} \item[(1)] $ \lim_{n \to \infty} |u_n - u|_{p(\cdot)} = 0$; \item[(2)] $ \lim_{n \to \infty} \rho_{p(\cdot)}(u_n - u) = 0$; \item[(3)] $ u_n$ converges to $u$ in $Q$ in measure and $ \lim_{n \to \infty} \rho_{p(\cdot)}(u_n) = \rho_{p(\cdot)}(u)$. \end{itemize} \end{lemma} Next we have a Lebesgue generalized convergence theorem. \begin{lemma} \label{lemmalebesgue} Let $ (f_n)_{n \in \mathbb{N}}$ be a sequence of measurable functions and $f$ a measurable function such that $ f_n \to f$ a.e. in $Q$. Let $ (g_n)_{n \in \mathbb{N}} \subset L^1(Q)$ such that for all $n \in \mathbb{N}$, $|f_n| \leq g_n$ a.e. in $Q$ and $ g_n \to g$ in $ L^1(Q)$. Then \begin{equation*} \iint_{Q} f_n \, dx \to \iint_{Q} f \, dx. \end{equation*} \end{lemma} Now, set $ f_{\epsilon} = |\nabla T_{k}(u_{\epsilon})|^{p(x)}$, $f = |\nabla T_{k}(u)|^{p(x)}$, $g_{\epsilon} = a(x, \nabla T_{k}(u_{\epsilon}))\cdot\nabla T_{k}(u_{\epsilon})$ and $g = a(x, \nabla T_{k}(u))\cdot\nabla T_{k}(u)$. We have: \begin{itemize} \item $f_{\epsilon}$ is a sequence of measurable functions, $f$ is a measurable function and according to $(ii)$, $ f_{\epsilon} \to f$ a.e. in $Q$. \item Using (iii), we have $ (g_{\epsilon})_{\epsilon > 0} \subset L^1(Q)$, $g_{\epsilon} \to g$ a.e. in $Q$, $g_{\epsilon} \to g$ in $ L^1(Q)$ and using \eqref{encadr}, we have $ |f_{\epsilon}| \leq C g_{\epsilon}$. \end{itemize} Then, by Lemma \ref{lemmalebesgue}, we have $\iint_{Q} f_{\epsilon}\,dx\,dt \to \iint_{Q} f \,dx\,dt$, which is equivalent to say \begin{equation*} \iint_{Q} |\nabla T_{k}(u_{\epsilon})|^{p(x)}\,dx\,dt \to \iint_{Q} |\nabla T_{k}(u)|^{p(x)}\,dx\,dt. \end{equation*} We deduce from (ii) that the sequence $ \big(\nabla T_{k}(u_{\epsilon})\big)_{\epsilon > 0}$ converges to $ \nabla T_{k}(u)$ in $Q$ in measure. Then, by Lemma \ref{lemmaconv} we deduce that \begin{equation*} \lim_{\epsilon \to 0} \iint_{Q} |\nabla T_{k}(u_{\epsilon}) - \nabla T_{k}(u)|^{p(x)}\,dx\,dt = 0, \end{equation*} which is equivalent to saying that $ \nabla T_{k}(u_{\epsilon}) \to \nabla T_{k}(u)$ in $ \big(L^{p(\cdot)}(Q)\big)^N$. \end{proof} \subsection{Existence of entropy solutions} For a given $a,k > 0$ defines the function $ T_{k,a}(s) = T_{a}(s - T_{k}(s))$. \begin{equation*} T_{k,a}(s) = \begin{cases} s - k\:\operatorname{sign}(s) & \text{if } k \leq |s| < k + a, \\ a\:\operatorname{sign}(s) & \text{if } |s| \geq k + a, \\ 0 & \text{if } |s| \leq k. \end{cases} \end{equation*} Let $ u_{\epsilon}$ be a weak solution of \eqref{pbmere}. Using $ \theta_{\epsilon}T_{k,a}(u_{\epsilon})$ as a test function in \eqref{solfaiblmere2} and letting $\epsilon$ goes to zero, we find \begin{equation}\label{test Tk,a} \begin{aligned} &\int_0^{t_1}\int_{\Omega} (u_{\epsilon})_{t} T_{k,a}(u_{\epsilon})\, dx\,dt + \int_0^{t_1}\int_{\Omega} a(x, \nabla u_{\epsilon}) \cdot\nabla T_{k,a}(u_{\epsilon})\, dx\,dt \\ &= \int_0^{t_1}\int_{\Omega} f_{\epsilon} T_{k,a}(u_{\epsilon})\, dx\,dt. \end{aligned} \end{equation} We have \begin{equation}\label{transf (ueps) t} \begin{aligned} &\int_0^{t_1}\int_{\Omega} (u_{\epsilon})_{t} T_{k,a}(u_{\epsilon})\, dx\,dt \\ & = \int_0^{t_1}\int_{\Omega} (u_{\epsilon})_{t} T_{a}(u_{\epsilon} - T_{k}(u_{\epsilon}))\, dx\,dt \\ & = \int_0^{t_1}\int_{\{|u_{\epsilon}| > k\}} (u_{\epsilon})_{t} T_{a}(u_{\epsilon} \mp k)\, dx\,dt \\ & = \int_0^{t_1}\int_{\{|u_{\epsilon}| > k\}} (u_{\epsilon} \mp k)_{t} T_{a}(u_{\epsilon} \mp k)\,dx\,dt \\ & = \int_{\{|u_{\epsilon}| > k\}} \Theta_{a}(u_{\epsilon} \mp k)(t_1)\, dx - \int_{\{|u_{0,\epsilon}| > k\}} \Theta_{a}(u_{0,\epsilon} \mp k)\, dx. \end{aligned} \end{equation} Using \eqref{encadr} and \eqref{transf (ueps) t}, from \eqref{test Tk,a} we obtain \begin{align*} &\int_{\{|u_{\epsilon}| > k\}} \Theta_{a}(u_{\epsilon} \mp k)(t_1)\, dx - \int_{\{|u_{0,\epsilon}| > k\}} \Theta_{a}(u_{0,\epsilon} \mp k)\, dx\\ &+ \frac{1}{C} \iint_{\{k \leq |u_{\epsilon}| \leq k+a\}} |\nabla u_{\epsilon}|^{p(x)}\,dx\,dt \\ &\leq \int_0^{t_1}\int_{\Omega} f_{\epsilon} T_{k,a}(u_{\epsilon})\,dx\,dt, \end{align*} which yields \begin{equation}\label{ineq nablaTk} \begin{aligned} &\iint_{\{k \leq |u_{\epsilon}| \leq k+a\}} |\nabla u_{\epsilon}|^{p(x)}\,dx\,dt \\ &\leq C'\Big( \iint_{\{|u_{\epsilon}| > k\}} |f_{\epsilon}|\,dx\,dt + \int_{\{|u_{0,\epsilon}| > k\}} |u_{0,\epsilon}|\, dx\Big). \end{aligned} \end{equation} Recalling that $ u_{\epsilon} \to u$ a.e. in $Q$, we have \begin{equation*} \lim_{k \to +\infty} \operatorname{meas} \{(t,x) \in Q: |u_{\epsilon}| > k\} = 0 \quad \text{uniformly with respect to } \epsilon. \end{equation*} Therefore, passing to the limit in \eqref{ineq nablaTk} with $\epsilon$ and $k$ tending to zero and infinity respectively, we conclude that \begin{equation*} \lim_{k \to +\infty} \iint_{\{(t,x) \in Q: k \leq |u_{\epsilon}| \leq k+a\}} |\nabla u|^{p(x)}\,dx\,dt = 0. \end{equation*} Choosing $a=1$, we obtain the renormalized condition \eqref{intprop}, i.e., \begin{equation*} \lim_{k \to +\infty} \iint_{\{(t,x) \in Q: k \leq |u_{\epsilon}| \leq k+1\}} |\nabla u|^{p(x)}\,dx\,dt = 0. \end{equation*} Now, let $\varphi \in \mathcal{D}(Q)$ with $\varphi (.,T) = 0$ and $S$ in $W^{2,\infty}(\mathbb{R})$ which is piecewise $C^1$ satisfying that $\operatorname{supp}S' \subset [-M, M]$ for some $M > 0$. Taking $ S'(u_{\epsilon})\varphi$ as a test function in \eqref{solfaiblmere2}, it yields \begin{equation}\label{test S'} \begin{aligned} &\int_0^{T}\int_{\Omega} (u_{\epsilon})_{t} S'(u_{\epsilon}) \varphi\,dx\,dt + \int_0^{T}\int_{\Omega} a(x, \nabla u_{\epsilon})\cdot \nabla( S'(u_{\epsilon})\varphi)\,dx\,dt \\ &= \int_0^{T}\int_{\Omega} f_{\epsilon} S'(u_{\epsilon}) \varphi\,dx\,dt. \end{aligned} \end{equation} We have $(u_{\epsilon})_{t} S'(u_{\epsilon}) \varphi = \big(S(u_{\epsilon})\big)_{t} \varphi$ and $ \nabla( S'(u_{\epsilon})\varphi) = S'(u_{\epsilon}) \nabla \varphi + S"(u_{\epsilon}) \nabla u_{\epsilon} \varphi$. Then, equality \eqref{test S'} becomes \begin{equation}\label{test S'2} \begin{aligned} &\int_0^{T}\int_{\Omega} \big(S(u_{\epsilon})\big)_{t} \varphi\,dx\,dt + \int_0^{T}\int_{\Omega} S'(u_{\epsilon}) a(x, \nabla u_{\epsilon}) \cdot\nabla \varphi \,dx\,dt \\ &\quad + \int_0^{T}\int_{\Omega} S"(u_{\epsilon}) a(x,\nabla u_{\epsilon})\cdot\nabla u_{\epsilon} \varphi \, dx\,dt \\ &= \int_0^{T}\int_{\Omega} f_{\epsilon} S'(u_{\epsilon})\, \varphi \, dx\,dt. \end{aligned} \end{equation} We consider the first term on the left-hand side of \eqref{test S'2}. Since $S$ is continuous, \eqref{convueps p.p.} implies that $ S(u_{\epsilon})$ converges to $S(u)$ a.e. in $Q$ and weakly$-*$ in $ L^{\infty}(Q)$. Then $ (S(u_{\epsilon}))_{t}$ converges to $ (S(u))_{t}$ in $\mathcal{D}'(Q)$ as $\epsilon \to 0$, that is \begin{equation*} \int_0^{T}\int_{\Omega} \big(S(u_{\epsilon})\big)_{t} \varphi\, dx\,dt \to \int_0^{T}\int_{\Omega} (S(u))_{t} \varphi\, dx\,dt. \end{equation*} For the other terms on the left-hand side of \eqref{test S'2}, as $\operatorname{supp}S' \subset [-M, M]$, we have \begin{gather*} S'(u_{\epsilon}) a(x, \nabla u_{\epsilon}) = S'(u_{\epsilon}) a(x, \nabla T_{M}(u_{\epsilon})), \\ S"(u_{\epsilon}) a(x, \nabla u_{\epsilon})\cdot\nabla u_{\epsilon} = S"(u_{\epsilon}) a(x, \nabla T_{M}(u_{\epsilon}))\cdot\nabla T_{M}(u_{\epsilon}). \end{gather*} Using \eqref{convueps p.p.} and Proposition \ref{propconv aTk}, we have \begin{gather*} S'(u_{\epsilon}) a(x, \nabla T_{M}(u_{\epsilon})) \to S'(u) a(x, \nabla T_{M}(u))\quad \text{in } \big(L^{p'(\cdot)}(Q)\big)^N, \\ S"(u_{\epsilon}) a(x, \nabla T_{M}(u_{\epsilon}))\cdot\nabla T_{M}(u_{\epsilon}) \to S"(u) a(x, \nabla T_{M}(u))\cdot\nabla T_{M}(u)\quad \text{in } L^1(Q). \end{gather*} For the right-hand side of \eqref{test S'2}, thanks to the strong convergence of $ f_{\epsilon}$, we have \begin{equation*} f_{\epsilon} S'(u_{\epsilon}) \to f S'(u) \quad \text{in } L^1(Q). \end{equation*} Therefore, we can pass to the limit in \eqref{test S'2} as $\epsilon \to 0$ to obtain \begin{equation}\label{limit test S'2} \begin{aligned} &\int_0^{T}\int_{\Omega} \big(S(u)\big)_{t} \varphi\, dx\,dt + \int_0^{T}\int_{\Omega} S'(u) a(x, \nabla u)\cdot\nabla \varphi \,dx\,dt \\ &+ \int_0^{T}\int_{\Omega} S"(u)a(x,\nabla u)\cdot\nabla u \varphi \, dx\,dt \\ &= \int_0^{T}\int_{\Omega} f S'(u)\, \varphi \,dx\,dt. \end{aligned} \end{equation} Employing the integration by parts formula for the evolution term, we obtain \begin{align*} &\int_0^{T}\int_{\Omega} \big(S(u)\big)_{t} \varphi\, dx\,dt \\ & = \int_{\Omega} S(u(T,x))\varphi(T,x)\, dx - \int_{\Omega} S(u_0)\varphi(0,x)\,dx - \int_0^{T}\int_{\Omega} S(u) (\varphi)_{t}\,dx\,dt\\ & = - \int_{\Omega} S(u_0)\varphi(0,x)\,dx - \int_0^{T}\int_{\Omega} S(u) (\varphi)_{t}\,dx\,dt \quad (\text{since } \varphi(T,x) = 0). \end{align*} Therefore, we deduce from \eqref{limit test S'2} that \begin{equation}\label{limit2 test S'2} \begin{aligned} &- \int_{\Omega} S(u_0)\varphi(0,x)\,dx - \int_0^{T}\int_{\Omega} S(u) (\varphi)_{t}\,dx\,dt \\ &+ \int_0^{T}\int_{\Omega} \big[S'(u) a(x, \nabla u)\cdot\nabla \varphi + S"(u)a(x,\nabla u)\cdot\nabla u \varphi\big] \,dx\,dt \\ &= \int_0^{T}\int_{\Omega} f S'(u) \varphi \,dx\,dt. \end{aligned} \end{equation} This complete the proof of the existence of a renormalized solution, and then of the entropy solution (cf. Theorem \ref{theorem equiv}). \subsection{Uniqueness of the entropy solution} Now, we prove the uniqueness of the entropy solution. By Theorem \ref{theorem equiv}, it is enough to prove the uniqueness of the renormalized solution. Let $u$ and $v$ be two renormalized solutions for problem \eqref{pbmere}. Let $ S_n$ be defined as in \eqref{def Sn}. We choose $ T_{k}\big(S_n(u) - S_n(v)\big)$ as a test function in both the equations solved by $u$ and $v$. Subtracting the equations, we have \begin{align} &\int_0^{T}\int_{\Omega} \big(S_n(u) - S_n(v)\big)_{t} T_{k}\big(S_n(u) - S_n(v)\big) \,dx\,dt \nonumber\\ &+ \int_0^{T}\int_{\Omega} \big(S'_n(u) a(x, \nabla u) - S'_n(v) a(x, \nabla v)\big)\cdot\nabla T_{k}\big(S_n(u) - S_n(v)\big) \,dx\,dt \nonumber\\ &+ \int_0^{T}\int_{\Omega} \big(S''_n(u) a(x, \nabla u)\cdot\nabla u - S''_n(v) a(x, \nabla v)\cdot\nabla v\big) T_{k}\big(S_n(u) - S_n(v)\big) \,dx\,dt \nonumber\\ &= \int_0^{T}\int_{\Omega} f\big( S'_n(u) - S'_n(v)\big) T_{k}\big(S_n(u) - S_n(v)\big) \,dx\,dt. \label{test soustr} \end{align} We set \begin{gather*} J_0 = \int_0^{T}\int_{\Omega} \big(S_n(u) - S_n(v)\big)_{t} T_{k}\big(S_n(u) - S_n(v)\big) \,dx\,dt \\ J_1 = \int_0^{T}\int_{\Omega} \big(S'_n(u) a(x, \nabla u) - S'_n(v) a(x, \nabla v)\big)\cdot\nabla T_{k}\big(S_n(u) - S_n(v)\big) \,dx\,dt \\ J_2 = \int_0^{T}\int_{\Omega} \big(S''_n(u) a(x, \nabla u)\cdot\nabla u - S''_n(v) a(x, \nabla v)\cdot\nabla v\big) T_{k}\big(S_n(u) - S_n(v)\big) \,dx\,dt \\ J_3 = \int_0^{T}\int_{\Omega} f\big( S'_n(u) - S'_n(v)\big) T_{k}\big(S_n(u) - S_n(v)\big) \,dx\,dt. \end{gather*} We estimate $ J_0, J_1, J_2$ and $ J_3$ one by one. Recalling the definition of $ \Theta_{k}(r), J_0$ can be written as \begin{equation*} J_0 = \int_{\Omega} \Theta_{k}\big(S_n(u) - S_n(v)\big)(T) \, dx - \int_{\Omega} \Theta_{k}\big(S_n(u) - S_n(v)\big)(0) \, dx. \end{equation*} Because $u$ and $v$ have the same initial condition, and by the properties of $ \Theta_{k}$, we obtain \begin{equation}\label{J0} J_0 = \int_{\Omega} \Theta_{k}\big(S_n(u) - S_n(v)\big)(T) \, dx \geq 0. \end{equation} We deal with $J_1$ splitting it as below \begin{align*} J_1 & = \iint_{\{|S_n(u)-S_n(v)| \leq k\} \cap \{|u| \leq n, |v| \leq n\}} \big(a(x, \nabla u) - a(x, \nabla v)\big)\cdot\nabla (u-v)\,dx\,dt \\ & \quad + \iint_{\{|S_n(u)-S_n(v)| \leq k\} \cap \{|u| \leq n, |v| > n\}} \big(a(x, \nabla u) - S'_n(v)a(x, \nabla v)\big)\\ &\quad \cdot\nabla (u - S_n(v)\,dx\,dt \\ & \quad + \iint_{\{|S_n(u)-S_n(v)| \leq k\} \cap \{|u| > n\}} \big(S'_n(u)a(x, \nabla u) - S'_n(v)a(x, \nabla v)\big) \\ &\quad \cdot\nabla (S_n(u) - S_n(v)\,dx\,dt := J_1^1 + J_1^2 + J_1^{3}. \end{align*} Since $ \{|S_n(u)-S_n(v)| \leq k, |u| > n\} \subset \{|u| > n, |v| > n - k\}$, we have, using the fact that $ S'_n(t) = 0$ if $|t| > n + 1$ and $|S'_n(t)| \leq 1$: \begin{equation}\label{J1,3} \begin{aligned} |J_1^{3}| & \leq \iint_{\{n \leq |u| \leq n+1\}} |a(x, \nabla u)\|\nabla u| \,dx\,dt \\ & \quad + \iint_{\{n \leq |u| \leq n+1\} \cap \{n-k \leq |v| \leq n+1\}} |a(x, \nabla u)\|\nabla v| \,dx\,dt \\ & \quad + \iint_{\{n \leq |u| \leq n+1\} \cap \{n-k \leq |v| \leq n+1\}} |a(x, \nabla v)\|\nabla u| \,dx\,dt \\ & \quad + \iint_{\{n-k \leq |v| \leq n+1\}} |a(x, \nabla v)\|\nabla v| \,dx\,dt. \end{aligned} \end{equation} Using assumption \eqref{C1} and Young's inequality, from the first integral in the right-hand side of \eqref{J1,3}, we obtain \begin{align*} & \iint_{\{n \leq |u| \leq n+1\}} |a(x, \nabla u)\|\nabla u| \,dx\,dt \\ &\leq \iint_{\{n \leq |u| \leq n+1\}} C_1(j(t,x) + |\nabla u|^{p(x)-1})|\nabla u| \,dx\,dt \\ &\leq \iint_{\{n \leq |u| \leq n+1\}} C_1j(t,x)|\nabla u| \,dx\,dt + \iint_{\{n \leq |u| \leq n+1\}} C_1|\nabla u|^{p(x)} \,dx\,dt \\ &\leq \iint_{\{n \leq |u_{\epsilon}| \leq n+1\}} \frac{C_1}{p'_{-}}|j(x)|^{p'(x)} \,dx\,dt + \iint_{\{n \leq |u| \leq n+1\}} \frac{C_1}{p_{-}}|\nabla u|^{p(x)} \,dx\,dt \\ &\quad + \iint_{\{n \leq |u| \leq n+1\}} C_1|\nabla u|^{p(x)} \,dx\,dt \\ &\leq C \iint_{\{n \leq |u| \leq n+1\}} |\nabla u|^{p(x)} \,dx\,dt + C' \iint_{\{n \leq |u| \leq n+1\}} |j(x)|^{p'(x)} \,dx\,dt. \end{align*} Function $j(x)$ is bounded in $ L^{p'_{-}}(0,T; W_0^{1,p(\cdot)}(\Omega))$ and $\operatorname{meas}\{n \leq |u_{\epsilon}| \leq n+1\}$ converges uniformly to zero as $n$ tends to infinity. Using the condition \eqref{intprop}, we can conclude that \begin{equation*} \lim_{n \to +\infty} \iint_{\{n \leq |u| \leq n+1\}} | a(x, \nabla u)\|\nabla u| \,dx\,dt = 0. \end{equation*} Similarly, we prove that all the other integrals in the right-hand side of \eqref{J1,3} converge to zero as $n \to +\infty$. Thus $ J_1^{3}$ converges to zero. Changing the roles of $u$ and $v$, the same arguments prove that $ J_1^2$ also converges to zero. We use Fatou's lemma to obtain \begin{equation}\label{J1} \liminf_{n \to +\infty} J_1 \geq \iint_{\{|u-v| \leq k\}} \big(a(x, \nabla u) - a(x, \nabla v)\big)\cdot\nabla (u-v)\,dx\,dt. \end{equation} Let us study the limit of $J_2$ now. We have \begin{align*} J_2 & = \int_0^{T}\int_{\Omega} S''_n(u) a(x, \nabla u) \cdot\nabla u T_{k}\big(S_n(u) - S_n(v)\big) \,dx\,dt \\ & \quad + \int_0^{T}\int_{\Omega} S''_n(v) a(x, \nabla v) \cdot\nabla v T_{k}\big(S_n(v) - S_n(u)\big) \,dx\,dt := J_2^1 + J_2^2. \end{align*} By symmetry between $ J_2^1$ and $ J_2^2$, it is sufficient to prove that $ J_2^1$ tends to zero. Since $|S''_n(s)| \leq 1$ and $S''_n(s) \neq 0$ only if $|s| \in [n,n+1]$, using \eqref{C1} we can write \begin{align*} \big|J_2^1\big| & \leq k \iint_{\{n \leq |u| \leq n+1\}} |a(x, \nabla u)\cdot\nabla u|\,dx\,dt \\ & \leq k \int_{\{n \leq |u| \leq n+1\}} C_1\big(j(x) + |\nabla u|^{p(x)-1}\big) |\nabla u| \,dx\,dt \\ & \leq k \int_{\Omega} C_1\Big(j(x) + |\nabla u|^{p(x)-1}\Big) |\nabla u| 1_{\{n \leq |u| \leq n+1\}}\,dx\,dt \to 0 \quad \text{as } n \to +\infty. \end{align*} We conclude that \begin{equation}\label{J2} \lim_{n \to +\infty} J_2 = 0. \end{equation} Let us recall that by definition of $ S_n$ we have that $ S'_n$ converges to $1$ for every $s$ in $\mathbb{R}$. Then \[ f(S'_n(u) - S'_n(v)) \to 0 \quad \text{strongly in } L^1(Q) \text{ as } n \to +\infty. \] Using the Lebesgue dominated convergence theorem, we deduce that \begin{equation}\label{J3} \lim_{n \to +\infty} J_3 = 0. \end{equation} Putting together \eqref{J0}, \eqref{J1}, \eqref{J2} and \eqref{J3}, from \eqref{test soustr}, we obtain that as $n$ tends to infinity, \begin{equation*} \iint_{\{|u-v| \leq k\}} \big(a(x, \nabla u) - a(x, \nabla v)\big)\cdot\nabla (u-v)\,dx\,dt \leq 0 \end{equation*} and then letting $k$ g{\oe}s to infinity (recall that $u$ and $v$ are finite a.e. in $Q$), we deduce that \begin{equation*} \iint_{Q} \big(a(x, \nabla u) - a(x, \nabla v)\big)\cdot\nabla (u-v)\,dx\,dt \leq 0. \end{equation*} The strict monotonicity assumption \eqref{coerc} then implies that $\nabla u = \nabla v$ a.e. in $Q$. Then, let $ \xi_n = T_1\big(T_{n+1}(u) - T_{n+1}(v)\big)$. We have $ \xi_n \in L^{p_{-}}(0,T; W_0^{1,p(\cdot)}(\Omega))$ and, since $\nabla u = \nabla v$ a.e. in $Q$, \begin{equation*} \nabla \xi_n = \begin{cases} 0 & \text{on } \{|u| \leq n+1, |v| \leq n+1\} \\ &\quad \cup \{|u| > n+1, |v| > n+1\} \\[3pt] 1_{\{|u - T_{n+1}(v)| \leq 1\}} \nabla u & \text{on } \{|u| \leq n+1, |v| > n+1\} \\[3pt] -1_{\{|v - T_{n+1}(u)| \leq 1\}} \nabla v & \text{on } \{|u| > n+1, |v| \leq n+1\}. \end{cases} \end{equation*} But, if $|s| > n+1, |t| \leq n+1$ and $|t - T_{n+1}(s)| \leq 1$, then $n \leq |t| \leq n+1$, which implies \begin{align*} \int_{Q} |\nabla \xi_n|^{p(x)}\,dx\,dt & \leq \int_{\{n \leq |u| \leq n+1\}} |\nabla u|^{p(x)}\,dx\,dt + \int_{\{n \leq |v| \leq n+1\}} |\nabla v|^{p(x)}\,dx\,dt \\ & \to 0 \quad \text{as } n \to +\infty. \end{align*} Then, $\xi_n \to 0$ in $ L^{p_{-}}(0,T; W_0^{1,p(\cdot)}(\Omega))$, and thus in $\mathcal{D}'(Q)$ as $n \to +\infty$. 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