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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 61, pp. 1--16.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2015 Texas State University - San Marcos.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2015/61\hfil Large time behavior]
{Large time behavior for $p(x)$-Laplacian equations with
irregular data}

\author[X. Chai, H. Li, W. Niu \hfil EJDE-2015/61\hfilneg]
{Xiaojuan Chai, Haisheng Li, Weisheng Niu}

\address{ 
School of Mathematical Sciences, Anhui University, Hefei
230601, China}
\email[Xiaojuan Chai]{chaixj@ahu.edu.cn}
\email[Haisheng Li]{squeensy@sina.com}
\email[Weisheng Niu]{niuwsh@ahu.edu.cn}

\thanks{Submitted November 24, 2014. Published March 11, 2015.}
\subjclass[2000]{35B40, 35K55}
\keywords{$p(x)$-Laplacian equation; large time behavior; irregular data}

\begin{abstract}
 We study the large time behavior of solutions to  $p(x)$-Laplacian
 equations with irregular data. Under proper assumptions, we show that
 the entropy solution of parabolic $p(x)$-Laplacian equations converges
 in $L^q(\Omega)$ to the unique stationary entropy solution as $t$ 
 tends to infinity.
\end{abstract}

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

\section{Introduction}

Let $\Omega$ be a bounded domain in $\mathbb{R}^N$ $(N\geq 2)$ with
smooth boundary $\partial \Omega$.  We consider the asymptotic
behavior of the following nonlinear initial-boundary value problem with irregular data,
\begin{equation} \label{e1.1}
\begin{gathered}
 u_t-\operatorname{div}(|\nabla u|^{p(x)-2}\nabla u) + |u|^{q-1}u =g
\quad \text{in } \Omega\times \mathbb{R}^+,\\
u=0 \quad \text{on } \partial \Omega\times\mathbb{R}^+, \\
u(x,0)=u_{0}(x) \quad \text{in } \Omega,
\end{gathered}
\end{equation}
where $ q\geq1$, $ p \in C( \overline{\Omega})$ with
 $1 < p^-=\min_{x\in \overline{\Omega}}p(x)\leq p^+
=\max_{x\in\overline{\Omega}}p(x) <\infty$.
By irregular data, we mean that $u_0,g \in  L^1(\Omega)$.


Equation in problem \eqref{e1.1} could be viewed as a generalization of the usual
$p$-Laplacian equations. It is a rather typical nonlinear problem
with variable exponents. Problems of this kind are interesting from
the purely mathematical point of view. Besides, they have potential
applications in various fields such as electrorheological fluids (an
essential class of non-Newtonian fluids) \cite{r,rr}, nonlinear
elasticity \cite{z1} flow through porous media \cite{as}, image
processing \cite{cl}, etc. Perhaps for these reasons,  such a
field has attracted more and more attention and has undergone an
explosive development in recent years, see the monograph \cite{dh}
and the large amounts of references therein.

As an essential model involving variable exponents, problem \eqref{e1.1}
has been studied in various contexts by  different authors. In
\cite{as1}, Antontsev and Shmarev investigated the existence and
uniqueness results for some anisotropic parabolic equations
involving variable exponents. Then with more general assumptions on
the variable exponents, in \cite{az,fp,lg}, some
existence results were obtained for the parabolic $p(x)$-Laplacian
equations in different frameworks. In \cite{bw,zz1}, existence and
uniqueness results were addressed for the parabolic $p(x)$-Laplacian
equations with $L^1$-data.

The asymptotic behavior for the parabolic $p(x)$-Laplacian equations
has also been studied largely. In \cite{as2,as3,as4}, the
extinction, decay and blow up of solutions for some anisotropic
parabolic equations with variable exponents  were investigated. In
\cite{ak}, Akagi and Matsuura studied the convergence to stationary
states for the solutions of the parabolic $p(x)$-Laplacian
equations. In \cite{ss4,n} and \cite{s}--\cite{ss3}, the large time behaviors for
certain kinds of $p(x)$-Lalacian equations were investigated and
described by means of global attractors.

The considerations in \cite{ss4,n}, \cite{s}--\cite{ss3} were mainly focused on
 $p(x)$-Laplacian equations with regular data (the initial data
and forcing terms were assumed to be $L^2$ integrable or even
essentially bounded). In \cite{cn}, we considered the existence of
global attractors for some $p(x)$-Laplacian equations involving
$L^1$ or even measure data. As we see, the less regularity of the
data influences the regularity of the solutions greatly, and which
in turn causes some crucial difficulties in investigating the
asymptotic behaviors of the solutions, see also
\cite{pe, n1, pe1,pe2,pe3,zn}.


In this article, we shall continue the study on the large time
behavior of solutions to $p(x)$-Laplacian equations with irregular
data as in \cite{cn}. But from a different point of view, here we
investigate the convergence of the solutions to the stationary
states as $t$ tends to infinity. Under proper assumptions, we shall
prove that the unique entropy solution $u(t)$   to  the
$p(x)$-Laplacian problem \eqref{e1.1} converges in $L^1(\Omega)$ to the
unique entropy solution $v$ of the corresponding elliptic problem
\eqref{e2.2} as $t$ tends to infinity.

Our work is largely motivated by the works of Petitta and his
coauthors in \cite{pe, pe1,pe2,pe3}. By using the comparison
principle and some compactness results successfully, the authors
have obtained the convergence of solutions to the stationary states
for several type of parabolic equations (with constant exponent)
involving irregular data. Yet, the variable exponent problem treated
here exhibits some stronger nonlinearity and inhomogeneity, which
require the analysis to be more delicate.

Next, we first provide some preliminaries in Section 2. Then in
Section 3, the last section, we investigate the large time behavior
of the entropy solution to problem \eqref{e1.1}. Throughout the paper, we
denote $ \Omega\times (0,T)$ by $Q_T$ for any $T>0$, and we use $C$
to denote some positive constant, which may distinguish with each
other even in the same line and that only depends on.

\section{Preliminaries}

Let us  begin with the definitions and some basic properties of the
generalized Lebesgue and Sobolev spaces. Interested readers may
refer to \cite{dh,fz,kr} for more details.

For a variable exponent $p \in C( \overline{\Omega})$ with
 $p^- > 1$, define the   Lebesgue space $L^{p(\cdot)}(\Omega)$ as
 $$
L^{p(\cdot)}(\Omega) = \{u :\Omega
 \to \mathbb{R}; u \text{ is measurable and }
\int_\Omega|u|^{p(x)}dx <\infty\}
$$
   with the Luxemburg norm
\[
\|u\|_{L^{p(\cdot)}(\Omega)} = \inf\{\lambda > 0 : \int_\Omega
|\frac{u(x)} {\lambda}|^{p(x)} dx\leq 1\}.
\]
 We have
\[
\min\{\|u\|^{p^+}_{L^{p(\cdot)}(\Omega)}, \|u\|^{p^-}_{L^{p(\cdot)}(\Omega)}\} \leq
\int_\Omega |u|^{p(x)}dx \leq \max\{
\|u\|^{p^+}_{L^{p(\cdot)}(\Omega)},
\|u\|^{p^-}_{L^{p(\cdot)}(\Omega)}\}.
\]
 As $p^->1$, the
space is a reflexive Banach space with dual $L^{p'(\cdot)}(\Omega)$,
where $\frac{1}{p(\cdot)} + \frac{1} {p'(\cdot)}=1$. Let
$r_i \in C(\overline{\Omega})$ with $r_i^- > 1$, $i = 1, 2$. Then if
$r_1(x)\leq r_2(x)$ for any $x \in \Omega$, the imbedding
$L^{r_2(\cdot)}(\Omega)\hookrightarrow L^{r_1(\cdot)}(\Omega)$ is
continuous, of which the norm does not exceed $|\Omega|+ 1$.
Besides, for any $u \in L^{p(\cdot)}(\Omega), v \in
L^{p'(\cdot)}(\Omega)$, we have  H\"{o}lder's inequality
\begin{align*}
\int_\Omega|uv|dx \leq \Big(\frac{1}{p^-}+ \frac{1}{(p^-)'}\Big)
\|u\|_{L^{p(\cdot)}(\Omega)}\|v\|_{L^{p(\cdot)}(\Omega)}.
\end{align*}

For a positive integer $k$, the generalized  Sobolev space
$W^{k,p(\cdot)}(\Omega)$ is defined as
$$
W^{k,p(\cdot)}(\Omega) =
\{u\in L^{p(\cdot)}(\Omega): D^\alpha u \in L^{p(\cdot)}(\Omega),
|\alpha| \leq k\}
$$
 with norm
$$ \|u\|_{W^{k,p(\cdot)}} =
\sum_{|\alpha|\leq k} \|D^\alpha u\|_{L^{p(\cdot)}(\Omega)}.
$$
Such a space is also a separable and reflexive Banach space.

For constant $ 1\leq m <\infty$, the time dependent
spaces  $L^{m}(0,T; W_0^{1,p(\cdot)}(\Omega))$ consists of all strongly measurable functions $u:[0,T]\to W_0^{1,p(\cdot)}(\Omega) $ with
$$
\| u\|_{ L^{m}(0,T; W_0^{1,p(\cdot)}(\Omega))}= (\int_0^T\|u\|^m_{W^{k,p(\cdot)}}dt)^{1/m}< \infty.
$$

 In this article, we assume that there exists a positive constant $C$
such that
\begin{align}\label{e2.1}
|p(x)-p(y)|\leq -\frac{C}{\log|x-y|}, \text{ for every } x, y \in
\Omega \text{ with } |x-y|<\frac{1}{2}.
\end{align}
This condition ensures that smooth functions are dense in the
generalized Sobolev spaces. Then $W^{k,p(\cdot)}_{0}(\Omega)$ can
naturally be defined as the completion of $C^{\infty}_c(\Omega)$ in
$W^{k,p(\cdot)}(\Omega)$ with respect to the norm
$\|\cdot\|_{W^{k,p(\cdot)}}$, and one has
 $W_0^{k,p(\cdot)}(\Omega)=W^{k,p(\cdot)}(\Omega) \cap
W_0^{1,1}(\Omega)$. For $u \in W^{1,p(\cdot)}_{0}(\Omega)$, the
Poincar\'{e} type inequality holds, i.e.,
$$
\|u\|_{L^{p(\cdot)}(\Omega)} \leq C\|\nabla
u\|_{L^{p(\cdot)}(\Omega)},
$$
where the positive constant $C$
depends on $p$ and $\Omega$. So $\|\nabla
u\|_{L^{p(\cdot)}(\Omega)}$ is an equivalent norm in
$W^{1,p(\cdot)}_{0}(\Omega)$.

Let $s(\cdot)$ be a measurable function on $\Omega$ such that
$\text{ess inf}_{x\in \Omega} s(x)>0$. Define the Marcinkiewicz
space $M^{s(\cdot)}(\Omega)$ as the set of measurable functions $v$
such that
$$
\int_{\Omega\cap \{|v|>k\}}k^{s(x)}dx<C,
$$
for some positive constant $C$ and all $k>0$ \cite{su}. It is obvious that if $s(x)\equiv s $
constant, the above definition coincides with the classical
definition of  Marcinkiewicz  spaces. Thanks to Proposition 2.5 in
\cite{su}, we have

\begin{lemma} \label{lem2.1}
Let $r(\cdot), s(\cdot)\in C(\overline{\Omega})$ such that $ s^->0,
(r-s)^->0$ and let $u(x,t)$ be a function defined on $Q_T$. If $u\in
M^{r(\cdot)}(Q_T)$, then $|u|^{s(x)} \in L^1(Q_T)$. In particular, $
M^{r(\cdot)}(Q_T)\subset L^{s(\cdot)}(Q_T) $ for all
$s(\cdot),r(\cdot)\geq1$ such that $(r-s)^->0$.
\end{lemma}

Consider the following elliptic equation corresponding to \eqref{e1.1}
\begin{equation}\label{e2.2}
\begin{gathered}
 -\operatorname{div}(|\nabla v |^{p(x)-2}\nabla v) +|v|^{q-1}v =g
\quad \text{in } \Omega,\\
v=0 \quad \text{on } \partial \Omega,
\end{gathered}
\end{equation}
where $g\in L^1(\Omega)$.
Let $ T_k(s)$ be the usual truncating function defined as
$T_k(\sigma)=\max\{-k, \min\{ k,\sigma\}\}$. Denote
$\Phi_k(\sigma)$ as its primitive function,
\[
\Phi_k(\sigma)=\int_0^\sigma T_k(r)dr
=\begin{cases}
\sigma^2/2  &\text{if }  |\sigma|<k,\\
k|\sigma| - k^2/2 &\text{if } |\sigma| \geq k.
\end{cases}
\]

\begin{definition}[\cite{cn1,su,zz}] \label{def2.1} \rm
A measurable function $v$ is called an entropy solution to problem
\eqref{e2.2}, if $v\in L^q(\Omega)$ and for every $k>0, T_k(v) \in
W_0^{1,p(\cdot)}(\Omega)$,
\begin{equation} \label{e31}
\int_{\Omega}  |\nabla v|^{p(\cdot)-2}\nabla v\nabla T_k(v-\varphi) dx
 +\int_\Omega|v|^{q-1}v
T_k(v-\varphi)dx \leq\int_{\Omega}T_k(v-\varphi)g dx
\end{equation}
 holds for any $\varphi \in W_0^{1, p(\cdot)}(\Omega)\cap L^\infty(\Omega)$.
\end{definition}

A function $v$ such that $T_k(v) \in W^{1,p(\cdot)}_0 (\Omega)$, for
all $k > 0$, does not necessarily belong to $W^{1,1}_0 (\Omega)$.
Thus $\nabla v$ in the equation is defined in a very weak sense
\cite{bb,su}:

\begin{quote}
For every measurable function $v: \Omega\to \mathbb{R}$ 
such that $T_k(v) \in W^{1,p(\cdot)}_0 (\Omega)$
for all $k> 0$, there exists a unique measurable
function $w: \Omega \to \mathbb{R}^N$, which we call
the very weak gradient of $v$ and denote $w=\nabla v$, such that
$$
\nabla T_k(v) = w\chi_{\{|v|<k\}}, \text{almost everywhere in
$\Omega$ and for every } k > 0,
$$
where $\chi_E$ denotes the characteristic function of a
measurable set $E$. Moreover, if $v$ belongs to
$ W^{1,1}_0 (\Omega)$, then $w$ coincides with the weak gradient of $v$.
\end{quote}

\begin{theorem}[\cite{cn1}] \label{thm2.1}
Assume that $g\in L^1(\Omega) $, and \eqref{e2.1} holds.  Then
problem \eqref{e2.2} admits a unique entropy solution $v$.
\end{theorem}



\begin{definition}[\cite{zz1}] \label{def2.2}\rm
 A function $u$ is called an entropy solution of  \eqref{e1.1}, 
if for any $T>0$,
$u\in C([0,T]; L^1(\Omega))\cap L^q(Q_T)$ such that
$ T_k(u)\in L^{p^-}(0,T; W_0^{1,p(\cdot)}(\Omega))$,
$\nabla T_k(u)  \in (L^{p(\cdot)} (Q_T))^N$, and
\begin{equation} \label{e2.4}
\begin{aligned}
& \int_{\Omega} \Phi_k(u-\varphi)(T)dx-\int_{\Omega} \Phi_k(u_0-\varphi(0))dx+
\int_0^T  \langle \varphi_t,T_k(u-\varphi)\rangle  dt   \\
&+\int_{Q_T} |\nabla u|^{p(x)-2} \nabla u\cdot\nabla
T_k(u-\varphi)\,dx\,dt +\int_{Q_T}
|u|^{q-1}uT_k(u-\varphi)\,dx\,dt\\
&\leq\int_{Q_T}gT_k(u-\varphi)dx,
\end{aligned}
\end{equation}
holds for any $ k>0$ and any $\varphi \in C^1(\overline{Q}_T)$ with
$\varphi=0$ on $\partial \Omega\times(0, T)$. Here $\langle \cdot,\cdot \rangle$ denotes the duality product between $W_0^{1,p(\cdot)}(\Omega)$ and its dual
space $ W^{-1,p'(\cdot)}(\Omega)$.
\end{definition}


\begin{remark} \label{rmk2.1} \rm
Similar to Definition \ref{def2.1}, the gradient of $u$ in 
Definition \ref{def2.1} is also defined in a very weak sense \cite{zz1}. 
 On the other hand, let
$$
X=\{\phi|~\phi\in L^{p^-}(0,T; W_0^{1,p(\cdot)}(\Omega)),
\nabla \phi  \in (L^{p(\cdot)} (Q_T))^N\}
$$
 with norm
$ \|\phi\|_{X}=\|\nabla \phi\|_{L^{p(\cdot)}(\Omega)}+\|\phi\|_{L^{p^-}(0,T;
W_0^{1,p(\cdot)}(\Omega))}$. We can choose
$\varphi\in X \cap L^{\infty}(Q_T)$ with $\varphi_t \in X^*+L^1(Q_T)$ as a test
function in the definition above, see \cite{bw}.
\end{remark}


\begin{remark} \label{rmk2.2}\rm
Let $v$ be an entropy solution to problem \eqref{e2.2}. Since it is
independent of time, we have, for any $\varphi \in C^1(\overline{Q}_T)$ with
$\varphi=0$ on $\partial \Omega\times(0, T)$,
\begin{align*}
\int_{\Omega}\Phi_k(v-\varphi)(T)dx
-\int_{\Omega}\Phi_k(v-\varphi(0))dx\\
&= \int_0^T \langle(v-\varphi)_t, T_k(v-\varphi) \rangle dt\\
&=-\int_0^T \langle\varphi_t, T_k(v-\varphi) \rangle dt.
\end{align*}
Thus  we find that  $v$  is actually an entropy solution to \eqref{e1.1}
with initial data $u_0=v$.
\end{remark}


\begin{theorem} \label{thm2.2}
Assuming  that $ u_0,g\in L^1(\Omega)$ and  \eqref{e2.1} holds,
problem \eqref{e1.1} admits a unique entropy solution $u$.
\end{theorem}

\begin{proof}
The proof is rather similar to \cite{zz1} (see also \cite{bg,pr}),
thus we just sketch it in a rather concise way. Consider the
approximate problem
\begin{equation} \label{e4.32}
\begin{gathered}
 u^n_t-\operatorname{div}(|\nabla u^n|^{p(x)-2}\nabla u^n) +|u^n|^{q-1}u^n
 =g^n \quad \text{in } \Omega\times \mathbb{R}^+,\\
u^n=0 \quad \text{on } \partial \Omega\times \mathbb{R}^+, \\
u^n(x,0)=u^n_{0} \quad \text{in } \Omega,
\end{gathered}
\end{equation}
where $\{g^n\}_{n\in \mathbb{N}}$, $\{u_0^n\}_{n\in \mathbb{N}}$ are
smooth approximations of the data $g$ and $u_0$ with
$$
\|u_{0}^{n}\|_{L^{1}(\Omega)}\leq \|u_{0}\|_{L^{1}(\Omega)},\quad
\|g^n\|_{L^{1}(\Omega)}\leq \|g\|_{L^{1}(\Omega)}.
$$
Similar to \cite[Lemma 2.5]{zz1}, with rather minor modifications, we can prove
that problem \eqref{e4.32} admits a unique weak solution $u^n$ for
each $n$.

Performing the calculations as in \cite[pp. 1384 Step 1]{zz1} (see also
\cite[Claim 1]{pr}), we  obtain that, up to a subsequence,
$\{u^n\}$ converges to a function $u$ in $ C([0,T]; L^1(\Omega))$,
and hence almost everywhere in $Q_T$, for any given $T>0$. Using
Vitali's convergence theorem, see for example \cite{cn}, we can
prove that $|u^n|^{q-1}u^n$ converges to $|u|^{q-1}u$  in
$L^1(Q_T)$. Performing the calculations as Step 2 in \cite{zz1}, we
can deduce that $\nabla T_k(u^n)$ converges to $\nabla T_k(u)$
strongly in $ (L^{p(\cdot)} (Q_T))^N$. Taking $T_k(u^n-\varphi)$ as
a test function in \eqref{e4.32} and passing to the limit, it is easy
to obtain that $u$ is an entropy solution to problem \eqref{e1.1}.   Thanks
to the monotonicity of the term $|u|^{q-1}u$, the uniqueness can be
proved in the same way as \cite[pp1396-1398]{zz1}.
\end{proof}

\begin{remark} \label{rmk2.3}\rm
Similar to \cite{zz1}, we can prove that \eqref{e2.4} actually can
hold as an equality. Yet, the inequality is enough to ensure the
uniqueness, see \cite{pr} for the constant exponent case.
\end{remark}

\section{Asymptotic behavior}

 In this section, we consider the asymptotic behavior of the entropy
 solution to \eqref{e1.1}. To state the main result, let us
 first adapt to our problem the definition of entropy subsolutions and entropy
 supersolutions, which were originally defined in \cite{pe2,pl}.
 Denote by $f^+, f^-$ the positive and negative parts of a function
$f$ with $f=f^+-f^-$.

\begin{definition} \label{def3.1} \rm
A function $\underline{v}(x)$ is an entropy subsolution of
\eqref{e2.2} if, for all  $ k>0$, we have $ \underline{v} \in
L^q(\Omega), T_k(\underline{v})\in
  W_0^{1,p(\cdot)}(\Omega)$, and
 it holds that
\begin{equation}
\int_{\Omega} |\nabla \underline{v}|^{p(x)-2} \nabla
\underline{v} \nabla T_k(\underline{v}-\varphi)^+dx +\int_{\Omega}
|\underline{v}|^{q-1}\underline{v}T_k(\underline{v}-\varphi)^+ dx
\leq\int_{\Omega}gT_k(\underline{v}-\varphi)^+dx,
\end{equation}
for any $\varphi \in    W_0^{1, p(\cdot)}(\Omega)\cap
L^\infty(\Omega) $.

On the other hand, a function $\overline{v}(x)$ is an entropy
supersolution of problem \eqref{e2.2} if, for all  $ k>0$, we have $
\overline{v} \in L^q(\Omega), T_k(\overline{v})\in
  W_0^{1,p(\cdot)}(\Omega)$, and
 it holds that
 \begin{equation}
\int_{\Omega} |\nabla \overline{v}|^{p(x)-2} \nabla
\overline{v} \nabla T_k(\overline{v}-\varphi)^-dx +\int_{\Omega}
|\overline{v}|^{q-1}\overline{v}T_k(\overline{v}-\varphi)^- dx
\geq\int_{\Omega}gT_k(\overline{v}-\varphi)^-dx ,
\end{equation}
for any $\varphi \in    W_0^{1, p(\cdot)}(\Omega)\cap
L^\infty(\Omega) $.
\end{definition}

\begin{definition} \label{def3.2} \rm
A function $\underline{u}(x,t)$ is an entropy subsolution of
\eqref{e1.1} if, for all  $T, k>0$, we have
$\underline{u}(x,t)\in C([0,T]; L^1(\Omega)) \cap L^q(Q_T), T_k(\underline{u})\in
L^{p^-}(0,T; W_0^{1,p(\cdot)}(\Omega))$,
 $ \nabla T_k(\underline{u})  \in (L^{p(\cdot)} (Q_T))^N$, and
 it holds that
 \begin{equation}
\begin{aligned}
&\int_{\Omega} \Phi_k((\underline{u}-\varphi)^+)(T)dx
-\int_{\Omega} \Phi_k((\underline{u}_0-\varphi(0))^+)dx\\
&+\int_{Q_T} |\nabla \underline{u}|^{p(x)-2} \nabla
\underline{u} \nabla T_k(\underline{u}-\varphi)^+\,dx\,dt
   \\
& +\int_0^T  \langle \varphi_t,T_k(\underline{u}-\varphi)^+\rangle
dt+\int_{Q_T}
|\underline{u}|^{q-1}\underline{u}T_k(\underline{u}-\varphi)^+ \,dx\,dt\\
&\leq\int_{Q_T}gT_k(\underline{u}-\varphi)^+\,dx\,dt,
\end{aligned}
\end{equation}
for any $\varphi \in C^1(\overline{Q}_T)$ with $\varphi=0$ on
$\partial \Omega\times(0, T)$  and
$\underline{u}(x,0)\equiv \underline{u}_0(x)\leq u_0(x)$ a.e. in
$\Omega$ with $\underline{u}_0\in L^1(\Omega)$.

On the other hand, a function $\overline {u}(x,t)$ is an entropy
supersolution of problem \eqref{e1.1} if, for all  $T, k>0$, we have
$\overline{u}(x,t)\in C([0,T]; L^1(\Omega)) \cap L^q(Q_T),
T_k(\overline{u})\in L^{p^-}(0,T; W_0^{1,p(\cdot)}(\Omega))$, $
\nabla T_k(\overline{u})
 \in (L^{p(\cdot)} (Q_T))^N$, and
 it holds that
 \begin{equation} \label{e3.4}
\begin{aligned}
& \int_{\Omega} \Phi_k((\overline {u}-\varphi)^-)(T)dx
-\int_{\Omega} \Phi_k((\overline {u}_0-\varphi(0))^-)dx\\
&+\int_{Q_T} |\nabla \overline {u}|^{p(x)-2} \nabla \overline
{u} \nabla T_k(\overline {u}-\varphi)^-\,dx\,dt  \\
& + \int_0^T  \langle \varphi_t,T_k(\overline {u}-\varphi)^-\rangle
dt +\int_{Q_T} |\overline {u}|^{q-1}\overline {u}T_k(\overline
{u}-\varphi)^- \,dx\,dt\\
&\geq\int_{Q_T}gT_k(\overline {u}-\varphi)^-\,dx\,dt,
\end{aligned}
\end{equation}
for any $\varphi \in C^1(\overline{Q}_T)$ with $\varphi=0$ on
$\partial \Omega\times(0, T)$  and
$\overline {u}(x,0)\equiv \overline {u}_0(x)\geq u_0(x)$ a.e. in
$\Omega$ with $\overline {u}_0\in L^1(\Omega)$.
\end{definition}


\begin{remark} \label{rmk3.1} \rm
Taking $T_k(u^n-\varphi)^+$,  $T_k(u^n-\varphi)^-$ as test functions 
in \eqref{e4.32} and passing to
 the limits, we obtain that an entropy solution to problem \eqref{e1.1} is
 both an entropy subsolution and an entropy supersolution of the same
 problem. In the same way, an entropy solution to the elliptic
 problem \eqref{e2.2} also turns out to be an entropy subsolution and an entropy
 supersolution to the problem.
\end{remark}

\begin{remark} \label{rmk3.2}\rm
Similar to the observation in Remark \ref{rmk2.2}, we may find that an
entropy subsolution (entropy supersolution)
 $\underline{v}$  (respectively $\overline{v} $ ) of the elliptic problem \eqref{e2.2} is automatically an entropy
subsolution (entropy supersolution) of  \eqref{e1.1} with itself
as initial data.
\end{remark}

\begin{lemma} \label{lem3.1}
Let $u_0, g\in L^1(\Omega)$, and  $\overline {u}, \underline{u} $ be
an entropy supersolution and an entropy  subsolution to problem
\eqref{e1.1} respectively. Let $u$ be the unique entropy solution to the
same problem. Then for any $t>0$, we have $ \underline{u} \leq
u\leq\overline {u}$ almost everywhere in $\Omega$.
\end{lemma}

The proof of the above lemma is almost the same as that of
\cite[Lemma 3.3]{pe2}, we omit it.

\begin{theorem} \label{thm3.1}
Let $\overline{v}$ and $\underline{v}$ be, respectively, an entropy supersolution
 and an entropy subsolution to problem \eqref{e2.2} respectively.
Assume \eqref{e2.1} holds, $ g, u_0\in  L^1(\Omega)$ and
 $\underline{v}\leq u_0 \leq \overline{v} $. If
$$
\theta(x) \doteq\max\{ \frac{p(x)q}{(q+1)},\quad
p(x)-\frac{N}{N+1} \}  >1  \text{ in } \overline{\Omega},
$$
then the unique entropy solution $u$ of problem \eqref{e1.1} converges in
$L^q(\Omega)$ to the unique entropy solution $v$ of problem
\eqref{e2.2} as $t$ tends to infinity.
\end{theorem}

\begin{corollary} \label{coro3.1}
Assume \eqref{e2.1} holds, $g\in  L^1(\Omega), u_0\equiv 0$.
If $\theta(x)>1$ in $ \overline{\Omega}$, then the unique entropy
 solution $u(t)$ of problem \eqref{e1.1} converges to the unique
entropy solution $v$ of problem \eqref{e2.2} in $L^q(\Omega)$ as $t$
tends to infinity.
\end{corollary}


\begin{proof}[Proof of  Theorem \ref{thm3.1}]
 Consider the  nonlinear problem
\begin{equation} \label{e355}
\begin{gathered}
 (u_m)_t-\operatorname{div}(|\nabla u_m|^{p(x)-2}\nabla u_m)  +|u_m|^{q-1}u_m=g
\quad \text{in } \Omega\times (0,1),\\
u_m=0 \quad \text{on } \partial \Omega\times (0,1), \\
u_m(x,0)=u(x,m) \quad \text{in } \Omega,
\end{gathered}
\end{equation}
where $m\in \mathbb{N}\cup \{0\}$, and $ u(x,0)=\overline{v}$. Let
$u(t)$ be the entropy solution for problem \eqref{e1.1} with $\overline{v}$
as initial data. Thanks to the uniqueness of entropy solutions and
the independency of $t$ for the data $g$, $u_m(t)$ is just the
restriction of $u(t)$ on the interval $[m, m+1)$. Note that $v$
 and $\overline{v}$ are the entropy subsolution and the entropy supersolution
respectively for problem \eqref{e1.1} with initial data $\overline{v}$.
Thanks to Lemma \ref{lem3.1}, $v\leq u(t)\leq \overline{v}$ for any $t>0$.
Similarly, let $u(s+t)$ be the solution  with $u(s)$ as initial
data, then we have $u(s+t)\leq u(t)$ for any $t,s>0$, which implies
that $u(t)$ is decreasing in $t$. Thus for $0<t<1$,
\begin{equation} \label{e35}
\overline{v}(x)\geq  u_m(x,0)\geq u_m(x,t)= u(x,m+t) \geq u_{m+1}(x,0)\geq v(x).
\end{equation}
Hence there must be a function $w(x)\geq v(x)$ such that $ u(x,t) $
converges to $w(x)$ almost everywhere in $\Omega$ as $t$ tends to
infinity. And then  by the dominated convergence theorem, we have
\begin{equation} \label{e3.66}
 u(x,t) \to w(x) \text{ ~in ~} L^1(\Omega) \quad \text{as~}  t  \to +\infty.
\end{equation}
Next, following the ideas of \cite{pe2} (see also \cite{pe3}), we
can perform some estimates for the sequence $\{u_m\}$, to prove that
$w(x)$ is actually the entropy solution $v$ to the elliptic problem
\eqref{e2.2}.

Thanks to \eqref{e35}, we have
\begin{equation}
\|u_m(t)\|_{L^1(\Omega)}=\|u(m+t)\|_{L^1(\Omega)}
 \leq \|\overline{v}\|_{L^1(\Omega)} + \|v\|_{L^1(\Omega)}\leq C,  \quad
0<t\leq 1,  \label{e366}
\end{equation}
where $C$ is obviously independent of $m$.
Thus the sequence $\{u_m\}$ is bounded in $L^\infty(0,1; L^1(\Omega))$.
Taking $u_0=u(x,m)$, $T=1$ and $\varphi=0$ in Definition \ref{def2.2}, we obtain
 \begin{equation} \label{e325}
\begin{aligned}
 &\int_{\Omega} \Phi_k(u_m)(1)dx +\int_0^1\int_{\Omega} |\nabla T_k(
 u_m)|^{p(x)}\,dx\,d\tau  + \int_0^1\int_{\Omega} |u_m|^q|T_k(u_m)| \,dx\,d\tau\\
&\leq k\|g\|_{L^1(\Omega)} +\int_{\Omega} \Phi_k(u(x,m))dx.
\end{aligned}
\end{equation}
Note that
\begin{equation} \label{e2.11}
0\leq \Phi_k(s) \leq k |s| \leq  \Phi_k(s) +\frac{k^2}{2} .
\end{equation}
Noticing \eqref{e366}, we deduce from  \eqref{e325} that
\begin{gather}
\int_0^1\int_{\Omega} |\nabla T_k(
 u_m)|^{p(x)}\,dx\,d\tau \leq C k,\label{e36}\\
\begin{aligned}
\int_0^1\int_{\Omega} |u_m|^q  \,dx\,d\tau
&\leq \int_0^1\int_{\Omega} |u_m|^q|T_k(u_m)| \,dx\,d\tau  +|\Omega|\\
&\leq \int_{\Omega} \Phi_1(u(x,m))dx+|\Omega|+\|g\|_{L^1(\Omega)}
\leq C.
\end{aligned} \label{e327}
\end{gather}
For a  given function $f(x,t)$ defined on $Q_T$, we set
$$
\{f\geq k\}=\{(x,t)\in Q_T: f(x,t)\geq k\} , \{f\leq k\}=\{(x,t)\in Q_T:
f(x,t)\leq k\}.
$$
Then setting
$ \alpha(\cdot)=  p(\cdot)/(q +1)$ in $\overline{\Omega} $  and using
\eqref{e327}, we  deduce that
\begin{equation} \label{e4.10}
\begin{aligned}
&\int_{\{|\nabla u_m|^{\alpha(x)}>k\}} k^q\,dx\,dt\\
&\leq \int_{\{|\nabla u_m|^{\alpha(x)}>k\}\cap \{|u_m|\leq k\}}k^q\,dx\,dt
 +\int_{\{|u_m|>k\}} k^q\,dx\,dt \\
& \leq\int_{\{|u_m|\leq k\}} k^q\Big(\frac{|\nabla
u_m|^{\alpha(x)}}{k}\Big)^{\frac{p(x)}{\alpha(x)}} \,dx\,dt
+\int_{Q_1}|u_m|^q \,dx\,dt \\
&\leq\frac{1}{k}\int_{Q_1}   |\nabla T_k( u_m)|^{p(x)} \,dx\,dt+C\leq C,
\end{aligned}
\end{equation}
which implies that $ |\nabla u_m|^{p(\cdot)/(q+1)}$ is bounded in
$ M^q(Q_1)$, and hence we conclude from Lemma \ref{lem2.1} that
\begin{equation}\label{e4.11}
\parbox{9cm}{ $|\nabla u_m|^{\beta(\cdot)}$ is bounded in 
$L^1(Q_1)$ for $\beta\in C(\overline{\Omega})$  satisfying 
$\beta(\cdot)< p(\cdot)q/(q+1)$  in $\overline{\Omega}$.}
\end{equation}


On the other hand, let $s\in C(\overline{\Omega})$ such that
$1< s(\cdot)<(N+1)p(\cdot)/N $ in $\overline{\Omega}$. From the
continuity of $s$ and $p$, for any $x\in \Omega$, there exists a
ball $B_\delta(x)$ of $x$, such that
$s^+(B_\delta(x)\cap\Omega)<\left((N+1)p(\cdot)/N \right)^-
 (B_\delta(x)\cap \Omega)$, where
\begin{gather*}
s^+(B_\delta(x)\cap\Omega)=\max\{s(y): y\in \overline{B_\delta(x)\cap\Omega}\}, \\
\left((N+1)p(\cdot)/N \right)^- (B_\delta(x)\cap\Omega)=\min\{
(N+1)p(y)/N: y\in \overline{B_\delta(x)\cap\Omega} \}.
\end{gather*}
It is obvious that $\cup_{x\in  \Omega}B_\delta(x) $ is an open
covering of $\overline{\Omega}$. Since $\overline{\Omega}$ is
compact, there is a finite sub-covering $B_{\delta_i}(x_i),
i=1,2,\dots,l$. For convenience, we denote  the set
$B_{\delta_i}(x_i)\cap\Omega$ by $U_i$ hereafter.
Assume that $\operatorname{meas}(U_i)>c>0, i=1,2,\dots, l$.
Denoting ${s_i}^+= s^+(U_i),{p_i}^-=p^-(U_i)$, we have
$$
\left((N+1)p(\cdot)/N\right)^-(U_i) = (N+ 1){p_i}^-/N.
$$
Setting $U_{i,1}=U_i\times (0,1)$, we deduce that
\begin{equation} \label{e4.12}
\begin{aligned}
&\int_{U_{i,1} \cap \{|u_m|>k\}}k^{\frac{(N+ 1){p_i}^-}{N}}\,dx\,dt\\
&\leq \int_0^1\int_{U_i}|T_k(u_m)|^{\frac{(N+ 1){p_i}^-}{N}}\,dx\,dt \\
&\leq 2^{\frac{(N+ 1){p_i}^-}{N}} \int_0^1\int_{U_i}|T_k(u_m)-(
T_k(u_m))_i|^{\frac{(N+ 1){p_i}^-}{N}}\,dx\,dt \\
&\quad +2^{\frac{(N+ 1){p_i}^-}{N}} \int_0^1\int_{U_i}|(
T_k(u_m))_i|^{\frac{(N+ 1){p_i}^-}{N}}\,dx\,dt,
\end{aligned}
\end{equation}
where
$$
(T_k(u_m))_i =\frac{1}{\operatorname{meas}(U_i)}
\int_{U_i}T_k(u_m)dx \text{ for almost all } t\in (0,1).
$$
Thanks to \eqref{e366}, we have
$$
|( T_k(u_m))_i|  \leq  \frac{1}{\operatorname{meas}(U_i)}
\int_{U_i}|T_k(u_m)|dx  \leq C,
$$
where $c$ is independent of $k, m$.
By the well-known Gagliardo-Nirenberg inequality, we have, for
almost all $t\in (0,1)$,
\begin{align*}
&\int_{U_i}|T_k(u_m)-( T_k(u_m))_i|^{\frac{(N+ 1){p_i}^-}{N}}dx\\
&\leq C  \int_{U_i} |\nabla T_k(u_m)|^{{p_i}^-}dx
\Big(\int_{U_i}|T_k(u_m)-( T_k(u_m))_i|dx\Big)^{\frac{{p_i}^-} {N}}.
\end{align*}
Integrating the above inequality over  $(0,1)$, we deduce that
\begin{align*}
&\int_0^1\int_{U_i}|T_k(u_m)-( T_k(u_m))_i|^{\frac{(N+1){p_i}^-}{N}}\,dx\,dt\\
&\leq  C \left(\|u_m\|_{L^\infty(0,1; L^1(\Omega)) }+C |\Omega|\right)
^{\frac{{p_i}^-} {N}} \int_0^1\int_{U_i} |\nabla  T_k(u_m)|^{{p_i}^-}\,dx\,dt.
\end{align*}
Taking the last inequality and  \eqref{e366}, \eqref{e36}
into \eqref{e4.12}, we obtain that
\begin{equation} \label{e4.13}
\int_{U_{i,1} \cap \{|u_m|>k\}}k^{\frac{(N+1){p_i}^--N}{N}}\,dx\,dt\leq C,
\end{equation}
where $C$ may depend on
$ \|g\|_{L^1(\Omega)}, \|\overline{v}\|_{L^1(\Omega)}, |\Omega|$,
but it is independent of $k,m$. Since ${s_i}^+<\frac{(N+ 1){p_i}^-}{N}$ in $U_i$,
\eqref{e4.13} implies that ($k\geq1$)
\begin{align*}
\int_{U_{i,1} \cap \{|u_m|>k\}}k^{s(x)-1}\,dx\,dt
&\leq \int_{U_{i,1} \cap \{|u_m|>k\}} k^{{s_i}^+-1}\,dx\,dt\\
& \leq\int_{U_{i,1} \cap \{|u_m|>k\}}k^{\frac{(N+1){p_i}^--N}{N}}\,dx\,dt
\leq C.
\end{align*}
Then
\begin{align}\label{e2.16}
\int_{\{|u_m|>k\}}k^{s(x)-1}dx\leq \sum_{i=1}^{l}\int_{U_i \cap
\{|u_m|>k\}}k^{s(x)-1}dx\leq l C.
\end{align}
Hence $\{u_m\}$ is  bounded in $M^{s(x)-1}(Q_1) $.
Similar calculations as \eqref{e4.10} help us to obtain that
$\{|\nabla u_m|^{\frac{p(x)}{s(x)}}\}$ is
 bounded in $M^{s(x)-1}(Q_1) $.
By Lemma \ref{lem2.1}, we have that 
\begin{equation} \label{e4.15}
\parbox{10cm}{the set $\{|\nabla u_m|^{\beta_1(x)}\}$
 is bounded in $L^1(Q_1)$  for $\beta_1 \in C(\overline{\Omega})$ satisfying
$\beta_1(\cdot) < p(\cdot)-N/(N+1)$.}
\end{equation}
Combining \eqref{e4.11} and \eqref{e4.15},
we obtain that $\{|\nabla u_m|^{\gamma(x)}\} $ is bounded in
$L^1(Q_1)$  for any $\gamma\in C(\overline{\Omega})$
satisfying
\[
 0< \gamma(x) <\theta(x)\doteq\max\{ p(x)q/(q+1), ~p(x)-N/(N+1) \} \quad
\text{in } \overline{\Omega}.
\]
 Thus if $\theta(x)>1$ in $\overline{\Omega}$ (for example, in the case
$p(x)>2-1/(N+1)$ or $q> 1/(p(x)-1)$ in $ \overline{\Omega}$),
we can choose a constant $1<r_0<\theta(x)$ such that
$$
\{u_m(t)\} \text{~ is  bounded in } L^{r_0}(0,1;W_0^{1,{r_0}}(\Omega)).
$$
On the other hand, note that
$$
p(x)-N/(N+1)>p(x)-1\text{~in~}\overline{\Omega}.
$$
The definition of $\theta(x)$ allows us to choose a positive function
$\gamma_0$ in $\overline{\Omega}$, such that
$$
p(x)-1<\gamma_0(x)<\theta(x) \quad \text{in }\overline{\Omega}.
$$

Hence $ \{(|\nabla u_m|^{(p(x)-1)})^{\gamma_0(x)/(p(x)-1)}\}$
(=$\{|\nabla u_m|^{\gamma_0(x)}\}$) is bounded in $L^1(Q_1)$. Therefore,
$\{|\nabla u_m|^{p(x)-1}\}$ is bounded in $L^{s_0}(Q_1)$ for
$1<s_0\leq\gamma_0(x)/(p(x)-1)$ in $\overline{\Omega}$, which
implies that $ \operatorname{div}(|\nabla u_m|^{p(x)-2}\nabla u_m) $ is
uniformly (with respect to $m$) bounded in
$L^{s_0}(0,1;W^{-1, {s_0}}(\Omega))$. Then we deduce from the equation that
$$
\{(u_m)_t\} \text{~ is bounded in ~}  L^{s_0}(0,1;W^{-1,
{s_0}}(\Omega))+L^1(0,1; L^1(\Omega)).
$$
So, thanks to the well-known compactness result of Aubin's type, see for example
\cite{si}, there is a subsequence of $\{u_m\}$, denoted by
$\{u_{m_k}\} $, which converges to a function $\widetilde{ u }$ in
$L^1(Q_1)$. Since $u_{m_k}(x,t)=u(x,t+m_k)$, we may conclude from
\eqref{e3.66} that $\widetilde{ u }=w(x)$ is
 independent of time.

Now, let us show that $\widetilde{ u }$ is actually the entropy
solution $v$ of the elliptic problem \eqref{e2.2}. From \eqref{e36},
we have
\begin{gather*}
T_k(u_m) \to   T_k(\widetilde{ u }) \quad \text{weakly in }
L^{p^-}(0,1; W_0^{1,p(\cdot)}(\Omega)),  \\
 \nabla T_k(u_m) \to  \nabla T_k(\widetilde{ u })\quad \text{weakly in }
(L^{p(\cdot)} (Q_1))^N.
\end{gather*}
From the estimate on $u_m$, we know that
$$
u_m \to \widetilde{u} \quad \text {weakly in }
L^{r_0}(0,1;W_0^{1,{r_0}}(\Omega)),  \text{ for some } 1< r_0< \theta(x).
$$
Furthermore,  with very minor modifications on the
proof of \cite[Theorem 3.3]{bg97}, one can prove that
$$
\nabla u_m \to \nabla \widetilde{ u }\quad\text{a.e.  in } Q_1.
$$

Since $u_m$ is the entropy solution of  \eqref{e355}, we have
\begin{align}
 & \int_{\Omega} \Phi_k(u_m-\varphi)(1)dx-\int_{\Omega} \Phi_k(u(x,m)-\varphi(0))dx+
\int_0^1  \langle \varphi_t,T_k(u_m-\varphi)\rangle  dt  \label{e3.11} \\
&+\int_{Q_1} |\nabla u_m|^{p(x)-2} \nabla u_m\cdot\nabla
T_k(u_m-\varphi)\,dx\,dt +\int_{Q_1} |u_m|^{q-1}u_mT_k(u_m-\varphi)\,dx\,dt
\label{e3.12}\\
&\leq\int_{Q_1}gT_k(u_m-\varphi)dx\label{e3.13}
\end{align}
for any $\varphi \in C^1(\overline{Q}_1)$ with $\varphi=0$ in 
$\partial \Omega\times(0, 1)$.
Using the convergence results above for $u_m$, and passing to the
limit, we deduce   that
 $$
\eqref{e3.11} \to
\int_{\Omega} \Phi_k(\widetilde{ u }-\varphi(1))dx-\int_{\Omega}
\Phi_k(\widetilde{ u } -\varphi(0))dx + \int_0^1  \langle
\varphi_t,T_k(\widetilde{ u } -\varphi)\rangle dt.
$$ 
Since $\widetilde{ u }=w(x) $ is independent of time, similar to Remark
\ref{rmk2.2} we have
\begin{equation} \label{e3.14}
\int_{\Omega} \Phi_k(\widetilde{ u }-\varphi(1))dx
-\int_{\Omega} \Phi_k(\widetilde{ u } -\varphi(0))dx
+ \int_0^1  \langle \varphi_t,T_k(\widetilde{ u } -\varphi)\rangle dt=0.
\end{equation}
Passing to the limit in \eqref{e3.13}, we obtain that
\begin{align}\label{e3.15}
\eqref{e3.13} \to \int_{Q_1}gT_k(\widetilde{ u } -\varphi)dx.
\end{align}
At last, passing to the limit in \eqref{e3.12}, we note that
\begin{align}
&\int_{Q_1} |\nabla u_m|^{p(x)-2} \nabla u_m\cdot\nabla
T_k(u_m-\varphi)\,dx\,dt \label{e3.16} \\
&= \int_{Q_1} (|\nabla u_m|^{p(x)-2} \nabla u_m- |\nabla
\varphi|^{p(x)-2} \nabla \varphi)\nabla T_k(u_m-\varphi)\,dx\,dt\label{e3.17}\\
&\quad +\int_{Q_1} |\nabla \varphi|^{p(x)-2} \nabla \varphi\cdot\nabla
T_k(u_m-\varphi)\,dx\,dt. \label{e3.18}
\end{align}
Using Fatou's lemma and the weak convergence of $\nabla T_k(u_m)$ we
obtain
\begin{equation}
\lim_{m\to \infty}\eqref{e3.16}\geq  \int_{Q_1}  |\nabla
\widetilde{ u } |^{p(x)-2} \nabla \widetilde{ u }  \nabla
T_k(\widetilde{ u } -\varphi)\,dx\,dt.\label{e3.19}
\end{equation}
Similarly, since
\begin{equation} \label{e3.20}
\begin{aligned}
\int_{Q_1} |u_m|^{q-1}u_mT_k(u_m-\varphi)\,dx\,dt  
&=\int_{Q_1} (|u_m|^{q-1}u_m-|\varphi|^{q-1}\varphi)T_k(u_m-\varphi)\,dx\,dt\\
&\quad +\int_{Q_1}|\varphi|^{q-1}\varphi T_k(u_m-\varphi) \,dx\,dt,
 \end{aligned}
\end{equation}
we deduce that
\begin{equation}
\lim_{m\to \infty}\eqref{e3.20}\geq  \int_{Q_1}  |
\widetilde{ u } |^{q-1}   \widetilde{ u }  \nabla T_k(\widetilde{u}
-\varphi)\,dx\,dt.\label{e3.21}
\end{equation}
We then conclude from \eqref{e3.14},  \eqref{e3.15},  \eqref{e3.19} and
\eqref{e3.21} that
\begin{align*}
&\int_{Q_1} |\nabla \widetilde{ u }|^{p(\cdot)
-2}\nabla \widetilde{ u } \nabla T_k(\widetilde{ u } -\varphi) \,dx\,dt
+\int_{Q_1}|\widetilde{ u } |^{q-1}\widetilde{ u }
T_k(\widetilde{ u } -\varphi)\,dx\,dt\\
&\leq\int_{Q_1}g T_k(\widetilde{ u} -\varphi)\,dx\,dt.
\end{align*}
Especially, for $ \phi \in C^1(\overline{\Omega}),
\phi\mid_{\partial \Omega}=0$, we have
\begin{align*}
\int_{\Omega}
 |\nabla \widetilde{ u } |^{p(\cdot)-2}\nabla \widetilde{ u }
\nabla T_k(\widetilde{ u } -\phi) \,dx
 +\int_{\Omega}|\widetilde{ u } |^{q-1}\widetilde{ u }
T_k(\widetilde{ u } -\phi)\,dx
\leq\int_{\Omega}T_k(\widetilde{ u }-\phi)g \,dx .
\end{align*}
Then from the density result, we conclude that $\widetilde{ u }$
satisfies the entropy formulation of problem \eqref{e2.2} and hence
it coincides with the unique entropy solution $v$. Performing a similar argument,
we can prove that the entropy solution $u^1(t)$ for problem \eqref{e1.1}
with $\underline{v}$ as initial data also converges in $L^1(\Omega)$
to the entropy solution $v$ of problem \eqref{e2.2}.

For the entropy solution $u^2(t)$ of  \eqref{e1.1} corresponding to
the initial data $u_0$ with  $\underline{v}\leq u_0\leq
\overline{v}$, thanks to the comparison result, we have
$$
\underline{v}\leq u^1(t)\leq u^2(t) \leq u(t)\leq \overline{v}.
$$ 
Thus we obtain that $u^2(t)$ converges to $v$ in $L^1(\Omega)$.
Since $\underline{v} , \overline{v}  $ all lie in $L^q(\Omega)$, 
we  obtain the convergence result in $L^q(\Omega)$.
\end{proof}


\begin{proof}[Proof of Corollary \ref{coro3.1}]
 Let $v_1, v_2$ be the entropy
solutions to the following two problems:
\begin{equation}\label{e330}
\begin{gathered}
 -\operatorname{div}(|\nabla v |^{p(x)-2}\nabla v) +|v|^{q-1}v =g^+
\quad \text{in } \Omega,\\
v=0 \quad \text{on } \partial \Omega,
\end{gathered}
\end{equation}
and
\begin{equation}\label{e331}
\begin{gathered}
 -\operatorname{div}(|\nabla v |^{p(x)-2}\nabla v) +|v|^{q-1}v =-g^-
\quad \text{in } \Omega,\\
v=0 \quad \text{on } \partial \Omega.
\end{gathered}
\end{equation}
Note that $0$ is an entropy subsolution of  \eqref{e330}, and
it is an entropy supersolution of \eqref{e331}. Since the entropy
solution can be obtained as the limit of the solutions for the
approximate problems, similar to Lemma \ref{lem3.1}, it is not difficult to
show the following comparison result, see \cite{pl,pe3}
for the constant exponents case,
$$ 
v_2\leq 0 \leq v_1 ~~a.e. \text{ in } \Omega.
$$
On the other hand, thanks to Remark \ref{rmk3.1}, $v_1$ is an entropy
supersolution of \eqref{e330}. And hence, it is an entropy
supersolution of \eqref{e2.2}. Similarly, $v_2$ is an entropy
subsolution of \eqref{e2.2}. Thus, the result of the
corollary  follows immediately from Theorem \ref{thm3.1}.
\end{proof}

\begin{remark}  \label{rmk3.3} \rm
Let $w(t)=u(t)-v$. We can prove that $w(t)$ converges to zero in $L^r(\Omega)$ 
for any $1\leq r<\infty$ as $t$ tends to infinity. Indeed, consider the
approximate problem for \eqref{e2.2},
 \begin{equation}\label{e4.33}
\begin{gathered}
 -\operatorname{div}(|\nabla v^n |^{p(x)-2}\nabla v^n) +|v^n|^{q-1}v^n =g^n
\quad \text{in } \Omega,\\
v^n=0 \quad \text{on } \partial \Omega,
\end{gathered}
\end{equation}
where $\{g^n\}$ is the same sequence as in \eqref{e4.32}. 
Problem \eqref{e4.33} admits a unique solution $v^n$ for each $n$, and up to
subsequences, $\{v^n\}$ converges to the unique entropy solution $v$
of  \eqref{e2.2} in $L^1(\Omega)$, see \cite{cn1,zz}.
Subtracting \eqref{e4.33} from \eqref{e4.32} and setting
$w^n=u^n-v^n$, we have
\begin{equation}\label{e4.34}
\begin{gathered}
w_{t}^{n}-\operatorname{div} (|\nabla u^n|^{p(x)-2}
\nabla u^n -|\nabla v^n|^{p(x)-2}\nabla v^n) 
+|u^n|^{q-1}u^n-|v^n|^{q-1}v^n=0 \\
 \text{in } \Omega\times\mathbb{R}^+,\\
w^{n}=0 \quad \text{on } \partial\Omega\times\mathbb{R}^+,\\
w^{n}(x,0)=u^{n}_{0}-v^{n} \quad \text{in } \Omega.
\end{gathered}
\end{equation}
Thanks to the convergence results for $u^n$ and $v^n$, we know that,
up to a subsequence, $w^n$ converges to $w=u-v$ in
$C([0,T];L^1(\Omega))$ for any $T>0$.

Taking $T_k(u^n)(k\geq1)$ as a test function in \eqref{e4.32}, we
deduce that
\begin{equation} \label{e3.5}
\begin{gathered}
 \frac{d}{dt}\int_{\Omega} \Phi_k(u^n)(t)dx
 + \int_{\Omega} |\nabla T_k(u^n)|^{p(x)}dx
+\int_{\Omega}|u^n|^q| T_k(u^n)|dx \leq
k\|g\|_{L^1(\Omega)}.
\end{gathered}
\end{equation}
 From the definition of $\Phi_k(\cdot )$ we obtain
\begin{equation} \label{e2.11b}
\int_{\Omega} \Phi_k(u^n)(t)dx
\leq k\|u^n(t)\|_{L^1(\Omega)}\leq \int_{\Omega}
\Phi_k(u^n)(t)dx+\frac{k^2}{2}|\Omega|,
\end{equation}
where $|\Omega|$ is the Lebesgue measure of  $\Omega$. Note that
$$
 \int_{\Omega} \Phi_1(u^n)(t)dx \leq \int_{\Omega}|u^n|^q| T_1(u^n)|dx+|\Omega|.
$$
We deduce from \eqref{e3.5} that
$$
\frac{d}{dt}\int_{\Omega} \Phi_1(u^n)(t)dx
 + \int_{\Omega} \Phi_1(u^n)(t)dx
\leq \|g\|_{L^1(\Omega)}+|\Omega|.
$$
Standard Gronwall type inequality implies that
\[
\int_{\Omega} \Phi_1(u^n)(t)dx \leq \|g\|_{L^1(\Omega)}+|\Omega|
 +e^{- t}\int_{\Omega}\Phi_1(u_0^n)dx , \quad t>0.
\]
Thanks to  \eqref{e2.11}, we have
\begin{equation}
\|u^n(t)\|_{L^1(\Omega)}\leq \|u_0\|_{L^1(\Omega)}
+\frac{3}{2}|\Omega| +   \|g\|_{L^1(\Omega)}  , \quad t>0.\label{e3.7}
\end{equation}
Integrating \eqref{e3.5} on $[t,t+1]$, we  obtain
\begin{equation} \label{e3.26}
\begin{aligned}
\int_t^{t+1}\int_{\Omega}|u^n|^q \,dx\,d\tau
&\leq\int_t^{t+1}\int_{\Omega} (|u^n|^q|T_1(u^n)|+1)\,dx\,d\tau\\
&\leq   \|g\|_{L^1(\Omega)} + \|u_0\|_{L^1(\Omega)} +|\Omega|.
\end{aligned}
\end{equation}
Multiplying \eqref{e4.33} by $T_1(v^n)$, we deduce that
\begin{equation} \label{e3.27}
 \int_{\Omega}|v^n|^q dx
\leq  \int_{\Omega} (|v^n|^q|T_1(v^n)|+1) dx
\leq   \|g\|_{L^1(\Omega)} +|\Omega|.
\end{equation}
Combining \eqref{e3.26}, \eqref{e3.27}, we have
\begin{equation} \label{e3.28}
 \int_t^{t+1} \int_{\Omega }
|w^n(\tau)|^q\,dx\,d\tau \leq2\|g\|_{L^1(\Omega)}
+2|\Omega|+\|u_0\|_{L^1(\Omega)} , \quad \text{for any } t\geq 0.
\end{equation}
Taking $|w^n|^{q-2} w^n  $ as a test function in \eqref{e4.34}
(if  $q< 2$, we can take $( (|w^n|+\epsilon)^{q-1}-\epsilon^{q-1}) sgn(w^n) $ as a
 test function and then let $\epsilon$ go to zero to justify this calculation.
Here for simplicity, we assume that $q\geq 2$.), we deduce that
\begin{align} \label{e3.29}
\frac{1}{q}\frac{d}{dt}\int_\Omega |w^n(t)|^q\,dx +C\int_\Omega
|w^n(t)|^{2q-1}dx\leq 0.
\end{align}
Integrating the above inequality from $s$ to $t+1(0\leq t \leq s
< t + 1)$,  yields
\[
\int_\Omega |w^n(t+1)|^q\,dx \leq   \int_\Omega |w^n(s)|^q\,dx.
\]
Integrating this inequality with respect to $s$ from $t$ to $t +
1$ and using \eqref{e3.28}, yields
\begin{equation} \label{e3.30}
\int_\Omega |w^n(t+1)|^q\,dx
\leq  \int_t^{t+1}\int_\Omega|w^n(\tau)|^q\,dx\,d\tau
\leq C, \quad \text{for any } t\geq 0,
\end{equation}
with $C $  independent of $n,t$. Integrating \eqref{e3.29} on
$[t,t+1]$ for any $t\geq 1$, and using \eqref{e3.30} we deduce that
\begin{equation} \label{e3.42}
\int_t^{t+1}\int_\Omega |w^n(\tau)|^{2q-1}\,dx\,d\tau
\leq C\int_\Omega |w^n(t)|^q\,dx\leq C
\end{equation}
Now setting $ q_{1}=2q-1$, and using $|w^n|^{q_1-2} w^n  $ as a test
function in \eqref{e4.34}, we obtain
\begin{equation} \label{e3.33}
\frac{1}{q_1} \frac{d}{dt}\int_{\Omega}|w^{n}|^{q_1}dx
+ C\int_{\Omega}| w^{n}|^{q_1+q-1}dx\leq 0.
\end{equation}
Integrating \eqref{e3.33} from $s$ to $t+1(1\leq t \leq s < t +
1)$,  yields
\[
\int_\Omega |w^n(t+1)|^{q_1}dx \leq  \int_\Omega |w^n(s)|^{q_1}dx.
\]
Integrating the above inequality with respect to $s$ from $t$ to
$t + 1$ and using \eqref{e3.42}, we obtain
\begin{equation} \label{e3.44}
\int_\Omega |w^n(t+1)|^{q_1}dx \leq
 \int_t^{t+1}\int_\Omega|w^n(\tau)|^{q_1}\,dx\,d\tau \leq C,\quad
 \text{for any } t\geq1,
\end{equation}
with $C $  independent of $n,t$. Now integrating \eqref{e3.33}  on
$[t,t+1]$ for $t\geq2$, and using \eqref{e3.44} we have
$$
\int_t^{t+1}\int_{\Omega}| w^{n}(\tau)|^{q_1+q-1}dx\tau
\leq  C\int_{\Omega}| w^{n}(t)|^{q_1}dx\leq C.
$$
Bootstrapping the above processes, we can deduce that
\[
\int_{\Omega}|w^{n}(t)|^{q_k}dx\leq C, \quad
\text{for } t\geq T_k,
\]
with $q_k=q_{k-1}+q-1$, $q_0=q$, $C$ being independent of $n$.
 Passing to the limit, we obtain the same estimate for $w $.
Combining this estimate with the
convergence result obtained in Theorem \ref{thm3.1}, we obtain  that $w(t)$
converges to zero in $L^r(\Omega)$ for any $1\leq r<\infty$ as $t$
tends to infinity.
\end{remark}


\begin{remark} \label{rmk3.4} \rm
Although we performed all the calculations under the assumption that
$g\in L^1(\Omega)$, with very minor modifications, we can show that
all the results can be extended to the case 
$g\in L^1(\Omega)+W^{-1,p'(x)}(\Omega)$.

If we replace the $p(x)$-Laplacian operator $ -\operatorname{div}(|\nabla
u|^{p(x)-2}\nabla u)$ by a more general Leray-Lions type operator
involving variable exponent $-\operatorname{div}(a(x,\nabla v))$,  where
$a:\Omega\times \mathbb{R}^N\to \mathbb{R}^N$ is a
Carath\'{e}odory function (i.e. $a(x,\xi)$ is measurable on 
$\Omega$ for all $\xi \in \mathbb{R}^N $,  and $a(x,\xi)$ is continuous on
$\mathbb{R}^N $ for a.e. $x\in \Omega$) such that
\begin{gather*}
 a(x,\xi)\xi \geq \alpha |\xi|^{p(x)},\\
 |a(x,\xi)|\leq \beta[b(x)+|\xi|^{p(x)-1}],\\
 (a(x,\xi)-a(x,\eta))(\xi-\eta)>0,
\end{gather*}
for almost every $x\in \Omega$ and for all 
$\xi, \eta \in \mathbb{R}^N$ with $\xi\neq \eta$, $\alpha, \beta$ being positive
constants, $b(x)$ being a nonnegative function in 
$L^{ p(\cdot)/ p(\cdot)-1} (\Omega)$,
then the results obtained above still hold.
\end{remark}

\section*{Acknowledgments}
This work was partially supported by the NSFC (11301003), 
by the Research Fund for Doctor Station of the Education Ministry 
of China (20123401120005), and by the NSF of Anhui Province (1308085QA02).

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\end{document}