\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2018 (2018), No. 121, pp. 1--36.\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-2018/121\hfil
  Musielak-Orlicz-Sobolev spaces]
{Nonlinear parabolic-elliptic system in Musielak-Orlicz-Sobolev spaces}

\author[F. Orteg\'on Gallego, M. Rhoudaf, H. Sabiki \hfil EJDE-2018/121\hfilneg]
{Francisco Orteg\'on Gallego, Mohamed Rhoudaf, Hajar Sabiki}

\address{Francisco Orteg\'on Gallego \newline
Departamento de Matem\'aticas,
Facultad de Ciencias, Universidad de C\'adiz,
Campus del R\'io San Pedro,
11510 Puerto Real, C\'adiz, Spain}
\email{francisco.ortegon@uca.es}

\address{Mohamed Rhoudaf \newline
Facult\'e des Sciences de Mekn\`es,
Universit\'e Moulay-Isma\"il - Mekn\`es,
\'Equipe: EDPs et Calcul Scientifique, Marocco}
\email{rhoudafmohamed@gmail.com}

\address{Hajar Sabiki \newline
Laboratoire d'Analyse,
G\'eom\'etrie et Applications,
Facult\'e des Sciences,
BP 133 K\'enitra 14000, Marocco}
\email{sabikihajar@gmail.com}

\dedicatory{Communicated by Jes\'us Ildefonso D\'iaz}

\thanks{Submitted December 26, 2017. Published June 15, 2018.}
\subjclass[2010]{35J70, 35K61, 46E30, 35M13}
\keywords{Parabolic-elliptic system; Musielak-Orlicz-Sobolev spaces;
\hfill\break\indent weak solutions; capacity solutions}

\begin{abstract}
 The existence of a capacity solution to the thermistor problem in the
 context of inhomogeneous Musielak-Orlicz-Sobolev spaces is analyzed.
 This is a coupled parabolic-elliptic system of nonlinear PDEs whose
 unknowns are the temperature inside a semiconductor material, $u$,
 and the electric potential, $\varphi$.
 We study the general case where the nonlinear elliptic operator in the
 parabolic equation is of the form
 $Au=-\operatorname{div} a(x,t,u,\nabla u)$, $A$ being a Leray-Lions operator
 defined on $W_0^{1,x}L_M(Q_T)$, where $M$ is a generalized $N$-function.
\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 the previous decade, there has been an increasing interest in the study of
various mathematical problems in modular spaces. These problems have many
consideration in applications \cite{11, RU, 29} and have resulted in a
renewal interest in Lebesgue and Sobolev spaces with variable exponent,
or the general Musielak-Orlicz spaces, the origins of which can be traced
back to the work of Orlicz in the 1930s. In the 1950s,
this study was carried on by Nakano \cite{18} who made the first systematic
study of spaces with variable exponent. Later on, Polish and Czechoslovak
mathematicians investigated the modular
function spaces (see, for instance, Musielak \cite{17},
Kovacik and Rakosnik \cite{15}).
The study of variational problems where the function $a$ satisfies
a nonpolynomial growth conditions instead of having the usual $p$-structure
arouses much interest with the development of applications to electro-rheological
fluids as an important class of non-Newtonian fluids
(sometimes referred to as smart fluids). The electro-rheological fluids are
characterized by their ability to drastically change the mechanical
 properties under the influence of an external electromagnetic field.
A mathematical model of electro-rheological fluids was proposed by
Rajagopal and Ruzicka (we refer to \cite{RA, RU} for more details).
Another important application is related to image processing \cite{PM}
where this kind of diffusion operator is used to underline the borders of
the distorted image and to eliminate the noise.

From a mathematical standpoint, it is a hard task to show the existence
of classical solutions, i.e., solutions which are continuously differentiable
as many times as the order of the differential equations under consideration.
However, the concept of weak solution is not enough to give a formulation
to all problems and may not provide existence or stability properties.
This is the case when we are dealing with nonuniformly elliptic problems,
as in the problem
\begin{equation}\label{P1}
\begin{gathered}
 \frac{\partial u}{\partial t}
-\operatorname{div}\Big( \frac{m\big( x, | \nabla u|
\big) }{| \nabla u| }\nabla u\Big)
= \rho(u)|\nabla \varphi|^2 \quad \text{in }Q_T=\Omega\times(0,T), \\
\operatorname{div} (\rho(u)\nabla \varphi) =0 \text{ in }Q_T,
 u=0,\quad\varphi=\varphi_0 \quad \text{on }\partial \Omega\times (0,T), \\
 u(x,0)=u_0(x)\quad \text{in } \Omega,
\end{gathered}
\end{equation}
where $\Omega\subset\mathbb{R}^d$, $d\ge2$, is an open and bounded set and
$\rho\in C(\mathbb{R})\cap L^\infty(\mathbb{R})$ is such that
 $\rho(s)>0$ for all $s\in\mathbb{R}$. In this situation, one readily realizes
that the search of weak solutions to problem~\eqref{P1} are not well suited.
Indeed, $\rho(s)$ may converge to zero as $|s|$ tends to infinity and as a result,
if $u$ is unbounded in
$\Omega\times(0,T)$, the elliptic equation becomes degenerate at points where $u$
is infinity and, therefore, no a priori estimates for $\nabla \varphi$ will
be available and thus, $\varphi$ may not belong to a Sobolev space. Instead of
$\varphi$, we may consider the function $\Phi=\rho(u)|\nabla \varphi|^2$ as
a whole and then show that it belongs to $L^2(\Omega)^d$.
This means that a new formulation of the original system is possible and the
solution to this new formulation will be called capacity solution.
 This concept was first introduced in the 1990s by Xu in \cite{XU} in the
analysis of a modified version of the thermistor problem where the monotone
mapping $a=a(\nabla u)$ is a Leray-Lions
operator from $L^2(H^1)$ to $L^2(H^{-1})$. The same author applied this
concept to more general settings where weaker assumptions \cite{XU1} or
mixed boundary conditions \cite{XU2} are considered.
Later, Gonz\'alez Montesinos and Orteg\'on Gallego \cite{MM} showed the
 existence of a capacity solution to problem~\eqref{P} where $a$ is
a Leray-Lions operator from $L^p(W^{1,p})$ into $L^{p'}(W^{-1,p'})$, $p\ge2$,
$1/p+1/p'=1$. In a recent paper, the existence of a capacity solution in the
context of Orlicz-Sobolev spaces has been established by Moussa,
Orteg\'on Gallego and
Rhoudaf \cite{mou-or-rhou}. The analysis developed in the present paper
is a generalization to that given in~\cite{mou-or-rhou}.
Our framework is the Musielak-Orlicz-Sobolev spaces.

This paper deals with the existence of a capacity solution to a coupled
system of parabolic-elliptic equations, whose unknowns are the temperature
inside a semiconductor material, $u$, and the electric potential, $\varphi$,
namely

\begin{equation}\label{P}
\begin{gathered}
 \frac{\partial u}{\partial t}
-\operatorname{div} a(x,t,u,\nabla u)= \rho(u)|\nabla \varphi|^2 \quad \text{in }
Q_T=\Omega\times(0,T), \\
 \operatorname{div} (\rho(u)\nabla \varphi) =0 \quad \text{in }Q_T,\\
 \varphi=\varphi_0 \quad \text{on } \partial \Omega\times (0,T), \\
 u(x,0)=u_0(x)\quad \text{in } \Omega ,\\
 u=0\quad \text{on } \partial \Omega \times (0,T),
\end{gathered}
\end{equation}
where $\Omega\subset\mathbb{R}^d$, $d\ge2$, is the space region occupied
 by the semiconductor, $T>0$ is the final time of observation,
$Au=-\operatorname{div} a(x,t,u,\nabla u)$ is a Leray-Lions operator
defined on $W_0^{1,x}L_M(\Omega)$, $M$ is a generalized $N$-function,
and the functions $\varphi_0$ and $u_0$ are given. The functional spaces
to deal with these problems are Musielak-Orlicz-Sobolev spaces.
In general, Orlicz-Sobolev spaces are neither reflexive nor separable.

Problem \eqref{P} may be regarded as a generalization of the so-called thermistor
problem arising in electromagnetism \cite{AC, GOG, MM}.

Our analysis makes extensively use of the notion of modular convergence in
Musielak-Orlicz spaces.
 The fundamental studies in this direction are due to Gossez for the case of
elliptic equations \cite{Go,go74}.
 The considerations of the problem with an $x$-dependent modular function
formulated in Musielak-Orlicz-Sobolev
spaces are due to Benkirane et al.\ \cite{3.11} where the authors
 formulate an approximation theorem with respect to the modular topology.
 A particular case of Musielak-Orlicz spaces with an $x$-dependent
 modular function are the variable exponent spaces $L^{p(x)}(\Omega)$ for
which $M(x,t)=|t|^{p(x)}$ \cite{Antontsev-Shmarev}.
Other possible choices are
\begin{gather*}
M(x,t)=|t|^{p(x)}\log(1+|t|),\\
M(x,t)=|t|\log(1+|t|)(\log(\tau_0+|t|))^{p(x)},\text{ for some } \tau_0\ge1,\\
M(x,t)=\exp\big(|t|^{p(x)}\big)-1.
\end{gather*}
 The reader is referred to~\cite{Antontsev-Shmarev} for an extensive analysis
on the theory of quasilinear of parabolic (and hyperbolic) equations related to
some variable exponent spaces, including the $L^{p(x)}(\Omega)$ spaces, and
 to \cite{10.1} for a comprehensive summary on these generalized modular spaces.

The main goal of this article is to prove the existence of a capacity solution
of \eqref{P} in the sense of Definition~\ref{def1} for a generalized $N$-function,
 $M$, along with the lack of reflexivity in this setting combined with
the nonuniformly elliptic character of the second differential equation.

This work is organized as follows. In Section~2 we recall some
well-known properties and results on Musielak-Orlicz-Sobolev spaces.
Section~3 is devoted to specify the assumptions on data.
In Section~4 we give the definition of a capacity solution
of~\eqref{P}. Finally, in Section 5 we present the existence result
and develop its proof.

\section{Preliminaries}

In this section we list some definitions and facts about Musielak-Orlicz-Sobolev
spaces. Standard reference is \cite{17}. We also include the definition of
inhomogeneous Musielak-Orlicz-Sobolev spaces and some preliminaries lemmas to
 be used later on this paper.

\subsection*{Musielak-Orlicz spaces}
Let $\Omega$ be a bounded domain in $\mathbb{R}^d$, $d\in\mathbb{N}$.

\begin{definition} \label{def2.1} \rm
 Let $M\colon\Omega \times \mathbb{R}\mapsto\mathbb{R}$ satisfying the
following conditions:
\begin{itemize}
 \item[(i)] For a.a.\ $x\in\Omega$, $M(x,\cdot)$ is an $N$-function, that is,
convex and even in $\mathbb{R}$, increasing in $\mathbb{R}^+$,
 $M(x,0)=0$,
 $ M(x,t)>0 $ for all $t>0$, $M(x,t)/t\to 0$ as
 $t\to 0$, $M(x,t)/t \to \infty$ as $t\to \infty$).

 \item[(ii)] For all $t\in\mathbb{R}$, $M(\cdot,t)$ is a measurable function.
\end{itemize}
A function $M(x,t)$ which satisfies the conditions (i) and (ii) is called a
 \emph{Musielak-Orlicz function}, a generalized $N$-function or a
generalized modular function.
\end{definition}

From now on, $M\colon\Omega \times \mathbb{R}\mapsto\mathbb{R}$ will stand
for a general Musielak-Orlicz function. Notice that
 \begin{equation}\label{inf}
 \operatorname{ess\,inf}_{x\in\Omega}\frac{M(x,t)}{t}\to \infty \quad
\text{as } t\to \infty .
 \end{equation}
Indeed, by the definition of $\operatorname{ess\,inf}_{x\in\Omega}M(x,t)$ we have
for all $\epsilon>0$
there exist a measurable $\Omega_\epsilon\subset \Omega$,
 $\operatorname{meas}(\Omega_\epsilon)>0$ such that
 \[
M(y,t)\leq \operatorname{ess\,inf}_{x\in\Omega}M(x,t)+ \epsilon, \quad
 \text{for all }y\in \Omega_\epsilon,
\]
dividing by $t$ we obtain
 \[
\frac{M(y,t)}{t}\leq \operatorname{ess\,inf}_{x\in\Omega}\frac{M(x,t)}{t}
+ \frac{\epsilon}{t}, \quad \text{for all }y\in \Omega_\epsilon,
\]
and letting $t\to\infty$, using (i), we obtain \eqref{inf}.

In some situations, the growth order with respect to $t$ of
two given Musielak-Orlicz functions $M$ and $P$ are comparable.
This concept is detailed in the next definition.

\begin{definition} \label{def2.2} \rm
Let $M, P\colon\Omega \times \mathbb{R}\mapsto\mathbb{R}$ be
 Musielak-Orlicz functions.
\begin{itemize}
 \item Assume that there exist two constants
 $\epsilon>0$ and $t_0\ge0$ such that for a.~a.~$x\in \Omega$ one has
 \[
P(x,t)\leq M(x,\epsilon t) \text{ for all } t\geq t_0,
\]
 then we write $P\prec M$ and we say that $M$ dominates $P$ globally if $t_0 = 0$
and near  infinity if $t_0 > 0$.

\item We say that $P$ grows essentially less rapidly than $M$ at
$t=0$ (respectively, near infinity), and we write $P\ll M$,
if for every positive constant $k$ we have
 \[
\lim_{t\to 0} \sup_{x\in \Omega} \frac{P(x,kt)}{M(x,t)}=0 \text{ (respectively, }
\lim_{t\to \infty} \sup_{x\in \Omega} \frac{P(x,kt)}{M(x,t)}=0).
\]
\end{itemize}
\end{definition}


We will also use the following notation:
 $M_x(t) = M(x,t)$, for a.~a.~$x\in\Omega$ and all $t\in\mathbb{R}$,
and we associate  its inverse function with respect to $t\ge0$, denoted
by $M^{-1}_{x} $, that is,
 \[
 M^{-1}_{x}(M(x,t))= M(x,M^{-1}_{x}(t))=t, \text{ for all }t\ge0.
\]

\begin{remark} \label{rmk2.3} \rm
 It is easy to check that
 $P\ll M$ near infinity if and only if
 \[
\lim_{t\to \infty} \frac{M^{-1}(x,kt)}{P^{-1}(x,t)}=0 \quad
\text{ uniformly for }x\in \Omega\setminus \Omega_0
\]
for some null subset $\Omega_0\subset\Omega$.
\end{remark}

We define the functional
 $$
\varrho_{M,\Omega}(u)=\int_{\Omega}M(x,u(x))\,{\rm d}x,
$$
for any Lebesgue measurable function
 $u\colon\Omega \mapsto \mathbb{R}$ is a Lebesgue measurable function.

 The set
$$
\mathcal{L}_M(\Omega)=\{u\colon\Omega \mapsto \mathbb{R}
\text{ measurable such that } \varrho_{M,\Omega}(u)<\infty\}
$$
is called the Musielak-Orlicz class related to $M$ in $\Omega$ or simply
the Musielak-Orlicz class.

The Musielak-Orlicz space $L_M(\Omega)$ is the vector space generated by
$\mathcal{L}_M(\Omega)$, that is, $L_M(\Omega)$ is the smallest linear space
containing the set $\mathcal{L}_M(\Omega)$. Equivalently,
 $$
L_M(\Omega)=\{ u\colon\Omega \mapsto \mathbb{R} \text{ measurable such that }
\varrho_{M,\Omega}(u/\alpha)<\infty, \text{ for some } \alpha>0 \}.
$$

For a Musielak-Orlicz function $M$, we introduce its complementary function,
denoted by $\bar{M}$, as
\[
\bar{M}(x,s)=\sup_{t\geq 0}\{st-M(x,t) \},
\]
that is $\bar{M}(x,s)$ is the complementary to $M(x,t)$ in the sense of
 Young with respect to the variable $s$. It turns out that $\bar{M}$ is
 another Musielak-Orlicz function and the following Young-Fenchel inequality holds
 \begin{equation}\label{eq:young}
|ts|\le M(x,t)+\bar{M}(x,s) \quad \text{for all $t,s\in\mathbb{R}$  and a.~a. }x\in \Omega.
\end{equation}

In the space $L_M(\Omega)$ we define the following two norms:
 \[
\| u \|_{M,\Omega}=\inf\big\{\lambda>0:
 \int_{\Omega}M(x,u(x)/\lambda)\,{\rm d}x\le 1 \big\},
\]
which is called the Luxemburg norm, and the so-called Orlicz norm,
namely
\[
\|u\|_{(M),\Omega}=\sup_{\varrho_{\bar{M},\Omega}(v)\leq
1}\int_{\Omega}u(x)v(x)\,{\rm d}x.
\]
where the supremum is taken over all $v\in E_{\bar{M}(\Omega)}$
such that $\varrho_{\bar{M},\Omega}(v)\leq 1$.
An important inequality in $L_{M}(\Omega)$ is the following:
 \begin{equation}\label{eq:ineq0}
 \int_\Omega M(x,u(x))\,{\rm d}x\leq \|u\|_{(M),\Omega}\quad
 \text{for all $u\in L_{M}(\Omega)$ such that $\|u\|_{(M),\Omega}\le1$,}
\end{equation}
from we readily deduce
\begin{equation}\label{eq:ineq1}
\int_\Omega M\Big(x,\frac{u(x)}{\|u\|_{(M),\Omega}}\Big)\,{\rm d}x\leq 1
\quad \text{for all $u\in L_{M}(\Omega)\setminus\{0\}$.}
\end{equation}

 It can be shown that the norm $\|\cdot\|_{(M),\Omega}$ is equivalent
to the Luxemburg norm  $\|\cdot\|_{M,\Omega}$. Indeed,
 \begin{equation}\label{eq:equivnorms}
 \|u\|_{M,\Omega}\leq \|u\|_{(M),\Omega}\leq 2\|u\|_{M,\Omega}
\quad \text{for all $u\in L_{M}(\Omega)$.}
 \end{equation}
Also, H\"older's inequality holds
\[
 \int_{\Omega}|u(x)v(x)|\,{\rm d}x \le
 \|u\|_{M,\Omega}\|v\|_{(\bar{M}),\Omega}
\quad \text{for all $u\in L_{M}(\Omega)$ and $v\in L_{\bar{M}}(\Omega)$,}
\]
in particular, if $\Omega$ has finite measure, H\"older's inequality
yields the continuous inclusion $L_{M}(\Omega)\subset L^1(\Omega)$.

 Strong convergence in $L_{M}(\Omega)$ is rather strict.
For most purposes, a mild concept of convergence will be enough, namely,
that of modular convergence. The closure in $L_M(\Omega)$ of the
bounded measurable functions with compact support in $\bar{\Omega}$
is denoted by $E_M(\Omega)$. The space $E_M(\Omega)$ is the largest linear
space such that $E_M(\Omega)\subset \mathcal{L}_M(\Omega)\subset L_M(\Omega)$,
 where the inclusion is in general strict.

\begin{definition} \label{def2.3} \rm
 We say that a sequence $(u_n)\subset L_M(\Omega)$ is modular convergent to
$u \in L_M(\Omega)$ if there exists a constant $\lambda>0$ such that
\[
\lim_{n\to \infty} \varrho_{M,\Omega}((u_n-u)/\lambda)=0.
\]
\end{definition}

\subsection*{Musielak-Orlicz-Sobolev spaces}
For any fixed nonnegative integer $m$ we define
 $$
W^m L_M(\Omega)=\{u\in L _M(\Omega): D^{\alpha}u\in L _M(\Omega)
\text{ for all }\alpha, |\alpha| \leq m \}
$$
where $\alpha=(\alpha_1 , \alpha_2 , \dots, \alpha_d )\in\mathbb{Z}$,
 $\alpha_j\ge0$, $j=1,\dots,d$, with
$|\alpha|=\alpha_1+\alpha_2+\dots+\alpha_d$ and $D^{\alpha}u$ denote the
distributional derivative of  multiindex $\alpha$. The space
$W^m L _M(\Omega)$ is called the  Musielak-Orlicz-Sobolev space (of order $m$).

 Let $u\in W^m L_M(\Omega)$, we define
\begin{gather*}
 \varrho_{M,\Omega}^{(m)}(u)=\sum_{|\alpha|\leq m}\varrho_{M,\Omega}(D^{\alpha}u),\\
\|u\|^{(m)}_{M,\Omega}
 =\inf \{ \lambda >0 :\varrho_{M,\Omega}^{(m)}(u/\lambda)\leq 1 \},\quad
 \| u\| _{m,M,\Omega }= \sum_{| \alpha | \leq m}\| D^\alpha u\|_{M,\Omega }.
\end{gather*}
The functional $\varrho_{M,\Omega}^{(m)}$ is convex
in $W^m L_M(\Omega)$, whereas the functionals $\|\cdot\|^{(m)}_{M,\Omega}$ and
 $\| \cdot\| _{m,M,\Omega }$ are equivalent norms on $W^m L_M(\Omega)$.
The pair $(W ^m L_M(\Omega) ,\| \cdot \|^{(m)}_{M,\Omega} )$
is a Banach space if there exists a constant $c>0$ such that
\begin{equation}\label{eq:Mc}
 \operatorname{ess\,inf}_{x\in \Omega}M(x,1)\geq c.
\end{equation}
From this point on we will assume that \eqref{eq:Mc} holds.
The space $W ^m L_M(\Omega)$
is identified to a subspace of the product
$\prod_{|\alpha|\leq m}L_M(\Omega)=\prod L_M$,
this subspace is $\sigma(\prod L_M,\prod E_{\bar{M}})$ closed.

Let $W_0^m L_M(\Omega)$ be the $\sigma(\prod L_M,\prod E_{\bar{M}})$
closure of $\mathcal{D}(\Omega)$ in $W^m L_M(\Omega)$. Let $W^m E_M(\Omega)$
be the space of functions $u$ such that $u$ and its
distribution derivatives up to order $m$ lie in $E_M(\Omega)$, and
$W_0^m E_M(\Omega)$ is the (norm) closure  of $\mathcal{D}(\Omega)$
in $W^m L_M(\Omega)$.

 \begin{lemma}[Poincar\'e's inequality \cite{Ait}] \label{poi1}
Let $\Omega$ be a bounded Lipchitz-continuous domain of $\mathbb{R}^d$.
Then there exists a constant $C_0=C_0(\Omega,M)>0$ such that
 \begin{equation}\label{poincare}
 \|u\|_{M,\Omega}\leq C_0 \|\nabla u\|_{M,\Omega},\quad
\text{for all }u\in W^1_0L_M(\Omega).
 \end{equation}
\end{lemma}

 \begin{remark} \label{rp} \rm
 Let $M$ be a Musielak-Orlicz function and $u\in W^1_0L_M(\Omega)$.
 Assume that, for some constant $C\ge0$, one has
 $\int_{\Omega} M(x,\nabla u)\,{\rm d}x\leq C$.
 Then we also have $\|u\|_{1,M,\Omega}\le C'$ where $C'=(C_0+1)\max(C,1)$.
 Indeed, since $\|u\|_{1,M,\Omega}=\|u\|_{M,\Omega}+\|\nabla u\|_{M,\Omega}$,
by using \eqref{poincare}, we obtain
 \[
\|u\|_{1,M,\Omega}\leq C_0\|\nabla u\|_{M,\Omega}+\|\nabla u\|_{M,\Omega}
\leq (C_0+1)\|\nabla u\|_{M,\Omega}.
\]
Now, if $C\geq1$, according to the convexity of $M(x,\cdot)$, it yields
 \[
\int_{\Omega} M\Big(x,\frac{\nabla u}{C}\Big)\,{\rm d}x
\leq \frac{1}{C}\int_{\Omega} M(x,\nabla u)\,{\rm d}x\leq \frac{C}{C}=1,
\]
this means that $C\in \{\lambda>0,\int_{\Omega} M(x,{\nabla u}/{\lambda})\,{\rm d}x
\leq 1 \}$, hence $\|\nabla u\|_{M,\Omega}\leq C$.
On the other hand, if $C<1$, then $\int_{\Omega} M(x,\nabla u)\,{\rm d}x\leq C<1$,
which yields $\|\nabla u\|_{M,\Omega}\leq 1$.
\end{remark}

Since we are going to work with two generalized $N$-functions, say $P$ and
$M$, such that $P\ll M$, we will consider the following assumptions for
both complementary functions $\bar{P}$ and $\bar{M}$:
 \begin{gather}\label{h1}
 \lim_{|\xi|\to\infty} \operatorname{ess\,inf}_{x\in\Omega}
\frac{\bar{M}(x,\xi)}{|\xi|}=\infty, \\
\label{p2}
 \lim_{|\xi|\to\infty} \operatorname{ess\,inf}_{x\in\Omega}\frac{\bar{P}(x,\xi)}{|\xi|}=\infty.
 \end{gather}

\begin{remark} \label{R1} \rm
 From \cite[Remark 2.1]{Gu} we have that the assumptions \eqref{h1} and \eqref{p2}
 provide the following:
 \begin{gather}\label{h2}
 \sup_{\xi\in B(0,R)} \operatorname{ess\,sup}_{x\in\Omega}M(x,\xi)<\infty
 \text{ for all } 0<R<+\infty, \\
\label{p1}
 \sup_{\xi\in B(0,R)} \operatorname{ess\,sup}_{x\in\Omega}P(x,\xi)<\infty
 \text{ for all } 0<R<+\infty.
 \end{gather}
 \end{remark}


\begin{definition} \label{def2.8} \rm
We say that a sequence $(u_n)\subset W^1L_M(\Omega)$ converges to
$u\in W^1L_M(\Omega)$ for the modular convergence in $W^1L_M(\Omega)$ if,
for some $h>0$,
\[
\lim_{n\to \infty}\bar{\varrho}^{(1)}_{M,\Omega}((u_n-u)/h)=0.
\]
\end{definition}

The following spaces of distributions will also be used:
\begin{gather*}
W^{-1} L_{\bar{M}}(\Omega)=\big\{ f\in \mathcal{D}'(\Omega):
f=\sum_{|\alpha|\leq 1}(-1)^{|\alpha|} {D}^{\alpha}f_{\alpha}
\text{ for some } f_{\alpha}\in L_{\bar{M}}(\Omega) \big\},
\\
W^{-1} E_{\bar{M}}(\Omega)=\big\{ f\in \mathcal{D}'(\Omega):
 f=\sum_{|\alpha|\leq 1}(-1)^{|\alpha|} {D}^{\alpha}f_{\alpha} \text{ for some }
f_{\alpha}\in E_{\bar{M}}(\Omega) \big\}.
\end{gather*}

\begin{lemma} \label{mod}
 If $P\ll M$ and $u_n\to u$ for the modular convergence in $L_M(\Omega)$,
then $u_n\to u$ strongly in $E_P(\Omega)$. In particular,
$L_M(\Omega)\subset E_P(\Omega)$ and
$L_{\bar{P}}(\Omega)\subset E_{\bar{M}}(\Omega)$ with continuous injections.
\end{lemma}

\begin{proof}
Let $\epsilon>0$ be given. Let $\lambda>0$ be such that
\[
\int_{\Omega}M\Big(x,\frac{u_n-u}{\lambda}\Big)\to 0, \quad \text{as } n\to \infty.
\]
Then, there exists $h\in L^1(\Omega)$ such that
\[
 M\Big(x,\frac{u_n-u}{\lambda}\Big)\leq h\quad \text{and}\quad u_n\to u
\text{ a.~e.~in }\Omega
\]
for a subsequence still denoted $(u_n)$.
 Since $P\ll M$, then for all $r>0$ there exists $t_0>0$ such that
\[
\frac{P(x,rt)}{M(x,t)}\leq 1, \quad \text{ a.~e.~in $\Omega$ and for all } t\geq t_0.
\]
For $r=\frac{\lambda}{\epsilon}$ and $t=\frac{t'}{\lambda}$, we obtain
\[
\frac{P(x,\frac{t'}{\epsilon})}{M(x,\frac{t'}{\lambda})}\leq 1, \quad
\text{when } t'\geq t_0\lambda.
\]
Then
\begin{align*}
 P\Big(x,\frac{u_n-u}{\epsilon}\Big)
 &\leq  M\Big(x,\frac{u_n-u}{\lambda}\Big)
 +\sup_{t'\in B(0,t_0\lambda) } \operatorname{ess\,sup}_{x\in\Omega}P(x,t'/\epsilon)
\\
 &\leq  h+\sup_{t'\in B(0,t_0) }
\operatorname{ess\,sup}_{x\in\Omega}P(x,t'/\epsilon)\text{ for a.~a.~}x\in\Omega.
 \end{align*}
Since $h+\sup_{t'\in B(0,t_0\lambda) } \operatorname{ess\,sup}_{x\in\Omega}
P(x,\frac{t'}{\epsilon})\in L^1(\Omega)$ (from Remark \ref{R1}),
 it yields, by the Lebesgue dominated convergence theorem,
\[
P\Big(x,\frac{u_n-u}{\epsilon}\Big)\to 0 \text{ in }L^1(\Omega),
\]
 hence, for $n$ big enough, we have $\|u_n-u\|_{P,\Omega}\leq \epsilon$.
That is, $u_n\to u$ in $L_P(\Omega)$.

The continuous injection $L_M(\Omega)\subset E_P(\Omega)$ is trivial since
the convergence in
$L_M(\Omega)$ implies the modular convergence in this space.
On the other hand, since
$P\ll M$ is equivalent to $\bar{M}\ll \bar{P}$, this yields the continuous
 injection $L_{\bar{P}}(\Omega)\subset E_{\bar{M}}(\Omega)$.
\end{proof}

\begin{lemma}[{\cite[Lemma 2.2]{mou-or-rhou}}] \label{mod1}
Let $(w_n)\subset L_M(\Omega)$, $w\in L_M(\Omega)$,
$(v_n)\subset L_{\bar{M}}(\Omega)$ and $v\in L_{\bar{M}}(\Omega)$.
If $w_n\to w$ in $L_M(\Omega)$ for the modular convergence and
$v_n\to v$ in $L_{\bar{M}}(\Omega)$ for the modular convergence, then
\[
\lim_{n\to\infty}\int_{\Omega}w_nv\,{\rm d}x
=\int_{\Omega}wv\,{\rm d}x\quad \text{and}\quad
\lim_{n\to\infty}\int_{\Omega}w_nv_n\,{\rm d}x
= \int_{\Omega}wv\,{\rm d}x.
\]
\end{lemma}

\begin{lemma}[\cite{KH, Val}] \label{lem-assum}
 Let $\Omega$ be a bounded and Lipschitz-continuous domain in
 $\mathbb{R}^d$ and let $M$ and $\bar{M}$ be two
 complementary Musielak-Orlicz functions which satisfy the following conditions:
 \begin{itemize}
 \item[(i)] There exists a constant $A > 0$ such that for all
 $x,y \in \Omega$ with $|x-y|\leq \frac{1}{2}$ one has
 \begin{equation}\label{eq:tA}
 \frac{M(x,t)}{M(y,t)}\leq t^{-A/\log|x-y|} \quad \text{for all } t\ge1.
 \end{equation}

 \item[(ii)] There exists a constant $C > 0$ such that
 \begin{equation}\label{eq:barM1}
  \bar{M}(x,1)\leq C \quad \text{a.~e. in $\Omega$.}
 \end{equation}
\end{itemize}
 Then the space $\mathcal{D}(\Omega)$ is dense in $L_M(\Omega)$ with respect to the
 modular convergence, $\mathcal{D}(\Omega)$
 is dense in $W_0^1 L_M(\Omega)$ for the modular convergence and
 $\mathcal{D}(\bar{\Omega})$ is dense in $W^1 L_M(\Omega)$ for the
 modular convergence.
\end{lemma}

\begin{remark}  \label{rem:M1c} \rm
By taking $t=1$ in \eqref{eq:tA} it yields that $M(x,1)=\text{constant}$
for a.~a.~$x\in\Omega$. In particular, the condition~\eqref{eq:Mc}
is obviously satisfied and also
\[
 \int_{\Omega} M(x,1)\,{\rm d}x < \infty.
\]
\end{remark}

\begin{remark}[\cite{Val}] \label{rmk} \rm
Let $p\colon \Omega\mapsto (1,\infty)$ be a measurable function such that
there exists a constant $A>0$ such that for all points $x, y \in\Omega$ with
$|x-y| < 1/2$, one has the inequality
\[
|p(x)-p(y)|\le -\frac{A}{\log|x-y|}.
\]
Then the following Musielak-Orlicz functions satisfy the assumption~\eqref{eq:tA}:
\begin{enumerate}
\item $M(x,t)=t^{p(x)}$;
\item $M(x,t)=t^{p(x)}\log(1+t)$;
\item $M(x,t)=t\log(1+t)(\log(e-1+t))^{p(x)}$.
\end{enumerate}
\end{remark}

\subsection*{Inhomogeneous Musielak-Orlicz-Sobolev spaces}

 Let $\Omega$ be a bounded and open subset of $\mathbb{R}^d$ and let
$Q_T = \Omega \times (0,T)$ with some given $T >	0$.
Let $M$ be a Musielak function. For each
$\alpha=(\alpha_1,\dots,\alpha_d) \in \mathbb{Z}^d$, $\alpha_j\ge0$, $j=1,\dots,d$,
we denote by $D^{\alpha}_x$
 the distributional	derivative on $Q_T$ of multiindex $\alpha$ with respect to
the variable $x \in \mathbb{R}^d$.
The inhomogeneous Musielak-Orlicz-Sobolev spaces of order one are defined as follows:
\begin{gather*}
W^{1,x}L_M(Q_T)= \{ u \in L_M(Q_T) :
D^{\alpha}_x u \in L_M(Q_T) \text{ for all }\alpha,\,\,
|\alpha| \leq 1 \} , \\
W^{1,x}E_M(Q_T)= \{ u \in E_M(Q_T) :
D^{\alpha}_x u \in E_M(Q_T) \text{ for all }\alpha,\,\,
|\alpha| \leq 1 \}
\end{gather*}
This last space is a subspace of the first one, and both are Banach spaces
under the  norm
$$
\|u\|= \sum_{|\alpha| \leq 1}\|D^{\alpha}_x u\| _{M,Q_T}.
$$
 These spaces are considered as subspaces of the product space $\varPi L_M(Q_T)$	
which has $(d +1)$ copies. We  also consider the
weak-$\ast$ topologies $\sigma(\varPi L_M	(Q_T),\varPi E_{\overline {M}}(Q_T) )$
and $\sigma(\varPi L_M	(Q_T),\varPi L_{\overline {M}}(Q_T) )$.
If $u \in W^{1,x}L_M(Q_T)$ then the function $t: \to u(t)$ is defined on $(0,T)$
with values in $W^{1}L_M(\Omega)$. If, further, $u \in W^{1,x}E_M(Q_T)$ then this
 function is a $W^{1}E_M(\Omega)$-valued and is strongly measurable.
The space $W^{1,x}	L_M(Q_T)$ is not in general separable.
If $u \in W^{1,x}	L_M(Q_T)$, we cannot conclude that the function
 $u(t)$ is measurable on $(0, T )$. However, the scalar function
$t \to \| u(t)\|_{M,\Omega}$	is in	$L^1(0, T )$.
 The space $W^{1,x}_0E_M	(Q_T)$ is defined as the (norm) closure in
$W^{1,x}E_M	(Q_T)$ of $\mathcal{D}(Q)$. We can easily show as in
\cite{AC} that when $\Omega$ is a Lipschitz-continuous domain
then each element $u$ of the closure of
 $\mathcal{D}(Q_T)$ with respect of the weak-$\ast$ topology
 $\sigma(\Pi L_M, \Pi E_{\bar{M}})$ is limit, in $W^{1,x}L_M(Q_T)$,
of some subsequence
 $(u_n) \subset \mathcal{D}(Q_T)$ for the modular convergence; i.~e.,
there exists $\lambda > 0$ such that for all $\alpha$ with
 $|\alpha|\leq 1$
\[\int_{Q_T}M\Big(x,\frac{D_x^{\alpha}u_n- D_x^{\alpha}u}{\lambda}\Big)
\,{\rm d}x\,{\rm d}t \to 0 \ as\ n\ \to \infty,\]
and, in particular, this implies that $(u_n)$ converges to $u$
in $W^{1,x}L_M(Q_T)$ for the weak-$\ast$ topology
$\sigma(\Pi L_M, \Pi L_{\bar{M}})$.
Consequently
\[
\overline{\mathcal{D}(Q_T)}^{\sigma(\Pi L_M, \Pi L_{\bar{M}})}
= \overline{\mathcal{D}(Q_T)}^{\sigma(\Pi L_M, \Pi E_{\bar{M}})}.
\]
This space will be denoted by $W_0^{1,x}L_M(Q_T)$.
Furthermore,
\[
W_0^{1,x}E_M(Q_T)= W_0^{1,x}L_M(Q_T)\cap \Pi E_{\bar{M}}(Q_T).
\]
Poincar\'e's inequality also holds in $W_0^{1,x}L_M(Q_T)$,
i.~e. there exists a constant $C > 0$
such that for all $u \in W_0^{1,x}L_M(Q_T)$ one has
\begin{equation}\label{Poincare}
 \sum_{| \alpha | \leq 1 }\| D_x^{\alpha}u\|_{M,Q_T}
 \leq C \sum_{| \alpha | = 1 }\| D_x^{\alpha}u\|_{M,Q_T}.
\end{equation}
The dual space of $W_0^{1,x}E_M(Q_T)$ will be denoted by
$W^{-1,x}L_{\bar{M}}(Q_T)$, and it can be shown that
\[
 W^{-1,x}L_{\bar{M}}(Q_T)
= \Big\{ f = \sum_{| \alpha| \leq 1}D_x^{\alpha}f_{\alpha}
 : f_{\alpha} \in L_{\bar{M}}(Q_T),\text{ for all }\alpha \Big\}.
\]
This space will be equipped with the usual quotient norm
\[
\|f \| = \inf \sum_{|\alpha|\leq 1}\|
 D_x^{\alpha}f_{\alpha}\|_{\bar{M}, Q_T}
\]
where the infimum is taken over all possible functions
$f_\alpha\in L_{\bar{M}}(Q_T)$ from which the decomposition
$f = \sum_{| \alpha| \leq 1}D_x^{\alpha}f_{\alpha}$ holds.

We also denote by $W^{-1,x}E_{\bar{M}}(Q_T)$ the subspace of
$W^{-1,x}L_{\bar{M}}(Q_T)$ consisting of those linear forms which are
$\sigma(\Pi L_M,\Pi E_{\bar{M}})$-continuous. It can be shown that
\[
W^{-1,x}E_{\bar{M}}(Q_T)= \big\{ f = \sum_{|\alpha| \leq 1}
D_x^{\alpha}f_{\alpha}: f_{\alpha} \in E_{\bar{M}}(Q_T) \big\}.
\]
The following Lemma will be needed later on this paper.

\begin{lemma}\label{injec}
Let $P$ be a Musielak function such that \eqref{p2} is satisfied.
Assume that $s^2\leq P(x,s)$,
for all a.~a.~$x\in \Omega$ and all $s\in\mathbb{R}$.
Then the following continuous inclusions hold:
\[
L_P(\Omega)\hookrightarrow L^2(\Omega) \hookrightarrow L_{\bar{P}}(\Omega).
\]
In particular,
$W^1_0L_P(\Omega)\hookrightarrow H^1_0(\Omega) $
 and $H^{-1}(\Omega)\hookrightarrow W^{-1}L_{\bar{P}}(\Omega)$.
Furthermore, if $M$ is a Musielak function
 verifying~\eqref{h1} and such that $P\ll M$, then the same continuous
inclusions hold for $M$; that is,
 \[
L_M(\Omega)\hookrightarrow L^2(\Omega) \hookrightarrow L_{\bar{M}}(\Omega),
\]
and also $W^1_0L_M(\Omega)\hookrightarrow H^1_0(\Omega) $
 and $H^{-1}(\Omega)\hookrightarrow W^{-1}L_{\bar{M}}(\Omega)$.
\end{lemma}

\begin{proof}
From the estimate on $P$ we have
\begin{equation}\label{eq:v2Pv}
\int_\Omega v^2 \, {\rm d}x \leq \int_\Omega P(x,v)\,{\rm d}x,\quad
\text{for all $v\in \mathcal{L}_P(\Omega)$}.
\end{equation}
Taking $v=u/\|u\|_{(P)}$ with $u\ne0$ in~\eqref{eq:v2Pv} and
using~\eqref{eq:ineq1} it yields
 \[
\|u\|_{L^2(\Omega)}\le \|u\|_{(P)}\quad  \text{for all $u\in L_P(\Omega)$,}
 \]
and	the first assertions of this Lemma are readily deduced.

Now let $P \ll M$. For $\varepsilon\in(0,1)$ there exists $t_0$ that
\begin{equation}\label{P<<M}
 P(x,t) \leq M(x,\varepsilon t) \quad \text{for all $t\geq t_0$
 and a.~a.~}x\in \Omega.
\end{equation}
  Then, taking $v\in \mathcal{L}_M(\Omega)$ and using
Remark~\ref{R1}, we deduce that for some constant $C_1=C_1(t_0)$,
\begin{align*}
\int_\Omega v^2 \, {\rm d}x
&\leq \int_{\{|v|< t_0\}} P(x,v)\,{\rm d}x+
 \int_{\{|v|\ge t_0\}} P(x,v)\,{\rm d}x\\
&\le C_1
+ \int_{\Omega} M(x,\varepsilon v)\,{\rm d}x\\
&\le C_1
+ \varepsilon\int_{\Omega} M(x,v)\,{\rm d}x.
\end{align*}
Making $v=u/\|u\|_{(M),Q_T}$, $u\ne0$, in this last inequality and
using~\eqref{eq:ineq1} we finally deduce
\[
\|u\|_{L^2(\Omega)}\le C_3\|u\|_{(M),Q_T}\quad \text{for all }u\in {L}_M(\Omega),
\]
where $C_3=(C_1+\varepsilon)^{1/2}$.
 \end{proof}

\begin{remark} \label{rem:injec} \rm
 Under the assumptions of Lemma~\ref{injec}, we have
 \[
L^2(0,T;H^{-1}(\Omega))\hookrightarrow W^{-1,x}L_{\bar{P}}(Q_T)\hookrightarrow
W^{-1,x}E_{\bar{M}}(Q_T).
\]
Indeed, let $f\in L^2(0,T;H^{-1}(\Omega))$. Then, for some
$f_\alpha\in L^2(Q_T)$, $f=\sum_{|\alpha|\le1}
 \nabla_x^\alpha f_\alpha$. But according to Lemma~\ref{mod}
 $L^2(Q_T)\subset L_{\bar{P}}(Q_T)\subset E_{\bar{M}}(Q_T)$ and thus
 \[
f\in W^{-1,x}L_{\bar{P}}(Q_T)\hookrightarrow W^{-1,x}E_{\bar{M}}(Q_T).
\]
\end{remark}

We will  use  truncations in the definition of our approximate problems.
To do so, for $K>0$, we introduce the truncation at height $K$, denoted by
$T_K\colon\mathbb{R}\mapsto\mathbb{R}$, as
\begin{equation}\label{eq:truncation}
T_K(s)=\min(K,\max(s,-K))=
\begin{cases}
 s & \text{if } |s|\leq K,\\
K s/|s| &\text{if } |s|>K,
\end{cases}
\end{equation}

\section{Compactness results}

In the sequel, we will  use the following results which concern
mollification with respect to time and space variables and some trace
results. Also, unless stated the contrary, $\Omega\subset\mathbb{R}^d$ is
 a bounded and open set with a Lipschitz-continuous boundary, and $M$
is Musielak function. We put $Q_T=\Omega\times(0,T)$.
For a function $u\in L^1(Q_T)$ we introduce the function
$\tilde{u}\in L^1(\Omega\times\mathbb{R})$ as
$\tilde{u}(x,s)=u(x,s)\chi_{(0,T)}$ and
define, for all $\mu>0$, $t\in[0,T]$ and a.e.~$x\in \Omega$,
the function $u_\mu$ given as follows
\begin{equation}\label{eq:timereg}
u_\mu(x,t) = \mu \int_{-\infty}^t\tilde{u}(x,s)\text{exp}(\mu(s-t))\,{\rm d}s.
\end{equation}

\begin{lemma}[\cite{BA}]  \label{lem3}
The following assertions hold:
\begin{enumerate}
\item Let $u\in L_M(Q_T)$. Then $u_\mu\in C([0,T]; L_M(\Omega))$ and
$u_\mu\to u$ as
$\mu\to +\infty$ in $L_M(Q_T)$ for the modular convergence.

\item Let $u\in W^{1,x}L_M(Q_T)$. Then $u_\mu\in C([0,T]; W^1L_M(\Omega))$ and
$u_\mu\to u$ as $\mu\to +\infty$ in $W^{1,x}L_M(Q_T)$ for the modular convergence.

\item Let $u\in E_M(Q_T)$ (respectively, $u\in W^{1,x}E_M(Q_T)$). Then $u_\mu\to u$ as $\mu\to +\infty$ strongly in $E_M(Q_T)$ (respectively, strongly in $W^{1,x}E_M(Q_T)$).

\item Let $u\in W^{1,x}L_M(Q_T)$ then $\frac{\partial u_\mu}{\partial
t} = \mu(u-u_\mu)\in W^{1,x}L_M(Q_T)$.

\item Let $(u_n)\subset W^{1,x}L_M(Q_T)$ and $u\in W^{1,x}L_M(Q_T)$ such that $u_n\to u$ strongly in $W^{1,x}L_M(Q_T)$ (respectively, for the modular convergence). Then, for all $\mu>0$,
$(u_n)_\mu\to u_\mu$ strongly in $W^{1,x}L_M(Q_T)$ (respectively, for the modular convergence).

\end{enumerate}
\end{lemma}


\begin{lemma}\label{lm_inj}
The following embedding holds with continuous injection
\begin{equation}\label{inj1}
E_M(Q_T)\subset L^1(0,T;E_M(\Omega))
 \end{equation}
\end{lemma}

\begin{proof}
Since $M(x,t)$ is convex with respect to $t$, then for every
$\lambda \geq 1$, $t \in [0,T]$ and a.~a.~$x\in \Omega$ we have
\begin{equation}\label{eq1}
 \alpha M(x,t)\leq M(x,\lambda t)\text{ and }\lambda M(x,t/\lambda)\leq M(x,t).
\end{equation}
Let $u \in E_M(Q_T)\setminus\{0\}$.
Owing to the definition of the space $E_M(Q_T)$, we have
$\int_{Q_T}M(x,\lambda u(x,t))\,{\rm d}x\,{\rm d}t < \infty$
for every $\lambda \geq 0$. Hence,	
$\int_{\Omega}M(x,\lambda u(x,t))\,{\rm d}x< \infty$ for
a.~a.~$t\in [0,T]$ and for all $\lambda \geq 0$.
Therefore the function $u(\cdot,t) \in E_M(\Omega)$
for a.~a.~$t\in[0,T]$. In particular,
\[
\int_{\Omega}M\Big(x,\frac{u(x,t)}{\|u(\cdot,t)\|_{M,\Omega}}\Big)\,{\rm d}x
 = 1 \quad \text{ for a.~a.~}t \in [0,T].
\]
Then, having in mind~\eqref{eq1},
\begin{align*}
&\int_{0}^{T}\|u\|_{M,\Omega}\,{\rm d}t \\
&=\int_{\{\|u(\cdot,t)\|_{M,\Omega}<1 \}}
\|u(\cdot,t)\|_{M,\Omega}\,{\rm d}t
+ \int_{\{ \|u(\cdot,t)\|_M\geq 1 \} }
\|u(\cdot,t)\|_{M,\Omega}\,{\rm d}t
\\
&\leq T + \int_{\{\|u(\cdot,t)\|_{M,\Omega}\geq 1 \} }
\|u(\cdot,t)\|_{M,\Omega}
\int_{\Omega}
M\Big(x,\frac{u(x,t)}{\|u(\cdot,t)\|_{M,\Omega}}\Big) \,{\rm d}x{\rm d}t
\\
&\leq T +\int_{\{\|u(\cdot,t)\|_{M,\Omega}\geq 1 \} }
\int_{\Omega}\|u(\cdot,t)\|_{M,\Omega}
 M\Big(x,\frac{u(x,t)}{\|u(\cdot,t)\|_{M,\Omega}}\Big) \,{\rm d}x{\rm d}t
\\
&\leq T + \int_{\{\|u(\cdot,t)\|_{M,\Omega}\geq 1 \} }
\int_{\Omega} M(x,u(x,t)) \,{\rm d}x{\rm d}t
\\
&\leq T + \int_{Q_T} M(x,u(x,t)) \,{\rm d}x{\rm d}t.
\end{align*}
By taking $u/\|u\|_{M,Q_T}$ instead of $u$ into the first and last terms
of this inequality, using \eqref{eq:ineq1} and \eqref{eq:equivnorms},
it follows that
$\int_{0}^{T}\|u\|_{M,\Omega}\,{\rm d}t \leq 2(T+1) \|u\|_{M,Q_T} $.
 \end{proof}

A straightforward consequence of Lemma~\ref{lm_inj} is given in the next
result.

\begin{lemma}\label{lm_inje}
 The following embeddings hold with continuous injections
\begin{gather}\label{inj2}
W^1E_M(Q_T) \subset L^1\big(0,T;W^1E_M(\Omega)\big), \\
\label{inj3}
W^{-1}E_{\bar{M}}(Q_T) \subset L^1\big(0,T;W^{-1}E_{\bar{M}}(\Omega)\big).
\end{gather}
\end{lemma}

The proof of the next three lemmas are straightforward adaptations
of the ones given in \cite[Lemmas 2, 5 and Theorem2]{Driss3}.

\begin{lemma}\label{rc2}
Let $Y$ be a Banach space such that $L^1(\Omega)\subset Y$ with
continuous embedding.
If $\mathcal{F}$ is bounded in $W^{1,x}_0L_M(Q_T)$ and
relatively compact in $L^1(0,T; Y)$
then $\mathcal{F}$ is relatively compact in $L^1(Q_T)$ and in $E_P(Q_T)$
for all $P\ll M$.
\end{lemma}

\begin{lemma}\label{lem5}
Let $\Omega$ be a bounded open subset of $\mathbb{R}^d$ with the segment
property. Consider the Banach space
\[
W=\Big\{u\in W_0^{1,x}L_{M}(Q_T):
\frac{\partial u}{\partial t}\in
W^{-1,x}L_{\bar{M}}(Q_T)+L^1(Q_T)\Big\}.
\]
Then the embedding $W\subset C([0,T];L^1(\Omega))$ holds and is
continuous.
\end{lemma}

\begin{lemma}\label{rc1}
 If $\mathcal{F}$ is bounded in $W^{1,x}_0L_M(Q_T)$
 and $\big\{\frac{\partial f}{\partial t}: f\in \mathcal{F}\big\}$
 is bounded in $W^{-1,x}L_{\bar{M}}(Q_T)$ then $\mathcal{F}$ is relatively
 compact in $L^1(Q_T)$.
\end{lemma}

The existence result given in Theorem~\ref{exist} will be useful in our analysis.
 It is related to a second-order partial differential operator
\[
 \mathbf{A}\colon D(\mathbf{A}) \subset W^{1,x}L_{M}(Q_{T})
\mapsto W^{-1,x}L_{\bar{M}}(Q_{T})
\]
in divergence form	$\mathbf{A}(u)=-\operatorname{div} \mathbf{a}(x,t, \nabla u)$,
where
\begin{equation}\label{a_catheo}
\mathbf{a}\colon \Omega \times (0,T) \times \mathbb{R}^{d} \mapsto \mathbb{R}^{d}
\text{ is a Carath\'{e}odory function}
\end{equation}
and for almost every $(x,t)\in Q_{T}$ and for all
$\xi, \xi' \in \mathbb{R}^{d}$, $\xi \neq \xi'$, one has
\begin{gather}\label{eq:croi}
|\mathbf{a}(x,t,\xi)| \leq \beta (c_{1}(x,t)+ \bar{M}_{x}^{-1}M(x,k_{1}|\xi|), \\
\label{eq:mono}
(\mathbf{a}(x,t,\xi) - \mathbf{a}(x,t,\xi'))(\xi - \xi') >0, \\
\label{coer1}
\mathbf{a}(x,t,\xi) \xi \geq \alpha M(x, | \xi |).
\end{gather}

For a function $f\in W^{-1,x}L_{\bar{M}}(Q_{T})$ and a function $u_0\in L^2(\Omega)$
we consider the parabolic problem given by
 \begin{equation} \label{1.1}
\begin{gathered}
 \frac{\partial u}{\partial t}
 -\operatorname{div} \mathbf{a}(x,t,\nabla u)= f \quad \text{in } Q_T, \\
  u(x,0)=u_0(x) \text{ in } \Omega, \\
  u=0 \text{ on }\partial \Omega\times (0,T).
 \end{gathered}
\end{equation}

\begin{theorem}[\cite{BA}] \label{exist}
 Under  assumptions \eqref{a_catheo}-\eqref{coer1} there exists at least one
 weak solution to problem~\eqref{1.1},
 $u\in D(\mathbf{A})\cap W^{1,x}_0L_{M}(Q_{T})\cap C([0,T];L^2(\Omega))$
 such that $\mathbf{a}(x,t,\nabla u)\in W^{-1,x}L_{\bar{M}}(Q_{T})$
and for all $v\in W^{1,x}_0L_{M}(Q_{T})$ with
 $\frac{\partial v}{\partial t}\in W^{-1,x}L_{\bar{M}}(Q_{T})$
and for all $\tau\in [0,T]$  one has
\begin{align*}
& -\langle\frac{\partial v}{\partial t},u\rangle_{Q_{\tau}}	
  + \int_\Omega u(x,\tau)v(x,\tau)\,{\rm d}x
  +\int_0^\tau\int_\Omega
 \mathbf{a}(x,t,\nabla u)\nabla v\,{\rm d}x{\rm d}t
 \\
& =  \langle f,v\rangle_{Q_{\tau}}	+ \int_\Omega u_0(x)v(x,0)\,{\rm d}x,
 \end{align*}
where the $\langle \cdot,\cdot\rangle_{Q_{\tau}}$ stands for the duality
pairing between the spaces $W^{-1,x}L_{\bar{M}}(Q_{\tau})$
 and $W^{1,x}_0L_{M}(Q_{\tau})$. Moreover, for all $\tau\in [0,T]$,
the following energy identity holds	
\[
\frac{1}{2}\int_\Omega |u(x,\tau)|^2\,{\rm d}x
 +\int_0^\tau   \int_\Omega \mathbf{a}(x,t,\nabla u)\nabla u\,{\rm d}x{\rm d}t
=	\langle f,u\rangle_{Q_{\tau}}	
+ \frac{1}{2}\int_\Omega |u_0(x)|^2\,{\rm d}x.
\]
\end{theorem}

\section{Notion of capacity solution}\label{s3}

In this section, we give the definition of a capacity solution for
problem \eqref{P} in the context of the Musielak-Orlicz-Sobolev spaces.

Let $\Omega$ be an open subset of $\mathbb{R}^d$ and let $M$ be
 a Musielak-Orlicz function satisfying the conditions of Lemma~\ref{lem-assum}.
We first consider the Banach space
\[
\mathbf{W}=\Big\{ v\in W^{1,x}_0L_M(Q_T):
\frac{\partial v}{\partial t}\in W^{-1,x}L_{\bar{M}}(Q_T) \Big\}
\]
provided with its standard norm
\[
\|v\|_{\mathbf{W}}=\|v\|_{W^{1,x}L_M(Q_T)}
+\|\frac{\partial v}{\partial t}\|_{W^{-1,x}L_{\bar{M}}(Q_T)}.
\]
Throughout this paper $\langle\cdot,\cdot\rangle$ stands for
 the duality pairing between the spaces
 $W^{1,x}L_M(Q_T)\cap L^2(Q_T)$
 and $W^{-1,x}L_{\bar{M}}(Q_T)+L^2(Q_T)$ or between
$W^{1,x}_0L_M(Q_T)$ and $W^{-1,x}L_{\bar{M}}(Q_T)$,
and we assume the following conditions:
\begin{equation}
P\ll M \text{ and } t^2\leq P(x,t) \text{ for a.~a.~}x\in\Omega
\text{ and all }t\in\mathbb{R},
\end{equation}
and their respective complementary functions, $\bar{M}$ and $\bar{P}$,
satisfy~\eqref{h1} and~\eqref{p2}, respectively.
We consider a second order partial differential operator
\[
A\colon D(A) \subset W^{1,x}L_{M}(Q_{T})\mapsto W^{-1,x}L_{\bar{M}}(Q_{T})
\]
in divergence form $Au=-\operatorname{div} a(x,t,u,\nabla u)$ where
$a\colon \Omega \times (0,T) \times \mathbb{R} \times \mathbb{R}^{d}
\mapsto \mathbb{R}^{d}$ is a Carath\'eodory function
(that is, $a=a(x,t,s,\xi)$ is measurable in $(x,t)$ for any value of
$(s,\xi)$ and continuous with respect to the arguments $(s,\xi)$ for
a.~a.~$(x,t)\in\Omega\times(0,T)$) satisfying the following assumptions,
 for a.~a.~$(x,t)\in Q_{T}$, all $s \in \mathbb{R}$, and all
 $\xi,\xi'\in \mathbb{R}^{d}$, $\xi \neq \xi'$,
\begin{gather}\label{croi}
|a(x,t,s,\xi)| \leq \zeta (c(x,t)+ \bar{M}_{x}^{-1}(P(x,k|s|))
+ \bar{M}_{x}^{-1}(M(x,k|\xi|)), \\
\label{mono}
[a(x,t,s,\xi) - a(x,t,s,\xi')][ \xi - \xi' ]
\geq \alpha (M(x,|\xi-\xi'|)+M(x,|s|)), \\
\label{28P}
|a(x,t,s_1,\xi)-a(x,t,s_2,\xi)|
\leq \zeta\Big[e(x,t) + |s_1|+|s_2|+P^{-1}(x,kM(|\xi|))\Big], \\
\label{30}
a(x,t,s,0)=0,
\end{gather}
with $ c(x,t) \in E_{\bar{M}}(Q_{T})$, $e\in E_{P}(Q_T)$ and $\alpha$, $\zeta$,
 $k>0$ are given real numbers.
 \begin{gather}\label{31}
\rho\in C(\mathbb{R}) \text{ and there exists $\bar{\rho}\in\mathbb{R}$ such that }
 0<\rho(s)\leq \bar{\rho}, \text{ for all } s\in \mathbb{R}, \\
\label{32}
\varphi_0 \in L^2(0,T; H^1(\Omega))\cap L^\infty (Q_T), \\
\label{33}
 u_0\in L^2(\Omega).
\end{gather}

\begin{remark}  \rm
Notice that from \eqref{mono} and \eqref{30} we obtain the elliptic condition
\begin{equation}\label{coer}
a(x,t,s,\xi) \xi \geq \alpha M(x, |\xi|),
\text{ for a.~a.~$(x,t)\in Q_{T}$, and all $(s,\xi) \in
\mathbb{R}\times\mathbb{R}^d$}.
\end{equation}
\end{remark}

The concept of capacity solution now follows.

\begin{definition}  \label{def1} \rm
 A triplet $(u,\varphi,\Phi)$ is called a capacity solution
 of~\eqref{P} if the following conditions are fulfilled:
\begin{itemize}	
\item[(1)] $u\in\mathbf{W}$, $a(x,t,u,\nabla u)\in L_{\bar{M}}(Q_T)^d,
\varphi\in L^\infty(Q_T)$ and $\Phi \in L^2(Q_T)^d$.

\item[(2)] $(u,\varphi,\Phi)$ satisfies the system of partial differential equations
 \begin{gather}
 \frac{\partial u}{\partial t}
 -\operatorname{div} a(x,t,u,\nabla u)= \operatorname{div}(\varphi\Phi) \text{ in }Q_T,
 \\
 \operatorname{div} \Phi=0 \text{ in }Q_T,
 \end{gather}


\item[(3)] For every $S\in C^1_0(\mathbb{R})$ (functions of $C^1(\mathbb{R})$
with compact support), one has
$S(u)\varphi-S(0)\varphi_0 \in L^2(0,T;H^1_0(\Omega))$, and
\begin{equation}\label{eq:Su}
S(u)\Phi=\rho(u)[\nabla (S(u)\varphi)-\varphi \nabla S(u)],
\end{equation}

\item[(4)] $u(\cdot,0)=u_0 \text{ in } \Omega$.
\end{itemize}
\end{definition}

Notice that, thanks to Lemma~\ref{lem5} and the regularity of $u$,
we obtain in particular $u\in C([0,T];L^1(\Omega))$ and thus the initial
condition in (4) makes sense at least in $L^1(\Omega)$.

\begin{remark}  \rm
The notion of capacity solution involves a triplet
$(u,\varphi,\Phi)$ whereas the original problem~\eqref{P} refers only to
two unknowns, $u$ and $\varphi$. Evidently, the vector function $\Phi$
is, in some way, related to $u$ and $\varphi$. For instance,
if we were allowed to take $S=1$ in~\eqref{eq:Su}, we would readily obtain
$\Phi=\rho(u)\nabla \varphi$. But the choice $S=1$ is not possible since
 it does not belong to the space $C^1_0(\mathbb{R})$.
To circumvent this situation, consider, for any $m>0$, a function
$S_m\in C^1_0(\mathbb{R})$ such that
$S_m(s)=1$ in $\{|s|\le m\}$. Using $S_m$ in~\eqref{eq:Su} and multiplying
this expression by $\chi_{\{|u|\le m\}}$ we obtain
\[
\chi_{\{|u|\le m\}}\Phi=\chi_{\{|u|\le m\}}\rho(u)\nabla (S_m(u)\varphi),\quad
\text{for all } m>0.
\]
This last expression provides a meaning, al least in a pointwise sense,
 to $\nabla\varphi$ so that $\Phi=\rho(u)\nabla\varphi$ almost everywhere
 in $Q_T$.
\end{remark}


\section{An existence result}\label{s5}

This section is devoted to establish the main theorem of this paper:

\begin{theorem}\label{t1}
Under the assumptions \eqref{h1}, \eqref{p2}, \eqref{eq:tA}, \eqref{eq:barM1}
and \eqref{croi}-\eqref{33}, the system~\eqref{P} admits a capacity solution
in the sense of Definition~\ref{def1}.
\end{theorem}

To prove this theorem, we need first to show the existence of a weak
solution to a similar problem but with stronger assumptions; namely,
there exists $c\in E_{\bar{M}}(Q_T)$, and two real numbers $\zeta>0$ and $k\ge0$,
such that for almost all $(x,t)\in Q_T$ and for all
$s\in \mathbb{R}$, $\xi\in \mathbb{R}^d$, we have
\begin{equation}\label{27bis}
|a(x,t,s,\xi)|\leq \zeta[c(x,t) +\bar{M}_{x}^{-1}(M(x,k|\xi|))],
\end{equation}
and
\begin{equation}\label{31bis}
\begin{gathered}
 \rho\in C(\mathbb{R}) \text{ and there exist $\rho_1$ and
$\rho_2 \in\mathbb{R}$ such that } \\
 0<\rho_1\leq \rho(s)\leq \rho_2, \text{ for all } s\in \mathbb{R}.
\end{gathered}
\end{equation}

\begin{theorem}\label{t2}
Assume \eqref{h1}, \eqref{p2}, \eqref{mono}-\eqref{30},
\eqref{32}, \eqref{33}, \eqref{27bis} and \eqref{31bis} are satified.
Then, there  exists a weak solution $(u,\varphi)$ to problem~\eqref{P}; that is,
 \begin{gather*}
 u\in W^{1,x}_0L_M(Q_T)\cap C([0,T];L^2(\Omega)),\quad
 a(x,t,u,\nabla u)\in L_{\bar{M}}(Q_T)^d, \\
\varphi-\varphi_0\in L^\infty(0,T; H^1_0(\Omega))\cap L^\infty(Q_T), \\
u(\cdot,0)=u_0 \text{ in } \Omega,\\
\int_0^t \big\langle \frac{\partial u}{\partial  t},\phi\big\rangle
+ \int_0^t   \int_\Omega a(x,t,u,\nabla u)\nabla \phi=
 -\int_0^t  \int_\Omega\rho(u)\varphi\nabla\varphi\nabla\phi, \\
 \text{for all } \phi\in W^{1,x}_0L_M(Q_T), \text{ for all } t\in [0,T],\\
\int_\Omega \rho(u)\nabla \varphi\nabla \psi=0, \quad \text{for all }
 \psi\in H_0^1(\Omega), \text{ a.e.\ } t\in (0,T).
\end{gather*}
\end{theorem}

\begin{proof}
To show the existence of a weak solution, Schauder's fixed point
theorem will be applied together with the existence and uniqueness result
of a weak solution to a parabolic equation.

For every $\omega\in E_{P}(Q_T)$ and almost everywhere $t\in (0,T)$,
we consider the elliptic problem
\begin{equation}\label{omega}
\begin{gathered}
\operatorname{div} (\rho(\omega)\nabla \varphi)=0 \quad
\text{in } \Omega\times(0,T), \\
\varphi=\varphi_0 \quad \text{on } \partial \Omega\times (0,T).
 \end{gathered}
\end{equation}

Thanks to Lax-Milgram's theorem, \eqref{omega} has an unique solution
$\varphi(t)\in H^1(\Omega)$, for almost all $t\in(0,T)$.
In fact, $\varphi $ is measurable in $t$
with values in $H^1(\Omega)$ \cite{AC}. In that case, it is
$\varphi\in L^\infty(0,T;H^1(\Omega))$.
Indeed, by the maximum principle we have
\begin{equation}\label{phii}
\|\varphi\|_{L^\infty(Q_T)}\leq \|\varphi_0\|_{L^\infty(Q_T)}.
\end{equation}
Using $\varphi-\varphi_0 \in H^1_0(\Omega)$
as a test function in~\eqref{omega} we obtain
\[
\int_{\Omega} \rho(\omega)\nabla \varphi \nabla
(\varphi-\varphi_0) =0,
\]
hence
\[
\rho_1\int_{\Omega}|\nabla \varphi|^2\, {\rm d}x
\leq \int_{\Omega} \rho(\omega) |\nabla \varphi\|
 \nabla \varphi_0|\,{\rm d}x
\leq \rho_2 \int_{\Omega}|\nabla \varphi|
|\nabla \varphi_0| \,{\rm d}x.
\]
By the Cauchy-Schwarz inequality, we obtain
\begin{equation}\label{varphii}
\int_{\Omega}|\nabla \varphi|^2 \,{\rm d}x\leq
C(\rho_1,\rho_2, \varphi_0 )=C, \quad \text{a.e. } t\in (0,T) .
\end{equation}

Note that the right-hand side in the original parabolic equation is
$\rho(u)|\nabla \varphi|^2\in L^1(\Omega\times(0,T))$.
Thanks to the elliptic equation, this term also belongs to the space
$L^2(0,T;H^{-1}(\Omega))$. Indeed, let $\phi\in \mathcal{D}(\Omega)$
and take $\xi=\phi\varphi$ as a test function in~\eqref{omega}.
We have, for a.e. $t\in[0,T]$,
\[
\int_\Omega \rho(\omega)\nabla\varphi\nabla(\phi\varphi)\,{\rm d}x=0,
\]
that is
\[
 \int_\Omega \rho(\omega)|\nabla\varphi|^2\phi \,{\rm d}x=
-\int_\Omega \rho(\omega)\varphi\nabla\varphi\nabla\phi \,{\rm d}x=
\langle \operatorname{div} (\rho(\omega)\varphi\nabla\varphi),
 \phi\rangle_{\mathcal{D}'(\Omega),\mathcal{D}(\Omega)}.
\]
This means that
\begin{equation}\label{eq:identity}
\rho(\omega)|\nabla\varphi|^2
= \operatorname{div} (\rho(\omega)\varphi\nabla\varphi)\text{ in }
\mathcal{D}'(\Omega) \text{ and a.e.\ in }[0,T].
\end{equation}
Since
$\rho(\omega)\varphi\nabla\varphi\in L^2(Q_T)^d$ we finally deduce the regularity
\[
\operatorname{div} (\rho(\omega)\varphi\nabla\varphi)\in L^2(0,T;H^{-1}(\Omega)).
\]
The identity~\eqref{eq:identity} is one of the keys that allows us to solve
the classical thermistor problem and the introduction of the notion of a
capacity solution as well.

Now we introduce the  parabolic problem
\begin{equation}\label{Para}
\begin{gathered}
\frac{\partial u}{\partial t}
-\operatorname{div} a(x,t,\omega,\nabla u)= \operatorname{div}
(\rho(\omega)\varphi\nabla\varphi) \quad \text{in } Q_T, \\
u=0 \quad \text{on } \partial \Omega\times (0,T), \\
u(\cdot,0)=u_0 \quad \text{in } \Omega.
\end{gathered}
\end{equation}
The variational formulation of the parabolic equation is given as follows.
\begin{equation}\label{Para1}
\begin{gathered}
 u\in W^{1,x}_0L_M(Q_T)\cap C([0,T];L^2(\Omega)),\quad
 a(x,t,\omega,\nabla u)\in L_{\bar{M}}(Q_T)^d,\\
\int_0^t \\langle \frac{\partial u}{\partial
 t},\phi\rangle + \int_0^t   \int_\Omega a(x,t,\omega,\nabla u)\nabla \phi
 =-\int_0^t   \int_{\Omega}
\rho(\omega)\varphi \nabla \varphi \nabla \phi, \\
 \quad\text{for all }\phi \in W^{1,x}_0L_M(Q_T),\quad \text{for all }t\in[0,T],\\
 u(\cdot,0)=u_0 \quad \text{in } \Omega.
 \end{gathered}
\end{equation}

Note that $\operatorname{div}(\rho(\omega)
\varphi\nabla\varphi) \in L^2(0,T;H^{-1}(\Omega)) \hookrightarrow
 W^{-1,x}E_{\bar{M}}(Q_T) $ due to \eqref{omega}, \eqref{phii}, \eqref{varphii},
Lemma~\ref{injec} and Remark~\ref{rem:injec}.

By Theorem~\ref{exist}, we have the existence of a solution to the problem
\eqref{Para1}. Now, we show that $|\nabla u|\in \mathcal{L}_M(Q_T)$,
and the estimates
\begin{gather}\label{p23}
\int_0^T   \int_\Omega M(x,|\nabla u|)\,{\rm d}x\,{\rm d}t
\leq C(u_0,\varphi_0, \alpha, T, \rho_2)=C_0, \\
\label{eq:aomegagradu}
\|a(x,t,\omega,\nabla u)\|_{\bar{M},Q_T}
\leq C_1,
\end{gather}
where $C_1$ only depends on data, but not on $\omega$.
Indeed, let $\lambda>0$ such that $|\nabla u|/\lambda\in \mathcal{L}_M(Q_T)$.
Since $\varphi \in L^2(0,T;H^1(\Omega))\subset W^{1,x}L_{\bar{M}}(Q_T)$,
there exists $\mu>0$ such that
$\frac{2}{\alpha\mu}\rho_2\|\varphi_0\|_{L^\infty(Q_T)}|\nabla
\varphi|\in \mathcal{L}_{\bar{M}}(Q_T)$.
 By taking $\phi=u$ as a test function in \eqref{Para}, from
\eqref{croi}, \eqref{30}, \eqref{31bis}, \eqref{phii} and Young's inequality,
we obtain
\begin{align*}
&\frac{\alpha}{\lambda\mu}\int_0^T \int_\Omega M(x,|\nabla u|)\,{\rm d}x\,{\rm d}t \\
&\le \frac{1}{\lambda\mu}\int_0^T \int_\Omega a(x,t,\omega,\nabla u)\nabla u
 \,{\rm d}x\,{\rm d}t \\
&\leq \frac{1}{2\lambda\mu}\|u_0\|^2_{L^2(\Omega)}
 +\frac{\alpha\mu}{2} \int_0^T  \int_\Omega \bar{M}(x,\frac{2}{\alpha\mu}
 \rho_2\|\varphi_0\|_{L^\infty(Q_T)}|\nabla \varphi|)\,{\rm d}x\,{\rm d}t \\
&\quad + \frac{\alpha}{2\mu}\int_0^T \int_\Omega M(x,|\nabla u|/\lambda)
 \,{\rm d}x\,{\rm d}t.
\end{align*}
This shows that $|\nabla u|\in \mathcal{L}_M(Q_T)$ and, consequently,
 estimate \eqref{p23} is derived by just taking $\lambda=1$ in this last
inequality. In order to obtain~\eqref{eq:aomegagradu}, first notice that
from the last inequality we also have
\begin{equation}\label{eq:aomegagradugradu}
\int_0^T   \int_\Omega a(x,t,\omega,\nabla u)\nabla u \,{\rm d}x\,{\rm d}t
\leq \alpha C_0.
\end{equation}
Then, owing to~\eqref{mono}, for any $\phi\in W^{1,x}_0E_M(Q_T)$
such that $\|\nabla \phi\|_{M,Q_T}=1/(k+1)$ it yields
\[
0\le \int_0^T    \int_\Omega
(a(x,t,\omega,\nabla u)-a(x,t,\omega,\nabla \phi))(\nabla u-\nabla\phi)
\,{\rm d}x\,{\rm d}t,
\]
and thus, using \eqref{eq:aomegagradugradu} and Young's inequality,
\begin{align*}
&\int_0^T    \int_\Omega a(x,t,\omega,\nabla u)\nabla\phi
\,{\rm d}x\,{\rm d}t\\
&\le \int_0^T    \int_\Omega a(x,t,\omega,\nabla u)\nabla u
\,{\rm d}x\,{\rm d}t
-\int_0^T    \int_\Omega a(x,t,\omega,\nabla \phi)(\nabla u-\nabla\phi)
\,{\rm d}x\,{\rm d}t\\
&\le \alpha C_0
+\int_0^T     \int_\Omega  |a(x,t,\omega,\nabla \phi)\nabla u|\, {\rm d}x\,{\rm d}t
+\int_0^T     \int_\Omega  a(x,t,\omega,\nabla \phi)\nabla \phi\, {\rm d}x\,{\rm d}t
\\
&\le \alpha C_0
+2\zeta \int_0^T     \int_\Omega
\Big[\bar{M}\Big(x,\frac{|a(x,t,\omega,\nabla \phi)|}{2\zeta}\Big)
 + M(x,|\nabla u|)\Big]\, {\rm d}x\,{\rm d}t \\
&\quad +2\zeta \int_0^T     \int_\Omega
\Big[\bar{M}\Big(x,\frac{|a(x,t,\omega,\nabla \phi)|}{2\zeta}\Big)
 + M(x,|\nabla \phi|)\Big]\, {\rm d}x\,{\rm d}t,
\end{align*}
where $\zeta$ is the constant appearing in \eqref{27bis}.
Since
\[
\bar{M}\Big(x,\frac{|a(x,t,\omega,\nabla \phi)|}{2\zeta}\Big)\le
\frac{1}{2}( \bar{M}(x,c(x,t)) + M(x,k|\nabla\phi|))
\quad \text{a.~e.\ in }Q_T,
\]
using~\eqref{eq:ineq0}, we have
\[
 \int_0^T     \int_\Omega
\bar{M}\Big(x,\frac{|a(x,t,\omega,\nabla \phi)|}{2\zeta}\Big) \,{\rm d}x\,{\rm d}t
\le \frac{1}{2} \int_0^T     \int_\Omega \bar{M}(x,c(x,t))\,{\rm d}x\,{\rm d}t
+\frac{1}{2}= C_2.
\]
Note that $C_2$ only depends on data (but not on $\omega$).
Therefore, gathering all these estimates, we deduce
for all $\phi\in W^{1,x}_0E_M(Q_T)$ such that $\|\nabla \phi\|_{M,Q_T}=1/(k+1)$
\[
\int_0^T    \int_\Omega a(x,t,\omega,\nabla u)\nabla\phi
\,{\rm d}x\,{\rm d}t \le C_1,
\]
which finally yields the estimate \eqref{eq:aomegagradu}
by considering the dual norm on $L_{\bar{M}}(Q_T)$.

Also from \eqref{croi}, \eqref{31bis}, \eqref{phii}, \eqref{varphii}
and \eqref{eq:aomegagradu} we obtain
\begin{equation}\label{eq:dudt}
\frac{\partial u}{\partial t}\in W^{-1,x}L_{\bar{M}}(Q_T)
 \text{ and }
\|\frac{\partial u}{\partial t}\|_{W^{-1,x}L_{\bar{M}}(Q_T)}\le
C_3,
\end{equation}
where, again, $C_3$ is a constant depending only on data,
but not on $\omega$.

We  define the operator
$G\colon\omega\in E_{P}(Q_T)\mapsto
 G(\omega)=u\in \mathbf{W}$,
with $u$ being the unique solution to
 \eqref{Para1}. From Lemma~\ref{rc1}, and Lemma~\ref{rc2} with $Y=L^1(\Omega)$,
 we have that $\mathbf{W} \hookrightarrow E_{P}(Q_T)$ with compact embedding.
Consequently, $G$ maps $E_{P}(Q_T)$ into itself and, due to the estimates
 \eqref{p23} and \eqref{eq:dudt}, $G$ is a compact operator.
Moreover, from~\eqref{p23} we have, for $R>0$ large enough
$G(B_R)\subset B_R$ where $B_R=\{ v\in E_{P}(Q_T): \|v\|_{L_{P}(Q_T)}\leq R \}$.

To complete the proof, it remains to show that $G$ is a continuous operator.
 Thus, let $(\omega_n)\subset B_R$ be a sequence such that
 $\omega_n\to \omega$ strongly in $E_{P}(Q_T)$ and consider the corresponding
functions to $\omega_n$, that is, $u_n=G(\omega_n)$ and
 $\varphi_n$ and put
 $F_n=\rho(\omega_n)\varphi_n \nabla \varphi_n$ and
$F=\rho(\omega)\varphi \nabla \varphi$. We have to show that
 \[
u_n \to u=G(\omega) \quad \text{strongly in } E_{P}(Q_T).
\]
Owing to $P\ll M$ and~\eqref{p23}, we have $\nabla u\in E_{P}(Q_T)^d$.
Since the inclusion $L_P(Q_T)\subset L^2(Q_T)$ is continuous, we also have
$\omega_n\to \omega$ strongly in $L^2(Q_T)$ and thus, we may extract a
subsequence, still denoted in the same way, such that $\omega_n\to \omega$
a.e.~in $Q_T$. Then, it is an easy task to show that $\varphi_n\to \varphi$
strongly in $L^2(0,T;H^1(Q_T))$ and, consequently, also for another
subsequence denoted in the same way, $F_n\to F$ strongly in
$L^2(Q_T)$.

On the other hand, since $(\omega_n)\subset L_P(Q_T)$ is bounded,
by the estimates obtained above, we deduce, again modulo a subsequence,
\begin{gather}
u_n\to U \text{ in } E_P(Q_T),\quad \text{for some $U\in E_P(Q_T),$} \label{untoU}
\\
\nabla u_n\to \nabla U \quad \text{weakly in } L^2(Q_T)^d, \label{graduntogradU}
\end{gather}
By subtracting the respective equations of \eqref{Para1} for $u_n$
and $u$, and taking $\phi =u_n-u$ as a test function,
for all $t\in [0,T]$, we obtain
\begin{align*}
&\frac{1}{2}\|u_n(t)- u(t)\|^2_{L^2(\Omega)}
+\int_0^t   \int_\Omega (a(x,s,\omega_n ,\nabla u_n)
 -a(x,s,\omega ,\nabla u))\nabla (u_n -u)\,{\rm d}x\,{\rm d}s\\
&=-\int_0^t   \int_\Omega (F_n-F)\nabla (u_n-u)\,{\rm d}x\,{\rm d}s.
\end{align*}
By using \eqref{mono}, we obtain
\begin{align*}
&(a(x,s,\omega_n ,\nabla u_n)-a(x,s,\omega ,\nabla u))\nabla  (u_n -u)\\
&\geq \alpha M(x,|\nabla (u_n -u)|)
 + (a(x,s,\omega_n ,\nabla u)-a(x,s,\omega ,\nabla u))\nabla (u_n -u).
\end{align*}
Let $h_n=a(x,s,\omega_n ,\nabla u)-a(x,s,\omega ,\nabla u)$ and
$g_n=\nabla(u_n-u)$. Then, $|h_n|\to 0$ a.e.\ in $Q_T$. For a given
positive number $\lambda_0$, to be chosen later, we have
\begin{equation}\label{eq:hngn}
\int_0^t   \int_\Omega |h_n g_n|
=\int_{\{|g_n|\le \lambda_0\}}|h_n g_n|
+\int_{\{|g_n|> \lambda_0\}}|h_n g_n| .
\end{equation}
For the first term of the right hand side of~\eqref{eq:hngn}, we have
\[
\int_{\{|g_n|\le \lambda_0\}}|h_n g_n| \le \lambda_0 \int_{Q_T}|h_n|
= \lambda_0 \int_{\{|h_n|\le 4\zeta\}}|h_n|
 + \lambda_0 \int_{\{|h_n|>4\zeta\}}|h_n|.
 \]
The first of these integrals converges trivially to zero.
As for the second one, using the fact that
$\frac{|h_n|}{4\zeta}>1$ on the set $\{|h_n|>4\zeta\}$ and
\eqref{eq:v2Pv}, it yields
\[
 \lambda_0 \int_{\{|h_n|>4\zeta\}}|h_n| \leq
 4\zeta\lambda_0 \int_{\{|h_n|>4\zeta\}} \Big(\frac{|h_n|}{4\zeta}\Big)^2 \leq
 4\zeta\lambda_0 \int_{Q_T} P\Big(x,\frac{|h_n|}{4\zeta}\Big).
 \]
 Bye \eqref{28P}, we deduce
\[
P\Big(x,\frac{|h_n|}{4\zeta}\Big)
\le  \frac{1}{4}\left(P(x,e)+ P(x,\omega_n)+ P(x,\omega)+ kM(x,|\nabla u|)\right),
\]
and since $P(x,\omega_n)\to P(x,\omega)$ strongly in $L^1(Q_T)$, by Lebesgue's
dominated theorem we have
\[
\lim_{n\to\infty} \int_{Q_T}P\Big(x,\frac{|h_n|}{4\zeta}\Big)=0,
\]
and consequently
\[
\lim_{n\to\infty}\int_{\{|g_n|\le \lambda_0\}}|h_n g_n|=0.
\]
As for the second term of the right-hand side of \eqref{eq:hngn}, we use Young's
inequality and \eqref{eq:v2Pv}. It yields,
\begin{align*}
\int_{\{|g_n|> \lambda_0\}}|h_n g_n|
&\le \frac{1}{2\alpha}\int_{Q_T}|h_n|^2
 + \frac{\alpha}{2}\int_{\{|g_n|> \lambda_0\}}|g_n|^2 \\
&\le \frac{(4\zeta)^2}{\alpha}\int_{Q_T}P\Big(x,\frac{|h_n|}{4\zeta}\Big)
 + \alpha\int_{\{|g_n|> \lambda_0\}} P(x,|g_n|).
\end{align*}
It has been already shown that the first of these terms converges to zero.
 As for the second one,
since $P\ll M$, we can take $\lambda_0$ large enough such that
$P(x,s)\le M(x,s)$ for $|s|>\lambda_0$, and then,
\[
\alpha\int_{\{|g_n|> \lambda_0\}} P(x,|g_n|)
\le \alpha \int_0^t   \int_\Omega M(x,|g_n|)=
 \alpha \int_0^t   \int_\Omega M(x,|\nabla(u_n-u)|).
\]
Consequently, for some sequence $(\epsilon_n)\subset\mathbb{R}$, $\epsilon_n\to 0$,
we have the estimate
\[
\frac{1}{2}\|u_n(t)- u(t)\|^2_{L^2(\Omega)}
 \le -\int_0^t   \int_\Omega (F_n-F)\nabla (u_n-u)\,{\rm d}x\,{\rm d}s
+ \epsilon_n,
\]
and integrating this inequality over $[0,T]$, we have
\begin{equation}\label{eq:desi1}
\frac{1}{2}\|u_n- u\|^2_{L^2(Q_T)}
 \le -\int_0^T   \int_\Omega (T-t)(F_n-F)\nabla (u_n-u)\,{\rm d}x\,{\rm d}t
+T\epsilon_n.
\end{equation}
The first term of the right hand side in \eqref{eq:desi1} converges to zero since
$F_n\to F$ strongly in $L^2(Q_T)^d$ and $(T-t)(\nabla u_n-\nabla u)$
is bounded in $L^2(Q_T)^d$.
In conclusion, $u_n\to u$ strongly in $L^2(Q_T)$. Since this limit does not
depend upon the subsequence one may extract, it is in fact the whole sequence
$(u_n)$ which converges to $u$ strongly in $L^2(Q_T)$.
On the other hand, in virtue of~\eqref{untoU}, we also have $u_n\to U$
strongly in $L^2(Q_T)$, so that $u=U$ and we can rewrite~\eqref{untoU} to
give $u_n\to u$ strongly in $E_P(Q_T)$. This shows that $G$ is continuous
and this ends the proof of Theorem~\ref{t2}.

\subsection*{Proof of Theorem \ref{t1}}
This stage is the main goal of this work. We start by introducing a sequence
of approximate problem and deriving a priori estimates of it and showing
two intermediate results, namely the strong convergence in $L^1(Q_T)$ of
both $\nabla u_n$ and $\varphi_n$, where
$(u_n,\varphi_n)$ is a weak solution to the approximate problem of~\eqref{P}.
\smallskip

\noindent\textbf{Step 1.}
For every $n\in \mathbb{N}$, we introduce the following
regularization of the data,
\begin{gather}\label{rho1n}
\rho_n(s)=\rho(s)+\frac{1}{n}, \\
a_n(x,t,s,\xi)=a(x,t,T_n(s),\xi),
\end{gather}
and consider the approximate system
\begin{gather}\label{55}
\frac{\partial u_n }{\partial t}-
\operatorname{div} a_n(x,t,u_n,\nabla u_n) =\rho_n(u_n)|\nabla \varphi_n|^2 \quad
\text{in } Q_T , \\
\label{551}
 \operatorname{div}(\rho_n(u_n)\nabla \varphi_n)=0\quad \text{in } Q_T , \\
\label{56}
u_n=0\quad\text{on } (0,T)\times{\partial\Omega}, \\
\label{561}
\varphi_n=\varphi_0\quad\text{on } (0,T)\times{\partial\Omega}, \\
\label{57}
u_n(\cdot,0)=u_{0}\quad\text{in } \Omega.
 \end{gather}
From~\eqref{croi} we deduce
\[
|a(x,t,T_n(s),\xi)|
\le \zeta\left[c(x,t)+ \bar{M}_{x}^{-1}(P(x,k|T_n(s)|))
+ \bar{M}_{x}^{-1}(M(x,k|\xi|))\right],
\]

by the Fenchel-Young inequality, we obtain
\begin{align*}
|a(x,t,T_n(s),\xi)|
&\leq \zeta\left[c(x,t)+ (P(x,k|T_n(s)|)+M(x,1))
 + \bar{M}_{x}^{-1}(M(x,k|\xi|))\right]\\
&\le \zeta\Big[c_n(x,t) +\bar{M}_{x}^{-1}(M(x,k|\xi|))\Big],
\end{align*}
where $c_n(x,t) = c(x,t) + \sup_{\xi\in B(0,kn)}
\operatorname{ess\,sup}_{x\in\Omega} P(x,\xi) + M(x,1)$.
Using Remark~\ref{rem:M1c}, $M(x,1)=\text{constant}$ for a.~a.~$x\in\Omega$.
Taking into account that
$L^\infty(Q_T)\subset L^2(Q_T)\subset L_{\bar{P}}(Q_T)\subset E_{\bar{M}}(Q_T)$
 (Lemma~\ref{mod}) and owing to \eqref{p1}, it yields that
$c_n\in E_{\bar{M}(Q_T)}$.

Also, in view of~\eqref{31}, we have =
 \begin{equation}\label{rho4}
 n^{-1}\leq \rho_n (s)\leq \rho_3+1=\rho_4,\quad \text{for all $s\in \mathbb{R}$.}
 \end{equation}
Thus, we can apply Theorem \ref{t2} to deduce the existence of a weak solution
$(u_n,\varphi_n)$  to  system \eqref{55}-\eqref{57}.

By the maximum principle we have
 \begin{equation}\label{p.1}
 \|\varphi_n\|_{L^\infty(Q_T)}\leq \|\varphi_0\|_{L^\infty(Q_T)},
 \end{equation}
hence there exists a function $\varphi\in L^\infty(Q_T)$ and a subsequence,
still denoted in the same way,  such that
 \begin{equation}\label{limit-varphi}
 \varphi_n\to \varphi \text{weak-* in }L^\infty(Q_T).
 \end{equation}
Now let multiply \eqref{551} by $\varphi_n-\varphi_0 \in L^2(0,T;H^1_0(\Omega))$
and integrate over  $Q_T$. We obtain
 \[
\int_{0}^T  \int_{\Omega} \rho_n(u_n)\nabla \varphi_n
 \nabla (\varphi_n-\varphi_0)\, {\rm d}x\, {\rm d}t =0,
\]
hence
 \begin{equation}\label{C1}
 \int_{0}^T  \int_{\Omega}\rho_n(u_n)|\nabla \varphi_n|^2\, {\rm d}x\, {\rm d}t
\leq C_1,
\quad \text{for all $n\ge1$.}
 \end{equation}
Where $C_1=C_1(\bar{\rho},\|\varphi_0\|_{L^2(0,T;H^1(\Omega))})$.
Consequently, the  sequence $(\rho_n(u_n)\nabla \varphi_n)$ is bounded in $L^2(Q_T)$.
Thus, there exists  a function $\Phi\in L^2(Q_T)^d$ and a subsequence,
still denoted the same way, such that
 \begin{equation}\label{limit-Phi}
 \rho_n(u_n)\nabla \varphi_n\to \Phi \text{ weakly in }L^2(Q_T)^d.
 \end{equation}
This weak limit function $\Phi\in L^2(Q_T)^d$ is in fact the third component of
 the triplet appearing in the Definition~\ref{def1} of a capacity solution.

Taking $u_n$ as a test function in \eqref{55}, for all $t\in [0,T]$, we obtain
 \begin{equation}\label{pati1}
 \begin{aligned}
&\frac{1}{2}\|u_n(t)\|^2_{L^2(\Omega)}+
 \int_0^t \int_\Omega a(x,t,T_n(u_n),\nabla u_n)\nabla u_n\,{\rm d}x\,
 {\rm d}t \\
&= \frac{1}{2}\|u_0\|^2_{L^2(\Omega)}
 -\int_0^t\int_{\Omega} \rho_n(u_n)\varphi_n\nabla \varphi_n
\nabla u_n\,{\rm d}x\,{\rm d}t.
 \end{aligned}
 \end{equation}

From \eqref{croi}, \eqref{30}, \eqref{p.1} and \eqref{rho4}, we obtain
 \begin{equation}\label{b23}
 \alpha\int_0^t \int_\Omega M(x,|\nabla u_n|)
 \leq \frac{1}{2}\|u_0\|^2_{L^2(\Omega)}+
 \int_0^t\int_{\Omega} \|\varphi_0\|_{L^\infty(Q_T)}\rho_2
\nabla \varphi_n \nabla u_n,
 \end{equation}
 and by Young's inequality, we may deduce that  for all $t\in [0,T]$,
 \begin{equation}\label{unb}
 \int_0^t \int_\Omega M(x,|\nabla u_n|)\,{\rm d}x\,{\rm d}t
 \leq C,
 \end{equation}
where $C$ is a positive constant not depending on $n$.
It follows from \eqref{rp} that the sequence $(u_n)$
is bounded in $W^{1,x}_0L_M(Q_T)$.
Consequently, there exist a subsequence of $(u_n)$, still denoted in the same way,
 and a function $u\in W^{1,x}_0L_M(Q_T)$ such that:
 \begin{equation}\label{un}
 u_n\rightharpoonup u \text{ in } W^{1,x}_0L_M(Q_T)
 \text{ for } \sigma(\Pi L_M,\Pi E_{\bar{M}}).
 \end{equation}

On the other hand, Let $\phi \in W^{1,x}_0E_M(Q_T)^d$ be an arbitrary
function such that $\|\nabla\phi\|_{(M),Q_T}=1/(k+1)$.
 In view of the monotonicity of $a_n$, one easily has
 \begin{equation}\label{eq:angradun}
\begin{aligned}
&\int_{Q_T} a_n (x,t,u_n,\nabla u_n)\nabla\phi \\
&\leq \int_{Q_T} a_n(x,t,u_n,\nabla u_n)\nabla u_n
 -\int_{Q_T} a_n(x,t,u_n,\nabla\phi)(\nabla u_n-\nabla\phi)
 \\
&\le C + \int_{Q_T} |a_n(x,t,u_n,\nabla\phi)\nabla u_n|
 + \int_{Q_T} a_n(x,t,u_n,\nabla\phi)\nabla \phi,
 \end{aligned}
 \end{equation}
We can show that the two last integrals in~\eqref{eq:angradun}
are bounded with respect to~$n$.
 Indeed, for the first one, by Young's inequality
 \[
\int_{Q_T}   |a_n(x,t,u_n,\nabla\phi)\nabla u_n|
 \le 3\zeta \int_{Q_T}
\Big[\bar{M}\Big(x,\frac{|a(x,t,T_n(u_n),\nabla \phi)|}{3\zeta}\Big)
 + M(x,|\nabla u_n|)\Big],
\]
 using~\eqref{croi} we have
 \[
 3\zeta\bar{M}\Big(x,\frac{|a(x,t,T_n(u_n),\nabla \phi)|}{3\zeta}\Big)
 \le \zeta\left(\bar{M}(x,c(x,t)) + P(x,kT_n(u_n)) + M(x,k\nabla\phi)\right),
\]
since $(u_n)$ is bounded in $W^{1,x}_0L_M(Q_T)$, and owing to Poincar\'e's
inequality, there exists $\lambda>0$ such that
 $\int_{Q_T} M(x,u_n/\lambda)\le 1$ for all $n\ge1$.
Also, since $P\ll M$, there exists $s_0>0$ such that
 $P(x,ks)\le P(x,ks_0) + M(x,s/\lambda)$ for all $s\in\mathbb{R}$.
 Consequently, using~\eqref{p1} it yields
\begin{align*}
&3\zeta\int_{Q_T}\bar{M}\Big(\frac{|a(x,t,T_n(u_n),\nabla \phi)|}{3\zeta}\Big)\\
&\le \zeta\Big(\int_{Q_T}\bar{M}(x,c(x,t)) + T\int_\Omega P(x,ks_0)\,{\rm d}x
+\int_{Q_T}M(x,u_n/\lambda) + \int_{Q_T}M(x,k\nabla\phi)\Big) \\
&\le C,
 \end{align*}
and thus $\int_{Q_T}   |a_n(x,t,u_n,\nabla\phi)\nabla u_n|  \le C$, for all
$n\ge1$ and $\phi \in W^{1,x}_0E_M(Q_T)^d$ such that
 $\|\nabla\phi\|_{(M),Q_T}=1/(k+1)$. On the other hand, the second integral
in~\eqref{eq:angradun}, namely
 $\int_{Q_T} a_n(x,t,u_n,\nabla\phi)\nabla \phi$ can be dealt in the same way
so that it is easy to check that it  is also bounded.
Gathering all these estimates, and using the dual norm, one easily deduce
 that
 \begin{equation}\label{bounded}
 (a_n(x,t,u_n,\nabla u_n))\text{ is bounded in }L_{\bar{M}}(Q_T)^d.
 \end{equation}
Thus, up to a subsequence, still denoted in the same way, there exists
$\delta\in L_{\bar{M}}(Q_T)^d$ such that
 \begin{equation}\label{eq:antodelta}
 a_n(x,t,u_n,\nabla u_n)\rightharpoonup \delta \quad
 \text{in } L_{\bar{M}}(Q_T)^d \text{ for } \sigma(\Pi L_{\bar{M}},\Pi E_M).
 \end{equation}

Finally, since both sequences $(\operatorname{div} a_n(x,t,u_n,\nabla u_n))$
and $(\operatorname{div}(\rho_n(u_n)\varphi_n\nabla \varphi_n))$ are bounded
in the space  $W^{-1,x}L_{\bar{M}}(Q_T)$, according to \eqref{55}, we have
 \begin{equation}\label{u'}
 \Big( \frac{\partial u_n}{\partial t}\Big) \quad \text{is bounded in }
 W^{-1,x}L_{\bar{M}}(Q_T).
 \end{equation}
 Consequently, $(u_n)\subset \textbf{W}$ is bounded and, since the embedding
 $\textbf{W}\hookrightarrow E_P(Q_T)$ is compact, for a subsequence,
still denoted in the same way, we have
 \begin{equation}\label{eq:unaeu}
 u_n\to u\quad \text{strongly in }E_P(Q_T)\text{ and a.e.~in }Q_T,
 \end{equation}
 where $u\in W^{1,x}_0L_M(Q_T)$ is also the limit function appearing in~\eqref{un}.
\smallskip

\noindent\textbf{Step 2.}
Introduction of regularized sequences and the almost everywhere convergence
 of the gradients.

First we introduce two smooth sequences, namely,
$(v_j)\subset {\mathcal D}(Q_T)$ and $(\psi_i)\subset {\mathcal D}(\Omega)$ such that
 \begin{enumerate}
 \item $v_j\to u$ in $W_0^{1,x}L_M(Q_T)$ for the modular convergence;
 \item $v_j\to u$ and $\nabla v_j\to \nabla u$ and almost everywhere in $Q_T$;
 \item $\psi_i\to u_0$ strongly in $L^2(\Omega)$;
 \item $\|\psi_i\|_{L^2(\Omega)}\le 2\|u_0\|_{L^2(\Omega)}$, for all $i\ge1$.
 \end{enumerate}
For a fixed positive real number $K$, we consider the truncation function at
height $K$, $T_K$, defined in \eqref{eq:truncation}. Then, for every
$K,\mu>0$ and $i,j\in\mathbb{N}$, we introduce the function
 $w_{\mu,j}^i\in W_0^{1,x}L_M(Q_T)$ (to simplify the notation,
we drop the index $K$) defined as
$w_{\mu,j}^i= T_K(v_j)_\mu + e^{-\mu t}T_K(\psi_i)$, where $T_K(v_j)_\mu$
is the mollification with respect to time of
 $T_K(v_j)$ given in \eqref{eq:timereg}. From Lemma~\ref{lem3}, we know that
\begin{gather}\label{81}
 \frac{\partial w_{\mu,j}^i }{ \partial t}= \mu(T_K(v_j) - w_{\mu,j}^i), \quad
w_{\mu,j}^i(\cdot,0)= T_K(\psi_i), \quad
 |w_{\mu,j}^i|\leq K\text{ a.e in }Q_T, \\
\label{82}
 w_{\mu,j}^i\to w_{\mu}^i\stackrel{\rm def}{=}
T_K(u)_\mu+e^{-\mu t}T_K(\psi_i)\text{ in }W_0^{1,x}L_M(Q_T),
 \end{gather}
for the modular convergence as $j\to \infty$.
\begin{equation}\label{83}
 T_K(u)_\mu +e^{-\mu
 t}T_K(\psi_i)\to T_K(u)\quad \text{in }W_0^{1,x}L_M(Q_T),
\end{equation}
 for the modular convergence as $\mu\to \infty$.
Since we may consider subsequences in~\eqref{81}-\eqref{83}, we will assume
without loss of generality that  the convergences \eqref{82} and \eqref{83}
also hold almost everywhere in $Q_T$.

We will establish the following proposition.

\begin{proposition}\label{prop12}
Let $(u_n,\varphi_n)$ be a solution of the  approximate problem
\eqref{55}-\eqref{57}. Then, for a suitable subsequence, still denoted
 in the same way, we have
\begin{equation}\label{78}
 \nabla u_n\to \nabla u\text{ a.e. in }Q_T,
\end{equation}
as $n$ tends to $+\infty$.
\end{proposition}

\begin{proof}
In the sequel and throughout this article,
 $\chi_s^j$ and $\chi_s$ will denote, respectively, the characteristic
functions of the sets
 $$
Q_s^j=\big\{(x,t)\in Q_T:|\nabla T_K(v_j)|\leq s\big\},\quad
 Q_s=\big\{(x,t)\in Q_T:|\nabla T_K(u)|\leq s\big\}.
$$
We also introduce the primitive of the truncation function $T_K$ vanishing
at the origin, $S_K$, that is,
 \begin{equation}\label{Sk}
 S_K(t)= \int_{0}^{t} T_K(s)ds
=\begin{cases}
 t^2/2 & \text{if }|t| \leq K,\\
 K|t|-K^2/2 & \text{if }|t|> K.
\end{cases}
\end{equation}
 It is straightforward to show that $0\le S_K(t) \leq K|t|$ for all
$t\in\mathbb{R}$.

We will also  use the following notation for vanishing sequences:
 $\epsilon(n)$ means a sequence such that
 $\lim_{n\to\infty} \epsilon(n) =0$ or $\limsup_{n\to\infty} \epsilon(n) =0$;
$\epsilon(n,j)$ is a term such that
 $\lim_{j\to\infty} \lim_{n\to\infty} \epsilon(n,j) =0$ where any occurrence
of $\lim$ may be substituted by
 $\limsup$. And so on for $\epsilon(n,j,\mu)$, etc.

For any $\mu,\nu>0$ and $i,j,n\ge1$ we may use the admissible test
function $\varphi_{n,j,\nu}^{\mu,i} =
 T_\nu(u_n-w_{\mu,j}^i)$ in \eqref{55}. This leads to
 \begin{equation}\label{84}
 \begin{aligned}
&\langle \frac{\partial u_n}{\partial
 t},\varphi_{n,j,\nu}^{\mu,i}\rangle + \int_{Q_T}
 a_n(x,t,u_n,\nabla u_n)\nabla T_\nu(u_n-w_{\mu,j}^i)\,{\rm d}x\,{\rm d}t\\
&= \int_{Q_T} \rho_n(u_n)|\nabla \varphi_n|^2
 \varphi_{n,j,\nu}^{\mu,i}\,{\rm d}x\,{\rm d}t.
\end{aligned}
\end{equation}
By using \eqref{C1}, we obtain
 \begin{equation}\label{n84}
 \big\langle \frac{\partial u_n}{\partial
 t},\varphi_{n,j,\nu}^{\mu,i}\big\rangle +\int_{Q_T}
 a_n(x,t,u_n,\nabla u_n)\nabla T_\nu(u_n-w_{\mu,j}^i)\,{\rm d}x\,{\rm d}t
\leq C_1\nu.
 \end{equation}

As far as the parabolic term is concerned, we have
 \begin{equation}\label{eq:parabolic}
 \begin{aligned}
 \big\langle \frac{\partial u_n}{\partial
 t},T_\nu(u_n-w_{\mu,j}^i)\big\rangle
 &=\big\langle \frac{\partial u_n}{\partial
 t}- \frac{\partial w_{\mu,j}^i}{\partial  t},T_\nu(u_n-w_{\mu,j}^i)\big\rangle\\
 &\quad + \big\langle\frac{\partial w_{\mu,j}^i}{\partial  t},
 T_\nu(u_n-w_{\mu,j}^i)\big\rangle.
 \end{aligned}
 \end{equation}
 The first term of the right-hand side in \eqref{eq:parabolic} can be written as
 \[
 \big\langle \frac{\partial u_n}{\partial
 t}- \frac{\partial w_{\mu,j}^i}{\partial
 t},T_\nu(u_n-w_{\mu,j}^i)\big\rangle =
 \int_\Omega  S_\nu(u_n(T)-w_{\mu,j}^i(T))
 -\int_\Omega  S_\nu(u_{0}-T_K(\psi_i)).
 \]
 Since
\begin{align*}
0&\le \int_\Omega S_\nu(u_{0}-T_K(\psi_i))
 \le \nu\int_\Omega |u_{0}-T_K(\psi_i)| \\
& \le \nu|\Omega|^{1/2}(\int_\Omega |u_{0}-T_K(\psi_i)|^2)^{1/2}\\
&\le 3\|u_0\|_{L^2(\Omega)}|\Omega|^{1/2}\nu = C_2\nu,
\end{align*}
 we deduce that for all $i,j,n\ge1$ and $\mu,n,K>0$,
 \begin{equation}\label{eq:Cnu}
 \big\langle \frac{\partial u_n}{\partial
 t}- \frac{\partial w_{\mu,j}^i}{\partial
 t},T_\nu(u_n-w_{\mu,j}^i)\big\rangle\ge -C_2\nu.
 \end{equation}
As for the second term of the right-hand side in \eqref{eq:parabolic} we have
 \begin{equation}
 \big\langle\frac{\partial w_{\mu,j}^i}{\partial
 t},T_\nu(u_n-w_{\mu,j}^i)\big\rangle
= \mu \int_{Q_T} (T_K(v_j) -
 w_{\mu,j}^i)T_\nu(u_n-w_{\mu,j}^i).
 \end{equation}
Passing to the limit first in $n\to\infty$, then in $j\to\infty$, it yields
 \[
 \lim_{j\to\infty}\lim_{n\to\infty}\big\langle\frac{\partial w_{\mu,j}^i}{\partial
 t},T_\nu(u_n-w_{\mu,j}^i)\big\rangle =
 \mu \int_{Q_T} (T_K(u) - w_{\mu}^i)T_\nu(u-w_{\mu}^i).
 \]
Owing to \eqref{81} and \eqref{82} we have $|w_{\mu}^i|\le K$ almost everywhere
in $Q_T$.  Also, since $sT_\nu(s)\ge 0$ for all $s\in\mathbb{R}$, we deduce, for all
 $\mu,\nu,K>0$ and $i\ge1$,
\begin{equation}\label{eq:limjn}
 \lim_{j\to\infty}\lim_{n\to\infty}\big\langle\frac{\partial w_{\mu,j}^i}{\partial
 t},T_\nu(u_n-w_{\mu,j}^i)\big\rangle \ge0.
\end{equation}
 Gathering \eqref{eq:parabolic}, \eqref{eq:Cnu} and \eqref{eq:limjn} we finally
obtain, for all $\mu,\nu,K>0$ and $i\ge1$, the following estimate for the
parabolic term
 \begin{equation}\label{eq:liminf}
 \liminf_{j\to\infty}\liminf_{n\to\infty}\big\langle\frac{\partial u_n}{\partial
 t},T_\nu(u_n-w_{\mu,j}^i)\big\rangle \ge -C_2\nu.
 \end{equation}

 It remains to analyze the diffusion term of~\eqref{84}. We have
\begin{align*}
&\int_{Q_T} a_n(x,t,u_n,\nabla u_n)\nabla  T_\nu(u_n-w_{\mu,j}^i)
 \,{\rm d}x\,{\rm d}t  \\
&=\int_{\{|u_n-w_{\mu,j}^i|\leq\nu \}} a_n(x,t,u_n,\nabla u_n)\nabla(u_n-w_{\mu,j}^i)
 \,{\rm d}x\,{\rm d}t\\
&=\int_{\{|u_n|>K\}\cap \{|u_n-w_{\mu,j}^i|\leq\nu \}}
 a_n(x,t,u_n,\nabla u_n)\nabla(u_n-w_{\mu,j}^i)\,{\rm d}x\,{\rm d}t\\
&\quad +\int_{\{|u_n|\leq K\}\cap \{|u_n-w_{\mu,j}^i|\leq\nu \}}
 a_n(x,t,u_n,\nabla u_n)\nabla(u_n-w_{\mu,j}^i)\,{\rm d}x\,{\rm d}t\\
&=\int_{\{|T_K(u_n)-w_{\mu,j}^i|\leq\nu \}} a_n(x,t,T_K(u_n),\nabla T_K(u_n))(\nabla
 T_K(u_n)-\nabla w_{\mu,j}^i)\,{\rm d}x\,{\rm d}t\\
&\quad +\int_{\{|u_n|>K\}\cap \{|u_n-w_{\mu,j}^i|\leq\nu \}}
 a_n(x,t,u_n,\nabla u_n)\nabla  u_n\,{\rm d}x\,{\rm d}t\\
&\quad -\int_{\{|u_n|>K\}\cap \{|u_n-w_{\mu,j}^i|\leq\nu \}}
 a_n(x,t,u_n,\nabla u_n)\nabla  w_{\mu,j}^i\,{\rm d}x\,{\rm d}t.
\end{align*}
By \eqref{coer} and \eqref{30} we have
\begin{align*}
&\int_{\{|u_n|>K\}\cap \{|u_n-w_{\mu,j}^i|\leq\nu \}} a_n(x,t,u_n,\nabla u_n)\nabla
 u_n\,{\rm d}x\,{\rm d}t\\
&\geq \alpha\int_{\{|u_n|>K\}\cap \{|u_n-w_{\mu,j}^i|\leq\nu \}}
 M(x,|\nabla u_n|) \,{\rm d}x\,{\rm d}t\geq 0,
 \end{align*}
which implies that
 \begin{equation}\label{ank2}
\begin{aligned}
&\int_{Q_T} a_n(x,t,u_n,\nabla u_n)\nabla  T_\nu(u_n-w_{\mu,j}^i)
 \,{\rm d}x\,{\rm d}t\\
&\geq\int_{\{|T_K(u_n)-w_{\mu,j}^i|\leq\nu \}} a_n(x,t,T_K(u_n),
 \nabla T_K(u_n))(\nabla  T_K(u_n)-\nabla w_{\mu,j}^i)\\
&\quad -\int_{\{|u_n|>K\}\cap \{|u_n-w_{\mu,j}^i|\leq\nu \}}
 a_n(x,t,u_n,\nabla u_n)\nabla  w_{\mu,j}^i\,{\rm d}x\,{\rm d}t.
\end{aligned}
 \end{equation}
On the one hand, let us observe that for any $K>0$, and for $n$ large enough,
 namely $n>K+\nu\geq K$, we have
 \begin{equation}\label{ank1}
a_n(x,t,T_{K}(u_n),\nabla T_{K}(u_n))=a(x,t,T_{K}(u_n),\nabla T_{K}(u_n)).
 \end{equation}
On the other hand, from \eqref{81}, we have $|w_{\mu,j}^i|\leq K$ a.e.~in $Q_T$,
 then in the set $\{|u_n-w_{\mu,j}^i|\leq\nu \}$, we have
 $|u_n|\leq |u_n-w_{\mu,j}^i|+|w_{\mu,j}^i|\leq \nu+K$.
 Then for $n>\nu+K$, we obtain
 \begin{equation}\label{ank}
\begin{aligned}
&\int_{\{|u_n|>K\}\cap \{|u_n-w_{\mu,j}^i|\leq\nu \}} a_n(x,t,u_n,\nabla u_n)\nabla
 w_{\mu,j}^i\,{\rm d}x\,{\rm d}t\\
&=\int_{\{|u_n|>K\}\cap \{|u_n-w_{\mu,j}^i|\leq\nu \}}
 a(x,t,T_{\nu+K}(u_n),\nabla T_{\nu+K}(u_n))\nabla
 w_{\mu,j}^i\,{\rm d}x\,{\rm d}t.
 \end{aligned}
\end{equation}
 From \eqref{ank1} and \eqref{ank}, inequality \eqref{ank2} becomes
 \begin{equation}\label{ank3}
\begin{aligned}
&\int_{Q_T} a_n(x,t,u_n,\nabla u_n)\nabla  T_\nu(u_n-w_{\mu,j}^i)
 \,{\rm d}x\,{\rm d}t\\
&\geq\int_{\{|T_K(u_n)-w_{\mu,j}^i|\leq\nu \}} a(x,t,T_K(u_n),
 \nabla T_K(u_n))(\nabla w_{\mu,j}^i
 T_K(u_n)-\nabla w_{\mu,j}^i)\\
&\quad -\int_{\{|u_n|>K\}\cap \{|u_n-w_{\mu,j}^i|\leq\nu \}}
 a(x,t,T_{\nu+K}(u_n),\nabla T_{\nu+K}(u_n))\nabla w_{\mu,j}^i.
\end{aligned}
\end{equation}
 We put
 \[
J_1=\int_{\{|u_n|>K\}\cap \{|u_n-w_{\mu,j}^i|\leq\nu \}}
a(x,t,T_{\nu+K}(u_n),\nabla T_{\nu+K}(u_n))\nabla w_{\mu,j}^i\,{\rm d}x\,{\rm d}t.
\]
Since $(a(x,t,T_{K+\nu}(u_n),\nabla T_{K+\nu}(u_n)))$ is bounded
 in $L_{\bar{M}}(Q_T)^d$, we have
 $$
a(x,t,T_{K+\nu}(u_n),\nabla T_{K+\nu}(u_n))\rightharpoonup  l_{K+\nu}
$$
weakly in $L_{\bar{M}}(Q_T)$ in $\sigma(\Pi
 L_{\bar{M}},\Pi E_M)$ as $n$ tends to infinity and since
 \[
\nabla w_{\mu,j}^i{\chi_{\{|u_n|>K\}\cap \{|u_n-w_{\mu,j}^i|\leq\nu \}}}\to
 \nabla w_{\mu,j}^i {\chi_{\{|u|>K\}\cap \{|u-w_{\mu,j}^i|\leq\nu \}}}
\]
 strongly in $E_M(Q_T)^d$ as $n$ tends to infinity, we have
\begin{align*}
&\int_{\{|u_n|>K\}\cap \{|u_n-w_{\mu,j}^i|\leq\nu \}} a(x,t,T_{\nu+K}(u_n),
 \nabla T_{\nu+K}(u_n))\nabla
 w_{\mu,j}^i\,{\rm d}x\,{\rm d}t\\
&\to \int_{\{|u|>K\}\cap \{|u-w_{\mu,j}^i|\leq\nu \}} l_{K+\nu} \nabla
 w_{\mu,j}^i\,{\rm d}x\,{\rm d}t
 \end{align*}
as $n$ approaches infinity.

Using Lemma \ref{mod} with the convergences \eqref{82}, \eqref{83},
together with the almost everywhere convergence, and letting first~$j$
then~$\mu$ tend to infinity, we obtain
(note that the index $i$ disappears in this process)
 $$
\int_{\{|u|>K\}\cap \{|u-w_{\mu,j}^i|\leq\nu \}} l_{K+\nu} \nabla
 w_{\mu,j}^i\to \int_{\{|u|>K\}\cap \{|u-T_K(u)|\leq\nu \}} l_{K+\nu}
\nabla T_K(u)=0
$$
since $\nabla T_K(u)=0$ in the set $\{|u|>K\}$.
 This gives
 \begin{equation}\label{j1}
 J_1=\epsilon(n,j,\mu,i).
 \end{equation}

Using $\eqref{eq:liminf}$, \eqref{ank3} and \eqref{j1} in \eqref{n84}, we obtain
\begin{equation}\label{eq.1}
 \begin{aligned}
&\int_{\{|T_K(u_n)-w_{\mu,j}^i|\leq\nu \}} a(x,t,T_K(u_n),\nabla T_K(u_n))(\nabla
 T_K(u_n)-\nabla w_{\mu,j}^i)\,{\rm d}x\,{\rm d}t\\
&\leq C\nu+\epsilon(n,j,\mu,i).
\end{aligned}
 \end{equation}
 where $C=(C_1+C_2)$.

On the other hand, note that
 \begin{equation}\label{94}
\begin{aligned}
& \int_{\{|T_K(u_n)-w_{\mu,j}^i|\leq\nu \}} a(x,t,T_K(u_n),\nabla T_K(u_n))(\nabla
 T_K(u_n)-\nabla w_{\mu,j}^i)\,{\rm d}x\,{\rm d}t
 \\
& =\int_{\{|T_K(u_n)-w_{\mu,j}^i|\leq\nu \}}   a(x,t,T_K(u_n),\nabla T_K(u_n))
 (\nabla T_K(u_n)-\nabla T_K(v_j)\chi_j^s)
 \\
&\quad +\int_{\{|T_K(u_n)-w_{\mu,j}^i|\leq\nu \}} a(x,t,T_K(u_n),\nabla T_K(u_n))
 (\nabla T_K(v_j)\chi_j^s-\nabla w_{\mu,j}^i)
 \\
&=J_2+J_3.
 \end{aligned}
 \end{equation}

The integral term $J_3$ tends to $0$ as first $n$, then $j$, $\mu$, $i$ and $s$
go to $\infty$. Indeed, since,
 \[
a(x,t,T_K(u_n),\nabla
 T_K(u_n))\rightharpoonup l_K \quad \text{weakly in }L_{\bar{M}}(Q_T)^d,
\]
 and since
 \[
(\nabla T_K(v_j)\chi_j^s-\nabla w_{\mu,j}^i)
\chi_{\{|T_K(u_n)-w_{\mu,j}^i|\leq\nu \}}
\to (\nabla T_K(v_j)\chi_j^s-\nabla w_{\mu,j}^i)
\chi_{\{|T_K(u)-w_{\mu,j}^i|\leq\nu \}}
\]
 strongly in $E_{\bar{M}}(Q_T)^d$ as $n\to\infty$, we have
 \[
\lim_{n\to\infty}J_3=\int_{\{|T_K(u)-w_{\mu,j}^i|\leq\nu \}} l_K\cdot
 (\nabla T_K(v_j)\chi_j^s-\nabla w_{\mu,j}^i)\,{\rm d}x\,{\rm d}t.
\]
Letting $j$, $\mu$, $i$ and $s$, in this order, tend to infinity we
readily deduce that
 \begin{equation}\label{j3}
 J_3=\epsilon(n,j,\mu,i,s).
 \end{equation}
Consequently, from \eqref{eq.1}, \eqref{94} and \eqref{j3}, one has
 \begin{equation}\label{J2}
 \begin{aligned}
J_2&=\int_{\{|T_K(u_n)-w_{\mu,j}^i|\leq\nu \}}  a(x,t,T_K(u_n),\nabla T_K(u_n))
 (\nabla T_K(u_n)-\nabla T_K(v_j)\chi_j^s)  \\
&\leq C\nu+\epsilon(n,i,j,\mu,s).
\end{aligned}
 \end{equation}
Let $M_n$ be the non-negative expression
 \[
M_n=(a(x,t,T_K(u_n),\nabla
 T_K(u_n))-a(x,t,T_K(u_n),\nabla T_K(u)))(\nabla T_K(u_n)-\nabla T_K(u)),
\]
 then for any $0<\theta<1$, we write
 \[
I_{n,r}= \int_{Q_r}M^\theta_n\,{\rm d}x\,{\rm d}t.
\]
We have
 \begin{equation}
 \int_{Q_r}M^\theta_n\,{\rm d}x\,{\rm d}t
 = \int_{Q_r}M^\theta_n\chi_{\{|T_K(u_n)-w_{\mu,j}^i|\leq\nu \}}
+ \int_{Q_r}M^\theta_n\chi_{\{|T_K(u_n)-w_{\mu,j}^i|>\nu \}}.
 \end{equation}
 Using H\"{o}lder's inequality the second term of the right-side hand is less than
 \[
\Big( \int_{Q_r}M_n\,{\rm d}x\,{\rm d}t\Big)^\theta
\Big( \int_{Q_r}\chi_{\{|T_K(u_n)-w_{\mu,j}^i|>\nu \}}\,{\rm d}x\,{\rm d}t
\Big)^{1-\theta}.
\]
Note that
\begin{align*}
 \int_{Q_r}M_n\,{\rm d}x\,{\rm d}t
&= \int_{Q_r}a(x,t,T_K(u_n),\nabla
 T_K(u_n))\nabla T_K(u_n)\,{\rm d}x\,{\rm d}t\\
&\quad - \int_{Q_r}a(x,t,T_K(u_n),\nabla T_K(u_n))\nabla
 T_K(u)\,{\rm d}x\,{\rm d}t\\
&\quad+ \int_{Q_r}a(x,t,T_K(u_n),\nabla
 T_K(u))\nabla T_K(u)\,{\rm d}x\,{\rm d}t\\
&\quad - \int_{Q_r}a(x,t,T_K(u_n),\nabla T_K(u))\nabla
 T_K(u_n)\,{\rm d}x\,{\rm d}t.
 \end{align*}
Since $(a(x,t,T_{K}(u_n),\nabla T_{K}(u_n)))$ is bounded in $L_{\bar{M}}(Q_T)^d$,
 $(\nabla T_K(u_n))$ is bounded in $L_{{M}}(Q_T)^d$ and
$(a(x,t,T_{K}(u_n),\nabla T_{K}(u)))$ is bounded in
 $L^\infty(Q_r)$, we have $(M_n)$ is bounded in $L^1(Q_r)$.

It follows that there exists a constant $C_3>0$ such that
 \begin{equation}\label{mnt1}
 \int_{Q_r}M^\theta_n\chi_{\{|T_K(u_n)-w_{\mu,j}^i|>\nu \}}\,{\rm d}x\,{\rm d}t
 \leq C_3 \mathop{\operatorname{meas}}\{|T_K(u_n)-w_{\mu,j}^i|>\nu\}^{1-\theta}.
 \end{equation}
Using again H\"{o}lder's inequality, we have
 \begin{equation}\label{mnt}
 \begin{aligned}
&\int_{Q_r}M^\theta_n\chi_{\{|T_K(u_n)-w_{\mu,j}^i|\leq\nu \}}\,{\rm d}x\,{\rm d}t
 \\
&\leq \Big(\int_{ Q_r}1\,{\rm d}x\,{\rm d}t\Big)^{1-\theta}
 \Big(\int_{\{|T_K(u_n)-w_{\mu,j}^i|\leq\nu \}\cap Q_r} M_n\,{\rm d}x\,{\rm d}t
 \Big)^\theta
 \\
&\leq C_4\Big( \int_{\{|T_K(u_n)-w_{\mu,j}^i|\leq\nu \}\cap Q_r}
  M_n\,{\rm d}x\,{\rm d}t\Big)^\theta.
 \end{aligned}
 \end{equation}
From \eqref{mnt1} and \eqref{mnt}, we obtain
 \begin{equation}\label{Mn1}
 \begin{aligned}
 I_{n,r}
&\leq C_3 \mathop{\operatorname{meas}}\{|T_K(u_n)-w_{\mu,j}^i|>\nu\}^{1-\theta} \\
&\quad +  C_4\Big( \int_{\{|T_K(u_n)-w_{\mu,j}^i|\leq\nu \}\cap Q_r}
  M_n \,{\rm d}x\,{\rm d}t\Big)^\theta.
 \end{aligned}
 \end{equation}

 On the other hand,  for every $s\geq r$ and $r>0$, we have
\begin{align*}
&\int_{\raisebox{-1ex}{$\{|T_K(u_n)-w_{\mu,j}^i|\leq\nu \}\cap Q_r$}}
  M_n\,{\rm d}x\,{\rm d}t \\
&\le \int_{\{|T_K(u_n)-w_{\mu,j}^i|\leq\nu \}\cap Q_s}
M_n\,{\rm d}x\,{\rm d}t \\
&= \int_{\{|T_K(u_n)-w_{\mu,j}^i|\leq\nu \}\cap Q_s}
 \cdot[a(x,t,T_K(u_n),\nabla T_K(u_n)) \\
&\quad -a(x,t,T_K(u_n),\nabla T_K(u)\chi_s)]
\cdot[\nabla T_K(u_n)-\nabla T_K(u)\chi_s] \,{\rm d}x\,{\rm d}t
\\
&\leq \int_{\{|T_K(u_n)-w_{\mu,j}^i|\leq\nu \}}
[a(x,t,T_K(u_n),\nabla T_K(u_n))\\
&\quad -a(x,t,T_K(u_n),\nabla T_K(u)\chi_s)]
  \cdot[\nabla T_K(u_n)-\nabla T_K(u)\chi_s] \,{\rm d}x\,{\rm d}t
 \\
&\leq \int_{\{|T_K(u_n)-w_{\mu,j}^i|\leq\nu \}}
  [a(x,t,T_K(u_n),\nabla T_K(u_n)) \\
&\quad -a(x,t,T_K(u_n),\nabla T_K(v_j)\chi_j^s)]
 \cdot[\nabla T_K(u_n)-\nabla T_K(v_j)\chi_j^s] \,{\rm d}x\,{\rm d}t
 \\
&\quad + \int_{\{|T_K(u_n)-w_{\mu,j}^i|\leq\nu \}}
  a(x,t,T_K(u_n),\nabla T_K(u_n))
 \cdot[\nabla T_K(v_j)\chi_j^s-\nabla T_K(u)\chi_s] \,{\rm d}x\,{\rm d}t
 \\
&\quad+ \int_{\{|T_K(u_n)-w_{\mu,j}^i|\leq\nu \}}
 \cdot [a(x,t,T_K(u_n),\nabla T_K(v_j)\chi_j^s) \\
&\quad -a(x,t,T_K(u_n),\nabla T_K(u)\chi^s)]
  \cdot\nabla T_K(u_n) \,{\rm d}x\,{\rm d}t \\
&\quad- \int_{\{|T_K(u_n)-w_{\mu,j}^i|\leq\nu \}}
 a(x,t,T_K(u_n),\nabla T_K(v_j)\chi_j^s)
 \cdot\nabla T_K(v_j)\chi_j^s \,{\rm d}x\,{\rm d}t
 \\
&\quad+ \int_{\raisebox{-1ex}{$\{|T_K(u_n)-w_{\mu,j}^i|\leq\nu \}$}}
  a(x,t,T_K(u_n),\nabla T_K(u)\chi_s)
 \cdot\nabla T_K(u)\chi_s \,{\rm d}x\,{\rm d}t
 \\
&=I_1+I_2+I_3+I_4+I_5.
\end{align*}
We will take the limit first in $n$ then in $j$, $\mu$, $i$ and $s$ as they
tend to infinity in these last five integrals.

 Starting with $I_1$, we have
\begin{align*}
 I_1&= \int_{\{|T_K(u_n)-w_{\mu,j}^i|\leq\nu \}} (a(x,t,T_K(u_n),\nabla
 T_K(u_n))-a(x,t,T_K(u_n),\nabla T_K(v_j)\chi_j^s))\\
 &\quad \cdot(\nabla T_K(u_n)-\nabla T_K(v_j)\chi_j^s) \,{\rm d}x\,{\rm d}t
 \\
 &= \int_{\{|T_K(u_n)-w_{\mu,j}^i|\leq\nu \}} a(x,t,T_K(u_n),\nabla
 T_K(u_n))\cdot(\nabla T_K(u_n)-\nabla T_K(v_j)\chi_j^s) \\
&\quad - \int_{\{|T_K(u_n)-w_{\mu,j}^i|\leq\nu \}} a(x,t,T_K(u_n),
 \nabla T_K(v_j)\chi_j^s)\cdot(\nabla T_K(u_n)-\nabla T_K(v_j)\chi_j^s) \\
&=J_2-J_3.
\end{align*}
Since the sequence $(a(x,t,T_K(u_n),\nabla T_K(v_j)\chi_j^s)
\chi_{\{|T_K(u_n)-w_{\mu,j}^i|\leq\nu \}})_n$
 converges to $a(x,t,T_K(u),\nabla T_K(v_j)\chi_j^s)\chi_{\{|T_K(u)-w_{\mu,j}^i|
\leq\nu \}}$
strongly in $E_{\bar{M}}(Q_T)^d$ and
 $(\nabla T_K(u_n))$ converges to $\nabla T_K(u)$ weakly in
$L_M(Q_T)^d$ for $\sigma(\Pi L_M,\Pi
 E_{\bar{M}})$, we then have
 \begin{align*}
J_3
&=\int_{\{|T_K(u)-w_{\mu,j}^i|\leq\nu \}}   a(x,t,T_K(u),\nabla T_K(v_j)\chi_j^s)
 \big(\nabla T_K(u)
-\nabla T_K(v_j)\chi_j^s\big) \,{\rm d}x\,{\rm d}t \\
&\quad +\epsilon(n).
\end{align*}
 Using the almost everywhere convergence of $w_{\mu,j}^i$ and since
 $(\nabla T_K(v_j)\chi_j^s)_j$ converges to $\nabla T_K(u)\chi_s$ strongly
 in $E_M(Q_T)^d$ and
 $(a(x,t,T_K(u),\nabla T_K(v_j)\chi_j^s))_j$ converges to
 $a(x,t,T_K(u),\nabla T_K(u)\chi_s)$ strongly in
 $L_{\bar{M}}(Q_T)^d$, we deduce
\begin{align*}
 J_3 &= \int_{Q_T}   a(x,t,T_K(u),\nabla T_K(u)\chi_s)
 (\nabla T_K(u)-\nabla T_K(u)\chi_s)\,{\rm d}x\,{\rm d}t +\epsilon(n,j,\mu,i)\\
 &=\epsilon(n,j,\mu,i,s).
\end{align*}
 Gathering all these estimates, taking into account~\eqref{J2}, we obtain
\begin{equation}
 I_1\leq C\nu+\epsilon(n,j,\mu,i,s)
 =\epsilon(n,j,\mu,i,s,\nu).
\end{equation}

 As for $I_2$, since $(a(x,t,T_K(u_n),\nabla T_K(u_n)))_n$ converges to $l_K$
 weakly in $L_{\bar{M}}(Q_T)^d$ for
 $\sigma(\Pi L_{\bar{M}},\Pi E_{M})$ and
 $((\nabla T_K(v_j)\chi_j^s-\nabla T_K(u)\chi_s)\chi_{\{|T_K(u_n)
-w_{\mu,j}^i|\leq\nu \}})_n$
 converges to $(\nabla T_K(v_j)\chi_j^s-\nabla T_K(u)\chi_s)
\chi_{\{|T_K(u)-w_{\mu,j}^i|\leq\nu \}}$ strongly
 in $E_M(Q_T)^d$, we obtain
 \[
 I_2= \int_{\{|T_K(u)-w_{\mu,j}^i|\leq\nu \}}l_K(\nabla T_K(v_j)\chi_j^s
-\nabla T_K(u)\chi_s) \,{\rm d}x\,{\rm d}t+\epsilon(n).
 \]
By letting now $j\to \infty$, and using Lebesgue's theorem, we deduce that
 \begin{equation}
 I_2=\epsilon(n,j).
 \end{equation}
Similar tools as above yield
\begin{gather}
 I_3=\epsilon(n,j), \\
 I_4=- \int_{Q_T} a(x,t,T_K(u),\nabla T_K(u)\chi_s)
 \nabla T_K(u)\chi_s+\epsilon(n,j,\mu,i,s) , \\
\label{Mn2}
 I_5= \int_{Q_T} a(x,t,T_K(u),\nabla T_K(u)\chi_s)
 \nabla T_K(u)\chi_s+\epsilon(n,j,\mu,i,s).
 \end{gather}
Combining \eqref{Mn1}-\eqref{Mn2}, we obtain
 \begin{equation}\label{Mn3}
 I_{n,r}\leq C_4\epsilon(n,j,\mu,i,s,\nu)^\theta
+C_3 \mathop{\operatorname{meas}}\{|T_K(u_n)-w_{\mu,j}^i|>\nu\}^{1-\theta}.
 \end{equation}
Consequently, when we take the limit superior  first in $n$, then in $j$,
$\mu$, $i$, $s$ and $\nu$ in 	\eqref{Mn3}, we obtain
 \begin{align*}
&\limsup_{n\to\infty}\int_{Q_r}\Big((a(x,t,T_K(u_n),\nabla
 T_K(u_n))-a(x,t,T_K(u_n),\nabla T_K(u)))
 \\
 &\quad\cdot(\nabla T_K(u_n)-\nabla T_K(u))\Big)^\theta{\rm d}x\,{\rm d}t = 0.
 \end{align*}
According to \eqref{coer} this last expression implies that			
 \[
 \lim_{n\to\infty}\int_{Q_r}M(x,\nabla T_K(u_n)-\nabla T_K(u))^\theta{\rm d}x
\,{\rm d}t = 0.
 \]
hence, for a subsequence, $\nabla T_K(u_n)\to \nabla T_K(u)$ almost everywhere
in $Q_r$.
 Since $r>0$ is arbitrary, we may deduce that, maybe for another subsequence,
 $\nabla T_K(u_n)\to \nabla T_K(u)$ almost everywhere in $Q_T$.
Finally, since $K>0$ is  arbitrary, it yields, still for a subsequence,
 \begin{equation}\label{aec}
 \nabla u_n\to \nabla u\quad \text{ a.e in }\, Q_T.
 \end{equation}
This completes the proof.
 \end{proof}


\begin{remark} \label{rmk5.4} \rm
A straightforward consequence of Proposition~\ref{prop12} is that,
owing to \eqref{eq:antodelta},
 $\delta= a(x,t,u,\nabla u)$; that is,
 \begin{equation}\label{eq:antoa}
 a_n(x,t,u_n\nabla u_n)\rightharpoonup a(x,t,u,\nabla u)\quad
 \text{in } L_{\bar{M}}(Q_T)^d \text{ for } \sigma(\Pi L_{\bar{M}},\Pi E_M).
 \end{equation}
 \end{remark}


\noindent\textbf{Step 3.}
In this step, we will show that $\varphi_n\to \varphi$ strongly in
$L^1(Q_T)$ modulo a subsequence.
The strongly convergence of $(\varphi_n)$ in $L^1(Q_T)$ is based in the next
result which generalizes that of  Gonz\'alez Montesinos and Orteg\'on Gallego
in \cite[Lemma 4]{MM}.

 \begin{lemma}\label{lempsi}
 Let $P$ be a Musielak function such that \eqref{p2} is satisfied.
Assume that $s^2\leq P(x,s)$,
for all a.~a.~$x\in \Omega$ and all $s\in\mathbb{R}$, and let $(u_n)$ be
a bounded sequence in $W^{1,x}L_M(Q_T)$ such that
 $u_n\to u$ strongly in $E_P(Q_T)$. Then there exists a subsequence
$(u_{n(k)})\subset (u_n)$ such that, for every $\epsilon>0$,
there exists a constant value $\textbf{M}=\textbf{M}(\epsilon)$ and a
function $\psi\in L^1(0,T;W^{1,1}(\Omega))$ satisfying the following properties:
 \begin{gather}\label{psi1}
0\leq \psi\leq 1. \\
\label{psi2} \|\psi -1\|_{L^1(Q_T)} +\|\nabla \psi\|_{L^1(Q_T)}
\leq \epsilon. \\
 \label{psi3} |u|,|u_{n(k)}|\leq \textbf{M} \quad \text{on }
 \{\psi >0 \}  \text{ for all } k\geq 1.
 \end{gather}
 \end{lemma}

\begin{proof}
 According to lemmas~\ref{mod} and \ref{injec} we deduce the
 the following continuous inclusions:
 \[
L_P(Q_T)\hookrightarrow L_{\bar{P}}(Q_T) \hookrightarrow L_{\bar{M}}(Q_T).
\]
 Since $(u_n)$ is relatively compact in $E_P(Q_T)$, we can extract
a subsequence $(u_{n(k)})\subset (u_n)$ such that
 \begin{equation}\label{psi4}
 \sum_{k=1}^{\infty} \|u_{n(k)}-u\|_{L_{\bar{M}}(Q_T)}\leq 1.
 \end{equation}
Fix $K>0$ to be chosen later big enough and introduce the function $\gamma$
given by
 \begin{equation}\label{gamma1}
 \gamma =(|u|-K)^+ +\sum_{k=1}^{\infty}(|u_{n(k)}-u|-K)^+.
 \end{equation}
Then putting $v_k = u_{n(k)}-u$, $k\geq 1$, and $v_0=u$, we have
\begin{align*}
&\int_{Q_T}     (|v_k| -K)^+ + \int_{Q_T}      |\nabla (|v_k|-K)^+| \\
&=  \int_{\{|v_k|>K\}} (|v_k| -K)^+ \frac{|v_k|}{|v_k|}
 +    \int_{\{|v_k|>K\}}  |\nabla (|v_k|-K)^+|\frac{|v_k|}{|v_k|} \\
&\leq \frac{1}{K}  (\|v_k\|_{L_{M}(Q_T)}+ \|\nabla v_k\|_{L_{M}(Q_T)})
 \|v_k\|_{L_{\bar{M}}(Q_T)}
\end{align*}
Summing these inequalities, bearing in mind that $(u_{n(k)})$ and $(v_k)$
 are bounded in $ W^{1,x}L_M(Q_T)$ and \eqref{gamma1}, we deduce
\begin{align*}
&\sum_{k=0}^{\infty} (\|(|v_k|-K)^+\|_{L^1(Q_T)} +
 \|(|\nabla v_k|-K)^+\|_{L^1(Q_T)} )\\
&\leq \frac{C_0}{K}  \sum_{k=0}^{\infty} \|v_k\|_{L_{\bar{M}}(Q_T)} \\
&=\frac{C_0}{K} \Big( \|u\|_{L_{\bar{M}}(Q_T)} +
 \sum_{k=1}^{\infty} \|u_{n(k)}-u\|_{L_{\bar{M}}(Q_T)} \Big)\\
&\leq \frac{C_0}{K} ( \|u\|_{L_{\bar{M}}(Q_T)}+1 )
 = \frac{C}{K}.
 \end{align*}
 Hence
 \[
\|\gamma\|_{L^1(0,T;W^{1,1}(\Omega))}\leq \frac{C}{K}.
\]

It is straightforward to check that the function $\psi=(1-\gamma)^+$
satisfies the asserted condition \eqref{psi1}-\eqref{psi3}
for $K \ge C/\epsilon$ and $\mathbf{M}=K+1$.
 \end{proof}

 The next two results analyze the behavior of certain subsequences of
$(\varphi_n)$. They will allow us, together with the convergences deduced
in the previous steps, to pass to the limit in the approximate
problems~\eqref{55}-\eqref{57} in order to show the existence of
a capacity solution to the system~\eqref{P}.

\begin{lemma}[\cite{MM}] \label{lem5.6}
 Let $(u_n,\varphi_n)$ be a weak solution to the system~\eqref{55}-\eqref{57},
 $u\in E_P(Q_T)$ and $\varphi\in L^\infty(Q_T)$ the limit functions appearing,
respectively,
 in~\eqref{limit-varphi} and~\eqref{eq:unaeu}.
Then, for any function $S\in C^1_0(\mathbb{R})$, there exists a subsequence,
still denoted in the same way, such that
 \begin{equation}\label{ss1}
 S(u_n)\varphi_n \rightharpoonup S(u)\varphi \quad \text{ weakly in }
 L^2(0,T;H^1(\Omega)).
 \end{equation}
 Moreover, if $0\leq S\leq 1$, then there exists a constant $C>0$,
independent of  $S$, such that
 \begin{equation}\label{ss2}
  \limsup_{n\to \infty}
  \int_{Q_T}\rho_n(u_n)|\nabla [S(u_n)\varphi_n-S(u)\varphi]|^2
 \leq C\|S'\|_\infty (1+\|S'\|_\infty).
 \end{equation}
 \end{lemma}

 \begin{lemma}\label{lemvarphi}
 There exists a subsequence $(\varphi_{n(k)})\subset (\varphi_n)$ such that
 \begin{equation}\label{varf}
  \lim_{k\to \infty} \int_{Q_T} |\varphi_{n(k)}-\varphi|=0.
 \end{equation}
 \end{lemma}

 \begin{proof}
The proof of this result is almost identical to that of \cite[Lemma 4.8]{MM}.
For the sake of completeness, we include it here.

Since the conditions of Lemma~\ref{lempsi} are fulfilled by a suitable
 subsequence $(u_{n(k)})$, we have for every $\epsilon>0$ there exists
 $\mathbf{M}>0$ and $\psi \in L^1(0,T;W^{1,1}(\Omega))$ such that
\eqref{psi1}-\eqref{psi3} are satisfied. By \eqref{psi3}, there exists
 $C_{\mathbf{M}}>0$ such that
 \begin{equation}\label{eq:xik}
 \xi_k\stackrel{{\rm def}}{=}
\rho_{n(k)}(u_{n(k)})\geq C_{\mathbf{M}}\quad \text{on } \{ \psi>0 \},
 \text{ for all } k\geq 1.
 \end{equation}
We consider a sequence of regular functions $(S_m)\subset C^1_0(\mathbb{R})$
such that
 \begin{gather}\label{s1}
 0\leq S_m \leq 1, \quad S_m=1 \quad\text{in }
 [-{\mathbf{M}},{\mathbf{M}}], \text{ for all $k\geq 1$},\\
\label{s2}
 \|S'_m\|_{L^\infty(\mathbb{R})}\leq \frac{1}{m},\quad \text{for all $m\geq 1$.}
 \end{gather}
From \eqref{psi3} and \eqref{s1}, we write
 \[
  \int_{Q_T} |\varphi_{n(k)}-\varphi|=
  \int_{\{\psi>0\}} |S_m(u_{n(k)})\varphi_{n(k)}-S_m(u)\varphi|
 + \int_{\{\psi=0\}} |\varphi_{n(k)}-\varphi|.
 \]
 Inserting $\pm \psi |S_m(u_{n(k)})\varphi_{n(k)}-S_m(u)\varphi|$ in the first
integral above and
 $-\psi |\varphi_{n(k)}-\varphi|=0$ in the second one, then owing to~\eqref{p.1},
 \eqref{limit-varphi}, \eqref{psi1} and using Poincar\'e's inequality, we obtain
\begin{align*}
&\int_{Q_T} |\varphi_{n(k)}-\varphi| \\
&=\int_{\{\psi>0\}} \psi|S_m(u_{n(k)})\varphi_{n(k)}-S_m(u)\varphi| \\
&\quad +\int_{\{\psi>0\}} (1-\psi)|S_m(u_{n(k)})\varphi_{n(k)}-S_m(u)\varphi|
 + \int_{\{\psi=0\}} (1-\psi)|\varphi_{n(k)}-\varphi|  \\
&\le C_0\int_{Q_T}|\nabla(\psi(S_m(u_{n(k)})\varphi_{n(k)}-S_m(u)\varphi))|
 +2\|\varphi_0\|_{L^\infty(Q_T)}\int_{Q_T}|1-\psi|  \\
&\le  2C_0\|\varphi_0\|_{L^\infty(Q_T)}\int_{Q_T}|\nabla\psi|
 + C_0\int_{Q_T}  |\nabla (S_m(u_{n(k)}))\varphi_{n(k)}-S_m(u)\varphi )| \\
&\quad +2\|\varphi_0\|_{L^\infty(Q_T)}\int_{Q_T}|1-\psi|,
 \end{align*}
Putting $C^*=2\|\varphi_0\|_{L^\infty(\Omega)}\max(C_0,1)$,
 $K_\mathbf{M}=C_0C_\mathbf{M}^{-1/2}|\Omega|^{1/2}T^{1/2}$ and taking
into account~\eqref{psi2} and \eqref{eq:xik}, we deduce
\begin{align*}
&\int_{Q_T} |\varphi_{n(k)}-\varphi|\le C^* \epsilon+ C_0\int_{Q_T}
 \xi_k^{-1/2}\xi_k^{1/2}|\nabla (S_m(u_{n(k)}))\varphi_{n(k)}-S_m(u)\varphi )|
 \\
&\leq  C^* \epsilon+  K_{\mathbf{M}}\left(\int_{Q_T} \xi_k
 |\nabla (S_m(u_{n(k)}))\varphi_{n(k)}-S_m(u)\varphi|^2 \right)^{1/2},	
\end{align*}
Owing to \eqref{ss2} and \eqref{s2}, we obtain
\begin{align*}
  \limsup_{k\to \infty} \int_{Q_T} |\varphi_{n(k)}-\varphi|
&\leq  C^*\epsilon+K_M\Big(C\|S'_m\|_\infty (1+\|S'_m\|_\infty)\Big)^{1/2}\\
&\leq C^*\epsilon+K_M C^{1/2}\\big[\frac{1}{m}\Big(1+\frac{1}{m}\Big)\big]^{1/2}.
\end{align*}
And since $\epsilon>0$ and $m\geq 1$ are arbitrary, we derive the desired result.
 \end{proof}


 \noindent\textbf{Step 5.}
Passing to the limit.
According to \eqref{limit-varphi}, \eqref{limit-Phi}, \eqref{un}, \eqref{bounded}
and \eqref{u'},  it is straightforward that the condition~$(C_1)$ of
Definition \ref{def1} is fulfilled.
 The convergences in Proposition \ref{prop12} and Lemma~\ref{lemvarphi}
lead us to $(C_2)$ of Definition \ref{def1}1, and in order to obtain the condition $(C_3)$,
 using Proposition~\ref{prop12} and Lemma~\ref{lemvarphi} again with \eqref{ss1},
it is sufficient to let $k$ goes to infinity in the  expression
 \[
 S(u_{n(k)})\rho_{n(k)}(u_{n(k)})\nabla \varphi_{n(k)}=
 \rho_{n(k)}(u_{n(k)})[\nabla (S(u_{n(k)})\varphi_{n(k)})-\varphi_{n(k)}\nabla
 S(u_{n(k)})]
 \]
\smallskip

\noindent\textbf{Step 6.} Regularity of $u$.
Finally, it remains to show the regularity of $u\in C([0,T];L^1(\Omega))$ \cite{BA}.
To this end,  we go back to the expression~\eqref{84} but the integration
in time happens in the interval $(0,\tau)$ for any
 $\tau\in(0,T]$, namely
 \begin{equation}\label{eq:Qtau}
 \begin{aligned}
\langle \frac{\partial u_n}{\partial
 t},T_\nu(u_n-w_{\mu,j}^i) \rangle_{Q_\tau}
&=   \int_{Q_\tau}
     a_n(x,t,u_n,\nabla u_n)(\nabla w_{\mu,j}^i-\nabla u_n)
\chi_{\{|u_n-w_{\mu,j}^i|\le\nu\}}\\
 &\quad -\int_{Q_\tau} \rho_n(u_n)\varphi_n\nabla \varphi_n\nabla
 T_\nu(u_n-w_{\mu,j}^i).
 \end{aligned}
 \end{equation}
where $\nu\in (0,1]$, $Q_\tau=(0,\tau)\times\Omega$ and
$\langle\cdot,\cdot\rangle_{Q_\tau}$ is the duality product between
 $W^{-1,x}L_{\bar{M}}(Q_\tau)$ and $W^{1,x}_0L_M(Q_\tau)$.
 We will consider the necessary subsequences to assure the almost
 everywhere convergence in $Q_T$ of $\varphi_n\to\varphi$, $u_n\to u$,
$\nabla u_n\to\nabla u$,
 and also for $(T_\nu(u_n-w_{\mu,j}^i))$, etc.

From \eqref{eq:antoa} we readily obtain
\begin{align*}
&\lim_{n\to\infty}
 \int_{Q_\tau}    a_n(x,t,u_n,\nabla u_n)\nabla w_{\mu,j}^i
\chi_{\{|u_n-w_{\mu,j}^i|\le\nu\}} \\
 &= \int_{Q_\tau}    a(x,t,u,\nabla u)\nabla w_{\mu,j}^i
 \chi_{\{|u-w_{\mu,j}^i|\le\nu\}}
\end{align*}
Also, by Fatou's lemma we obtain
 \[
\int_{Q_\tau}  a(x,t,u,\nabla u)\nabla u\chi_{\{|u-w_{\mu,j}^i|\le\nu\}}
\le  \liminf_{n\to\infty}\int_{Q_\tau}
 a_n(x,t,u_n,\nabla u_n)\nabla u_n\chi_{\{|u_n-w_{\mu,j}^i|\le\nu\}}
\]
 then, passing to the limit in these two expressions, first in $j$, then in $\mu$,
$i$ and $K$, we deduce, uniformly in $\tau$, that
 \begin{equation}\label{eq:epsan}
 \int_{Q_\tau}  a_n(x,t,u_n,\nabla u_n)(\nabla w_{\mu,j}^i-\nabla u_n)
\chi_{\{|u_n-w_{\mu,j}^i|\le\nu\}}
 \le  \epsilon(n,j,\mu,i,K)
 \end{equation}

The analysis of the term $\int_{Q_\tau} \rho_n(u_n)\varphi_n\nabla \varphi_n
 \nabla T_\nu(u_n-w_{\mu,j}^i)\,{\rm d}x\,{\rm d}t$ is more involved.
Here the difficulty relies on the fact
 that the sequence $(\rho_n(u_n)|\nabla \varphi_n|^2)$ does not converge,
in general, strongly in $L^1(Q_T)$.
To deal with this situation, we are going to make use of the properties already
shown for a capacity solution. Indeed, we first notice that
 $\nabla T_\nu(u_n-w_{\mu,j}^i)=0$ in the set
$\{|u_n|\le K+\nu\}\subset \{|u_n|\le K+1\}$. Then we consider a
 sequence of functions $S_K\subset C^1_0(\mathbb{R})$ such that
\begin{gather*}
 0\leq S_K \leq 1, \quad
S_K=1 \quad \text{in } [-(K+1),K+1], \text{ for all $K>0$},\\
 \|S'_K\|_{L^\infty(\mathbb{R})}\leq \frac{1}{K+1}, \quad \text{for all $K>0$.}
 \end{gather*}
We have
 \begin{align*}
&\int_{Q_\tau} \rho_n(u_n)\varphi_n\nabla \varphi_n\nabla
 T_\nu(u_n-w_{\mu,j}^i)\,{\rm d}x\,{\rm d}t \\
& =
 \int_{Q_\tau} \rho_n(u_n)\varphi_n\nabla [S_K(u_n)\varphi_n]\nabla
 T_\nu(u_n-w_{\mu,j}^i)\,{\rm d}x\,{\rm d}t  \\
&=  \int_{Q_\tau} \rho_n(u_n)\varphi_n\nabla [S_K(u_n)\varphi_n-S(u)\varphi]\nabla
 T_\nu(u_n-w_{\mu,j}^i)\,{\rm d}x\,{\rm d}t \\
&\quad  + \int_{Q_\tau} \rho_n(u_n)\varphi_n\nabla [S_K(u)\varphi]\nabla
 T_\nu(u_n-w_{\mu,j}^i)\,{\rm d}x\,{\rm d}t = I_1+I_2.
 \end{align*}
According to the almost everywhere convergence of $(u_n)$ and
$(\varphi_n)$ together with \eqref{p.1}
 and \eqref{un}, we readily deduce that
 \[
 \lim_{n\to\infty} I_2 =\int_{Q_\tau} \rho(u)\varphi\nabla [S_K(u)\varphi]
 \nabla T_\nu(u-w_{\mu,j}^i)\,{\rm d}x\,{\rm d}t,
 \]
 and using the identity $(C_3)$, already shown in the previous step, namely,
 \[
\rho(u)\nabla[S_K(u)\varphi]=S_K(u)\Phi+\varphi\nabla S_K(u),
\]
 we can easily obtain the estimate
 \[
 I_2=\epsilon(n,j,\mu,i,K).
 \]
 As for the term $I_1$, we use \eqref{ss2} to get, for some constant $C>0$,
 \begin{align*}
 |I_1|^2
&\leq \Big(\int_{Q_\tau} \rho_n(u_n)|\nabla [S_K(u_n)\varphi_n-S(u)\varphi]|^2
 \,{\rm d}x\,{\rm d}t \Big)\\
&\quad\times
 \Big(\int_{Q_\tau} \rho_n(u_n)|\varphi_n|^2 |\nabla T_\nu(u_n-w_{\mu,j}^i)
 |^2\,{\rm d}x\,{\rm d}t \Big) \\
&\le\frac{C}{K+1},
 \end{align*}
and thus it is also
 \[
 I_1=\epsilon(n,j,\mu,i,K).
 \]
Consequently, we obtain, for any fixed $\nu\in(0,1]$ and uniformly in $\tau\in[0,T]$,
 \begin{equation}\label{eq:epsrhon}
 \int_{Q_\tau} \rho_n(u_n)\varphi_n\nabla \varphi_n\nabla
 T_\nu(u_n-w_{\mu,j}^i)\,{\rm d}x\,{\rm d}t\le
 \epsilon(n,j,\mu,i,K),
 \end{equation}
Gathering \eqref{eq:Qtau}, \eqref{eq:epsan} and \eqref{eq:epsrhon}
 we obtain the estimate
 \begin{equation}\label{eq:partialun}
\langle \frac{\partial u_n}{\partial
 t},T_\nu(u_n-w_{\mu,j}^i)\rangle_{Q_\tau}
\le \epsilon(n,j,\mu,i,K).
 \end{equation}
Then we write, as in \eqref{eq:parabolic}-\eqref{eq:limjn},
\begin{align*}
&\int_\Omega S_\nu(u_n(x,\tau) - w_{\mu,j}^i(x,\tau))\,{\rm d}x \\
&= \big\langle \frac{\partial (u_n-w_{\mu,j}^i)}{\partial
 t},T_\nu(u_n-w_{\mu,j}^i) \big\rangle_{Q_\tau}
 +\int_\Omega S_\nu(u_0 - T_K(\psi_i))\,{\rm d}x\\
 &=\big\langle \frac{\partial u_n}{\partial  t},T_\nu(u_n-w_{\mu,j}^i)
\big\rangle_{Q_\tau}
 - \big\langle \frac{\partial w_{\mu,j}^i}{\partial
 t},T_\nu(u_n-w_{\mu,j}^i)\big\rangle_{Q_\tau}
+\int_\Omega  S_\nu(u_0 - T_K(\psi_i))\,{\rm d}x.
 \end{align*}
Consequently, owing to \eqref{eq:limjn} and \eqref{eq:partialun}, it yields,
for every fixed $\nu\in(0,1]$ and uniformly in $\tau\in[0,T]$,
 \[
 \int_\Omega S_\nu(u_n(x,\tau) - w_{\mu,j}^i(x,\tau))\,{\rm d}x\le \epsilon(n,j,
\mu,i,K),
 \]
 and using the convexity of the function $S_\nu$ we may also derive the
estimate
\begin{align*}
&\int_\Omega S_\nu\Big(\frac{1}{2}(u_n(x,\tau)
 - u_m(x,\tau))\Big)\,{\rm d}x
 \\
&\le\frac{1}{2}\int_\Omega S_\nu(u_n(x,\tau) - w_{\mu,j}^i(x,\tau))\,{\rm d}x
 + \frac{1}{2}\int_\Omega S_\nu(u_m(x,\tau) - w_{\mu,j}^i(x,\tau))\,{\rm d}x\\
&\le \epsilon(n,j,\mu,i,K)+\epsilon(m,j,\mu,i,K),
 \end{align*}
and thus, for any fixed $\nu>0$ and uniformly in $\tau\in[0,T]$, we have
 \begin{equation}\label{eq:Snunm}
 \int_\Omega S_\nu\Big(\frac{1}{2}(u_n(x,\tau) - u_m(x,\tau))\Big)\,{\rm d}x
 \le \epsilon(n)+\epsilon(m).
 \end{equation}
 Consequently, using the definition of $S_\nu$ and \eqref{eq:Snunm},
for all $\tau\in[0,T]$, we have
 \begin{align*}
&\int_\Omega {\frac{1}{2}}|u_n(x,\tau) - u_m(x,\tau)|\,{\rm d}x\\
&\le  \int_{\{|u_n(x,\tau) - u_m(x,\tau)|\le 2\nu\}}
 {\frac{1}{2}}|u_n(x,\tau) - u_m(x,\tau)|\,{\rm d}x
 \\
 &\quad  +\int_{\{|u_n(x,\tau) - u_m(x,\tau)|> 2\nu\}}
 {\frac{1}{2}}|u_n(x,\tau) - u_m(x,\tau)|\,{\rm d}x
 \\
 &\le |\Omega|\nu + \frac{1}{\nu}\int_{\{|u_n(x,\tau) - u_m(x,\tau)|> 2\nu\}}
 {\frac{\nu}{2}}|u_n(x,\tau) - u_m(x,\tau)|\,{\rm d}x
 \\
 &= |\Omega|\nu + \frac{1}{\nu}\int_{\{|u_n(x,\tau) - u_m(x,\tau)|> 2\nu\}}
 \big[S_\nu\Big({\frac{1}{2}}|u_n(x,\tau) - u_m(x,\tau)|\Big)
 +\frac{\nu^2}{2}\big]\,{\rm d}x
 \\
&= \frac{3}{2}|\Omega|\nu + \frac{1}{\nu}(\epsilon(n)+\epsilon(m)).
 \end{align*}
This last estimate shows that $(u_n)$ is a Cauchy sequence in the space
$C([0,T];L^1(\Omega))$ and, in particular, its limit $u$ lies in this space.
 This completes the proof.
\end{proof}

 \begin{remark} \label{rmk5.8} \rm
The previous result given in Theorem~\ref{t1} gives just the existence
of a capacity solution. The uniqueness of the capacity solution is an open
problem, even in a Hilbertian context.
Other interesting questions on this capacity solution are concerned with
the establishment of certain qualitative properties \cite{Antontsev-Shmarev}
as the derivation of some energy estimate, the analysis of large time
behavior or even the occurrence of a blow-up situation.
\end{remark}


\subsection*{Acknowledgements}
The authors wish to thank the anonymous referees for the comments and suggestions
that has led us to improve the presentation of this article.

This research was partially supported by Ministerio de
Econom\'{\i}a y Competitividad of the Spanish Government under grant
 TEC2014-54357-C2-2-R with the participation of FEDER.


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