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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 100, pp. 1--17.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2013 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2013/100\hfil Weak solutions]
{Weak solutions for nonlocal evolution variational inequalities
 involving  gradient constraints and variable exponent}

\author[M. Xiang, Y. Fu \hfil EJDE-2013/100\hfilneg]
{Mingqi Xiang, Yongqiang Fu}  % in alphabetical order

\address{Mingqi Xiang  \newline
Department of Mathematics, Harbin Institute of Technology,
 Harbin, 150001, China}
\email{xiangmingqi\_hit@163.com}

\address{Yongqiang Fu  \newline
Department of Mathematics, Harbin Institute of Technology,
 Harbin, 150001, China}
\email{fuyqhagd@yahoo.cn}


\thanks{Submitted March 7, 2013. Published April 19, 2013.}
\subjclass[2000]{35K30, 35K86, 35K59}
\keywords{Nonlocal evolution variational inequality;
variable exponent space;  \hfill\break\indent
Galerkin approximation; penalty method}

\begin{abstract}
 In this article, we study a class of nonlocal quasilinear parabolic
 variational inequality involving $p(x)$-Laplacian operator and gradient
 constraint on a bounded domain. Choosing a special penalty functional
 according to the gradient constraint, we transform the variational
 inequality to a parabolic equation. By means of Galerkin's approximation
 method, we obtain the existence of weak solutions for this equation,
 and then through a priori estimates, we obtain the weak solutions of
 variational inequality.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks


\section{Introduction}

In this article, we are concerned with the existence of weak solutions
for nonlocal (Kirchhoff type) parabolic variational inequality involving
variable exponent. More precisely, we shall find a function
 $u\in\mathscr{K}=\{w(x,t)\in V(Q_T)\cap L^{\infty}(0,T;L^2(\Omega)):
 w(x,0)=0,  |\nabla w(x,t)|\leq 1\text{ a.e. } (x,t)\in Q_T\}$
satisfying the follow inequality
\begin{equation}
\begin{aligned}
& \int_{Q_T}\frac{\partial v}{\partial t}(v-u)\,dx\,dt
+\int_0^Ta\Big(t,\int_\Omega |\nabla u|^{p(x)}dx\Big)
\int_\Omega |\nabla u|^{p(x)-2}\nabla u\nabla(v-u)\,dx\,dt\\
&\geq \int_{Q_T}f(v-u)\,dx\,dt, 
\end{aligned} \label{e1.1}
\end{equation}
for all $v\in V(Q_T)$ with $\frac{\partial v}{\partial t}\in V'(Q_T)$,
 $v(x,0)=0$, $|\nabla v(x,t)|\leq1$ a.e. $(x,t)\in Q_T$, where $V'(Q_T)$
is the dual space of variable exponent Sobolev space $V(Q_T)$
(see Definition \ref{def2.2} below).


In recent years, the research of nonlinear problems
with variable exponent growth conditions has been an interesting topic.
$p(\cdot)$-growth problems can be regarded as a kind of nonstandard
growth problems and these problems possess very complicated nonlinearities,
for instance, the $p(x)$-Laplacian operator
$-\operatorname{div}(|\nabla u|^{p(x)-2}\nabla u)$ is inhomogeneous.
 And these problems have many important applications in nonlinear elastic,
electrorheological fluids and image restoration (see \cite{c3,r1,z2,z3,z4}).
Many results have been obtained on this kind of problems,
see \cite{a1,a2,b1,b2,d1,d2,f2,f3,f4,m1}.
 Especially, in \cite{b2,m1}, the authors studied
the existence and uniqueness of weak solutions for anisotropy
parabolic variation inequalities in the framework of variable
exponent Sobolev spaces. Motivating by their works, we study a class
variational inequalities with gradient constrain and variable exponent.
To the best of our knowledge, there are no papers dealing with parabolic
equalities involving variable growth and gradient constraints.
For the fundamental theory about variable exponent Lebesgue and
Sobolev spaces, we refer to \cite{f1,k1}. The basic theory about
Variational inequalities, we refer the reader to \cite{c1,m2} for the details.


The study of Kirchhoff-type problems has  received considerable attention
in recent years, see \cite{a3,a4,c4,g3,g2,h1,t1,z1}.
This interest arises from their contributions to the modeling of many
physical and biological phenomena. We refer the reader to \cite{g1,l3}
for some interesting results and further references.
In \cite{a3,a4}, the authors discussed the asymptotic stability for
Kirchhoff systems with variable exponent growth conditions
\begin{gather*}
u_{tt}-M(\mathscr{F}u(t))\Delta_{p(x)}u+Q(t,x,u,u_t)+f(x,u)=0\quad\text{in }
 \mathbb{R}_0^{+}\times\Omega\\
u(t,x)=0\quad\text{on } \mathbb{R}_0^{+}\times\partial\Omega,
\end{gather*}
where $M(\tau)=a+b\tau^{\gamma-1}$, $\tau\geq0$ with $a,b\geq0, a+b>0$
and $\gamma>1$, and $\mathscr{F}u(t)=\int_\Omega\{|Du(x,t)|^{p(x)}/p(x)\}dx$,
$\Delta _{p(x)}=\operatorname{div}(|Du|^{p(x)-2}Du)$.

On the one hand, our motivation for investigating \eqref{e1.1} arises
from reaction-diffusion equations that model population density or
 heat propagation (see \cite{c2}). The following equation describes the
density of a population (for instance of bacteria) subject to spreading
\begin{gather*}
u_t={a(u)}\Delta u+F(u)\quad\text{in }\Omega\times(0,T),\\
u(x,t)=0\quad\text{on }\partial\Omega\times(0,T),\\
u(x,0)=u_0(x)\quad\text{in }\Omega.
\end{gather*}
The diffusion coefficient $a$ depends on a nonlocal quantity
related to the total population in the domain $\Omega$;
that is, the diffusion of individuals is guided by the global state
of the population in the medium. From an experimentalist point view,
it certainly makes sense to introduce nonlocal quantities,
since measurements are often averages. The function $F$ describes
the reaction or growth of the population.

On the other hand, we can use problem \eqref{e1.1} to describe
the motion of a nonstationary fluid or gas in a nonhomogeneous
and anisotropic medium and the nonlocal term $a$ appearing in
\eqref{e1.1} can describe a possible change in the global state
of the fluid or gas caused by its motion in the considered medium.

This article is organized as follows. In section 2, we will give
some necessary definitions and properties of variable exponent Lebesgue
spaces and Sobolev spaces. Moreover, we introduce the space $V(Q_{T})$
and give some necessary properties, which provides a basic framework
 to solve our problem. In section 3, using the penalty method,
we consider class of parametrized parabolic equations, and obtain 
weak solutions by Galerkin's approximation.
In section 4, we give the proof of main theorem to this paper.

\section{Preliminaries}

In this section, we first recall some important properties of variable
exponent Lebesgue spaces and Sobolev spaces, see \cite{d2,f1,k1}
 for details.


\subsection{Variable exponent Lebesgue space and Sobolev space}


Let $\Omega\subset\mathbb{R}^N$ be a domain.
A measurable function $p:\Omega\to [1,\infty)$ is called a
variable exponent and we define $p^{-}=\operatorname{ess\,inf}_{x\in \Omega}p(x)$
 and $p^{+}=\operatorname{ess\,sup}_{x\in \Omega}p(x)$. If $p^{+}$
is finite, then the exponent $p$ is said to be bounded.
The variable exponent Lebesgue space is
\[
L^{p(x)}(\Omega)=\{u:\Omega\to \mathbb{R}
\text{ is a measurable function}; \rho_{p(x)}(u)
=\int_{\Omega}|u(x)|^{p(x)}dx<\infty\}
\]
with the Luxemburg norm
\[
\|u\|_{L^{p(x)}(\Omega)}=\inf\{\lambda>0 : \rho_{p(x)}(\lambda^{-1}u)\leq1\},
\]
then $L^{p(x)}(\Omega)$ is a Banach space, and when $p$ is bounded,
we have the following relations
\begin{align*}
\min\{\|u\|^{p^-}_{L^{p(x)}(\Omega)},\|u\|^{p^+}_{L^{p(x)}(\Omega)}\}
\leq\rho_{p(x)}(u)
\leq\max\{\|u\|^{p^-}_{L^{p(x)}(\Omega)},\|u\|^{p^+}_{L^{p(x)}(\Omega)}\}.
\end{align*}
That is, if $p$ is bounded, then norm convergence is equivalent to
convergence with respect to the modular $\rho_{p(x)}$.
For bounded exponent the dual space $(L^{p(x)}(\Omega))'$
can be identified with $L^{p'(x)}(\Omega)$, where the conjugate
exponent $p'$ is defined by $p'=\frac{p}{p-1}$.
If $1<p^{-}\leq p^{+}<\infty$, then the variable exponent Lebesgue space
$L^{p(x)}(\Omega)$ is separable and reflexive.


In the variable exponent Lebesgue space, H\"older's inequality
is still valid. For all $u\in L^{p(x)}(\Omega),\ v\in L^{p'(x)}(\Omega)$
with $p(x)\in (1,\infty)$ the following inequality holds
\[
\int_\Omega|uv|dx\leq\big(\frac{1}{p^-}+\frac{1}{(p')^-}\big)
\|u\|_{L^{p(x)}(\Omega)}\|v\|_{L^{p'(x)}(\Omega)}
\leq2\|u\|_{L^{p(x)}(\Omega)}\|v\|_{L^{p'(x)}(\Omega)}.
\]

\begin{definition}[\cite{d1,d2}]  \label{def2.1} \rm
We say a bounded exponent $p: \Omega\to \mathbb{R}$ is globally
log-H\"older continuous if $p$ satisfies the following two conditions:
\begin{itemize}
\item[(1)] there is a constant $c_1>0$ such that
 \[
|p(y)-p(z)|\leq\frac{c_1}{\log(e+|y-z|^{-1})}
\]
for all points $y, z\in \Omega$;

\item[(2)] there exist constants $c_{2}>0$ and $p_{\infty}\in\mathbb{R}$
such that
\[
|p(y)-p_{\infty}|\leq\frac{c_{2}}{\log(e+|y|^{-1})}
\]
for all $y\in \Omega$.
\end{itemize}
\end{definition}


The Variable exponent Sobolev space $W^{1,p(x)}(\Omega)$ is defined as
\[
W^{1,p(x)}(\Omega)=\{u\in L^{p(x)}(\Omega):  |\nabla u|\in L^{p(x)}(\Omega)\}
\]
and equipped with the norm
\[
\|u\|_{W^{1,p(x)}(\Omega)}=\|u\|_{L^{p(x)}(\Omega)}
+\|\nabla u\|_{L^{p(x)}(\Omega)},
\]
then the space $W^{1,p(x)}(\Omega)$ is a Banach space.
 The space $W^{1,p(x)}_{0}(\Omega)$ is defined as the closure of
 $C^{\infty}_{0}(\Omega)$ with the norm of $\|\cdot\|_{W^{1,p(x)}(\Omega)}$.
If $1<p^{-}\leq p^{+}<\infty$, then the space $W^{1,p(x)}(\Omega)$ is
separable and reflexive.

\begin{theorem}[\cite{d1,d2}] \label{thm2.1}
Let $\Omega\subset\mathbb{R}^N$ be a bounded domain and assume that
 $p: \mathbb{R}^N\to (1, \infty)$ is a bounded globally log-H\"older
continuous exponent such that $p^{-}>1$, then for every
 $u\in W^{1, p(x)}_{0}(\Omega)$ we have
\[
\|u\|_{L^{p(x)}(\Omega)}\leq c \operatorname{diam}(\Omega)\|\nabla u\|_{L^{p(x)}(\Omega)},
\]
where the constant $c$ only depends on the dimension $N$ and the
log-H\"older constant of $p$.
\end{theorem}


\subsection{Variable exponent Sobolev space $V(Q_T)$}

\begin{definition} \label{def2.2} \rm
Let $\Omega\subset\mathbb{R}^N$ be a bounded domain with smooth boundary.
Denote $Q_T=\Omega\times(0,T),0<T<\infty$. Suppose that $p(x)$ is a
bounded globally log-H\"older continuous function on $\overline{\Omega}$
with $p^{-}>1$, we set
\[
V(Q_{T})=\{u\in L^{2}(Q_{T}) : |\nabla u|\in L^{p(x)}(Q_{T}),
u(\cdot,t)\in W_0^{1,p(x)}(\Omega)\quad\text{a.e. }t\in(0,T)\},
\]
with the norm
 \[
\|u\|=\|u\|_{L^{2}(Q_{T})}+\|\nabla u\|_{L^{p(x)}(Q_{T})}.
\]
\end{definition}


\begin{remark}\label{rmk2.1} \rm
Following the standard proof for Sobolev spaces, we can prove that
$V(Q_{T})$ is a Banach space, and it's easy to check that $V(Q_{T})$
can be continuously embedded into the space
$L^r(0,T; W^{1, p^{-}}_{0}(\Omega)\cap L^2(\Omega))$, where
$r=\min\{p^-,2\}$.
It is worth to mention the paper \cite{b2} where the space $V(Q_T)$
is defined in a similar way.
\end{remark}

By the same method in \cite{d1}, we have the following theorem.

\begin{theorem}[\cite{d1}] \label{thm2.2}
The space $C^{\infty}_{0}(Q_{T})$ is dense in $V(Q_{T})$.
\end{theorem}

Since $C^{\infty}_{0}(Q_{T})\subset C^{\infty}(0,T; C^{\infty}_{0}(\Omega))$,
we have the following result.

\begin{lemma} \label{lem2.1}
The space $ C^{\infty}(0,T; C^{\infty}_{0}(\Omega))$ is dense in $V(Q_{T})$.
\end{lemma}

Let $V'(Q_{T})$ denote the dual space of $V(Q_{T})$.

\begin{theorem}[\cite{b2,d1}] \label{thm2.3}
A function $g\in V'(Q_{T})$ if and only if there exist
$\bar{g}\in L^{2}(Q_{T})$ and $\bar{G}\in (L^{p'(x)}(Q_{T}))^N$ such that
\begin{equation}
\int_{Q_{T}}g\varphi \,dx\,dt
=\int_{Q_{T}}\bar{g}\varphi \,dx\,dt 
+ \int_{Q_{T}}\bar{G}\nabla \varphi \,dx\,dt.\label{e2.4}
\end{equation}
\end{theorem}

\begin{remark} \label{rmk2.2} \rm
It follows from the proof of Theorem \ref{thm2.3} that $V(Q_{T})$ is reflexive and
\[
V'(Q_{T})\hookrightarrow L^{s'}(0,T; W^{-1,(p^{+})'}(\Omega)+L^{2}(\Omega)),
\quad\text{where } s=\max\{p^{+},2\}.
\]
\end{remark}

Similar to that in \cite{d1}, we give the following definition.

\begin{definition} \label{def2.3} \rm
We define the space $W(Q_{T})=\{u\in V(Q_{T}):
\frac{\partial u}{\partial t}\in V'(Q_{T})\}$ with the norm
\[
\|u\|_{W(Q_{T})}=\|u\|_{V(Q_{T})}+\big\|\frac{\partial u}{\partial t}
\big\|_{V'(Q_{T})},
\]
where $\frac{\partial u}{\partial t}$ is the weak derivative of $u$
with respect to time variable $t$ defined by
\[
\int_{Q_{T}}\frac{\partial u}{\partial t}\varphi \,dx\,dt
=-\int_{Q_{T}}u\frac{\partial \varphi}{\partial t}\,dx\,dt,\quad
\text{for all } \varphi\in C^{\infty}_{0}(Q_{T}).
\]
\end{definition}

\begin{lemma}[\cite{d1}] \label{lem2.2}
The space $W(Q_{T})$ is a Banach space.
\end{lemma}

By the method in \cite{d1}, we have the following result.

\begin{theorem} \label{thm2.4}
The space $C^{\infty}(0,T;C^{\infty}_{0}(\Omega))$ is dense in $W(Q_{T})$.
\end{theorem}

The following theorem can be proved similarly to that in \cite{d1},
 thus we omit its proof.

\begin{theorem}[\cite{a1,d1}] \label{thm2.5}
 $W(Q_{T})$ can be embedded continuously  in $C(0,T;L^{2}(\Omega))$.
 Furthermore, for all $u,v \in W(Q_T)$ and $s,t\in [0,T]$ the following rule for integration by parts is valid
\[
\int^t_s\int_{\Omega}\frac{\partial u}{\partial t}v\,dx\,d\tau
=\int_{\Omega}u(x,t)v(x,t)dx-\int_{\Omega}u(x,s)v(x,s)dx
-\int^t_s\int_{\Omega}u\frac{\partial v}{\partial t}\,dx\,d\tau.
\]
\end{theorem}


The following theorem gives a relation between almost everywhere
convergence and weak convergence.

\begin{theorem}[\cite{c3}] \label{thm2.6}
Let $p(x): Q_{T}\to  \mathbb{R}$ be a bounded globally
log-H\"older continuous function, with $p^{-}>1$.
If $\{u_{n}\}^{\infty}_{n=1}$ is bounded in $L^{p(x)}(Q_{T})$ and
$u_{n}\to u$ a.e. $(x,t)\in Q_{T}$ as $n\to\infty$, then there
exists a subsequence of $\{u_n\}$ still denoted by $\{u_n\}$ such
that  $u_{n}\to u$ weakly in $L^{p(x)}(Q_{T})$ as $n\to\infty$.
\end{theorem}

We will give a compact embedding for $V(Q_T)$ in the following.

\begin{theorem}[\cite{l2}] \label{thm2.7}
Let $B_0\subset B\subset B_1 $ be three Banach spaces, where $B_0$,
$B_1$ are reflexive, and the embedding $B_0\subset B$ is compact.
Denote $W=\{v:v\in L^{p_0}(0,T;B_0),
 \frac{\partial v}{\partial t}\in L^{p_1}(0,T;B_1)\}$, where $T$
is a fixed positive number, $1<p_i<\infty$, $i=0,1$, then $W$ can
be compactly embedded into $L^{p_0}(0,T;B)$.
\end{theorem}

\begin{theorem} \label{thm2.8}
Let $F$ be a bound subset in $V(Q_T)$ and
$\{\frac{\partial u}{\partial t}:u\in F\}$ be bounded in $V'(Q_T)$,
then $F$ is relatively compact in $L^r(0,T;L^2(\Omega))$.
\end{theorem}

\begin{proof}
Since $p^- > \frac{2N}{N+2}$ $(N\geq2)$, the embedding
$W_0^{1,p^-}(\Omega)\hookrightarrow L^2(\Omega)$ is compact.
By Remarks \ref{rmk2.1} and \ref{rmk2.2},  the embeddings
$V(Q_T)\hookrightarrow L^r(0,T;W_0^{1,p^-}(\Omega)\cap L^2(\Omega))$ and
\begin{align*}
V'(Q_{T})\hookrightarrow L^{s'}(0,T; W^{-1,(p^{+})'}
(\Omega)+L^2(\Omega))\hookrightarrow L^{s'}(0,T; W^{-1,\lambda}(\Omega))
\end{align*}
are continuous, where $\lambda=\min\{2,(p^+)'\}$. As the embedding
$L^2(\Omega)\hookrightarrow W^{-1,\lambda}(\Omega)$ is continuous,
by Theorem \ref{thm2.7}, $F$ is relatively compact in $L^r(0,T;L^2(\Omega))$.
\end{proof}

\section{Existence of solutions for parabolic equations}

In this section, for $\varepsilon\in(0,1)$, we consider the following
nonlocal parabolic equation with Diriclet boundary-value conditions:
\begin{equation} \label{e3.1}
\begin{gathered}
\begin{aligned}
&\frac{\partial u}{\partial t}-a\Big(t,\int_\Omega |\nabla u|^{p(x)}dx\Big)
\operatorname{div}(|\nabla u|^{p(x)-2}\nabla u)\\
&-\frac{1}{\varepsilon}\operatorname{div} \big((|\nabla u|^{p(x)-2}-1)^+
\nabla u\big) 
=f(x,t), \quad (x,t)\in \Omega\times(0,T),
\end{aligned}\\
u(x,t)=0, \quad (x,t)\in\partial\Omega\times(0,T),\\
u(x,0)=0,\quad  x\in\Omega,
\end{gathered}
\end{equation}
where $(|\nabla u|^{p(x)-2}-1)^+=\max\{|\nabla u|^{p(x)-2}-1,0\}$.
We assume that
\begin{itemize}
\item[(H1)] $a(t,s):[0,\infty)\times[0,\infty)\to(0,\infty)$
 is a continuous function and there exists two positive constants
 $a_0$ and $a_1$ such that
\[
a_0\leq a(t,s)\leq a_1\quad \text{for each } (t,s)\in[0,\infty)\times[0,\infty).
\]

\item[(H2)] $p(x): \Omega\to (1,\infty)$ is a global log-H\"older
continuous function. Denote $p^-=\inf_{x\in\overline{\Omega}}p(x)$,
$p^+=\sup_{x\in\overline{\Omega}}p(x)$. And there holds
\[
2<p^-\leq p(x)\leq p^+<\infty\quad\text{for each } x\in \Omega.
\]

\item[(H3)] $f\in V'(Q_T)$.
\end{itemize}


\begin{definition} \label{def3.1}\rm
 A function $u_\varepsilon\in V(Q_{T})\cap C(0,T;L^2(\Omega))$ with
$\frac{\partial u_\varepsilon}{\partial t}\in V'(Q_{T})$ is called a
weak solution of \eqref{e3.1}, if
\begin{align*}
&\int_{Q_T}\frac{\partial u_\varepsilon}{\partial t}\varphi \,dx\,dt
+\int_{0}^{T} a\Big(t,\int_\Omega|\nabla u_\varepsilon|^{p(x)}dx\Big)
\int_\Omega|\nabla u_\varepsilon|^{p(x)-2}\nabla u_\varepsilon
 \nabla\varphi \,dx\,dt\\
&+ \int_{Q_T}\frac{1}{\varepsilon}(|\nabla u_\varepsilon|^{p(x)-2}-1)^+
\nabla u_\varepsilon\nabla\varphi \,dx\,dt= \int_{Q_T}f\varphi \,dx\,dt,
\end{align*}
for all $\varphi\in V(Q_{T})$.
\end{definition}

Since $f\in V'(Q_T)$, there exists a sequence $f_n\in C^{\infty}_{0}(Q_T)$
such that $\lim_{n\to \infty} f_n= f$ in $V'(Q_T)$.
Similar to that in \cite{f2,f3}, we choose a sequence
$\{w_j\}_{j=1}^\infty\subset C_0^\infty(\Omega)$ such that
$C_0^\infty(\Omega)\subset \overline{\cup_{n=1}^\infty V_n}^{C^{1}(\bar{\Omega})}$
and $\{w_j\}_{j=1}^\infty$ is a standard orthogonal basis in $L^2(\Omega)$,
where $V_n=\mathrm{span}\{w_1, w_2, \dots, w_n\}$.

\begin{theorem} \label{thm3.1}
 Let assumptions {\rm (H1)--(H3)} hold and let $\varepsilon\in(0,1)$ be fixed.
Then there exists a weak solution for equation \eqref{e3.1}.
\end{theorem}

\begin{proof}
(i) \textbf{Galerkin approximation.}
 For each $n\in \mathbb{N}$, we want to find the approximate
solutions to problem \eqref{e3.1} in the form
\[
u_n(x,t)=\sum_{j=1}^n (\eta_n(t))_j w_j(x).
\]
First we define a vector-valued function
$P_n(t,\upsilon): [0,1]\times \mathbb{R}^n \to \mathbb{R}^n$ as
\begin{align*}
(P_n(t,\nu))_i
&=a\Big(t,\int_\Omega|\sum_{j=1}^n\nu_j\nabla\omega_j|^{p(x)}dx\Big)
\int_\Omega |\sum_{j=1}^n\nu_j\nabla w_j|^{p(x)-2}
\big(\sum_{j=1}^n\nu_j\nabla w_j\big)\nabla w_idx\\
&\quad +\int_\Omega\frac{1}{\varepsilon}
\big(\big|\sum_{j=1}^n\nu_j\nabla w_j\big|^{p(x)-2}-1\big)^+
\big(\sum_{j=1}^n \nu_j\nabla w_j\big)\nabla w_i dx,
\end{align*}
where $\nu=(\nu_1,\cdot\cdot\cdot,\nu_n)$. Since $a$ and $p$ are
continuous, from the definition of $P_n(t,\nu)$, $P_n(t,\nu)$
is continuous with respect to $t$ and $\nu$.

We consider the following ordinary differential systems
\begin{equation} \label{e3.2}
\begin{gathered}
\eta'(t)+P_n(t,\eta(t))=F_n,\\
\eta(0)=0,
\end{gathered}
\end{equation}
where
$(F_n)_i=\int_\Omega f_nw_i dx$.

Multiplying \eqref{e3.2} by $\eta(t)$, we arrive at the equality
 \[
\eta'(t)\eta(t)+P_n\left(t,\eta(t)\right)\eta(t)=F_n\eta(t).
\]
Since
\begin{align*}
P_n(t,\eta)\eta
&=a\Big(t,\int_\Omega|\sum_{j=1}^n\eta_j\nabla\omega_j|^{p(x)}dx\Big)
\int_\Omega |\sum_{j=1}^n\eta_j\nabla w_j|^{p(x)}dx\\
&\quad +\int_\Omega\frac{1}{\varepsilon}(|\sum_{j=1}^n\nu_j\nabla w_j|^{p(x)-2}-1)^+
|\sum_{j=1}^n \nu_j\nabla w_j|^2 dx\geq0,
\end{align*}
by Young's inequality, there holds
\begin{align*}
\frac{1}{2}\frac{\partial |\eta(t)|^2}{\partial t}\leq |F_n||\eta(t)|\leq \frac{1}{2}|F_n|^2+\frac{1}{2}|\eta(t)|^2.
\end{align*}
Then integrating with respect to $t$ from $0$ to $t$, we obtain
\begin{align*}
|\eta(t)|^2\leq C_n+\int_0^t |\eta(s)|^2ds.
\end{align*}
By Gronwall's inequality, we obtain that
$|\eta(t)|\leq C_n(T)$.
We denote
\[
L_n=\max_{(t,\eta)\in [0, T]\times B(\eta(0),C_n(T))}|F_n-P_n(t, \eta)|,\quad
 \tau_n=\min\{T, \frac{C_n(T)}{L_n}\},
\]
where $B(\eta(0),C_n(T))$ is the ball of radius $C_n(T)$ with the center
at the point $\eta(0)$ in $\mathbb{R}^n$. By Peano's Theorem we know
that \eqref{e3.2} admits a $C^1$ solution in $[0, \tau_n]$.
Let $\eta(\tau_n)$ be a initial value, then we can repeat the above
process and get a $C^1$ solution in $[t_n, 2\tau_n]$.
Without lost of generality, we assume that
$T=[\frac{T}{\tau_n}]\tau_n+(\frac{T}{\tau_n})\tau_n, 0<(\frac{T}{\tau_n})<1$,
where $[\frac{T}{\tau_n}]$ is the integer part of $\frac{T}{\tau_n}$,
$(\frac{T}{\tau_n})$ is the decimal part of $\frac{T}{\tau_n}$.
 We can divide $[0, T]$ into $[(i-1)\tau_n,i\tau_n]$, $i=1,\dots,L$ and
$[L\tau_n,T]$ where $L=[\frac{T}{\tau_n}]$, then there exist
$C^1$ solution $\eta^{i}_{n}(t)$ in $[(i-1)\tau_n,i\tau_n]$, $i=1,\dots,L$
and $\eta^{L+1}_n(t)$ in $[L\tau_n,T]$. Therefore, we obtain a solution
 $\eta_n(t)$ $\in C^1[0, T]$ defined by
\[
\eta_{n}(t)=
\begin{cases}
\eta^{1}_{n}(t), &\text{if } t\in [0,\tau_n],\\
\eta^{2}_{n}(t), &\text{if } t\in (\tau_n,2\tau_n],\\
\dots\\
\eta^{L}_{n}(t), &\text{if } t\in ((L-1)\tau_n,L\tau_n],\\
\eta^{L+1}_n(t), &\text{if } t\in (L\tau_n,T].
\end{cases}
\]
Thus, we obtain the approximate solutions sequence
 $u_n=\sum_{j=1}^n(\eta_n(t))_jw_j(x)$.
From \eqref{e3.2}, for $1\leq i\leq n$, we have
\begin{equation}
\begin{aligned}
&\int_\Omega\frac{\partial u_n}{\partial t}w_idx
 +a\Big(t,\int_\Omega|\nabla u_n|^{p(x)}dx\Big)
 \int_\Omega|\nabla u_n|^{p(x)-2}\nabla u_n\nabla w_idx\\
& +\int_\Omega\frac{1}{\varepsilon}(|\nabla u_n|^{p(x)-2}-1)^+
 \nabla u_n\nabla w_i dx\\
&=\int_\Omega f_nw_idx.
\end{aligned} \label{e3.3}
\end{equation}
Multiplying  by $(\eta_n(t))_i$, summing up $i$ from $1$ to $n$, and
integrating with respect to $t$ from $0$ to $\tau$, where $\tau\in(0,T]$,
we obtain
\begin{equation}
\begin{aligned}
&\int_0^{\tau}\int_\Omega\frac{\partial u_n}{\partial t}u_n\,dx\,dt
+\int_0^\tau a\Big(t,\int_\Omega|\nabla u_n|^{p(x)}dx\Big)
 \int_\Omega|\nabla u_n|^{p(x)}\,dx\,dt\\
&+\int_0^\tau\int_\Omega\frac{1}{\varepsilon}(|\nabla u_n|^{p(x)-2}-1)^+
 |\nabla u_n|^2 \,dx\,dt\\
&=\int_0^\tau\int_\Omega f_nu_n\,dx\,dt.
\end{aligned} \label{e3.4}
\end{equation}

\begin{remark} \label{rmk3.1} \rm
The approximate solutions $u_n$ depends on $\varepsilon$;
For convenience, we omit the $\varepsilon$.
For all $\varphi \in C^{1}(0,T;V_{k})$, $k\leq n$, there holds
\begin{align*}
&\int_0^{\tau}\int_\Omega\frac{\partial u_n}{\partial t}\varphi \,dx\,dt
+\int_0^\tau a\Big(t,\int_\Omega|\nabla u_n|^{p(x)}\nabla u_ndx\Big)
\int_\Omega|\nabla u_n|^{p(x)-2}\nabla u_n\nabla\varphi \,dx\,dt\\
& +\int_0^\tau\int_\Omega\frac{1}{\varepsilon}(|\nabla u_n|^{p(x)-2}-1)^+
\nabla u_n\nabla \varphi \,dx\,dt\\
&=\int_0^\tau\int_\Omega f_n\varphi \,dx\,dt.
\end{align*}
\end{remark}



(ii) \textbf{A priori estimates.}
By \eqref{e3.4}, assumption (H1) and integration by parts, we arrive
at the inequality
\[
\frac{1}{2}\int_\Omega|u_n(x,\tau)|^2-|u_n(x,0)|^2dx
+a_0\int_0^{\tau}\int_\Omega|\nabla u_n|^{p(x)}\,dx\,dt
\leq\|f_n\|_{V'(Q_\tau)}\|u_n\|_{V(Q_\tau)},
\]
where $Q_{\tau}=\Omega\times (0,\tau)$, $\tau\in (0,T]$.
Since $u_n(x,0)=0$ and $f_n \to f$ in $V'(Q_T)$, $\|f_n\|_{V'(Q_T)}\leq C$,
where $C$ independent of $\tau$ and $n$. Thus, we obtain
\begin{equation}
\int_\Omega u_n^2(x,\tau)dx+a_0\int_0^{\tau}
\int_\Omega |\nabla u_n|^{p(x)}\,dx\,dt
\leq
C(\|u_n\|_{L^{2}(Q_{\tau})}+\|\nabla u_n\|_{L^{p(x)}(Q_\tau)}).\label{e3.5}
\end{equation}
Without lost generality, we assume that
$\|\nabla u_n\|_{L^{p(x)}(Q_\tau)}\geq1$. Then
\[
\|\nabla u_n\|^{p^-}_{L^{p(x)}(Q_\tau)}
\leq \int_{Q_\tau}|\nabla u_n|^{p(x)}\,dx\,dt.
\]
By \eqref{e3.5} and Young's inequality, there holds
\begin{align*}
\int_\Omega u_n^2(x,\tau)dx+\frac{a_0}{2}
\int_0^{\tau} \int_\Omega |\nabla u_n|^{p(x)}\,dx\,dt
\leq C(\|u_n\|_{L^{2}(Q_{\tau})}+1).
\end{align*}
By Gronwall's inequality, we obtain $\|u_n\|_{L^{\infty}(0,T;L^2(\Omega))}\leq C$.
Therefore,
\begin{equation}
\|u_n\|_{L^\infty(0,T;L^2(\Omega))}+\|u_n\|_{V(Q_T)}\leq C.\label{e3.6}
\end{equation}
Combining assumption (H1), with \eqref{e3.6}, we have
\begin{gather*}
\int_{Q_T}\Big|a\Big(t,\int_\Omega|\nabla u_n|^{p(x)}dx\Big)
|\nabla u_n|^{p(x)-2}\nabla u_n\Big|^{p'(x)}\,dx\,dt\leq C.\\
\int_{Q_T}|(|\nabla u_n|^{p(x)-2}-1)^+\nabla u_n|^{p'(x)}\,dx\,dt
\leq C(\varepsilon),
\end{gather*}
where $C(\varepsilon)$ is a constant independent of $n$ on $\varepsilon$
and $C(\varepsilon)\to\infty$ as $\varepsilon\to\infty$.
Thus we obtain
\begin{equation}
\begin{gathered}
\big\|a(t,\int_\Omega|\nabla u_n|^{p(x)}dx)| \nabla u_n|^{p(x)-2}\nabla u_n
\big\|_{L^{p'(x)}(Q_T)} \leq C, \\
\big\|(|\nabla u_n|^{p(x)-2}-1)^+\nabla u_n\big\|_{L^{p'(x)}(Q_T)}
\leq C(\varepsilon).
\end{gathered}\label{e3.7}
\end{equation}

By Lemma \ref{lem2.1}, for all $\varphi\in V(Q_{T})$,  there exists a
sequence $\varphi_{n}{\rm \in} C^{1}(0,T;V_n) $ such that
 $\varphi_n\to\varphi$ strongly in $V(Q_{T})$. By Remark \ref{rmk3.1}, we have
\begin{align*}
&\Big|\int_{Q_T}\frac{\partial u_n}{\partial t}\varphi_n \,dx\,dt\Big|\\
&=\Big|-\int_{Q_T}a\Big(t,\int_\Omega|\nabla u_n|^{p(x)}dx\Big)
 |\nabla u_n|^{p(x)-2}\nabla u_n\nabla\varphi_n \,dx\,dt\\
&\quad -\int_{Q_T}\frac{1}{\varepsilon}(|\nabla u_n|^{p(x)-2}-1)^+
 \nabla u_n\varphi_n\,dx\,dt+\int_{Q_T}f_n\varphi_n\,dx\,dt\Big|\\
&\leq C\Big(\Big\|a\Big(t,\int_\Omega|\nabla u_n|^{p(x)}dx\Big)
 |\nabla u_n|^{p(x)-2}\nabla u_n\Big\|_{L^{p'(x,t)}(Q_T)}
 \big\|\nabla\varphi_n\big\|_{L^{p(x)}(Q_T)}\\
&\quad +\frac{1}{\varepsilon}\|(|\nabla u_n|^{p(x)-2}-1)^+
 \nabla u_n\|_{L^{p'(x,t)}Q_T}\|\nabla \varphi_n\|
 +\|f_n\|_{V'(Q_T)}\|\varphi_n\|_{V(Q_T)}\Big)\\
&\leq C(\varepsilon)\|\varphi_n\|_{V(Q_T)},
\end{align*}
where $C(\varepsilon)$ is a constant independent of $n$ on $\varepsilon$.
We immediately get that
\begin{equation}
\|\frac{\partial u_n}{\partial t}\|_{V'(Q_T)}\leq C(\varepsilon).\label{e3.8}
\end{equation}


(iii) \textbf{Passage to the limit.}
From \eqref{e3.6}-\eqref{e3.8}, we obtain a subsequence of $\{u_n\}$
 (still denoted by $\{u_n\}$) such that
\begin{gather*}
u_n\rightharpoonup u_\varepsilon\quad  \text{weakly* in } 
 L^\infty(0, T; L^2(\Omega)), \\
u_n\rightharpoonup u_\varepsilon\quad\text{weakly in }  V(Q_T),\\
a\big(t,\int_\Omega|\nabla u_n|^{p(x)}dx\big)|\nabla u_n|^{p(x)-2}
\nabla u_n\rightharpoonup \xi\quad \text{weakly in }
 \big(L^{p'(x)}(Q_T)\big)^N,\\
(|\nabla u_n|^{p(x)-2}-1)^+\nabla u_n\rightharpoonup \eta\quad
\text{weakly in }  (L^{p'(x)}(Q_T))^N\\
\frac{\partial u_n}{\partial t}\rightharpoonup
\frac{\partial u_\varepsilon}{\partial t} \quad\text{weakly in } V'(Q_T).
\end{gather*}

Since $\int_\Omega u^{2}_{n}(x,T)dx\leq C$, there exist a subsequence
of $\{u_n(x,T)\}$ (still denoted by $\{u_n(x,T)\}$) and a function
$\tilde{u}$ in $L^2(\Omega)$ such that $u_{n}(x,T)\to \tilde{u}$
weakly in $L^{2}(\Omega)$. Then for any $\varphi(x)\in C_0^{\infty}(\Omega)$
and $\eta(t)\in C^1[0,T]$, there holds
\begin{align*}
&\int^{T}_0\int_{\Omega}\frac{\partial u_{n}}{\partial t}\varphi\eta \,dx\,dt\\
&=\int_{\omega}u_{n}(x,T)\varphi\eta(T)dx-\int_{\Omega}u_{n}(x,0)\varphi\eta(0)dx
-\int_0^T\int_\Omega u_n\varphi\eta'(t)\,dx\,dt.
\end{align*}
Letting $n\to\infty$, by integration by parts, we obtain
\[
\int_{\Omega}(\tilde{u}-u_\varepsilon(x,T))\eta(T)\varphi dx
+\int_{\Omega}u_\varepsilon(x,0)\eta(0)\varphi dx=0.
\]
Choosing $\eta(T)=1,\eta(0)=0$ or $\eta(T)=0,\eta(0)=1$, by the density
of $C_0^{\infty}(\Omega)$ in $L^2(\Omega)$, we have
 $\tilde{u}=u_\varepsilon(x,T)$ and $u_\varepsilon(x,0)=0$ for
almost every $x\in\Omega$. That is $u_{n}(x,T)\to u_\varepsilon(x,T)$
weakly in $L^{2}(\Omega)$, as $n\to\infty$, thus
\begin{equation}
\int_{\Omega}u_\varepsilon^{2}
(x,T)dx\leq\liminf_{n\to\infty}\int_{\Omega}u^{2}_{n}(x,T)dx.\label{e3.9}
\end{equation}

In view of Remark \ref{rmk3.1}, for all $\varphi\in C^1(0,T; V_k)$ where $k\leq n$,
letting $n\to\infty$ there holds
\begin{equation}
\int_{Q_{T}}\frac{\partial u_\varepsilon}{\partial t}\varphi
+\xi\nabla\varphi+\frac{1}{\varepsilon}\eta\nabla\varphi \,dx\,dt
=\int_{Q_{T}} f\varphi \,dx\,dt,\label{e3.10}
\end{equation}
since $C^1\big(0,T;\cup^{\infty}_{k=1}V_k\big)$ is dense in
$C^1(0,T;C^1(\overline{\Omega}))$, the above equality is valid
for all $\varphi\in C^1(0,T;C_0^{\infty}(\Omega))$.
 Moreover, for all $\varphi\in V(Q_T)$, the above equality is valid.
Thus, we can take $\varphi=u_\varepsilon$. By integration by parts, we have
\begin{equation}
\frac{1}{2}\int_\Omega|u_\varepsilon(x,T)|^2dx+\int_{Q_T}
\xi\nabla u_\varepsilon+\frac{1}{\varepsilon}\eta \nabla u_\varepsilon \,dx\,dt
=\int_{Q_T} fu_\varepsilon \,dx\,dt.\label{e3.11}
\end{equation}
We denote
\begin{align*}
Y_n&=\int_{Q_T} a\Big(t,\int_\Omega|\nabla u_n|^{p(x)}dx\Big)
 \big(|\nabla u_n|^{p(x)-2}\nabla u_n-|\nabla u_\varepsilon|^{p(x)-2}
 \nabla u_\varepsilon\big)\\
&\quad\times \big(\nabla u_n-\nabla u_\varepsilon\big)\,dx\,dt\\
&\quad +\frac{1}{\varepsilon}\int_{Q_T}\big((|\nabla u_n|^{p(x)-2}-1)^+
 \nabla u_n-(|\nabla u_\varepsilon|^{p(x)-2}-)^+
 \nabla u_\varepsilon\big)
\big(\nabla u_n-\nabla u_\varepsilon)\,dx\,dt.
\end{align*}
By \eqref{e3.4}, we obtain
\begin{equation}
\begin{aligned}
0\leq Y_n
&=\int_{Q_T}f_nu_n-\frac{1}{2}\int_{\Omega}|u_n(x,T)|^2-|u_n(x,0)|^2dx\\&
-\int_{Q_T}a\Big(t,\int_\Omega|\nabla u_n|^{p(x)}dx\Big)|
 \nabla u_n|^{p(x)-2}\nabla u_n\nabla u_\varepsilon \,dx\,dt\\
&-\int_{Q_T}a\Big(t,\int_\Omega|\nabla u_n|^{p(x)}dx\Big)|
 \nabla u_\varepsilon|^{p(x)-2}\nabla u_\varepsilon(\nabla u_n
 -\nabla u_\varepsilon)\,dx\,dt\\
&-\frac{1}{\varepsilon}\int_{Q_T}(|\nabla u_n|^{p(x)-2}-1)^+
 \nabla u_n\nabla u_\varepsilon \,dx\,dt\\
&-\frac{1}{\varepsilon}\int_{Q_T}(|\nabla u_\varepsilon|^{p(x)-2}-1)^+
 \nabla u_\varepsilon(\nabla u_n-\nabla u_\varepsilon)\,dx\,dt
\end{aligned}\label{e3.12}
\end{equation}
By assumption (H1), the sequence
$\{a\big(t,\int_\Omega|\nabla u_n|^{p(x)}dx\big)\}_{n=1}^\infty$
is equi-integrable and uniformly bounded in $L^1(0,T)$. Therefore,
there exist a subsequence of $\{u_n\}$ (still denoted by $\{u_n\}$)
 and $\bar{a}(t)$ such that
\[
a\Big(t,\int_\Omega|\nabla u_n|^{p(x)}dx\Big)\to \bar{a}(t)
\quad \text{a.e. } t\in[0,T].
\]
As
\[
\Big|a\Big(t,\int_\Omega|\nabla u_n|^{p(x)}dx\Big)|\nabla
 u_\varepsilon|^{p(x)-2}\nabla u_\varepsilon\Big|^{p'(x)}
\leq C|\nabla u_\varepsilon|^{p(x)}\in L^1(Q_T),
\]
by the Lebesgue dominated convergence theorem, we obtain
\[
\int_{Q_T}\Big|\Big[a\Big(t,\int_\Omega|\nabla u_n|^{p(x)}dx\Big)-\bar{a}(t)
\Big]|\nabla u_\varepsilon|^{p(x)-2}\nabla u_\varepsilon\Big|^{p'(x)}\,dx\,dt
\to0.
\]
That is,
\begin{equation}
a\Big(t,\int_\Omega|\nabla u_n|^{p(x)}dx\Big)|\nabla u_\varepsilon|^{p(x)-2}\nabla u_\varepsilon
\to \bar{a}(t)|\nabla u_\varepsilon|^{p(x)-2}\nabla u_\varepsilon\quad\text{in }
\big(L^{p'(x)}(Q_T)\big)^N. \label{e3.13}
\end{equation}
Thus, from \eqref{e3.9}, \eqref{e3.11}-\eqref{e3.13}, we obtain
\begin{align*}
0\leq\mathop{\lim\sup}_{n\to\infty}Y_n
&\leq \int_{Q_T}fu\,dx\,dt-\frac{1}{2}\int_\Omega|u(x,T)|^2dx\\
&\quad -\int_{Q_T}\xi\nabla u_\varepsilon \,dx\,dt
-\frac{1}{\varepsilon}\int_{Q_T}\eta\nabla u_\varepsilon \,dx\,dt=0;
\end{align*}
therefore $\lim_{n\to\infty} Y_n=0$. Furthermore, by
 assumption (H1), there holds
\[
\int_{Q_T}\big(|\nabla u_n|^{p(x)-2}\nabla u_n-|\nabla u_\varepsilon|^{p(x)-2}
\nabla u_\varepsilon\big)\big(\nabla u_n-\nabla u_\varepsilon\big)\,dx\,dt\to0.
\]
Since $p(x,t)\geq p^->2$, as $n\to\infty$, there holds
\begin{equation}
\begin{aligned}
&\int_{Q_T}|\nabla u_n-\nabla u_\varepsilon|^{p(x)}\,dx\,dt\\
&\leq C\int_{Q_T}(|\nabla u_n|^{p(x)-2}\nabla u_n
-|\nabla u_\varepsilon|^{p(x)-2}\nabla u_\varepsilon)
(\nabla u_n-\nabla u_\varepsilon)\,dx\,dt
\to 0.
\end{aligned} \label{e3.14}
\end{equation}
Therefore, from \eqref{e3.14}, we obtain
$\nabla u_n\to\nabla u_\varepsilon \quad\text{in }(L^{p(x)}(Q_T))^N$.
Thus, there exists a subsequence of $\{u_n\}$ still denoted by $\{u_n\}$
such that
\begin{equation}
\int_{\Omega}|\nabla u_n-\nabla u_\varepsilon|^{p(x)}dx\to0\quad
\text{a.e. }t\in [0,T].\label{e3.15}
\end{equation}
Since
\begin{align*}
&\Big|\int_\Omega|\nabla u_n|^{p(x)}-|\nabla u_\varepsilon|^{p(x)}dx\Big|\\
&\leq \int_\Omega p(x)\big||\nabla u_n|
 +\theta(|\nabla u_n|-|\nabla u_\varepsilon|)\big|^{p(x)-1}\big||\nabla u_n|
 -|\nabla u_\varepsilon|\big|dx\\
&\leq C\big\||\nabla u_n|^{p(x)-1}+|\nabla u_\varepsilon|^{p(x)-1}
 \big\|_{L^{p'(x)}(\Omega)}\|\nabla u_n
 -\nabla u_\varepsilon\|_{L^{p(x)}(\Omega)},
\end{align*}
where $0\leq\theta\leq1$,
by \eqref{e3.15}, we have
\[
\int_\Omega|\nabla u_n|^{p(x)}\to\int_\Omega|\nabla u_\varepsilon|^{p(x)}dx
\quad \text{a.e. } t\in[0,T].
\]
Thus, by the continuity of $a$, we obtain that
\[
\bar{a}(t)=a\Big(t,\int_{\Omega}|\nabla u_\varepsilon|^{p(x)}dx\Big)\quad
\text{a.e. }t\in[0,T]
\]
Since $\nabla u_n\to\nabla u_\varepsilon \quad\text{in }(L^{p(x)}(Q_T))^N$,
there exists a subsequence of $\{u_n\}$ (still labeled by $\{u_n\}$) such that
$\nabla u_n\to \nabla u_\varepsilon$ for a.e. $(x,t)\in\ Q_T$, then
\begin{align*}
&a\Big(t,\int_{\Omega}|\nabla u_n|^{p(x)}dx\Big)|\nabla u_n|^{p(x,t-2)}
\nabla u_n\\
&\to  a\Big(t,\int_{\Omega}|\nabla u_\varepsilon|^{p(x)}dx\Big)|
\nabla u_\varepsilon|^{p(x)-2}\nabla u_\varepsilon\quad\text{a.e. }
 (x,t)\in Q_T.
\end{align*}
By Theorem \ref{thm2.6}, we obtain
$\xi=a\big(t,\int_{\Omega}|\nabla u_\varepsilon|^{p(x)}dx\big)
|\nabla u_\varepsilon|^{p(x)-2}\nabla u_\varepsilon$.
Similarly, $\eta=(|\nabla u_\varepsilon|^{p(x,t)-2}-1)^+\nabla u_\varepsilon$.

It follows from \eqref{e3.10} that
\begin{align*}
&\int_{Q_T}\frac{\partial u_\varepsilon}{\partial t}\varphi \,dx\,dt
+\int_{0}^{T}a\Big(t,\int_{\Omega}|\nabla u_n|^{p(x)}dx\Big)
\int_\Omega|\nabla u|^{p(x)-2}\nabla u_\varepsilon \nabla\varphi\\
&+\frac{1}{\varepsilon}(|\nabla u_\varepsilon|^{p(x,t)-2}-1)^+
\nabla u_\varepsilon\nabla\varphi \,dx\,dt
=\int_{Q_T}f\varphi \,dx\,dt,
\end{align*}
for all $\varphi\in V(Q_T)$. Since $u\in V(Q_T)$ and
$\frac{\partial u}{\partial t}\in V'(Q_T)$, by Theorem \ref{thm2.5},
 up to a set of measure zero, we have $u\in C(0,T;L^2(\Omega))$.
\end{proof}

\section{Existence of solutions for the variational inequality}

In this section, we  prove our main theorem.

\begin{theorem} \label{thm4.1}
Under assumptions {\rm (H1)--(H3)} there exists a function
 $u(x,t)\in \mathscr{K} $ such that
\begin{align*}
&\int_{Q_T}\frac{\partial v}{\partial t}(v-u)\,dx\,dt
 +\int_0^T a\Big(t,\int_\Omega|\nabla u|^{p(x)}dx\Big)
 \int_\Omega|\nabla u|^{p(x)-2}\nabla u\nabla (v-u)\,dx\,dt \\
&\geq \int_{Q_T} f(v-u)\,dx\,dt
\end{align*}
for all $v\in V(Q_T)$ with $\frac{\partial v}{\partial t}\in V'(Q_T)$,
$v(x,0)=0$, $|\nabla v|\leq 1$ a.e. $(x,t)\in Q_T$.
\end{theorem}

\begin{proof}
We will prove this theorem in three steps.

\noindent\textbf{(Step 1) A priori estimates.}
In Definition \ref{def3.1}, we take $\varphi=u_\varepsilon\chi_{(0,\tau)}$
as a test function,  where $\chi_{(0,\tau)}$ is defined as the
 characteristic function of $(0,\tau)$, $\tau\in (0,T]$, then
\begin{align*}
&\int_{Q_\tau}\frac{\partial u_\varepsilon}{\partial t}u_\varepsilon \,dx\,dt
+\int_{Q_\tau}a\Big(t,\int_\Omega| \nabla u_\varepsilon|^{p(x)}dx\Big)
|\nabla u_\varepsilon|^{p(x)}\\
& +\frac{1}{\varepsilon}
(|\nabla u_\varepsilon|^{p(x)-2}-1)^+|\nabla u_\varepsilon|^2 \,dx\,dt\\
&= \int_{Q_\tau}f(x,t)u_\varepsilon \,dx\,dt,
\end{align*}
where $Q_\tau=\Omega\times (0,\tau)$.
Similar to Section 3, we have
\[
\int_\Omega|u_\varepsilon(x,\tau)|^2dx+\int_{Q_\tau}
|\nabla u_\varepsilon|^{p(x)}\,dx\,dt\leq C,\quad\text{for all } \tau\in[0,T].
\]
Therefore,
\begin{equation}
\frac{1}{\varepsilon}\int_{Q_T}(|\nabla u_\varepsilon|^{p(x)-2}-1)^+
|\nabla u_\varepsilon|^2 \,dx\,dt
+ \big\|u_\varepsilon\big\|_{L^\infty(0,T;L^2(\Omega))}
+\big\|u_\varepsilon\big\|_{V(Q_T)}\leq C.\label{e4.1}
\end{equation}
Since
\begin{align*}
&\int_{Q_T}\big|a\Big(t,\int_\Omega| \nabla u_\varepsilon|^{p(x,t)}dx\Big)
|\nabla u_\varepsilon|^{p(x)-2}\nabla u_\varepsilon\big|^{p'(x)}\,dx\,dt\\
&\leq C\int_{Q_T} |\nabla u_\varepsilon|^{p(x)}\,dx\,dt\\
&\leq C\max\{\big\|\nabla u_\varepsilon\big\|^{p_-}_{L^{p(x)}(Q_T)},
 \big\|\nabla u_\varepsilon\big\|^{p_+}_{L^{p(x)}(Q_T)}\} \leq C,
\end{align*}
there holds
\[
\Big\| \big|a\Big(t,\int_\Omega|\nabla u_\varepsilon|^{p(x)}dx\Big)
 |\nabla u_\varepsilon|^{p(x)-2}\nabla u_\varepsilon\big|
 \Big\|_{L^{p'(x)}(Q_T)}
\leq C.
\]

\noindent\textbf{(Step 2) Passage to the limit.}
By \eqref{e4.1}-\eqref{e4.2}, there exists a subsequence of
$\{u_\varepsilon\}_{\varepsilon>0}$, still denoted by
$\{u_{\varepsilon}\}_{\varepsilon>0}$, such that
\begin{equation}
\begin{gathered}
u_{\varepsilon}\stackrel{\ast}\rightharpoonup u\quad\text{weakly * in }
  L^\infty(0, T; L^2(\Omega)),\\
u_{\varepsilon}\rightharpoonup u\quad \text{weakly in }  V(Q_T),\\
a\Big(t,\int_\Omega| \nabla u_\varepsilon|^{p(x)}dx\Big)
|\nabla u_{\varepsilon}|^{p(x)-2}\nabla u_{\varepsilon}\rightharpoonup A\quad
\text{weakly in } (L^{p'(x)}(Q_T))^N.
\end{gathered}\label{e4.3}
\end{equation}

For all $\varphi\in V(Q_T)$, there holds
\[
\int_{Q_T}[(|\nabla u_\varepsilon|^{p(x)-2}-1)^+
\nabla u_\varepsilon-(|\nabla \varphi|^{p(x)-2}-1)^+
\nabla \varphi](\nabla u_\varepsilon-\nabla \varphi)\,dx\,dt\geq0.
\]
Since
\[
\int_{Q_T}|(|\nabla u_\varepsilon|^{p(x)-2}-1)^+
 \nabla u_\varepsilon|^{p'(x)}\,dx\,dt
\leq\int_{Q_T}(|\nabla u_\varepsilon|^{p(x)-2}-1)^+
|\nabla u_\varepsilon|^2\,dx\,dt,
\]
by \eqref{e4.1}, we obtain that
$\int_{Q_T}|(|\nabla u_\varepsilon|^{p(x)-2}-1)^+
\nabla u_\varepsilon|^{p'(x)} \,dx\,dt\to0$ as $\varepsilon\to 0$;
that is, $\|(|\nabla u_\varepsilon|^{p(x)-2}-1)^+
\nabla u_\varepsilon\|_{L^{p'(x)}(Q_T)}\to0$.
From $\nabla u_\varepsilon\rightharpoonup u$ weakly
in $(L^{p(x)}(Q_T))^N$, we have
\[
\int_{Q_T}(|\nabla\varphi|^{p(x)-2}-1)^+
\nabla\varphi(\nabla u-\nabla \varphi)\,dx\,dt\leq 0
\]
We take $\varphi=u+\lambda w$, where $0<\lambda<1$ and $w\in V(Q_T)$, then
\[
\int_{Q_T}(|\nabla(u+\lambda w)|^{p(x)-2}-1)^+
\nabla(u+\lambda w)\nabla w\,dx\,dt\leq0.
\]
Since $|(|\nabla(u+\lambda w)|^{p(x)-2}-1)^+
\nabla(u+\lambda w)\nabla w|\leq C(|\nabla u|^{p(x)}+|\nabla w|^{p(x)})\in L^1(Q_T)$ and $(|\nabla(u+\lambda w)|^{p(x)-2}-1)^+
\nabla(u+\lambda w)\nabla w\to (|\nabla u|^{p(x)-2}-1)^+
\nabla u\nabla w$ as $\lambda\to0$,  by the Lebesgue Dominated Convergence
Theorem and the arbitrariness of $w$, we obtain
\begin{align*}
\int_{Q_T}(|\nabla u|^{p(x)-2}-1)^+ |\nabla u|^2\,dx\,dt=0
\end{align*}
Thus, $|\nabla u|\leq 1$ a.e. $(x,t)\in Q_T$.


Taking $\varphi=v-u_{\varepsilon}$ as a test function in
\eqref{e3.1}, where $v\in V(Q_T)$, $\frac{\partial v}{\partial t}\in V'(Q_T)$,
$v(x,0)=0$ and $|\nabla v|\leq 1$ a.e. $(x,t)\in Q_T$, then
\begin{align*}
&\int_{Q_T}\frac{\partial v}{\partial t}(v-u_{\varepsilon})
+a\Big(t,\int_\Omega| \nabla u_\varepsilon|^{p(x)}dx\Big)
|\nabla u_{\varepsilon}|^{p(x)-2}\nabla u_{\varepsilon}
\nabla (v-u_{\varepsilon})\\
& -f(x,t)(v-u_{\varepsilon})\,dx\,dt\\
&=\int_{Q_T}\frac{\partial u_{\varepsilon}}{\partial t}(v-u_{\varepsilon})
+a\Big(t,\int_\Omega|\nabla u_\varepsilon|^{p(x)}dx\Big)
|\nabla u_{\varepsilon}|^{p(x)-2}\nabla u_{\varepsilon}
\nabla (v-u_{\varepsilon})\\
&\quad -f(x,t)(v-u_{\varepsilon})\,dx\,dt
+\int_{Q_T}\frac{\partial (v-u_{\varepsilon})}{\partial t}(v-u_{\varepsilon})
 \,dx\,dt\\
&=\frac{1}{\varepsilon}\int_{Q_T}\big((|\nabla v|^{p(x)-2}-1)^+
\nabla v-(|\nabla u_\varepsilon|^{p(x)-2}-1)^+\nabla u_\varepsilon\big)
(\nabla v-\nabla u_{\varepsilon})\,dx\,dt\\
&\quad +\int_{Q_T}\frac{\partial (v-u_{\varepsilon})}{\partial t}
(v-u_{\varepsilon})\,dx\,dt
\geq 0,
\end{align*}
and further
\begin{equation}
\begin{aligned}
&\int_{Q_T}a\Big(t,\int_\Omega| \nabla u_\varepsilon|^{p(x)}dx\Big)
|\nabla u_{\varepsilon}|^{p(x)}\,dx\,dt\\
&\leq \int_{Q_T}a\Big(t,\int_\Omega|\nabla u_\varepsilon|^{p(x)}dx\Big)
|\nabla u_{\varepsilon}|^{p(x)-2}\nabla u_{\varepsilon}\nabla u\,dx\,dt\\
&\quad +\int_{Q_T}\frac{\partial v}{\partial t}(v-u_{\varepsilon})\,dx\,dt
-\int_{Q_T}f(x,t)(v-u_{\varepsilon})\,dx\,dt.
\end{aligned} \label{e4.4}
\end{equation}
For $k>0$, we denote
\[
u^{(k)}=\begin{cases}
 k, &u<-k,\\
u,  &|u|\leq k,\\
k,  &u>k,
\end{cases}
\]
and $u^{(k)}_\mu(x,t)=\mu\int_0^t e^{\mu(s-t)}u^{(k)}(x,s)ds$.
It's easy to check that
$\frac{\partial u^{(k)}_\mu}{\partial t}=\mu(u^{(k)}-u^{(k)}_\mu)$.
From that in \cite{b2}, we obtain $u^{(k)}_\mu\to u^{(k)}$ strongly
in $L^2(Q_T)$ and weakly in $V(Q_T)$ as $\mu\to\infty$.
Denote $A_k=\{(x,t)\in Q_T:|u|\leq k\}$,
then $u^{(k)}=u$ in $A_k$ and
$\operatorname{sgn}(u^{(k)}-u^{(k)}_\mu)=\operatorname{sgn}(u-u^{(k)}_\mu)$
in $ Q_T\setminus A_k$ (because $|u^{(k)}_\mu|\leq k$). Thus,
\begin{align*}
&\int_{Q_T}\frac{\partial u^{(k)}_\mu}{\partial t}(u^{(k)}_\mu-u)\,dx\,dt\\
&=\mu\int_{Q_T}(u^{(k)}-u^{(k)}_\mu)(u^{(k)}_\mu-u)\,dx\,dt\\
&=-\mu\int_{A_k}(u-u^{(k)}_\mu)^2\,dx\,dt
-\mu\int_{Q_T\setminus A_k }(u^{(k)}-u^{(k)}_\mu)(u-u^{(k)}_\mu)\,dx\,dt\leq0.
\end{align*}
 By a diagonal rule, we obtain a sequence denoted by $v_k$ such that
 $v_k\to u$ strongly in $L^2(Q_T)$ and weakly in $V(Q_T)$ as $k\to\infty$,
 and $\limsup_{k\to\infty}\int_{Q_T}\frac{\partial v_k}{\partial t}(v_k-u)\,dx\,dt\leq0$.
Taking $v=v_k$ in \eqref{e4.4}, we obtain
\begin{align*}
&\limsup_{\varepsilon\to0}\int_{Q_T}a\Big(t,\int_\Omega| \nabla u_\varepsilon|^{p(x)}dx\Big)|\nabla u_{\varepsilon}|^{p(x)}\,dx\,dt\\
&\leq \int_{Q_T}A\nabla u\,dx\,dt+\int_{Q_T}\frac{\partial v_k}{\partial t}(v_k-u)\,dx\,dt -\int_{Q_T}f(x,t)(v_k-u)\,dx\,dt.
\end{align*}
Letting $k\to\infty$, we have
\begin{align*}
&\mathop{\lim\sup}_{\varepsilon\to 0}\int_{Q_T}a
 \Big(t,\int_\Omega| \nabla u_\varepsilon|^{p(x)}dx\Big)|\nabla u_{\varepsilon}|^{p(x)}\,dx\,dt\\
&\leq \int_{Q_T}A\nabla u\,dx\,dt=\lim_{\varepsilon\to 0}
\int_{Q_T}a\Big(t,\int_\Omega|\nabla u_\varepsilon|^{p(x)}dx\Big)
 |\nabla u_{\varepsilon}|^{p(x)-2}\nabla u_{\varepsilon}\nabla u\,dx\,dt;
\end{align*}
that is,
\begin{equation}
\limsup_{\varepsilon\to 0}\int_{Q_T}a
\Big(t,\int_\Omega| \nabla u_\varepsilon|^{p(x)}dx\Big)|
\nabla u_{\varepsilon}|^{p(x)-2}\nabla u_{\varepsilon}
\nabla (u_{\varepsilon}-u)\,dx\,dt\leq 0.\label{e4.5}
\end{equation}
As the sequence
$\Big\{a\Big(t,\int_\Omega| \nabla u_\varepsilon|^{p(x)}dx\Big)\Big\}_\varepsilon$
is  uniformly bounded and equi-integrable in $L^1(Q_T)$, there exist a
subsequence of $\{u_\varepsilon\}$ (for convenience still relabeled by
$\{u_\varepsilon\}$ ) and $a^*$ such that
$a\Big(t,\int_\Omega|\nabla u_\varepsilon|^{p(x)}dx\Big)\to a^*$ for almost
every $t\in [0,T]$. Since
\[
\Big|\Big(a\Big(t,\int_\Omega|\nabla u_\varepsilon|^{p(x)}dx\Big)-a^*\Big)
|\nabla u|^{p(x)-2}\nabla u\Big|^{p'(x)}
\leq C|\nabla u|^{p(x)}\in L^1(Q_T),
\]
by the Lebesgue dominated convergence theorem,  we obtain
\begin{align*}
a\Big(t,\int_\Omega|\nabla u_\varepsilon|^{p(x)}dx\Big)
|\nabla u|^{p(x)-2}\nabla u\to  a^*|\nabla u|^{p(x)-2}\nabla u\quad
\text{strongly in } L^{p'(x)}(Q_T).
\end{align*}
Since
\begin{align*}
0&\leq \int_{Q_T}a\Big(t,\int_\Omega|\nabla u_\varepsilon|^{p(x)}dx\Big)
(|\nabla u_{\varepsilon}|^{p(x)-2}\nabla u_{\varepsilon}-|\nabla u|^{p(x)-2}
\nabla u)(\nabla u_{\varepsilon}-\nabla u)\\
&=\int_{Q_T}a\Big(t,\int_\Omega|\nabla u_\varepsilon|^{p(x)}dx\Big)
|\nabla u_{\varepsilon}|^{p(x)-2}\nabla u_{\varepsilon}(\nabla u_{\varepsilon}
-\nabla u)\\
&\quad -a\Big(t,\int_\Omega|\nabla u_\varepsilon|^{p(x)}dx\Big)
|\nabla u|^{p(x)-2}\nabla u(\nabla u_{\varepsilon}-\nabla u)\,dx\,dt,
\end{align*}
we have
\begin{equation}
\liminf_{\varepsilon\to 0}\int_{Q_T}a\Big(t,\int_\Omega
| \nabla u_\varepsilon|^{p(x)}dx\Big)|\nabla u_{\varepsilon}|^{p(x)-2}
\nabla u_{\varepsilon}\nabla(u_{\varepsilon}-u)\,dx\,dt\geq0. \label{e4.6}
\end{equation}
From \eqref{e4.5}--\eqref{e4.6} and
$\nabla u_\varepsilon\rightharpoonup\nabla u$ weakly in $(L^{p(x)}(Q_T))^N$,
there holds
\[
\lim_{\varepsilon\to 0}\int_{Q_T}a\Big(t,\int_\Omega
| \nabla u_\varepsilon|^{p(x)}dx\Big)(|\nabla u_{\varepsilon}|^{p(x)-2}
\nabla u_{\varepsilon}-|\nabla u|^{p(x)-2}\nabla u)\nabla(u_{\varepsilon}-u)
\,dx\,dt=0.
\]
Similar to Section 3, we have
$\nabla u_{\varepsilon}\to\nabla u\ {\rm strongly\ in}\ (L^{p(x)}(Q_T))^N$
as $\varepsilon\to 0$.
 Thus there exists a subsequence of $\{u_{\varepsilon}\}$,
still labeled by $\{u_{\varepsilon}\}$ such that
$\nabla u_{\varepsilon}\to\nabla u\ {\rm a.e.}\ (x,t)\in Q_T$ and
$\int_{\Omega}|\nabla u_\varepsilon|^{p(x)}dx\to\int_\Omega|\nabla u|^{p(x)}dx$
a.e. $t\in[0,T]$. Thus, we obtain that
\[
A=a\Big(t,\int_\Omega|\nabla u|^{p(x)}dx\Big)|\nabla u|^{p(x)-2}\nabla u.
\]

\noindent\textbf{(Sep 3) Existence of weak solutions.}
By Fatou's Lemma, 
\begin{align*}
&\liminf_{\varepsilon\to 0}\int_{Q_T}
a\Big(t,\int_\Omega|\nabla u_\varepsilon|^{p(x)}dx\Big)|
\nabla u_\varepsilon|^{p(x)}\,dx\,dt\\
&\geq \int_{Q_T} a\Big(t,\int_\Omega|\nabla u|^{p(x)}dx\Big)
|\nabla u|^{p(x)}\,dx\,dt.
\end{align*}
For all $v\in V(Q_T)$ with $\frac{\partial v}{\partial t}\in V'(Q_T)$,
$v(x,0)= 0$, $|\nabla v|\leq 1$ a.e. $(x,t)\in Q_T$, we take
$\varphi=v-u_{\varepsilon}$ as a test function in  \eqref{e3.1}, then
\begin{align*}
&\int_{Q_T}\frac{\partial v}{\partial t}(v-u_\varepsilon)
+a\Big(t,\int_\Omega| \nabla u_\varepsilon|^{p(x)}dx\Big)
|\nabla u_{\varepsilon}|^{p(x)-2}\nabla u_{\varepsilon}
\nabla(v-u_{\varepsilon})\\
&-f(x,t)(v-u_{\varepsilon})\,dx\,dt\\
&=\frac{1}{\varepsilon}\int_{Q_T}\big((|\nabla v|^{p(x)-2}-1)^+\nabla v-(|\nabla u_\varepsilon|^{p(x)-2}-1)^+\nabla u_\varepsilon\big)(\nabla v-\nabla u_{\varepsilon})\,dx\,dt\\
&\quad+\int_{Q_T}\frac{\partial (v-u_{\varepsilon})}{\partial t}
(v-u_{\varepsilon})\,dx\,dt\geq0,
\end{align*}
and furthermore,
\begin{align*}
&\liminf_{\varepsilon\to0}\int_{Q_T}\frac{\partial v}{\partial t}(v-u_\varepsilon)
+a\Big(t,\int_\Omega|\nabla u_\varepsilon|^{p(x)}dx\Big)
|\nabla u_{\varepsilon}|^{p(x)-2}\nabla u_{\varepsilon}
\nabla v\\
&-f(x,t)(v-u_{\varepsilon})\,dx\,dt\\
&\geq\int_{Q_T} a\Big(t,\int_\Omega|\nabla u|^{p(x)}dx\Big)
|\nabla u|^{p(x)}\,dx\,dt.
\end{align*}
Since
\[
 a\Big(t,\int_\Omega|\nabla u_\varepsilon|^{p(x)}dx\Big)
|\nabla u_{\varepsilon}|^{p(x)-2}\nabla u_{\varepsilon}\rightharpoonup
a\Big(t,\int_\Omega|\nabla u|^{p(x)}dx\Big)|\nabla u|^{p(x)-2}\nabla u
\]
weakly in $(L^{p'(x)}(Q_T))^N$,
and $u_{\varepsilon}\rightharpoonup  u$ weakly in  $V(Q_T)$,
there holds
\begin{align*}
&\int_{Q_T}\frac{\partial v}{\partial t}(v-u)\,dx\,dt+
\int_0^T a\Big(t,\int_\Omega|\nabla u|^{p(x,t)}dx\Big)
\int_\Omega|\nabla u|^{p(x)-2}\nabla u\nabla(v-u)\,dx\,dt\\
&\geq \int_{Q_T} f(x,t)(v-u)\,dx\,dt.
\end{align*}
Thus we have proved our main theorem.
\end{proof}


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\end{document}