\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2018 (2018), No. 132, pp. 1--29.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu} \thanks{\copyright 2018 Texas State University.} \vspace{8mm}} \begin{document} \title[\hfilneg EJDE-2018/132\hfil Renormalized solutions] {Renormalized solutions for nonlinear parabolic equations with general measure data} \author[M. Abdellaoui, E. Azroul \hfil EJDE-2018/132\hfilneg] {Mohammed Abdellaoui, Elhoussine Azroul} \address{Mohammed Abdellaoui \newline University of Fez, Faculty of Sciences Dhar El Mahraz, Laboratory LAMA, Department of Mathematics, B.P. 1796, Atlas Fez, Morocco} \email{mohammed.abdellaoui3@usmba.ac.ma} \address{Elhoussine Azroul \newline University of Fez, Faculty of Sciences Dhar El Mahraz, Laboratory LAMA, Department of Mathematics, B.P. 1796, Atlas Fez, Morocco} \email{elhoussine.azroul@usmba.ac.ma} \dedicatory{Communicated by Jerome A. Goldstein} \thanks{Submitted July 15, 2017. Published June 27, 2018.} \subjclass[2010]{35R06, 41A30, 35B45, 37K45, 32U20} \keywords{Nonlinear parabolic problems; $p$-capacity; renormalized solution; \hfill\break\indent stability; general measure} \begin{abstract} We prove the existence of parabolic initial boundary value problems of the type \begin{gather*} u_t-\operatorname{div}(a_{\epsilon}(t,x,u_{\epsilon},\nabla u_{\epsilon})) =\mu_{\epsilon}\quad\text{in }Q:=(0,T)\times\Omega,\\ u_{\epsilon}=0\quad\text{on }(0,T)\times\partial \Omega,\quad u_{\epsilon}(0)=u_{0,\epsilon}\quad\text{in }\Omega, \end{gather*} with respect to suitable convergence of the nonlinear operators $a_{\epsilon}$ and of the measure data $\mu_{\epsilon}$. As a consequence, we obtain the existence of a renormalized solution for a general class of nonlinear parabolic equations with right-hand side measure. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{definition}[theorem]{Definition} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \allowdisplaybreaks \section{Introduction} In this article we consider the parabolic problem \begin{equation} \label{e1.1} \begin{gathered} u_t-\operatorname{div}(a(t,x,u,\nabla u))=\mu \quad\text{in }Q:=(0,T)\times\Omega,\\ u=0 \quad \text{on }(0,T)\times\partial \Omega,\\ u(0)=u_0 \quad\text{in }\Omega, \end{gathered} \end{equation} where $\Omega$ is an open bounded subset of $\mathbb{R}^{N}$, $N\geq 2$, $T>0$ and $Q$ is the cylinder $(0,T)\times\Omega$, $(0,T)\times\partial \Omega$ being its lateral surface, the operator of Leray-Lions $u\mapsto -\operatorname{div}(a(t,x,u,\nabla u))$ is pseudo-monotone defined on the space $L^p(0,T;W^{1,p}_0(\Omega))$ with values in its dual $L^{p'}(0,T;W^{-1,p'}(\Omega))$, $p>1$ and $\frac{1}{p}+\frac{1}{p'}=1$. We assume that $u_0\in L^{2}(\Omega)$ and the data $\mu$ is a Radon measure with bounded variation on $Q$. Under some assumptions on $a$, If $\mu\in L^{p'}(Q)$ the existence and unicity of a weak solution $u$ of \eqref{e1.1} belonging to suitable energy space and to $C([0,T;L^{2}(\Omega)])$ was proved in \cite{L}. In the case of linear operators the difficulty can be overcome by defining the solution through the adjoint operator, this method is used in \cite{S} and yields a formulation having a unique solution. For nonlinear operators, the authors in \cite{BM} and \cite{P} extends the results in two different directions, assuming that $\mu\in L^{1}(Q)$ and $u_0\in L^{1}(\Omega)$, they prove existence of a renormalized solution, and of entropy solution, the same notions of solutions are used to ensure existence and uniqueness of equations with bounded Radon measures on $Q$ that does not charge the sets of zero parabolic $p$-capacity (See \cite{BM,DPP,Po1}), the authors show in \cite{DP} that these two notions of solution actually coincide. Here we use the notion of renormalized solution, introduced in \cite{DMOP,Ma,Pe2}. Roughly speaking, a renormalized solution to \eqref{e1.1} is a measurable function with all the truncations in the space $L^p(0,T;W^{1,p}_0(\Omega))\cap L^{\infty}(0,T;L^{1}(\Omega)) $ and such that for every $S\in W^{2,\infty}(\mathbb{R})(S(0)=0)$ with $S'$ has compact support on $\mathbb{R}$, we have \begin{equation} \label{e1.2} \begin{aligned} &-\int_{\Omega}S(u_0)\varphi(0)\,dx -\int_0^{T}\langle\varphi_{t}, S(u-g)\rangle\,dt\\ & +\int_QS'(u-g)a(t,x,u,\nabla u)\cdot\nabla \varphi\,dx\,dt \\ &+\int_QS''(u-g)a(t,x,u,\nabla u)\cdot\nabla (u-g) \varphi\,dx\,dt \\ &=\int_QS'(u-g)\varphi\,d\tilde{\mu}_0, \end{aligned} \end{equation} for every function $\varphi\in L^p(0,T;W^{1,p}_0(\Omega))\cap L^{\infty}(Q)$, $\varphi_{t}\in L^{p'}(0,T;W^{-1,p'}(\Omega))$, with $\varphi(T,x)=0$, such that $S'(u-g)\varphi\in L^p(0,T;W^{1,p}_0(\Omega))$, $g_{t}$ is the time derivative part of $\mu_0$ and $\hat{\mu}_0=\mu-g_{t}-\mu_{s}=f-\operatorname{div}(G)$. Moreover, for every $\psi\in C(\overline{Q})$ we have \begin{gather*} \lim_{n\to +\infty}\frac{1}{n}\int_{\{ n\leq v< 2n\}}a(t,x,u,\nabla u) \cdot\nabla v \psi\,dx\,dt=\int_Q\psi d\mu_{s}^{+}, \\ \lim_{n\to +\infty}\frac{1}{n}\int_{\{ -2n< v\leq -n\}}a(t,x,u,\nabla u) \cdot\nabla v \psi\,dx\,dt=\int_Q\psi d\mu_{s}^{-}, \end{gather*} where $\mu_{s}^{+}$ and $\mu_{s}^{-}$ are respectively the positive and the negative part of the singular part of the measure $\mu$ w.r.t. the $p$-capacity. In the proof of \cite[Theorem 2]{Pe2}, they used the fact that the approximating sequences $\mu_{\epsilon}$ having a splitting converging to $\mu$, the estimate concerning $u_{\epsilon}$ and $u_{\epsilon}-g_{\epsilon}^{t}$, next they prove the strong convergence of $T_k(u_{\epsilon}-g_{\epsilon})$ in $L^p(0,T;W^{1,p}_0(\Omega))$. To obtain this result, they use the same technique as in \cite{DMOP} adapted to the parabolic case. In the present paper we generalize this existence result to renormalized solutions of problems depending on $u$ and $\nabla u$ \begin{equation} \label{pba} \begin{gathered} (u_{\epsilon})_t -\operatorname{div}(a(t,x,u_{\epsilon},\nabla u_{\epsilon})) =\mu_{\epsilon} \quad\text{in }Q:=(0,T)\times\Omega, \\ u_{\epsilon}=0 \quad\text{on }(0,T)\times\partial \Omega,\\ u_{\epsilon}(0)=u_0 \quad\text{in }\Omega, \end{gathered} \end{equation} where $(\mu_{\epsilon})$ is a sequences of measures with splitting converging to $\mu$, and \[ \lim_{\epsilon\to 0} a_{\epsilon}(t,x,s_{\epsilon},\zeta_{\epsilon}) =a_0(t,x,s,\zeta), \] for every sequence $(s_{\epsilon},\zeta_{\epsilon})\in \mathbb{R}\times\mathbb{R}^{N}$ converging to $(s,\zeta)$ and for a.e. $(t,x)\in Q$. The main point which allows to go further the previous works, is the proof of the almost everywhere convergence of gradients in Proposition \ref{prop5.2} using the technique developed in \cite{Po1,Pr1}. To underline the importance of this tool, we have chosen to plan the paper in the following way: in Sect. 2, we recall some basic notations and we investigate the link between measures in $Q$ and the notion of parabolic capacity, this notion can be obtained from the result of the "elliptic capacity" contained in \cite{D}, which can be slightly adapted to this context of parabolic spaces, and we show the decomposition method for more general measures with bounded total variation in order to find a sense of solution to Cauchy-Dirichlet problems. In Sect. 3, we introduce and study a special type of approximating sequences of measures obtained via convolution arguments. In Sect. 4 we show the interest of cut-off functions and intermediary lemmas. In the last two sections, we establish the fundamental a priori estimates and we use the proof of strong convergence of truncates to obtain our main result. \section{Preliminaries} \subsection{Assumptions on the operator} Throughout this paper $\Omega$ will be a bounded open subset of $\mathbb{R}^{N}$, $N\geq 2$, $p$ and $p'$ will be real numbers, with $p> 1$ and $\frac{1}{p}+\frac{1}{p'}=1$. In what follows, $|\zeta|$ and $\zeta\cdot\zeta'$ will denote respectively the Euclidean norm of a vector $\zeta\in \mathbb{R}^{N}$ and the scalar product between $\zeta$ and $\zeta'\in \mathbb{R}^{N}$. Fixed three positive constants $c_0,c_1,c_2$, and a non-negative function $b_0=b(t,x)\in L^{p'}(Q)$, we say that a function $a:(0,T)\times\Omega\times\mathbb{R}\times\mathbb{R}^{N}\to \mathbb{R}^{N}$ satisfies the assumptions $H(c_0,c_1,c_2,b_0)$ if $a$ is a Carath\'eodory function (that is, $a(\cdot,\cdot,s,\zeta)$ is measurable on $Q$ for every $(s,\zeta)$ in $\mathbb{R}\times\mathbb{R}^{N}$, and $a(t,x,\cdot)$ is continuous on $\mathbb{R}\times \mathbb{R}^{N}$ for almost every $(t,x)$ in $Q$) such that, for every $s\in \mathbb{R}$, $\zeta,\zeta'\in \mathbb{R}^{N}$ with $\zeta\neq \zeta'$, satisfying the following properties. \begin{gather} a(t,x,s,\zeta)\cdot\zeta \geq c_0|\zeta|^p, \label{e2.1}\\ |a(t,x,s,\zeta)|\leq b_0(t,x)+c_1|s|^{p-1}+ c_2|\zeta|^{p-1}, \label{e2.2}\\ (a(t,s,s,\zeta)-a(t,x,s,\zeta'))\cdot(\zeta-\zeta')> 0. \label{e2.3} \end{gather} Notice that, as a consequence of \eqref{e2.1} and of the continuity of $a$ with respect to $\zeta$, we have that $a(t,x,s,0)=0$ for a.e. $(t,x)$ in $Q$ and for every $s\in \mathbb{R}$. Thanks to assumptions $H(c_0,c_1,c_2,b_0)$, the map $u\mapsto - \operatorname{div} (a(t,x,u,\nabla u))$ is a coercive, continuous, bounded and monotone operator defined on $L^p(0,T;W^{1,p}_0(\Omega))$ with values into its dual space $L^{p'}(0,T;W^{-1,p'}(\Omega))$; hence by the standard theory of monotone operators (see, e.g.,\cite{L}), for every $F$ in $L^{p'}(Q)$ and $u_0\in L^{2}(\Omega)$ there exists a variational solution $u$ of the problem \begin{gather*} u_t-\operatorname{div}(a(t,x,v,\nabla v))=F \quad \text{in }Q:=(0,T)\times\Omega, \\ v=0 \quad \text{on } (0,T)\times\partial \Omega,\\ v(0)=u_0 \quad \text{in }\Omega, \end{gather*} in the sense that $v$ belongs to $W\cap C(0,T;L^{2}(\Omega))$ (where $W=\{ u\in L^p(0,T;V), u_{t}\in L^{p'}(0,T;V')\}$ with $V= W^{1,p}_0(\Omega)\cap L^{2}(\Omega)$), and \begin{equation} \label{e2.4} \begin{aligned} &-\int_{\Omega} u_0\varphi(0)\,dx - \int_0^{T}\langle\varphi_{t},v\rangle\,dt +\int_Qa(t,x,v,\nabla v)\cdot\nabla \varphi\,dx\,dt\\ &=\int_0^{T}\langle F,\varphi\rangle_{W^{-1,p'}(\Omega),W^{1,p}_0(\Omega)}dt, \end{aligned} \end{equation} for all $\varphi \in W \text{ such that } \varphi(T)=0$. (Here and in the following $\langle\cdot,\cdot\rangle$ denotes the duality pairing between $W^{-1,p'}(\Omega)$ and $W^{1,p}_0(\Omega)$). \subsection{Capacity and measures} For every set $B\subseteq Q$, its $p$-capacity $\operatorname{cap}_{p}(B,Q)$ with respect to $Q$ is defined by \[ \inf \{ \| u\|_{W}\} \] where the infimum is taken over all the functions $u\in W$ such that $u\geq 1$ almost everywhere in a neighborhood of $B$. We say that a property $\mathcal{P}(t,x)$ holds $\operatorname{cap}_{p}$-quasi everywhere if $\mathcal{P}(t,x)$ holds for every $(x,t)$ outside a subset of $Q$ of zero $p$-capacity. A function $u$ defined on $Q$ is said to be $\operatorname{cap}_{p}$-quasi continuous if for every $\epsilon>0$ there exists $B\subseteq Q$ with $\operatorname{cap}_{p}(B,Q)< \epsilon$ such that the restriction of $u$ to $Q\backslash B$ is continuous. It is well known that every function in $W$ has a unique, up to sets of $p$-capacity zero, $\operatorname{cap}_{p}$-quasi continuous representative, whose values are defined $\operatorname{cap}_{p}$-quasi everywhere in $Q$. In what follows we always identify a function $u\in W$ with its $\operatorname{cap}_{p}$-quasi continuous representative. We define $\mathcal{M}_b(Q)$ as the space of all Radon measures on $Q$ with bounded total variation, and $C_b(Q)$ as the space of all bounded, continuous functions on $Q$, so that $\int_Q\varphi d\mu$ is defined for $\varphi\in C_b(Q)$ and $\mu$ in $\mathcal{M}_b(Q)$. The positive part, the negative part, and the total variation of a measure $\mu$ in $\mathcal{M}_b(Q)$ are denoted by $\mu^{+}$, $\mu^{-}$, and $|\mu|$, respectively. We recall that for a measure $\mu$ in $\mathcal{M}_b(Q)$, and a Borel set $E\subseteq Q$, the measure $\mu\perp E$ is defined by $(\mu\perp E)(Q)=\mu(E\cap B)$ for any Borel set $B\subseteq Q$. In the sequel we suppose that $p$ satisfies $p>2-\frac{1}{N+1}$. Then the embedding $W^{1,p}_0(\Omega)\subset L^{2}(\Omega)$ is valid, i.e., \[ X=L^p((0,T);W^{1,p}_0(\Omega)),\quad X'=L^{p'}((0,T);W^{-1,p'}(\Omega)). \] We say that a sequence $(\mu_n)$ of measures in $\mathcal{M}_b(Q)$ converges in the narrow topology to a measure $\mu$ in $\mathcal{M}_b(Q)$ if \begin{equation} \label{e2.5} \lim_{n\to +\infty}\int_Q\varphi d\mu_n=\int_Q\varphi d\mu \end{equation} for every $\varphi \in C(\overline{Q})$. If \eqref{e2.5} holds only for all the continuous functions $\varphi$ with compact support in $Q$, then we have the usual weak\_* convergence in $\mathcal{M}_b(Q)$. We define $\mathcal{M}_0(Q)$ as the set of all measures $\mu$ in $\mathcal{M}_b(Q)$ which satisfy $\mu(B)=0$ for every Borel set $B\subseteq Q$ such that $\operatorname{cap}_{p}(B,Q)=0$, while $\mathcal{M}_{s}(Q)$ will be the set of all measures $\mu$ in $\mathcal{M}_b(Q)$ for which there exists a Borel set $B\subset Q$, with $\operatorname{cap}_{p}(B,Q)=0$, such that $\mu=\mu\perp E$. For every $\mu\in \mathcal{M}_b(Q)$ there exist a unique pair $(\mu_0,\mu_{s})$ such that $\mu=\mu_0+\mu_{s}$, $\mu_0\in \mathcal{M}_0(Q)$, $\mu_{s}\in \mathcal{M}_{s}(Q)$ (see \cite[Lemma 2.1]{FST}). In addition, a measure $\mu$ belongs to $\mathcal{M}_0(Q)$ if and only if $\mu$ belongs to $L^{1}(Q)+L^{p'}(0,T;W^{-1,p'}(\Omega))+ L^p(0,T;V)$ (see \cite[Theorem 1.1]{DPP}). Hence a measure $\mu\in \mathcal{M}_b(Q)$ can be decomposed (not in a unique way) as \begin{equation} \label{e2.6} \mu=f+F+g_{t}+\mu_{s}^{+}-\mu_{s}^{-} \end{equation} with $f\in L^{1}(Q)$, $F\in L^{p'}(0,T;W^{-1,p'}(\Omega))$, $g_{t}\in L^p(0,T;V)$ and $\mu_{s}\perp p$-capacity. \subsection{Definition of renormalized solution} For any $k>0$, we define the truncation function $T_k:\mathbb{R}\to \mathbb{R}$ by \[ T_k(t)=\text{max}(-k,\text{min}(k,t)),\quad t\in \mathbb{R} \] Let us consider the space of all measurable functions, finite a.e. in $Q$ such that $T_k(u)$ belongs to $L^p(0,T;W^{1,p}_0(\Omega))$ for every $k>0$. We can see that every function $u$ in this space has a $\operatorname{cap}_{p}$-quasi continuous representative, that will always be identified with $u$. Moreover, there exists a measurable function $v:Q\to \mathbb{R}^{N}$, which is unique up to almost everywhere equivalence, such that $\nabla T_k(u)=v\chi_{\{ |u|0$, (see \cite[Lemma 2.1]{B6}). Hence it is possible to define a generalized gradient $\nabla u$ of $u$, setting $\nabla u=v$. If $u\in L^{1}(0,T;W^{1,1}_0(\Omega))$, this gradient coincide with the usual gradient in distributional sense. Let $T_k(t)$ be the Lipschitz continuous function $T_k:\mathbb{R}\to\mathbb{R}$, so that we can define the auxiliary functions \[ \Theta_n(s)=T_1(s-T_n(s)),\quad h_n(s)=1-(\Theta_n(s)),\quad S_n(s) =\int_0^{s}h_n(r)dr,\; \forall s\in \mathbb{R}. \] We are now in a position to introduce (following \cite{Pe2}) the notion of renormalized solution. To simplify the notation, let us define $v=u-g$, where $u$ is the solution and $g$ is the time-derivative part of $\mu_0$, and $\hat{\mu}_0=\mu-g_{t}-\mu_{s}=f-\operatorname{div}(G)$. \begin{definition}\label{def2.1} \rm Let $u_0\in L^{1}(\Omega)$, $\mu\in \mathcal{M}_b(Q)$. A measurable function $u$ is a renormalized solution of problem \eqref{e1.1} if there exists a decomposition $(f,G,g)$ of $\mu_0$ such that \begin{equation} \label{e2.7} \begin{gathered} v=u-g\in L^{q}(0,T;W^{1,q}_0(\Omega))\cap L^{\infty}(0,T;L^{1}(\Omega))\quad \forall q0, \end{gathered} \end{equation} and, for every $S\in W^{2,\infty}(\mathbb{R})$ such that $S'$ has compact support on $\mathbb{R}$, and $S(0)=0$, \begin{equation} \label{e2.8} \begin{aligned} &-\int_{\Omega}S(u_0)\varphi(0)\,dx -\int_0^{T}\langle \varphi_{t},S(v)\rangle\,dt + \int_QS'(v)a(t,x,u,\nabla u)\cdot\nabla \varphi\,dx\,dt\\ &+\int_QS''(v)a(t,x,u,\nabla u)\cdot\nabla v\varphi\,dx\,dt =\int_QS'(v)\varphi\,d\tilde{\mu}_0, \end{aligned} \end{equation} for any $\varphi\in X\cap L^{\infty}(Q)$ such that $\varphi_{t}\in X'+L^{1}(Q)$ and $\varphi(\cdot,T)=0$; for any $\psi\in C(\overline{Q})$ \begin{equation} \label{e2.9} \begin{aligned} \lim_{n\to +\infty}\frac{1}{n}\int_{\{ n\leq v< 2n\}}a(t,x,u,\nabla u) \cdot\nabla v \psi\,dx\,dt &=\int_Q\psi d\mu_{s}^{+},\\ \lim_{n\to +\infty}\frac{1}{n}\int_{\{ -2n< v\leq -n\}}a(t,x,u,\nabla u) \cdot\nabla v \psi\,dx\,dt&=\int_Q\psi d\mu_{s}^{-}, \end{aligned} \end{equation} \end{definition} \begin{remark} \label{rmk2.2} \rm Notice that, if $u$ is a renormalized solution of \eqref{e1.1}, then \begin{equation} \label{e2.10} \begin{aligned} &(S(u-g))_{t} - \operatorname{div} (a(t,x,u,\nabla u)S'(u-g)) + S''(u-g)a(t,x,u,\nabla u)\cdot\nabla (u-g) \\ &= S'(u-g)f + S''(u-g)G\cdot\nabla (u-g) - \operatorname{div} (GS'(u-g)) \end{aligned} \end{equation} is satisfied in the sense of distributions. Hence we can put as test functions not only functions in $C^{\infty}_0(Q)$ but also in $L^p(0,T;W^{1,p}_0(\Omega))\cap L^{\infty}(Q)$. \end{remark} \section{Statement of results} In what follows the variable $\epsilon$ will belong to a sequence of positive numbers converging to zero. Let $a_{\epsilon}:Q\times\mathbb{R}\times\mathbb{R}^{N}\to\mathbb{R}^{N}$ be a sequence of functions satisfying the hypothesis $H(c_0,c_1,c_2,b_0)$. Assume that there exists a function $a_0:Q\times\mathbb{R}\times\mathbb{R}^{N}\to \mathbb{R}^{N}$ satisfying the hypothesis $H(c_0,c_1,c_2,b_0)$, and such that \begin{equation} \label{e3.1} \lim_{\epsilon\to 0} a_{\epsilon}(t,x,s_{\epsilon},\zeta_{\epsilon}) = a_0(t,x,s,\zeta), \end{equation} for every sequence $(s_{\epsilon},\zeta_{\epsilon})\in \mathbb{R}\times \mathbb{R}^{N}$ which converges to $(s,\zeta)$ and for almost $(t,x)\in Q$. Fixed $\mu\in \mathcal{M}_b(Q)$, we consider a special type of approximating sequence $\mu_{\epsilon}$, defined as follows. \begin{definition}\label{def3.1} \rm Let $\mu\in \mathcal{M}_b(Q)$ be decomposed as $\mu=f+F+g_{t}+\mu_{s}^{+}-\mu_{s}^{-}$, with $f\in L^{1}(Q)$, and $F=-\operatorname{div}(G)$, $G\in (L^{p'}(Q))^{N}$, $g_{t}\in L^{p'}(0,T;W^{-1,p'}(\Omega))$. Let $(\mu_{\epsilon})$ be a sequence of measures in $\mathcal{M}_b(Q)$, we say that $(\mu_{\epsilon})$ has a splitting $(f_{\epsilon},F_{\epsilon},g^{t}_{\epsilon},\lambda_{\epsilon}^{\oplus}, \lambda_{\epsilon}^{\ominus})$ converging to $\mu$. If for every $\epsilon$ the measure $\mu_{\epsilon}$ can be decomposed as \begin{equation} \label{e3.2} \mu_{\epsilon}=f_{\epsilon}+F_{\epsilon}+g^{t}_{\epsilon} +\lambda_{\epsilon}^{\oplus}-\lambda_{\epsilon}^{\ominus}, \end{equation} and the following holds \begin{itemize} \item[(i)] $(f_{\epsilon})$ is a sequence of $C^{\infty}_{c}(Q)$ functions converging to $f$ weakly in $L^{1}(Q)$; \item[(ii)]$(G_{\epsilon})$ is a sequence of functions in $(C^{\infty}_{c}(Q))^{N}$ that converges to $g$ strongly in $(L^{p'}(Q))^{N}$; \item[(iii)] $(g_{\epsilon}^{t})$ is a sequence of functions in $(C^{\infty}_{c}(Q))^{N}$ that converges to $g_{t}$ in $L^p(0,T;V)$; \item[(iv)] $(\lambda_{\epsilon}^{\oplus})$ is a sequence of non-negative measures in $\mathcal{M}_b(Q)$ such that $\lambda_{\epsilon}^{\oplus}=\lambda_{\epsilon,0}^{1,\oplus}- \operatorname{div} (\lambda_{\epsilon,0}^{2,\oplus})+\lambda_{\epsilon,s}^{\oplus}$ with ($\lambda_{\epsilon,0}^{1,\oplus}\in L^{1}(Q)$, $\lambda_{\epsilon,0}^{2,\oplus}\in (L^{p'}(Q))^{N}$ and $\lambda_{\epsilon,s}^{\oplus}\in \mathcal{M}_{s}^{+}(Q)$) that converges to $\mu_{s}^{+}$ in the narrow topology of measures; \item[(v)] $(\lambda_{\epsilon}^{\ominus})$ is a sequence of non-negative measures in $\mathcal{M}_b(Q)$ such that $\lambda_{\epsilon}^{\ominus} =\lambda_{\epsilon,0}^{1,\ominus}- \operatorname{div} (\lambda_{\epsilon,0}^{2,\ominus})+\lambda_{\epsilon,s}^{\ominus}$ with ($\lambda_{\epsilon,0}^{1,\ominus}\in L^{1}(Q)$, $\lambda_{\epsilon,0}^{2,\ominus}\in (L^{p'}(Q))^{N}$ and $\lambda_{\epsilon,s}^{\ominus}\in \mathcal{M}_{s}^{+}(Q)$) that converges to $\mu_{s}^{-}$ in the narrow topology of measures. \end{itemize} Moreover, let $u_0^{\epsilon}\in C^{\infty}_0(\Omega)$ that approaches $u_0$ in $L^{1}(\Omega)$, notice that this approximation can be easily obtained via a standard convolution arguments and we can also assume \[ \| \mu_{\epsilon}\|_{L^{1}(Q)}\leq C|\mu|;\quad \| u_{0,\epsilon}\|_{L^{1}(\Omega)}\leq C\| u_0\|_{L^{1}(\Omega)}. \] \end{definition} \begin{remark}\label{rmk3.2} \rm Let us introduce the following function that we will often use in the following \[ H_n(r)=\chi_{[-n,n]}(r)+\frac{2n-|s|}{n}\chi_{\{ n<|s|\leq 2n\}}(r), \quad \overline{H}_n(r)=\int_0^{r}H_n(\tau)d\tau, \] and another auxiliary function introduced in terms of $H_n(s)$ \[ B_n(s)=1-H_n(s). \] \end{remark} \begin{proposition}\label{prop2.1} Let $v=u-g$ be a renormalized solution of problem \eqref{e1.1}. Then, for every, $k>0$, we have \[ \int_Q|\nabla T_k(v)|^pdx\,dt\leq C(k+1), \] where $C$ is a positive constant not depending on $k$. \end{proposition} For a proof of the above proposition see \cite[Proposition 2]{Pe2}. \begin{remark} \label{rmk3.4} \rm If we decompose the measures, $\mu_{\epsilon}$, $\lambda_{\epsilon}^{\oplus}$, $\lambda_{\epsilon}^{\ominus}$ respectively as $\mu_{\epsilon}=\mu_{\epsilon,0}+\mu_{\epsilon,s}$, $\lambda_{\epsilon}^{\oplus} =\lambda_{\epsilon,0}^{\oplus}+\lambda_{\epsilon,s}^{\oplus}$ $(\lambda_{\epsilon,0}^{\oplus}=\lambda_{\epsilon,0}^{1,\oplus} - \operatorname{div} (\lambda_{\epsilon,0}^{2,\oplus}))$, $\lambda_{\epsilon}^{\ominus}=\lambda_{\epsilon,0}^{\ominus} +\lambda_{\epsilon,s}^{\ominus}$ ($\lambda_{\epsilon,0}^{\ominus}=\lambda_{\epsilon,0}^{1,\ominus} - \operatorname{div} (\lambda_{\epsilon,0}^{2,\ominus})$), with $\mu_{\epsilon,0},\lambda_{\epsilon,0}^{\oplus}$, $\lambda_{\epsilon,0}^{\ominus}$ in $\mathcal{M}_0(Q)$, and $\mu_{\epsilon,s}$, $\lambda_{\epsilon,s}^{\oplus}$, $\lambda_{\epsilon,s}^{\ominus}$ in $\mathcal{M}_{s}(Q)$, then clearly $\lambda_{\epsilon,0}^{\oplus}$, $\lambda_{\epsilon,0}^{\ominus}$, $\lambda_{\epsilon,s}^{\oplus}$, $\lambda_{\epsilon,s}^{\ominus}$ are non-negative, $\mu_{\epsilon,0}=f_{\epsilon}+ F_{\epsilon}+g_{\epsilon} +\lambda_{\epsilon,0}^{\oplus}-\lambda_{\epsilon,0}^{\ominus}$ and $\mu_{\epsilon,s}=\lambda_{\epsilon,s}^{\oplus}-\lambda_{\epsilon,s}^{\ominus}$. In particular we have \begin{equation} \label{e3.3} 0\leq \mu_{\epsilon,s}^{+}\leq \lambda_{\epsilon,s}^{\oplus},\quad 0\leq \mu_{\epsilon,s}^{-}\leq \lambda_{\epsilon,s}^{\ominus}. \end{equation} \end{remark} We are interested in the asymptotic behaviour of a sequence of renormalized solutions $(u_{\epsilon})$ to the problem \begin{equation} \label{e3.4} \begin{gathered} (u_{\epsilon})_t -\operatorname{div}(a(t,x,u_{\epsilon},\nabla u_{\epsilon})) =\mu_{\epsilon} \quad \text{in }Q:=(0,T)\times\Omega, \\ u_{\epsilon}=0 \quad \text{on } (0,T)\times\partial \Omega,\\ u_{\epsilon}(0)=u_0 \quad \text{in }\Omega, \end{gathered} \end{equation} in the sense of Definition \ref{def2.1}. Our main result reads as follows. \begin{theorem}\label{thm3.5} Let $(a_{\epsilon}), a_0$ be functions satisfying $H(c_0,c_1,c_2,b_0)$ and \eqref{e3.1}. Let $\mu\in \mathcal{M}_b(Q)$ be decomposed as $f+F+g_{t}+\mu_{s}^{+}-\mu_{s}^{-}$, and let $(\mu_{\epsilon})$ a sequence of measures in $\mathcal{M}_b(Q)$ which have a splitting $(f_{\epsilon},F_{\epsilon},g_{\epsilon},\lambda_{\epsilon}^{\oplus}, \lambda_{\epsilon}^{\ominus})$ converging to $\mu$. Assume that $u_{\epsilon}$ is a renormalized solution of \eqref{e3.4}. Then there exists a subsequence, still denoted by $(u_{\epsilon})$, and a renormalized solution $u$ to the problem \begin{equation} \label{e3.5} \begin{gathered} u_t-\operatorname{div}(a_0(t,x,u,\nabla u))=\mu \quad \text{in }Q:=(0,T)\times\Omega,\\ u=0 \quad \text{on } (0,T)\times\partial \Omega,\\ u(0)=u_0 \quad\text{in }\Omega, \end{gathered} \end{equation} such that $(u_{\epsilon})$ converges to $u$ a.e. in $Q$, and $(v_{\epsilon})=(u_{\epsilon}-g_{\epsilon})$ converges to $v=u-g$ a.e. in $Q$. \end{theorem} \begin{remark} \label{rmk3.6} \rm The convergence of $u_{\epsilon}$ to $u$ is not merely pointwise. The kind of converges obtained are listed in Proposition \ref{prop5.2}, where the existence of the limit function $u$ is obtained. \end{remark} \begin{remark} \label{rmk3.7}\rm Let $z_{\nu}$ be a sequence of functions such that \begin{gather*} z_{\nu}\in W^{1,p}_0(\Omega)\cap L^{\infty}(\Omega),\quad \| z_{\nu}\|_{L^{\infty}(\Omega)}\leq k, \\ z_{\nu}\to T_k(u_0)\text{ a.e. in } \Omega \text{ as } \nu \text{ tends to infinity}, \\ \frac{1}{\nu}\| z_{\nu}\|_{W^{1,p}_0(\Omega)}^p\to 0 \text{ as $\nu$ tends to infinity}. \end{gather*} Then, for fixed $k>0$, and $\nu>0$, we denote by $(T_k(v))_{\nu}$ (Landes-time regularization of the truncate function $T_k(v)$ introduced in \cite{La} and used in several articles (see \cite{BDGO,BP,DO}) the unique solution of the problem \begin{gather*} \frac{dT_k(v)_{\nu}}{dt}=\nu(T_k(v)-T_k(v)_{\nu}) \quad \text{in the sense of distributions},\\ T_k(v)_{\nu}=z_{\nu} \quad \text{ in } \Omega, \end{gather*} therefore, $T_k(v)_{\nu}\in L^p(0,T;W^{1,p}_0(\Omega)\cap L^{\infty}(Q))$ and $\frac{dT_k(v)}{dt}\in L^p(0,T;W^{1,p}_0(\Omega))$, and it can be proved that, up to a subsequences, as $\nu$ diverges \begin{gather*} T_k(v)_{\nu}\to T_k(v)\quad \text{ strongly in $L^p(0,T;W^{1,p}_0(\Omega))$ and a.e. in }Q, \\ \| T_k(v)_{\nu}\|_{L^{\infty}(Q)}\leq k\quad \forall \nu>0 \end{gather*} Then choosing this approximation in parabolic case with fact that $(\mu_{\epsilon})$ approximates $\mu$ in the sense of Definition \ref{def3.1}. Hence we obtain, as consequence of the strong convergence of truncates the existence of renormalized solution of \eqref{e3.5} obtained as stated in the following theorem. \end{remark} \begin{theorem}\label{thm3.8} Let $a_0$ be a function satisfying $H(c_0,c_1,c_2,b_0)$ and $u_0\in L^{1}(\Omega)$, $\mu\in \mathcal{M}_b(Q)$. Then there exists a renormalized solution $u$ to problem \begin{gather*} u_t-\operatorname{div}(a_0(t,x,u,\nabla u))=\mu \quad \text{in }Q:=(0,T)\times\Omega, \\ u=0 \quad \text{on } (0,T)\times\partial \Omega,\\ u(0)=u_0 \quad \text{in }\Omega. \end{gather*} \end{theorem} \section{Some remarks on measures} We recall that a sequence $(\mu_{\epsilon})$ of non-negative measures converges to $\mu$ in the narrow topology if and only if $(\mu_{\epsilon}(Q))$ converges to $\mu(Q)$ and \eqref{e2.5} holds for every $\varphi\in C^{\infty}_{c}(Q)$. In particular a sequence $(\mu_{\epsilon})$ of non-negative measures converges to $\mu$ in the narrow topology if and only if \eqref{e2.5} holds for every $\varphi\in C_{c}(\overline{Q})$. The following lemma states a consequence result of the Dunford-pettis theorem. \begin{lemma}\label{lem4.1} Let $(\rho_{\epsilon})$ be a sequence in $L^{1}(Q)$ converging to $\rho$ weakly in $L^{1}(Q)$ and $(\sigma_{\epsilon})$ a bounded sequence in $L^{\infty}(Q)$ converging to $\sigma$ a.e. in $Q$. Then \[ \lim_{\epsilon\to 0} \int_Q\rho_{\epsilon}\sigma_{\epsilon}dx\,dt =\int_Q\rho\sigma\,dx\,dt \] \end{lemma} Next we need to localize some integrals near the support of $\mu_{s}\in \mathcal{M}_{s}(Q)$ (singular measure with respect to $p$-capacity). This will be done in terms of the following cut-off functions (see \cite[Lemma 5]{Pe2}). \begin{lemma}\label{lem4.2} Let $\mu_{s}$ be a measure in $\mathcal{M}_{s}(Q)$, and let $\mu_{s}^{+},\mu_{s}^{-}$ be respectively the positive and the negative part of $\mu_{s}$. Then for every $\delta>0$, there exists two functions $\psi_{\delta}^{+},\psi_{\delta}^{-}$ in $C^{1}_0(Q)$, such that the following hold \begin{itemize} \item[(i)] $0\leq \psi_{\delta}^{+}\leq 1$ and $ 0\leq \psi_{\delta}^{-}\leq 1$ on $Q$; \item[(ii)] $\lim_{\delta\to 0} \psi_{\delta}^{+} =\lim_{\delta\to 0} \psi_{\delta}^{-}=0$ strongly in $L^p(0,T;W^{1,p}_0(\Omega))$ and weakly\_* in $L^{\infty}(Q)$; \item[(iii)] $\lim_{\delta\to 0} (\psi_{\delta}^{+})_{t} =\lim_{\delta\to 0} (\psi_{\delta}^{-})_{t}=0$ strongly in $L^{p'}(0,T;W^{-1,p'}(\Omega))+L^{1}(Q)$; \item[(iv)] $\int_Q\psi_{\delta}^{-}d\mu_{s}^{+}\leq \delta$ and $\int_Q\psi_{\delta}^{+}d\mu_{s}^{-}\leq \delta$; \item[(v)] $\int_Q(1-\psi_{\delta}^{+}\psi_{\eta}^{+})d\mu_{s}^{+} \leq \delta+ \eta$ and $\int_Q(1-\psi_{\delta}^{-}\psi_{\eta}^{-})d\mu_{s}^{-}\leq\delta+\eta$ for all $\eta>0$. \end{itemize} \end{lemma} \begin{lemma}\label{lem4.3} Let $\mu_{s}$ be a measure in $\mathcal{M}_{s}(\Omega)$, decomposed as $\mu_{s}=\mu_{s}^{+}-\mu_{s}^{-},$ with $\mu_{s}^{+}$ and $\mu_{s}^{-}$ concentrated on two disjoint subsets $E^{+}$ and $E^{-}$ of zero $p$-capacity. Then, for every $\delta>0$, there exists two compact sets $K_{\delta}^{+}\subseteq E^{+}$ and $K_{\delta}^{-}\subseteq E^{-}$ such that \begin{equation} \label{e4.1} \mu_{s}^{+}(E^{+}\backslash K_{\delta}^{+})\leq \delta,\quad \mu_{s}^{-}(E^{-}\backslash K_{\delta}^{-})\leq \delta, \end{equation} and there exists $\psi_{\delta}^{+}$, $\psi_{\delta}^{-}\in C_0^{1}(Q)$, such that \begin{gather} \label{e4.2} \psi_{\delta}^{+}, \psi_{\delta}^{-}\equiv 1\quad \text{respectively on } K_{\delta}^{+}, K_{\delta}^{-}, \\ \label{e4.3} 0\leq \psi_{\delta}^{+}, \psi_{\delta}^{-} \leq 1, \\ \label{e4.4} \operatorname{supp}(\psi_{\delta}^{+})\cap\operatorname{supp} (\psi_{\delta}^{-})\equiv \emptyset. \end{gather} Moreover \begin{equation} \label{e4.5} \| \psi_{\delta}^{+}\|_{S}\leq \delta, \quad \| \psi_{\delta}^{-}\|_{S}\leq \delta, \end{equation} and, in particular, there exists a decomposition of $(\psi_{\delta}^{+})_{t}$ and a decomposition of $(\psi_{\delta}^{-})_{t}$ such that \begin{gather} \label{e4.6} \| (\psi_{\delta}^{+})_{t}^{1}\|_{L^{p'}(0,T;W^{-1,p'}(\Omega))} \leq \frac{\delta}{3},\quad \| (\psi_{\delta}^{+})_{t}^{2}\|_{L^{1}(Q)}\leq \frac{\delta}{3}, \\ \label{e4.7} \| (\psi_{\delta}^{-})_{t}^{1}\|_{L^{p'}(0,T;W^{-1,p'}(\Omega))} \leq \frac{\delta}{3},\quad \| (\psi_{\delta}^{-})_{t}^{2}\|_{L^{1}(Q)}\leq \frac{\delta}{3}, \end{gather} and both $\psi_{\delta}^{+}$ and $\psi_{\delta}^{-}$ converges to zero *\_weakly in $L^{\infty}(Q)$, in $L^{1}(Q)$, and up to subsequences, almost everywhere as $\delta$ vanishes. Moreover, if $\lambda^{\oplus}_{\epsilon}$ and $\lambda^{\ominus}_{\epsilon}$ are as in \eqref{e3.2} we have \begin{gather} \label{e4.8} \int_Q\psi_{\delta}^{-}d\lambda^{\oplus}_{\epsilon}=\omega(\epsilon,\delta),\quad \int_Q\psi_{\delta}^{-}d\mu_{s}^{+}\leq \delta, \\ \label{e4.9} \int_Q\psi_{\delta}^{+}d\lambda^{\ominus}_{\epsilon}=\omega(\epsilon,\delta),\quad \int_Q\psi_{\delta}^{+}d\mu_{s}^{-}\leq \delta, \\ \label{e4.10} \int_Q(1-\psi_{\delta}^{+}\psi_{\eta}^{+})d\lambda^{\oplus}_{\epsilon} =\omega(\epsilon,\delta,\eta),\quad \int_Q(1-\psi_{\delta}^{+}\psi_{\eta}^{+})d\mu_{s}^{+}\leq \delta+\eta, \\ \label{e4.11} \int_Q(1-\psi_{\delta}^{-}\psi_{\eta}^{-})d\lambda^{\ominus}_{\epsilon} =\omega(\epsilon,\delta,\eta),\quad \int_Q(1-\psi_{\delta}^{-}\psi_{\eta}^{-})d\mu_{s}^{-}\leq \delta+\eta. \end{gather} \end{lemma} For a proof of the above lemma see \cite[Lemma 5]{Pe2}. \begin{remark}\label{rmk4.4} \rm If $\lambda_{\epsilon}^{\oplus}$ and $\lambda_{\epsilon}^{\ominus}$ satisfy (iii) and (iv) of Definition \ref{def3.1}, respectively, and $\psi_{\delta}^{-}$ and $\psi_{\delta}^{+}$ are the functions defined in Lemma \ref{lem4.2}, as an easy consequence of the narrow convergence we obtain \begin{gather} \label{e4.12} \lim_{\delta\to 0} \lim_{\epsilon\to 0} \int_Q\psi_{\delta}^{-}d\lambda_{\epsilon}^{\oplus}= 0,\quad \lim_{\delta\to 0} \lim_{\epsilon\to 0} \int_Q\psi_{\delta}^{+}d\lambda_{\epsilon}^{\ominus}= 0, \\ \label{e4.13} \lim_{\eta \to 0} \lim_{\delta\to 0} \lim_{\epsilon\to 0} \int_Q(1-\psi_{\delta}^{+}\psi_{\eta}^{+})d\lambda_{\epsilon}^{\oplus}=0, \quad \lim_{\eta \to 0} \lim_{\delta\to 0} \lim_{\epsilon\to 0} \int_Q(1-\psi_{\delta}^{-}\psi_{\eta}^{-})d\lambda_{\epsilon}^{\ominus}=0. \end{gather} \end{remark} \section{Existence of a limit function} The following lemma is the main tool in order to establish the fundamental a priori estimates for the sequence $(u_{\epsilon})$. \begin{lemma}\label{lem5.1} Let $u,v$ as defined before, and assume that there exists $C>0$ such that \begin{equation} \label{e5.1} \begin{gathered} \| u\|_{L^{\infty}(0,T;L^{1}(\Omega))}\leq C;\quad \| v\|_{L^{\infty}(0,T;L^{1}(\Omega))}\leq C,\\ \int_Q|\nabla T_k(u)|^pdx\,dt\leq Ck;\quad \int_Q|\nabla T_k(v)|^pdx\,dt \leq C(k+1), \end{gathered} \end{equation} for every $k>0$. Then there exists $C=C(N,M,p)>0$ such that \begin{itemize} \item[(i)] $\operatorname{meas}\{ |u|\geq k\} \leq Ck^{-(p-1+\frac{p}{n})}$, $\operatorname{meas}\{ |v|\geq k\} \leq Ck^{-(p-1+\frac{p}{n})} $, \item[(ii)] $\operatorname{meas}\{ |\nabla u|\geq k\} \leq Ck^{-(p-\frac{N}{N+1})}$, $\operatorname{meas}\{ |\nabla v|\geq k\} \leq Ck^{-(p-\frac{N}{N+1})} $. \end{itemize} \end{lemma} \begin{proof} (i) We can improve this kind of estimate by using a suitable Gagliardo-Niremberg typeinequality (see \cite[Proposition 3.1]{DiB}) which asserts that is $w\in L^{q}(0,T;W^{1,q}_0(\Omega))\cap L^{\infty}(0,T;L^{2}(\Omega))$, with $q\geq 1$, $\sigma\geq 1$. Then $w\in L^{\sigma}(Q)$ with $\sigma=q\frac{N+\rho}{N}$ and \[ \int_Q|w|^{\sigma}dx\,dt \leq C\| w\|_{L^{\infty}(0,T;L^{\rho}(\Omega))}^{\frac{\rho q}{N}} \int_Q|\nabla w|^{q}dx\,dt. \] Indeed, in this way we obtain \[ \int_Q|T_k(u)|^{p+\frac{p}{N}}dx\,dt \leq Ck, \] and so, we can write \[ K^{p+\frac{p}{N}}meas \{ |u|\geq k\} \leq \int_{\{ |u|\geq k\}}|T_k(u)|^{p+\frac{p}{N}}\,dx\,dt \leq \int_Q|T_k(u)|^{p+\frac{p}{N}}\,dx\,dt \leq Ck, \] Then, \[ \operatorname{meas}\{ |u|\geq k\} \leq \frac{C}{k^{p-1+\frac{p}{N}}}. \] \noindent(ii) We are interested about a similar estimate on the gradients of functions $u$; let us emphasize that these estimates hold true. First of all, observe that \[ \operatorname{meas}\{ |\nabla u|\neq \lambda\} \leq \operatorname{meas} \{ |\nabla u|\neq \lambda;|u|\leq k\} + \operatorname{meas} \{ |\nabla u|\neq \lambda;|u|> k\} \] with regard to the first term in the right hand side, we have \begin{equation} \label{e5.2} \begin{gathered} \operatorname{meas}\{ |\nabla u|\neq \lambda;|u|\leq k\} \leq \frac{1}{\lambda^p}\int_{\{|\nabla u|\geq \lambda; |u|\leq k\}} |\nabla u|^pdx \\ \frac{1}{\lambda^p}\int_{\{|u|\leq k\}}|\nabla u|^pdx =\frac{1}{\lambda^p}\int_Q|\nabla T_k(u)|^pdx\leq \frac{Ck}{\lambda^p}; \end{gathered} \end{equation} while for the last term, thanks to (i), we can write \[ \operatorname{meas}\{ |\nabla u|\geq \lambda; |u|>k\} \leq \operatorname{meas}\{ |u|\geq k\}\leq \frac{\overline{C}}{K^{\sigma}}, \] with $\sigma=p-1+\frac{p}{N}$. So, finally, we obtain \[ \operatorname{meas}\{ |\nabla u|\geq \lambda\} \leq \frac{\overline{C}}{k^{\sigma}}+\frac{Ck}{\lambda^p}, \] and we obtain a better estimate by taking the minimum over $k$ of the right-hand side; the minimum is achieved for the value \[ k_0=\big(\frac{\sigma C}{\overline{C}}\big)^{\frac{1}{\sigma +1}} \lambda^{\frac{p}{\sigma+1}} \] and so we obtain the desired estimate \[ \operatorname{meas}\{ |\nabla u|\geq \lambda\} \leq C\lambda^{-\gamma} \] with $\gamma=p(\frac{\sigma}{\sigma+1})=\frac{Np+p-N}{N+1}=p-\frac{N}{N+1}$. Then, we found that $u$ (resp $v$) is uniformally bounded in the Marcinkiewicz space $\mathcal{M}^{p-1+\frac{p}{N}}(Q)$ and $\nabla u$ (resp $\nabla v$) is equibounded in $\mathcal{M}^{\gamma}(Q)$, with $\gamma=p-\frac{N}{N+1}$. \end{proof} From now we always assume that $(a_{\epsilon})$, $a_0$ are functions satisfying $H(c_0,c_1,c_2,b_0)$ and \eqref{e3.1}, that $\mu\in\mathcal{M}_b(Q)$ is decomposed as $f+F+g_{t}+\mu_{s}$, $f\in L^{1}(Q)$, $F\in L^{p'}(0,T;W^{-1,p'}(\Omega))$, $g_{t}\in L^p(0,T;V)$, $\mu_{s}\in \mathcal{M}_{s}(Q)$, and that $(\mu_{s})$ is a sequence of measure in $\mathcal{M}_b(Q)$, which have a splitting $(f_{\epsilon},F_{\epsilon},g_{\epsilon},\lambda_{\epsilon}^{\oplus}, \lambda_{\epsilon}^{\ominus})$ converging to $\mu$. We shall denotes by $u_{\epsilon}$ a renormalized solution of \eqref{e3.4} with $\mu_{\epsilon}$ as datum. Hence it satisfies: \begin{equation} \label{e5.3} \begin{aligned} &\int_0^{T}\langle(v_{\epsilon})_{t},\varphi \rangle\,dt +\int_Qa(t,x,u_{\epsilon},\nabla u_{\epsilon})\cdot\nabla \varphi\,dx\,dt\\ &= \int_Q f_{\epsilon}\varphi\,dx\,dt +\int_0^{T}\langle F_{\epsilon},\varphi\rangle\,dx\,dt+\int_Q\varphi \;d(\lambda_{\epsilon}^{\oplus}-\lambda_{\epsilon}^{\ominus}), \end{aligned} \end{equation} for all $\varphi\in L^p(0,T;W^{1,p}_0(\Omega))\cap L^{\infty}(Q)$, $\varphi_{t}\in L^{p'}(0,T;W^{-1,p'}(\Omega))$, with $\varphi(T,0)=0$. As a first step, we find a function $u\in L^{\infty}(0,T;L^{1}(\Omega))$ such that $T_k(u)\in L^p(0,T;W^{1,p}_0(\Omega))$ which is the limit, up to a subsequence, of $(u_{\epsilon})$ in suitable topologies. \begin{proposition}\label{prop5.2} Let $\mu_{\epsilon}\in \mathcal{M}_b(Q)$, $(u_{0,\epsilon})\in L^{1}(\Omega)$, with $\text{sup}_{\epsilon}|\mu_{\epsilon}(Q)|< \infty$ and $\| u_{0,\epsilon}\|_{1,\Omega}< \infty$. Let $(u_{\epsilon})$ be a sequence of renormalized solutions of \eqref{e3.4}, and let $v_{\epsilon}=u_{\epsilon}-g_{\epsilon}$. Then there exists $C> 0$ such that \begin{equation} \label{e5.4} \begin{gathered} \| u_{\epsilon}\|_{L^{\infty}(0,T;L^{1}(\Omega))} \leq C, \quad \int_Q|\nabla T_k(u_{\epsilon})|^pdx\,dt \leq Ck, \\ \| v_{\epsilon}\|_{L^{\infty}(0,T;L^{1}(\Omega))} \leq C,\quad \int_Q|\nabla T_k(v_{\epsilon})|^pdx\,dt \leq C(k+1), \end{gathered} \end{equation} for every $\epsilon$ and for every $k>0$. Moreover there exists a subsequence, still denoted by $u_{\epsilon}$ (resp $v_{\epsilon}$) and a measurable function $u$ (resp $v$) such that the following convergence hold. \begin{itemize} \item[(i)] $u_{\epsilon}$ (resp $(v_{\epsilon})$) converges to $u$ (resp $v$) a.e. in $Q$; \item[(ii)] $u$ (resp $v$) belongs to $L^{\infty}(0,T;L^{1}(\Omega))$ and for every $k>0$, the sequence $(T_k(u_{\epsilon}))$ (resp $T_k(v_{\epsilon})$) converges to $T_k(u)$ (resp $T_k(v)$) $\in L^p(0,T;W^{1,p}_0(\Omega))$ in the weak topology of $L^p(0,T;W^{1,p}_0(\Omega))$; \item[(iii)] $\nabla u_{\epsilon}$ (resp $(\nabla v_{\epsilon})$) converges to $\nabla u$ (resp $\nabla v$) a.e. in $Q$; \item[(iv)]$a_{\epsilon}(t,x,u_{\epsilon},\nabla u_{\epsilon})$ converges to $a_0(t,x,u,\nabla u)$ in the strong topology of the space $L^{q}(0,T;W^{1,q}_0(\Omega))$ for every $q< p-\frac{N}{N+1}$, while $a_{\epsilon}(t,x,u,\nabla T_k(u_{\epsilon}))$ converges to $a_0(t,x,u,\nabla T_k(u))$ in the weak topology of $(L^{p'}(Q))^{N}$ for every $k>0$. \end{itemize} \end{proposition} \begin{proof} \textbf{Step 1.} a priori estimates. Let us choose $T_k(u_{\epsilon})$ as test function in \eqref{e5.3} and we integrate in $]0,t[$ to obtain \begin{equation} \label{e5.5} \begin{aligned} &\int_{\Omega}\Theta_k(u_{\epsilon}(t))\,dx + \int_0^{t}\int_{\Omega}a(t,x,u_{\epsilon},\nabla u_{\epsilon}) \nabla T_k(u_{\epsilon})\,dx\,dt \\ &=\int_0^{t}\int_{\Omega}T_k(u_{\epsilon})d\mu_{\epsilon} +\int_{\Omega}\Theta_k(u_{0,\epsilon})\,dx \end{aligned} \end{equation} using \eqref{e3.1} and the fact that $\| u_{0,\epsilon}\|_{L^{1}(\Omega)}$ and $\| \mu_{\epsilon}\|_{L^{1}(Q)}$ are bounded: \[ \int_{\Omega}\Theta_k(u_{\epsilon})(t)\,dx +\int_0^{t}\int_{\Omega}\vert \nabla T_k(u_{\epsilon})\vert^pdx\,dt \leq Ck \] Since $\Theta_k(s)\geq 0$ and $\vert \Theta_1(s)\vert \geq \vert s\vert-1$, we obtain \[ \int_{\Omega}\vert u_{\epsilon}(t)\vert\,dx + \int_0^{t}\int_{\Omega}\vert \nabla T_k(u_{\epsilon})\vert^pdx\,dt\leq C(k+1), \quad \forall k>0,\forall t\in [0,T]. \] Taking the supremum on $(0,T)$. As a consequence we obtain the estimate of $u_{\epsilon}$ in $L^{\infty}(0,T;L^{1}(\Omega))$ \[ \| u_{\epsilon}\|_{L^{\infty}(0,T;L^{1}(\Omega))}\leq C, \] We repeat here the same argument to get the estimate on $v_{\epsilon}$: let us choose $T_k(v_{\epsilon})$ as test function in \eqref{e5.3}. by integration by parts (recall that $g_{\epsilon}$ has compact support in $Q$, so that $(v_{\epsilon}(0)=u_{\epsilon}(0)=u_{0,\epsilon})$) and using \eqref{e3.1} \begin{align*} &\int_{\Omega}\Theta(v_{\epsilon})(t)\,dx +\alpha\int_0^{t}\int_{\Omega}|\nabla u_{\epsilon}|^p \chi_{\{ |v_{\epsilon}\leq k|\}}\,dx\,ds\\ &\leq \int_{\Omega}\Theta_k(u_{0,\epsilon})\,dx +\int_Qf_{\epsilon}T_k(v_{\epsilon})\,dx\,dt +\int_0^{t}\int_{\Omega}G_{\epsilon}\cdot\nabla u_{\epsilon} \chi_{\{|v_{\epsilon}\leq k|\}}dx ds\\ &\quad-\int_0^{t}\int_{\Omega}G_{\epsilon}\cdot\nabla g_{\epsilon} \chi_{\{ |v_{\epsilon}\leq k|\}}dx ds +\int_0^{t}\int_{\Omega} a(s,x,u_{\epsilon},\nabla u_{\epsilon}) \nabla g_{\epsilon} \chi_{\{|v_{\epsilon}|\leq k\}}ds ds\\ &\quad +\int_QT_k(v_{\epsilon})d\lambda_{\epsilon}^{\oplus} -\int_Q T_k(v_{\epsilon})d\lambda_{\epsilon}^{\ominus}. \end{align*} thanks to \eqref{e3.2} and young's inequality, \begin{align*} &\int_{\Omega}\Theta(v_{\epsilon})(t)\,dx +\frac{\alpha}{2}\int_0^{t}\int_{\Omega}|\nabla u_{\epsilon}|^p \chi_{\{ |v_{\epsilon}\leq k|\}}\,dx\,ds \\ &\leq \int_Q|f_{\epsilon}|dx\,dt +C\int_Q|G_{\epsilon}|^{p'}dx\,dt + C\int_Q|\nabla g_{\epsilon}|^pdx\,dt \\ &\quad + C\int_Q|b(t,x)|^{p'}dx\,dt + k\int_{\Omega}|u_{0,\epsilon}|dx +k\int_Qd\lambda_{\epsilon}^{\oplus} + k\int_Qd\lambda_{\epsilon}^{\ominus}. \end{align*} Using that $G_{\epsilon}$ is bounded in $L^{p'}(Q)$, $g_{\epsilon}$ is bounded in $L^p(0,T;W^{1,p}_0(\Omega))$, $f_{\epsilon}$, $\lambda_{\epsilon}^{\oplus}$ and $\lambda_{\epsilon}^{\ominus}$ are bounded in $L^{1}(Q)$ and $u_{0,{\epsilon}}$ is bounded in $L^{1}(\Omega)$, we have \[ \int_{\Omega}\Theta_1(v_{\epsilon})\,dx \leq C\quad \forall t\in [0,T], \] In this way the same estimate of $u_{\epsilon}$ follows for $v_{\epsilon}$ in $L^{\infty}(0,T;L^{1}(\Omega))$: \begin{gather*} \| v_{\epsilon}\|_{L^{\infty}(0,T;L^{1}(\Omega))}\leq C, \\ \int_Q|\nabla u_{\epsilon}|^p\chi_{\{|v_{\epsilon}|\leq k\}}dx\,dt\leq C(k+1), \end{gather*} which yields that $T_k(v_{\epsilon})$ is bounded in $L^p(0,T;W^{1,p}_0(\Omega))$ for any $k>0$ (recall that $g_{\epsilon}$ itself is bounded in $L^p(0,T;W^{1,p}_0(\Omega))$). Then \[ \int_Q|\nabla T_k(v_{\epsilon})|^pdx\,dt\leq C(k+1). \] \noindent\textbf{Step 2.} Up to a subsequence, $u_{\epsilon}$ is a Cauchy sequence in measure. We are going to prove now that, up to subsequences, $u_{\epsilon}$ converges almost everywhere in $Q$ towards a measurable function $u$. Lemma \ref{lem5.1} gives the usual estimates for parabolic equation with measure data, that is to say $u_{\epsilon}$ is bounded in $L^{q}(0,T;W^{1,q}_0(\Omega))$ for every $qk\}=0 \quad \text{uniformly with respect to } u. \] From \eqref{e5.4} we have that $T_k(u_{\epsilon})$ is bounded in $L^p(0,T;W^{1,p}_0(\Omega))$ for every $k>0$. Now, if we multiply the approximating equation by $\mathcal{T}'_k(v_{\epsilon})$, where $\mathcal{T}_k(s)$ is a $C^{2}(\mathbb{R})$, nondecreasing function such that $\mathcal{T}_k(s)=s$ for $|s|\leq \frac{k}{2}$ and $\mathcal{T}_k(s)=k$ for $|s|>k$, we obtain \begin{align*} &(\mathcal{T}_k(v_{\epsilon}))_{t} - \operatorname{div}(a(t,x,u_{\epsilon}, \nabla u_{\epsilon})\mathcal{T}'_k(v_{\epsilon})) + a(x,t,u_{\epsilon},\nabla u_{\epsilon})\cdot\nabla v_{\epsilon} \mathcal{T}''_k(v_{\epsilon})\\ &=\mathcal{T}'_k(v_{\epsilon})f_{\epsilon}+\mathcal{T}''_k(v_{\epsilon}) G_{\epsilon}\cdot\nabla v_{\epsilon}- \operatorname{div} (G_{\epsilon}\mathcal{T}'_k(v_{\epsilon}))+(\lambda_{\epsilon}^{\oplus} -\lambda_{\epsilon}^{\ominus})\mathcal{T}'_k(v_{\epsilon}). \end{align*} in the sense of distributions. This implies, thanks to the last equality and to the fact that $\mathcal{T}'_k$ has compact support, that $\mathcal{T}_k(v_{\epsilon})$ is bounded in $L^p(0,T;W^{1,p}_0(\Omega))$ while its time derivative $(\mathcal{T}_k(v_{\epsilon}))_{t}$ is bounded in $L^p(0,T;W^{-1,p'}(\Omega))+L^{1}(Q)$, hence a classical compactness result (see \cite{Si}) allows us to conclude that $\mathcal{T}_k(v_{\epsilon})$ is compact in $L^{2}(Q)$. Thus for a subsequence, it also converges in measure, and almost everywhere in $Q$. Since we have, for $\sigma>0$, \begin{align*} &\operatorname{meas}\{ (x,t): |v_n-v_{m}|>\sigma\} \\ &\leq \operatorname{meas}\{ (x,t): |v_n|>\frac{k}{2}\} +\operatorname{meas}\{ (x,t): |v_n|>\frac{k}{2}\} \\ &\quad +\operatorname{meas}\{ (x,t): |\mathcal{T}_k(v_n)-\mathcal{T}_k(v_{m})|>\sigma,\} \end{align*} by \eqref{e5.4} for every fixed $\epsilon>0$ we can choose $\overline{k}$ large enough to have \begin{equation} \label{e5.6} \operatorname{meas}\{ (x,t): |v_n-v_{m}|>\sigma\} \leq \operatorname{meas}\{ (x,t):|\mathcal{T}_k(v_n) -\mathcal{T}_{\overline{k}}(v_{m})|>\sigma\}+\epsilon, \end{equation} for all $n,m\in \mathbb{N}$. The fact that $\mathcal{T}_k(v_{\epsilon})$ converges in measure for every $k>0$ implies, using \eqref{e2.8}, that, up to subsequences, $v_{\epsilon}$ also converges in measure and almost everywhere in $Q$. In particular, we have found out that there exists a measurable function $v$ in $L^{\infty}(0,T;L^{1}(\Omega))\cap L^{q}(0,T;W^{1,q}_0(\Omega))$ for every $q0$, and for a subsequences, not relabeled, \[ T_k(v_{\epsilon})\to T_k(v)\text{ weakly in }L^p(0,T;W^{1,p}_0(\Omega)), \text{ strongly in }L^p(Q)\text{ and a.e. in }Q. \] We deduce that \[ v_{\epsilon}\to v \text{ a.e. in }Q, \] and since $g_{\epsilon}$ strongly converges to $g$ in $L^p(0,T;W^{1,p}_0(\Omega))$, there exists a measurable function $u$ such that \[ u_{\epsilon}\to u\text{ a.e. in }Q, \] The estimate \eqref{e5.4} also imply that $u\in L^{\infty}(0,T;L^{1}(\Omega))$. Indeed, using Fatou's Lemma on the first term of the left-hand of \[ \int_{\Omega}\vert u_{\epsilon}(t)\vert\,dx + \int_0^{t}\int_{\Omega}\vert \nabla T_k(u_{\epsilon})\vert^pdx\,dt\leq C(k+1), \quad \forall k>0,\forall t\in [0,T]. \] where \[ T_k(u_{\epsilon})\rightharpoonup T_k(u)\quad \text{weakly in }L^p(0,T;W^{1,p}_0(\Omega)) \] and in addition \begin{equation} \label{e5.7} \int_Q|\nabla T_k(u)|^pdx\,dt \leq Ck,\quad\int_Q|\nabla T_k(v)|^pdx\,dt \leq C(k+1), \end{equation} that is property (ii) holds. \smallskip \noindent\textbf{Step 3.} $\nabla u_{\epsilon}$ is a Cauchy sequence in measure. Let us show that $\nabla u_{\epsilon}$ is a Cauchy sequence in measure, which will yields $\nabla u_{\epsilon}\to \nabla u$ almost everywhere, for a convenient subsequence. Given $\delta>0$ for every $\eta>0$ and $k>0$ one has \begin{equation} \label{e5.8} \begin{aligned} &\{ (t,x), |\nabla u_n-\nabla u_{m}|\geq \delta\} \\ &\subseteq \{ (t,x),|u_n|>k\}\cup\{ (t,x),|u_{m}|>k\}\\ &\quad \cup \{ (t,x),|\nabla u_n|>k\}\cup \{ (t,x),|\nabla u_{m}|>k\} \cup\{ (t,x),|u_n-u_{m}|>\eta\}\\ &\quad \cup\big\{(t,x),|\nabla u_n-\nabla u_{m}|\geq \delta, |u_n\leq k|, |\nabla u_n|\leq k,\\ &\quad\quad |u_n|\leq k, |\nabla u_{m}|\leq k, |u_n-u_{m}|\leq \eta \big\}. \end{aligned} \end{equation} We will denote $A_1$ to $A_6$ the six sets of the right hand side. One could remark, in the sequel of the proof, that only the upper bound of the measure of $A_6$ uses the equation of which $u_n$ and $u_{m}$ are solutions. The other bounds use the boundedness of $(u_n)$ and $(\nabla u_n)$. Let us bound $\operatorname{meas}(A_1)$ and $\operatorname{meas}(A_2)$, we have \[ k\operatorname{meas}(A_1)\leq \int_{A_1}|\nabla u_n|dx\,dt \leq \int_0^{T}\int_{\Omega}|\nabla u_n|\,dx\,dt \] hence \[ \operatorname{meas}(A_1)\leq \frac{1}{k}\int_0^{T}\int_{\Omega}|\nabla u_n|dx\,dt \leq \frac{C}{k}\leq \varepsilon, \] for $k$ large enough, because $(\nabla u_n)$ is bounded in $L^{q}((0,T)\times\Omega)$ for $q0$ for $\zeta_1-\zeta_2\neq 0$. Since the set of $(\zeta_1,\zeta_2)$ such that: $\{ (t,x),|s|\leq k, |\zeta_1|\leq k,|\zeta_2|\leq k$ and $|\zeta_1-\zeta_2|\geq \delta\}$ is compact and $a$ is continuous with respect to $\zeta$ for almost all $t$ and $x$, $[a(t,x,s,\zeta_1)-(a(t,x,s,\zeta_2)](\zeta_1-\zeta_2)$ reaches on this compact its minimum that we will denotes $\gamma(t,x)$, and that verifies $\gamma(t,x)>0$ a.e. Since $\gamma(t,x)>0$ a.e., there exists $\epsilon'>0$ such that, for all measurable set $A\subset(0,T)\times\Omega$, \[ \int_{A}\gamma \leq \varepsilon'\Longrightarrow \operatorname{meas}(A)\leq \varepsilon \] hence, to obtain $\operatorname{meas}(A_4)\leq \varepsilon$, it is sufficient to show that \begin{equation} \label{e5.9} \int_{A_4}\gamma \leq \varepsilon' \end{equation} By definition of $\gamma$ and $A_4$, we have \[ \int_{A_4}\gamma \leq\int_{A_4}(a(t,x,u_n,\nabla u_{m})-a(t,x,u_{m},\nabla u_{m})) (\nabla u_n-\nabla u_{m})\chi_{\{|u_n-u_{m}|\leq \eta\}}. \] Moreover the term to be integrated is non negative and $\nabla T_{\eta}(u_n-u_{m})=(\nabla u_n-\nabla u_{m})\chi_{\{|u_n-u_{m}|\leq \eta\}}$, hence we have \[ \int_{A_4}\gamma \leq \int_0^{T}(a(t,x,u_n,\nabla u_n)-a(t,x,u_{m},\nabla u_{m})) \cdot\nabla T_{\eta}(u_n-u_{m}), \] if one chooses $\varphi=T_{\eta}(u_n-u_{m})\in L^p(0,T;W^{1,p}(\Omega)) \cap L^{\infty}(0,T;L^{1}(\Omega))$, which satisfies $T_{\eta}(u_n-u_{m})_{t}\in L^{p'}(]0,T[;W^{-1,p'}(\Omega))$, in equation in the sense of distributions written successively with $u_n$ and $u_{m}$ one gets \begin{align*} &\int_0^{T}\langle (u_n-u_{m})_{t},T_{\eta}(u_n-u_{m})\rangle\\ &+\int_0^{T}\int_{\Omega}(a(t,x,u_n,\nabla u_n)-a(t,x,u_{m},\nabla u_{m}))\nabla T_{\eta}(u_n-u_{m})\\ &=\int_0^{T}\int_{\Omega}(\mu_n-\mu_{m})T_{\eta}(u_n-u_{m}). \end{align*} that is (using $\Theta_{\eta}$ the primitive of $T_{\eta}$) \begin{align*} &\int_{\Omega}\Theta_{\eta}(u_n-u_{m})(T)-\int_{\Omega}\Theta_{\eta}(u_n-u_{m})(0)\\ &+\int_0^{T}\int_{\Omega}(a(t,x,u_n,\nabla u_n)-a(t,x,u_{m},\nabla u_{m})) \nabla T_{\eta}(u_n-u_{m})\\ &=\int_0^{T}\int_{\Omega}(\mu_n-\mu_{m})T_{\eta}(u_n-u_{m}) \end{align*} Since the first term is non-negative ($\Theta_{\eta}(x)\geq 0$), and $\Theta_{\eta}(x)\leq \eta|x|$ one has \begin{align*} &\int_0^{T}\int_{\Omega}(a(t,x,u_n,\nabla u_n)-a(t,x,u_{m},\nabla u_{m})) \cdot\nabla T_{\eta}(u_n-u_{m})\\ &\leq \eta\int_0^{T}\int_{\Omega}|\mu_n-\mu_{m}|+\eta\int_{\Omega}|u_0^n-u_0^{m}| \\ &\leq 2\eta(\| \mu(Q)\|+\| u_0\|_{1,\Omega})\,. \end{align*} Then for $\eta$ small enough, one has $\int_{A_4}\gamma\leq\varepsilon'$ and thus $\operatorname{meas}(A_4)\leq\varepsilon$ and therefore for all $n,m\geq n_0$ we have \[ \operatorname{meas}(\{ |(\nabla u_n-\nabla u_{m})(x)|\geq\delta\}) \leq 4\varepsilon, \] thus, we obtain that $\nabla u_{\epsilon}$ is a Cauchy sequence in measure. Passing to a subsequence, we assume that \[ \nabla u_{\epsilon}\to \nabla u\quad \text{almost everywhere in }Q. \] Similarly, we obtain the convergence a.e of $v_{\epsilon}$, this gives \[ \nabla v_{\epsilon}\to\nabla v \text{ almost everywhere in }Q. \] that is property (iii) holds. It remains to prove (iv). By \eqref{e5.5}, Lemma \ref{lem5.1}, and \eqref{e2.2}, $a(t,x,u_{\epsilon},\nabla u_{\epsilon})$ is bounded in $L^{q}(0,T;W^{1,q}_0(\Omega))$ for every $q0$ the sequence $(T_k(u_{\epsilon}))$ converges strongly in $L^p(0,T;W^{1,p}_0(\Omega))$ to $T_k(u)$ as $\epsilon$ goes to 0. \end{proposition} \begin{proof} It is sufficient to follow the lines of the long and not easy proof of the same result, for a fixed operator independent of $u$, for the elliptic case in \cite[sections 5--8]{DMOP}, for the parabolic case in \cite[section 7]{Pe2}. The assumptions on $a_{\epsilon}$ allow to obtain some estimates for varying operators explicitly depending on $u$. For any $\delta,\eta >0$, let $\psi_{\delta}^{+},\psi_{\eta}^{+},\psi_{\delta}^{-}$ and $\psi_{\eta}^{-}$ as in Lemma \ref{lem4.3} and let $E^{+}$ and $E^{-}$ be the sets where, respectively, $\mu_{s}^{+},\mu_{s}^{-}$ are concentrated; setting \[ \Phi_{\delta ,\eta}=\psi_{\delta}^{+}\psi_{\eta}^{+}+\psi_{\delta}^{-}\psi_{\eta}^{-}. \] Suppose that, the estimate near $E$, \begin{equation} \label{e6.2} I_1= \int_{\{|v_{\epsilon}|\leq k\}}\Phi_{\delta,\eta} a(x,t,u_{\epsilon},\nabla u_{\epsilon})\cdot\nabla (v_{\epsilon} -T_k(v)_{\nu})\leq \omega (\epsilon,\nu,\delta,\eta), \end{equation} and far from $E$, \begin{equation} \label{e6.3} I_2=\int_{\{|v_{\epsilon}|\leq k\}}(1-\Phi_{\delta,\eta}) a(t,x,u_{\epsilon},\nabla u_{\epsilon})\cdot\nabla (v_{\epsilon} -T_k(v)_{\nu})\leq \omega (\epsilon,\nu,\delta,\eta). \end{equation} Putting these statements together we obtain \begin{equation} \label{e6.4} \limsup_{ \nu\to 0\,,\epsilon\to 0} \int_{\{|v_{\epsilon}|\leq k\}}a(t,x,u_{\epsilon},\nabla u_{\epsilon}) \cdot\nabla (v_{\epsilon}-T_k(v)_{\nu}) \leq 0, \end{equation} so that using the convergence of $(T_k(v)_{\nu})$ to $T_k(v)$ in $X$ we deduce \begin{equation} \label{e6.5} \limsup_{\epsilon\to 0} \int_{\{|v_{\epsilon}|\leq k\}}a(t,x,u_{\epsilon}, \nabla u_{\epsilon})\cdot\nabla (v_{\epsilon}-T_k(v))\leq 0, \end{equation} since by the weak convergence of $T_k(v_{\epsilon})$ to $T_k(v)$ in $X$, Proposition \ref{prop5.2} implies that \begin{equation} \label{e6.6} \int_{\{ |v_{\epsilon}|\leq k\}}a(t,x,u,\nabla (T_k(v)+g_{\epsilon})) \cdot\nabla (T_k(v_{\epsilon})-T_k(v))=\omega(\epsilon). \end{equation} then we obtain \[ \int_{\{ |v_{\epsilon}|\leq k \}}(a(t,x,u_{\epsilon},\nabla u_{\epsilon}) - a(t,x,u,\nabla (T_k(v)+g_{\epsilon})))\cdot\nabla (u_{\epsilon} -(T_k(v)+g_{\epsilon}))=\omega(\epsilon). \] we also have, using the convergence of $\nabla u_{\epsilon}$ to $\nabla u$ a.e. in $Q$ \begin{equation} \label{e6.7} (a(t,x,u_{\epsilon},\nabla u_{\epsilon}))\rightharpoonup a(t,x,u,\nabla u) \quad \text{in }(L^{p'}(Q))^{N}, \end{equation} then we obtain \[ \limsup_{\epsilon\to 0} \int_Q a(t,x,u_{\epsilon},\nabla u_{\epsilon}) \cdot\nabla T_k(v_{\epsilon})\leq \int_Q a(t,x,u,\nabla u)\cdot\nabla T_k(v). \] so that by Proposition \ref{prop5.2}, since $(a(t,x,u_{\epsilon},\nabla (T_k(v_{\epsilon}+g_{\epsilon}))$ converges weakly in $(L^{p'}(Q))^{N}$ to some $F_k$, it follows that $F_k=a(t,x,u,\nabla(T_k(u)+g))$. We get \begin{align*} &\limsup_{\epsilon\to 0} \int_Qa(t,x,u_{\epsilon},\nabla (T_k(v_{\epsilon}) +g_{\epsilon}))\cdot\nabla (T_k(v_{\epsilon})+g_{\epsilon})\\ &\leq \limsup_{\epsilon\to 0} \int_Qa(t,x,u_{\epsilon},\nabla v_{\epsilon}) \cdot\nabla T_k(v_{\epsilon}) + \limsup_{\epsilon\to 0} \int_Qa(t,x,\nabla (T_k(v_{\epsilon}) +g_{\epsilon}))\cdot\nabla g_{\epsilon}\\ &\leq \int_Qa(t,x,u,\nabla (T_k(v)+h))\cdot\nabla (T_k(v)+g). \end{align*} We finally deduce \begin{equation} \label{e6.8} (T_k(v_{\epsilon}))\quad \text{converges to $T_k(v)$ strongly in $X$ for all } k>0. \end{equation} \end{proof} The next Lemma is devoted to establish the preliminary essential estimate. \begin{lemma}\label{lem6.2} Near $E$ we have the estimate \[ I_1=\int_{\{ |v_{\epsilon}|\leq k\}} \Phi_{\delta,\eta}a(t,x,u_{\epsilon}, \nabla u_{\epsilon})\cdot\nabla (v_{\epsilon}-T_k(v)_{\nu}) \leq \omega (\epsilon,\nu,\delta,\eta). \] \end{lemma} \begin{proof} We have \[ I_1=\int_Q\Phi_{\delta,\eta}a(t,x,u_{\epsilon},\nabla u_{\epsilon}) \cdot\nabla T_k(v_{\epsilon})- \int_{\{|v_{\epsilon}|\leq k\}} \Phi_{\delta,\eta} a(t,x,u_{\epsilon},\nabla u_{\epsilon}) \cdot\nabla (T_k(v))_{\nu}. \] so that, from Proposition \ref{prop5.2} (iv) and since $a(t,x,u_{\epsilon},\nabla T_k(v_{\epsilon})+g_{\epsilon})\nabla T_k(v)_{\nu}$ converges weakly in $L^{1}(Q)$ to $F_k\nabla (T_k(v))_{\nu}$, $\chi_{\{ |v_{\epsilon}|\leq k\}}$ converges to $\chi_{\{ |v|\leq k\}}$ a.e in $Q$, $\Phi_{\delta,\eta}$ converges to 0 a.e. in $Q$ as $\delta\to 0$ and $\Phi_{\delta,\eta}$ takes its values in $[0,1]$, using Lemma \ref{lem4.1}, we have the first integral \begin{align*} &\int_{\{ |v_{\epsilon}|\leq k\}}\Phi_{\delta,\eta}a(t,x,u_{\epsilon}, \nabla u_{\epsilon})\cdot\nabla (T_k(v))_{\nu} \\ &=\int_Q \chi_{\{|v_{\epsilon}|\leq k\}}\Phi_{\delta,\eta}a(t,x,u_{\epsilon}, \nabla (T_k(v_{\epsilon})+g_{\epsilon}))\cdot\nabla (T_k(v))_{\nu}\\ &=\int_Q\chi_{\{ |v|\leq k\}}\Phi_{\delta,\eta}F_k\cdot\nabla (T_k(v))_{\nu} +\omega(\epsilon)\\ &=\omega(\epsilon,\nu,\delta). \end{align*} To obtain the second integral, we set, for any $n>k>0$, and any $s\in \mathbb{R}$ \[ \hat{S}_{n,k}(s)=\int_0^{s}(k-T_k(r))H_n(r)dr \] where $H_n$ is defined at Remark \ref{rmk3.2}. We take $(S,\varphi)=(\hat{S}_{n,k},\psi_{\delta}^{+}\psi_{\eta}^{+})$ as test function in \eqref{e6.1}, and we obtain \[ A_1+A_2+A_3+A_4+A_5+A_6=0, \] where \begin{gather*} A_1=-\int_Q(\psi_{\delta}^{+}\psi_{\eta}^{+})_{t}\hat{S}_{n,k}(v_{\epsilon})\,dx\,dt,\\ A_2=\int_Q(k-T_k(v_{\epsilon})) H_n(v_{\epsilon}) a(t,x,u_{\epsilon}, \nabla u_{\epsilon})\cdot\nabla (\psi_{\delta}^{+}\psi_{\eta}^{+})\,dx\,dt ,\\ A_3=-\int_Q\psi_{\delta}^{+}\psi_{\eta}^{+} a(t,x,u_{\epsilon},\nabla u_{\epsilon}) \cdot\nabla T_k(v_{\epsilon})\,dx\,dt,\\ A_4=\frac{2k}{n}\int_{\{ -2n< v_{\epsilon}\leq -n\}} \psi_{\delta}^{+} \psi_{\eta}^{+}a(t,x,u_{\epsilon},\nabla u_{\epsilon})\cdot\nabla v_{\epsilon} \,dx\,dt, \\ A_5=-\int_Q(k-T_k(v_{\epsilon})) H_n(v_{\epsilon})\psi_{\delta}^{+}\psi_{\eta}^{+} d\hat{\mu}_{0,\epsilon} ,\\ A_6=\int_Q(k-T_k(v_{\epsilon}))H_n(v_{\epsilon})\psi_{\delta}^{+}\psi_{\eta}^{+} d(\lambda_{\epsilon}^{\oplus}+\lambda_{\epsilon}^{\ominus}) \end{gather*} Therefore, as in \cite{Pe2}, using the fact that $(\hat{S}_{n,k}(v_{\epsilon}))$ weakly converges to $\hat{S}_{n,k}(v)$ in $X$, $\hat{S}_{n,k}(v)\in L^{\infty}(Q)$ and \eqref{e4.6} we obtain \[ A_1 =-\int_Q(\psi_{\delta}^{+})_{t}\psi_{\eta}^{+}\hat{S}_{n,k}(v) -\int_Q\psi_{\delta}^{+}(\psi_{\eta}^{+})_{t}\hat{S}_{n,k}(v) + \omega(\epsilon)=\omega(\epsilon,\delta). \] Now since $v_{\epsilon}=T_{2n}(v_{\epsilon})$ on $\operatorname{supp}(H_n(v_{\epsilon}))$ it follows from Proposition \ref{prop5.2}, (iv) that sequence $(a(t,x,u_{\epsilon},\nabla (T_{2n}(v_{\epsilon})+g_{\epsilon}))) \cdot\nabla (\psi_{\delta}^{+}\psi_{\eta}^{+})$ weakly converges to $F_{2n}\cdot\nabla(\psi_{\delta}^{+}\psi_{\eta}^{+})$ in $L^{1}(Q)$. From Lemma \ref{lem4.1} and the convergence of $\psi_{\delta}^{+}\psi_{\eta}^{+}$ in $X$ to 0 as $\delta$ tends to 0, we obtain \[ A_2=\int_Q(k-T_k(v_{\epsilon}))H_n(v_{\epsilon})F_{2n}\cdot\nabla (\psi_{\delta}^{+}\psi_{\eta}^{+})+ \omega(\epsilon)=\omega(\epsilon,\delta). \] Because $0\leq \psi_{\delta}^{+}\leq 1$ (resp $0\leq \psi_{\delta}^{-}\leq 1$). we then deduce \begin{align*} A_4 &= \frac{2k}{n}\int_{-2n< v_{\epsilon}\leq -n} a(t,x,u_{\epsilon}, \nabla (T_{2n}(v_{\epsilon})+g_{\epsilon}))\cdot \nabla (T_{2n}(v_{\epsilon}) +g_{\epsilon}) \\ &\quad -\nabla g_{\epsilon}] \psi_{\delta}^{+}\psi_{\eta}^{+}\,dx\,dt\\ &\leq \frac{2k}{n} \int_{\{ -2n2k>0$, we define \[ w_{\epsilon}=T_{2k}(v_{\epsilon}-T_{h}(v_{\epsilon}) +T_k(v_{\epsilon})-T_k(v)_{\nu}), \] Note that $\nabla w_{\epsilon}=0$ if $|v_{\epsilon}|>h+4k$. As a consequence of the estimate on $T_k(v_{\epsilon})$ in Proposition \ref{prop5.2} we have $w_{\epsilon}$ is bounded in $L^p(0,T;W^{1,p}_0(\Omega))$; we easily obtain \[ w_{\epsilon}\to T_{2k}(v-T_{h}(v)+T_k(v)-T_k(v)_{\nu})) \] since $\| T_k(v)_{\nu}\|_{L^{\infty}(Q)}\leq k$, we have also \begin{gather*} w_{\epsilon}=2k\text{ sign}(v_{\epsilon}), \text{ in }\{|v_{\epsilon}|>h+2k\}, \quad |w_{\epsilon}|\leq 4k,\quad w_{\epsilon}=w(\epsilon,\nu,h)\text{ a.e. in }Q,\\ \lim_{\epsilon} w_{\epsilon}=T_{h+k}(v-(T_k(v))_{\nu})-T_{h-k}(v-T_k(v)), \text{ a.e. in }Q \text{ and weakly in X}. \end{gather*} Let us take $w_{\epsilon}(1-\Phi_{\delta,\eta})$ as test functions in \eqref{e5.3}. We obtain \[ A_1+A_2+A_3= A_4+A_5, \] where \begin{gather*} A_1= \int_0^{T}\langle v_{t,\epsilon},w_{\epsilon}(1-\Phi_{\delta,\eta})\rangle\,dt,\\ A_2=\int_Qa(t,x,u_{\epsilon},\nabla u_{\epsilon})\cdot\nabla w_{\epsilon} (1-\Phi_{\delta,\eta}), \\ A_3=-\int_Qa(t,x,u_{\epsilon},\nabla u_{\epsilon}) \cdot\nabla\Phi_{\delta,\eta}w_{\epsilon}dx\,dt, \\ A_4=w_{\epsilon}(1-\Phi_{\delta,\eta}) d\hat{\mu_0} ,\\ A_5=\int_Qw_{\epsilon}(1-\Phi_{\delta,\eta}) d(\lambda_{\epsilon}^{\oplus} -\lambda_{\epsilon}^{\ominus}) \end{gather*} Using the weak convergence of $f_{\epsilon}$, again from the decomposition \eqref{e3.2} \[ A_4=\int_Qf_{\epsilon}w_{\epsilon}(1-\Phi_{\delta,\eta})\,dx\,dt +\int_QG_{\epsilon}\cdot\nabla (w_{\epsilon}(1-\Phi_{\delta,\eta}))\,dx\,dt, \] since $f_{\epsilon}$ converges to $f$ weakly in $L^{1}(Q)$, from Lemma \ref{lem4.1}, we obtain \[ \int_Qf_{\epsilon}w_{\epsilon}(1-\Phi_{\delta,\eta})\,dx\,dt =\omega(\epsilon,\nu,h). \] \begin{lemma}\label{lem6.5} Let $h,k>0$, and $u_{\epsilon}$ and $\Phi_{\delta,\eta}$ as before, then \[ \int_{\{ h\leq |v_{\epsilon}|< h+k \}}|\nabla u_{\epsilon}|^p (1-\Phi_{\delta,\eta})=\omega(\epsilon,h,\delta,\eta). \] \end{lemma} For a proof of the above lemma see \cite[Lemma 7]{Pe2}. Note that $(g_{\epsilon})$ converges to $g$ strongly in $(L^{p'}(Q))^{N}$, and $T_k(v)_{\nu}$ converges to $T_k(v)$ strongly in $X$. Then we deduce from Young's inequality and Lemma \ref{lem6.5}, \begin{align*} &\int_QG_{\epsilon}\cdot\nabla (w_{\epsilon}(1-\Phi_{\delta,\eta}))\,dx\,dt\\ &=\int_Q(1-\Phi_{\delta,\eta})G\cdot\nabla (T_{h+k}(v-T_k(v))-T_{h-k}(v-T_k(v))) \,dx\,dt+\omega(\epsilon,\nu)\\ &=\int_{\{ h\leq v< h+2k\}}(1-\Phi_{\delta,\eta})G\cdot\nabla v\,dx\,dt +\omega(\epsilon,\nu,h)\\ &=\omega(h,\delta,\eta). \end{align*} Then \[ A_4=\omega(\epsilon,\nu,h,\delta,\eta). \] To estimate of $A_5$, we have $|w_{\epsilon}|\leq 2k$ and reasoning as in the proof of Lemma \ref{lem6.5}, and thanks to \eqref{e4.8} - \eqref{e4.11}; we obtain \[ A_5=\omega(\epsilon,\delta,\eta). \] To estimate of $A_1$, we observe that, since $|T_k(v)_{\nu}|\leq k$, $w_{\epsilon}$ can be written in the following way: \[ w_{\epsilon}=T_{h+k}(v_{\epsilon}-T_k(v)_{\nu}) - T_{h-k}(v_{\epsilon}-T_k(v_{\epsilon})). \] Hence, setting $G(t)=\int_0^{t}T_{h-k}(s-T_k(s))ds$, we have \begin{align*} &\int_0^{t}\langle (v_{\epsilon})_{t},w_{\epsilon}(1-\Phi_{\delta,\eta})\rangle\,dt\\ &=\int_0^{t}\langle (T_k(v)_{\nu})_{t}, T_{h+k}(v_{\epsilon}-T_k(v)_{\nu}) (1-\Phi_{\delta,\eta})\rangle\,dt\\ &\quad +\int_QS_{h+k}(v_{\epsilon}-T_k(v)_{\nu})_{t}(1-\Phi_{\delta,\eta})\,\,dx\,dt -\int_QG(v_{\epsilon})_{t}(1-\Phi_{\delta,\eta})\,dx\,dt \end{align*} and since $|T_k(v)_{\nu}|\leq k$, using the definition of $T_k(v)_{\nu}$ we obtain \begin{align*} &\int_0^{t}\langle (T_k(v)_{\nu})_{t}, T_{h+k}(v_{\epsilon}-T_k(v)_{\nu})(1-\Phi_{\delta,\eta})\rangle\,dt\\ &= \nu\int_Q(T_k(v)-T_k(v)_{\nu})T_{h+k}(v_{\epsilon}-T_k(v)_{\nu})\,dx\,dt, \end{align*} so that as $\epsilon$ tends to infinity, we have \begin{align*} &\int_0^{t}\langle (T_k(v))_{t}, T_{h+k}(v_{\epsilon}-T_k(v)_{\nu}) (1-\Phi_{\delta,\eta})\rangle\,dt\\ &= \omega(\epsilon)+\nu\int_Q(T_k(v)-T_k(v)_{\nu})T_{h+k}(v-T_k(v)_{\nu}) (1-\Phi_{\delta,\eta})\,dx\,dt\\ &=\omega(\epsilon)+\nu\int_{\{ |v|\leq k\}}(v-T_k(v)_{\nu})T_{h+k} (v-T_k(v)_{\nu})(1-\Phi_{\delta,\eta})\,dx\,dt\\ &\quad +\int_{\{ v>k\}}(k-T_k(v)_{\nu})T_{h+k}(v-T_k(v)_{\nu}) (1-\Phi_{\delta,\eta})\,dx\,dt\\ &\quad +\int_{\{ v<-k\}}(-k-T_k(v)_{\nu})T_{h+k}(v-T_k(v)_{\nu}) (1-\Phi_{\delta,\eta})\,dx\,dt. \end{align*} since $|T_k(v)_{\nu}|\leq k$, last three terms are positives, hence we deduce by letting $\epsilon$ and $\nu$ to $\infty$, \begin{align*} &\int_0^{t}\langle (v_{\epsilon})_{t},w_{\epsilon}(1-\Phi_{\delta,\eta})\rangle\,dt\\ &= \omega(\epsilon)+\int_QS_{h+k}(v_{\epsilon}-T_k(v)_{\nu})_{t} (1-\Phi_{\delta,\eta})\,dx\,dt -\int_QG(v_{\epsilon})_{t} (1-\Phi_{\delta,\eta})\,dx\,dt\\ &=\omega(\epsilon)+\int_QS_{h+k}(v_{\epsilon}-T_k(v)_{\nu}) \frac{\partial \Phi_{\delta\eta}} {dt}dx\,dt -\int_QG(v_{\epsilon})\frac{\partial \Phi_{\delta\eta}}\,\,dx\,dt\\ &\quad +\int_{\Omega} S_{h+k}(v_{\epsilon}-T_k(v)_{\nu})(T)\,dx -\int_{\Omega}S_{h+k}(u_{0,\epsilon}-z_{\nu})\,dx\\ &\quad -\int_{\Omega}G(v_{\epsilon})(T)\,dx +\int_{\Omega}G(u_{0,\epsilon})\,dx. \end{align*} Now we define the function $R(y)=S_{h+k}(y-z)\cdot G(y)$, with $|z|\leq k$. Then \begin{gather*} R(y)=S_{h+k}(y+z)\geq 0 ,\quad |y|\leq k,\\ R'(y)=T_{h+k}(y-z)-T_{h-k}(y-T_k(y))\geq 0, \quad y\geq k\geq z,\\ R'(y)\leq 0, \quad y\leq -k\leq z. \end{gather*} Hence for every $z, |z|\leq k$, we have $R(y)\geq 0$ for every $y$ in $\mathbb{R}$, we obtain \[ \int_{\Omega}S_{h+k}(v_{\epsilon}-T_k(v)_{\nu})(T)\,dx - \int_{\Omega} G(v_{\epsilon})(T)\,dx \geq 0, \] letting $\epsilon$ and $\nu$ go to their limits, \[ \int_{\Omega}G(u_{u_{0,\epsilon}})\,dx -\int_{\Omega}S_{h+k}(u_{0,\epsilon}-z_{\nu})\,dx =\int_{\Omega}G(u_0)-\int_{\Omega}S_{h+k}(u_0-T_k(u_0))+\omega (\epsilon,\nu), \] Since we have $|G(u_0)-S_{h+k}(u_0-T_k(u_0))|\leq 2k|u_0|\chi_{\{ |u_0|>k\}}$, it follows that by letting $h$ to $+\infty$, \[ \int_{\Omega}G(u_{0,\epsilon})\,dx -\int_{\Omega}S_{h+k}(u_{0,\epsilon}-z_{\nu})\,dx = \omega(\epsilon,\nu,h)\,. \] By the definition of $T_k(v)_{\nu}$, \begin{align*} &\int_QS_{h+k}(v_{\epsilon}-T_k(v)_{\nu})\frac{d\Phi_{\delta\eta}}{dt}dx\,dt -\int_QG(v_{\epsilon})\frac{d\Phi_{\delta\eta}}{dt}dx\,dt\\ &=\int_Q(S_{h+k}(v-T_k(v)-G(v))\frac{d\Phi_{\delta\eta}}{dt}dx\,dt +\omega(\epsilon,\nu). \end{align*} So, if $|v|\leq h-k$, $S_{h+k}(v-T_k(v))- G(v)=0$, then $S_{h+k}(v-T_k(v))-G(v)$ converges a.e. to $0$ on $Q$, and since $v\in L^{1}(Q)$, by dominated convergence theorem \[ \int_QS_{h+k}(v_{\epsilon}- T_k(v)_{\nu})\frac{d\Phi_{\delta\eta}}{dt}dx\,dt - \int_Q(v_{\epsilon})\frac{d\Phi_{\delta\eta}}{dt}dx\,dt \geq \omega(\epsilon,\nu,h), \] and so \[ \int_0^{T}\langle (v_{\epsilon})_{t},w_{\epsilon}(1-\Phi_{\delta\eta})\rangle \geq \omega(\epsilon,\nu,h). \] Now we estimate of $A_2$. Note that $\nabla w_{\epsilon}=0$ if $|v_{\epsilon}|>h+4k$; then if we set $M=h+4k$, splitting the integral $(A_2)$ on the sets $\{ |v_{\epsilon}|>k\}$ and $\{ |v_{\epsilon}|\leq k\}$, using the fact that $T_{h}(v_{\epsilon})=T_k(v_{\epsilon})=v_{\epsilon}$ in $\{ |v_{\epsilon}|\leq k\}$ and $\nabla T_k(v_{\epsilon})\chi_{|v_{\epsilon}|>k}=0$. Then for $\{ |v_{\epsilon}|\leq M\}$ and $h\geq 2k$, we have \begin{align*} A_2 &=\int_Qa(t,x,u_{\epsilon},\nabla u_{\epsilon})\cdot\nabla w_{\epsilon} (1-\Phi_{\delta\eta})\,dx\,dt\\ &=\int_{\{ |v_{\epsilon}|\leq k\}}a(t,x,u_{\epsilon},\nabla u_{\epsilon}) \cdot\nabla (v_{\epsilon}-T_k(v)_{\nu})(1-\Phi_{\delta\eta})\,dx\,dt\\ &\quad+\int_{\{ |v_{\epsilon}|>k\}}a(t,x,u_{\epsilon},\nabla u_{\epsilon}) \cdot\nabla [(v_{\epsilon}-T_{h}(v_{\epsilon}))-(T_k(v)_{\nu})] (1-\Phi_{\delta\eta})\,dx\,dt\\ &=\int_{\{ |v_{\epsilon}|\leq k\}}a(t,x,u_{\epsilon},\nabla u_{\epsilon}) \cdot\nabla (v_{\epsilon}-T_k(v)_{\nu})(1-\Phi_{\delta\eta})\,dx\,dt\\ &\quad+\int_{\{ |v_{\epsilon}|>h\}}a(t,x,u_{\epsilon},\nabla u_{\epsilon}) \cdot\nabla [(v_{\epsilon}-T_{h}(v_{\epsilon}))(1-\Phi_{\delta\eta})\,dx\,dt\\ &\quad+\int_{\{ |v_{\epsilon}|>k\}}a(t,x,u_{\epsilon},\nabla u_{\epsilon}) \cdot\nabla (T_k(v)_{\nu}-T_k(v))+\nabla T_k(v)(1-\Phi_{\delta\eta})\,dx\,dt\\ &=\int_{\{ |v_{\epsilon}|\leq k\}}a(t,x,u_{\epsilon},\nabla u_{\epsilon}) \cdot\nabla (v_{\epsilon}-T_k(v)_{\nu})(1-\Phi_{\delta\eta})\,dx\,dt\\ &\quad+\int_{\{ h<|v_{\epsilon}|>h+4k\}}a(t,x,u_{\epsilon},\nabla u_{\epsilon}) \cdot\nabla v_{\epsilon}(1-\Phi_{\delta\eta})\,dx\,dt\\ &\quad+\int_{\{ |v_{\epsilon}|>k\}}a(t,x,u_{\epsilon},\nabla u_{\epsilon}) \cdot\nabla (T_k(v)_{\nu}-T_k(v))(1-\Phi_{\delta\eta})\,dx\,dt\\ &\quad +\int_{\{ |v_{\epsilon}|>k\}}a(t,x,u_{\epsilon},\nabla u_{\epsilon}) \cdot\nabla T_k(v)(1-\Phi_{\delta\eta})\,dx\,dt\,. \end{align*} Using assumption \eqref{e2.2}, young's inequality, equi-integrability and Lemma \ref{lem6.5}, we see that for some $C=C(p,c_2)$, \begin{align*} &\int_{\{ h\leq |v_{\epsilon}|k\}}a(t,x,u_{\epsilon},\nabla u_{\epsilon}) \cdot\nabla T_k(v)(1-\Phi_{\delta\eta})\,dx\,dt=\omega(\epsilon), \\ \int_{\{ |v_{\epsilon}|>k\}}a(t,x,u_{\epsilon},\nabla u_{\epsilon}) \cdot\nabla (T_k(v)_{\nu}-T_k(v))(1-\Phi_{\delta\eta})\,dx\,dt =\omega(\epsilon,\nu), \end{gather*} Therefore, \[ A_2=\int_{\{ |v_{\epsilon}|\leq k\}}a(t,x,u_{\epsilon},\nabla u_{\epsilon}) \cdot\nabla (v_{\epsilon}-T_k(v)_{\nu})(1-\Phi_{\delta\eta})\,dx\,dt + \omega(\epsilon,\nu,h,\delta,\eta). \] \end{proof} Next we conclude the proof of Theorem \ref{thm3.5}. \begin{lemma}\label{lem6.6} The function $u$ is a renormalized solution of \eqref{e1.1}. \end{lemma} \begin{proof} (i) Let $\varphi\in X\cap L^{\infty}(Q)$ such that $\varphi_{t}\in X'+L^{1}(Q)$, $\varphi(\cdot,T)=0$, and $S\in W^{2,\infty}(\mathbb{R})$, such that $S'$ has compact support on $\mathbb{R}$, $S(0)=0$. Let $M>0$ such that $\operatorname{supp} S'\subset [-M,M]$. Taking successively $(\varphi,S)$, $(\varphi,\psi_{\delta}^{+})$ and $(\varphi,\psi_{\delta}^{-})$ as test functions in \eqref{e6.1} applied to $u_{\epsilon}$, we can write \begin{gather*} A_1+A_2)+A_3+A_4=A_5+A_6+A_7, \\ (A_2)_{\delta}^{+}+(A_3)_{\delta}^{+}+(A_4)_{\delta}^{+} =(A_5)_{\delta}^{+}+(A_6)_{\delta}^{+}+(A_7)_{\delta}^{+}, \\ (A_2)_{\delta}^{-}+(A_3)_{\delta}^{-}+(A_4)_{\delta}^{-} =(A_5)_{\delta}^{-}+(A_6)_{\delta}^{-}+(A_7)_{\delta}^{-} \end{gather*} where \begin{gather*} A_1=-\int_{\Omega}\varphi(0)S(u_{0,\epsilon})dt,\quad A_2=-\int_Q\varphi_{t}S(v_{\epsilon})\,dx\,dt, \\ A_3=\int_QS'(v_{\epsilon})a(t,x,u_{\epsilon},\nabla u_{\epsilon}) \cdot\nabla\varphi\,dx\,dt,\\ A_4=\int_QS''(v_{\epsilon})\varphi a(t,x,u_{\epsilon}, \nabla u_{\epsilon})\cdot\nabla v_{\epsilon}dx\,dt,\\ A_5=\int_QS'(v_{\epsilon})\varphi \hat{\mu}_{\epsilon},\quad A_6=\int_QS'(v_{\epsilon})\varphi d\lambda_{\epsilon}^{\oplus}\\ A_7=-\int_QS'(v_{\epsilon})\varphi d\lambda_{\epsilon}^{\ominus}, \end{gather*} and \begin{gather*} (A_2)_{\delta}^{+}=-\int_Q(\varphi\psi_{\delta}^{+})_{t}S(v_{\epsilon})\,dx\,dt, \\ (A_3)_{\delta}^{+}=\int_QS'(v_{\epsilon})a(t,x,u_{\epsilon},\nabla u_{\epsilon}) \cdot\nabla (\varphi\psi_{\delta}^{+})\,dx\,dt,\\ (A_4)_{\delta}^{+}=\int_QS''(v_{\epsilon})\varphi\psi_{\delta}^{+} a(t,x,u_{\epsilon},\nabla u_{\epsilon})\cdot\nabla v_{\epsilon}dx\,dt,\\ (A_5)_{\delta}^{+}=\int_QS'(v_{\epsilon})\varphi\psi_{\delta}^{+} d\lambda_{\epsilon}^{\oplus},\\ (A_6)_{\delta}^{+}=-\int_QS'(v_{\epsilon})\varphi\psi_{\delta}^{+} d\lambda_{\epsilon}^{\ominus}\,. \end{gather*} Since $(u_{0,\epsilon})$ converges to $u_0$ in $L^{1}(\Omega)$, and $(S(v_{\epsilon}))$ converges to $S(v)$, strongly in $X$, and weak\_* in $L^{\infty}(Q)$, it follows that \begin{gather*} A_1=\int_{\Omega}\varphi(0)S(u_0)\,dx +\omega(\epsilon),\quad A_2=-\int_Q\varphi_{t}S(v)+\omega(\epsilon),\\ (A_2)_{\delta}^{+}=\omega(\epsilon,\delta),\quad (A_2)_{\delta}^{-}=\omega(\epsilon,\delta)\,. \end{gather*} Moreover, $T_{M}(v_{\epsilon})$ converges to $T_{M}(v)$, then $T_{M}(v_{\epsilon})+h_{\epsilon}$ converges to $T_k(v)+h$ strongly in $X$. Therefore, \begin{align*} A_3&=\int_QS'(v_{\epsilon})a(t,x,u_{\epsilon},\nabla (T_{M}(v_{\epsilon}) +h_{\epsilon})\cdot\nabla \varphi,\\ &=\omega(\epsilon)+\int_QS'(v)a(t,x,u_{\epsilon},\nabla (T_{M}(v)+h)) \cdot\nabla \varphi, \\ &=\omega(\epsilon)+\int_QS'(v)a(t,x,u,\nabla u)\cdot\nabla \varphi, \end{align*} and \begin{align*} A_4&=\int_QS''(v_{\epsilon})\varphi a(t,x,u_{\epsilon}, \nabla (T_{M}(v_{\epsilon})+h_{\epsilon}))\cdot\nabla T_{M}(v_{\epsilon})\\ &=\omega(\epsilon)+\int_QS''(v)\varphi a(t,x,u,\nabla (T_{M}(v)+h)) \cdot\nabla T_{M}(v)\\ &=\omega(\epsilon)+\int_QS''(v)\varphi a(t,x,u,\nabla u)\cdot\nabla v\,. \end{align*} In the same way, since $\psi_{\delta}^{+},\psi_{\delta}^{-}$ converges to $0$ in $X$, \begin{gather*} (A_3)_{\delta}^{+}=\omega(\epsilon)+\int_QS'(v)a(t,x,u,\nabla u)\cdot\nabla (\varphi\psi)_{\delta}^{+}=\omega(\epsilon,\delta), \\ (A_3)_{\delta}^{-}=\omega(\epsilon)+\int_QS'(v)a(t,x,u,\nabla u)\cdot\nabla (\varphi\psi_{\delta}^{-})=\omega(\epsilon,\delta), \\ (A_4)_{\delta}^{+}=\omega(\epsilon)+\int_QS''(v)\varphi\psi_{\delta}^{+} a(t,x,u,\nabla u)\cdot\nabla v=\omega(\epsilon,\delta), \\ (A_4)_{\delta}^{-}=\omega(\epsilon)+\int_QS''(v)\varphi\psi_{\delta}^{-} a(t,x,u,\nabla u)\cdot\nabla v=\omega(\epsilon,\delta), \end{gather*} and $(g_{\epsilon})$ strongly converges to $g$ in $(L^{p'}(\Omega))^{N}$. Therefore, \begin{align*} (A_5)&=\int_QS'(v_{\epsilon})\varphi f_{\epsilon} +\int_QS'(v_{\epsilon})g_{\epsilon}\cdot\nabla \varphi +\int_QS''(v_{\epsilon})\varphi g_{\epsilon}\cdot\nabla T_{M}(v_{\epsilon})\\ &=\omega(\epsilon)+\int_QS'(v)\varphi f+\int_QS'(v)g\cdot\nabla \varphi +\int_QS''(v)\varphi g\cdot\nabla T_{M}(v)\\ &=\omega(\epsilon)+\int_QS'(v)\varphi d\hat{\mu}_0 \end{align*} Now, thanks to Proposition \ref{prop5.2} and the proprieties of $\psi_{\delta}^{+}$ and $\psi_{\delta}^{-}$, we readily have \begin{gather*} (A_5)_{\delta}^{+}=\omega(\epsilon) +\int_QS'(v)\varphi \psi_{\delta}^{+}d\hat{\mu}_{\epsilon}=\omega(\epsilon,\delta),\\ (A_5)_{\delta}^{-}=\omega(\epsilon)+\int_QS'(v)\varphi \psi_{\delta}^{-}d\hat{\mu}_{\epsilon} =\omega(\epsilon,\delta). \end{gather*} Then \[ (A_6)_{\delta}^{+}+(A_7)_{\delta}^{+}=\omega(\epsilon,\delta), \] and thanks to \eqref{e4.9}, \begin{gather*} (A_7)_{\delta}^{+}\leq |\int_QS'(v_{\epsilon})\varphi \psi_{\delta}^{+} d\lambda_{\epsilon}^{\ominus}| \leq c\int_Q\psi_{\delta}^{+}d\lambda_{\epsilon}^{\ominus}=\omega(\epsilon,\delta),\\ (A_7)_{\delta}^{-}=\omega(\epsilon,\delta)\,. \end{gather*} Then \[ (A_6)_{\delta}^{+}=\int_QS'(v_{\epsilon})\varphi\psi_{\delta}^{+} d\lambda_{\epsilon}^{\oplus}= \omega(\epsilon,\delta). \] Moreover, \begin{align*} A_6 &=\int_QS'(v_{\epsilon})\varphi d\lambda_{\epsilon}^{\ominus}\\ &=\int_QS'(v_{\epsilon})\varphi \psi_{\delta}^{+}d\lambda_{\epsilon}^{\oplus} +\int_QS'(v_{\epsilon})\varphi(1-\psi_{\delta}^{+})d\lambda_{\epsilon}^{\oplus}\\ &\leq \omega(\epsilon,\delta)+\int_Q|S'(v_{\epsilon})\varphi|(1-\psi_{\delta}^{+}) d\lambda_{\epsilon}^{\oplus}\\ &\leq \omega(\epsilon,\delta)+ \| S\|_{W^{2,\infty}(\mathbb{R})} \| \varphi\|_{L^{\infty}(Q)}\int_Q(1-\psi_{\delta}^{+}) d\lambda_{\epsilon}^{\oplus}\\ &\leq \omega(\epsilon,\delta)\,. \end{align*} Hence \[ A_6=\omega(\epsilon)\text{ and } (A_7)=\omega(\epsilon). \] Therefore, we finally obtain \begin{align*} &-\int_{\Omega}\varphi(0)S(u_0)\,dx -\int_Q\varphi_{t}S(v) +\int_QS'(v)a(t,x,u,\nabla u)\cdot\nabla \varphi \\ &+\int_QS''(v)\varphi a(t,x,u,\nabla u)\cdot\nabla v\\ &=\int_QS'(v)\varphi d\hat{\mu}_0 \end{align*} with $\varphi\in C^{1}_0([0,T]\times\Omega)$. By density argument we have \eqref{e2.8} for any $\varphi\in X\cap L^{\infty}(Q)$ such that $\varphi_{t}\in X'+L^{1}(Q)$ and $\varphi(\cdot,T)=0$. \smallskip (ii) Next, we prove \eqref{e2.9}. We take $\varphi\in C^{\infty}_{c}(Q)$ and $(\varphi,S)=((1-\psi_{\delta}{-})\varphi,\overline{H}_n)$ as test functions in \eqref{e2.8} and the same test functions in \eqref{e6.1} applied to $u_{\epsilon}$, we can write \begin{gather*} B_1^n+B_2^n=B_3^n+B_4^n+B_5^n, \\ B_{1,\epsilon}^n+B_{2,\epsilon}^n=B_{3,\epsilon}^n+B_{4,\epsilon}^n +B_{5,\epsilon}^n, \end{gather*} where \begin{gather*} B_1^n=-\int_Q((1-\psi_{\delta}^{-})\varphi)_{t}\overline{H}_n(v)\,dx\,dt, \\ B_2^n=\int_QH_n(v)a(t,x,u,\nabla u)\cdot\nabla ((1-\psi_{\delta}^{-})\varphi)\,dx\,dt,\\ B_3^n=\int_QH_n(v)(1-\psi_{\delta}^{-})\varphi d\hat{\mu}_0,\\ B_4^n=\frac{1}{n}\int_{\{ n