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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 209, pp. 1--15.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2016 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2016/209\hfil Parabolic equations with $p(x)$-growth]
{Weak solutions for parabolic equations with $p(x)$-growth}

\author[N. Pan, B. Zhang, J. Cao \hfil EJDE-2016/209\hfilneg]
{Ning Pan, Binlin Zhang, Jun Cao}

\address{Ning Pan \newline
Department of Mathematics,
Northeast Forestry University,
Harbin 150040, China}
\email{hljpning@163.com}

\address{Binlin Zhang (corresponding author)\newline
Department of Mathematics,
Heilongjiang Institute of Technology,
Harbin 150050, China}
\email{zhangbinlin2012@163.com}

\address{Jun Cao \newline
College of Mechanical and Electrical Engineering,
Northeast Forestry University,
Harbin 150040, China}
\email{zdhcaojun@163.com}

\thanks{Submitted  April 2, 2016. Published August 2, 2016.}
\subjclass[2010]{35K15, 35K20, 35K55}
\keywords{Parabolic equation; $W^{1,x}L^{p(x)}(Q)$ space; $p(x)$-growth condition}

\begin{abstract}
 In this article we  study nonlinear parabolic equations with
 $p(x)$-growth in the space $W^{1,x}L^{p(x)}(Q)\cap L^\infty(0,T; L^2(\Omega))$.
 By using the method of parabolic regularization, we prove the existence and
 uniqueness of weak solutions for the equation
 $$
 \frac{\partial u}{\partial t}=\operatorname{div}(a(u)
 |\nabla u|^{p(x)-2}\nabla u)+f(x,t).
 $$
 Also, we study the localization property of weak solutions for the
 above equation.
\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 and statement of main results}

Let $N\geq2$ be an integer and $\Omega$ be a bounded simply connected domain
in $\mathbb{R}^N$. Let $Q$  be $\Omega\times(0,T)$ where $T>0$ is given.
We consider the  parabolic initial boundary-value problem
\begin{equation}
\begin{gathered}
\frac{\partial u}{\partial t}=\operatorname{div}(a(u)
|\nabla u|^{p(x)-2}\nabla u)+f(x,t),\quad (x,t)\in Q,\\
u(x,t)=0, \quad  (x,t)\in \Gamma,\\
u(x,0)=u_0(x),\quad  x\in \Omega.
\end{gathered} \label{e1.1}
\end{equation}
where $\Gamma$ denotes the lateral boundary of  the  cylinder $Q$, and
$a(u)=u^\sigma+d_0$ with  $\sigma$ and $d_0$ two positive constants
to be defined later.

For the case   $p$ constant, there are many results about the existence,
uniqueness and the qualitative properties of the solutions,
we refer the reader to \cite{2, 4, 1, 3}.

In recent years, the research of variational problems
with nonstandard growth conditions has been an interesting topic,
see for examples \cite{7, 8, 13, 5, 19, 11, 17, 6, 12, 9, 16, 15, 18, 10}
and the references therein.
In \cite{5}, the authors studied the nonlinear parabolic equations with
nonstandard anisotropic growth conditions:
\begin{align}\label{eq1}
  u_t-\sum_i\frac{d}{dx_i}[a_i(z,u)|D_iu|^{p_i(z)-2}D_iu+b_i(z,u)]+d(z,u)=0
\end{align}
where $z=(x,t)$. They proved the existence and uniqueness of weak solutions
by applying Galerkin's method in the  Orlicz-Sobolev spaces $W(Q)$ with the
norm $\|u\|_{W(Q)}=\sum_i\|D_iu\|_{p_i(z),Q}+\|u\|_{2,Q}$.
 Note that the coefficient of nonlinearity in \cite{5} is allowed to depend
on $x$ and $t$ and is assumed to be the Caratheodory function, and so
problem \eqref{eq1} is called the evolutional $p(x, t)$-Laplacian.
In \cite{6}, the authors considered the quasilinear degenerate parabolic
problem with nonstandard growth:
\begin{equation} \label{eq2}
\begin{gathered}
\frac{\partial u}{\partial t}=\operatorname{div}(a(u)|\nabla u|^{p(x,t)-2}\nabla u)
+f(x,t),\quad (x,t)\in Q_T,\\
u(x,t)=0,\quad  (x,t)\in \Gamma_T,\\
u(x,0)=u_0(x),\quad x\in \Omega.
\end{gathered}
\end{equation}
and  studied the existence, uniqueness and localization property of weak solutions
for \eqref{eq2}. It is worthy pointing out that
they used the Banach spaces $L^{p(x,t)}(Q_T)$ and $W(Q_T)$ which appeared
in \cite{6} as solution space.
Indeed, many authors dedicated to studying the variable exponent problems,
in which $p(x,t)$ depends on $x$ and $t$, see for instance
\cite{5, 6, 13, 15}.
But for some important problems, the solution spaces only depending on variable
$x$ for parabolic equations are needed.
Note that $p(x)$-growth problems can be regarded as a kind of problems with
nonstandard growth, which appear in nonlinear elastic,
electrorheological fluids and other physics phenomena. For
a recent overview of variable exponent spaces with applications to nonlinear
partial differential equations we refer to \cite{9} and the references therein.

To illustrate the significance of variable exponent spaces independent of the
time variable $t$, we would like to mention a paper \cite{20},  which has been
an excellent reference as the applications of variable exponent spaces.
More precisely, the authors in \cite{20} studied the  Dirichlet problem
\begin{equation} \label{eq3}
\frac{\partial u}{\partial t}-\operatorname{div}(\phi_r(x,Du))
+\lambda(u-I)=0,\quad  (x,t)\in\Omega\times[0,T],
\end{equation}
which is a model for image denoising, enhancement, and restoration,
where $\lambda\geq0$ is a constant,
$$
\phi(x,r)=\begin{cases}
\frac{1}{q(x)}|r|^{q(x)},& |r|\leq\beta,\\
|r|-\frac{\beta q(x)-\beta^{q(x)}}{q(x)}, & |r|>\beta,
\end{cases}
$$
where $q(x)$ satisfies $1\leq q(x)\leq2$.
They proved the existence and uniqueness of weak solutions and also
discussed the behavior of weak solutions for \eqref{eq3} as $t\to \infty$.
Notice that the direction and speed of diffusion at each location
depend on the local behavior, hence $q(x)$ only depends on the location
$x$ in the image. Thanks to this fact, the authors gave the above model
which can study  the  denoising, enhancement, and restoration for the  image well.
Based on the above reason, we thus seek for a kind of space in which the
variable exponent only depend on $x$ for problem \eqref{eq1}.
Considering that the space $W^{1,x}L^{p(x)}(Q)$, which is different
from the space $W(Q_T)$ in \cite{5, 6},
can provide a suitable framework to
discuss the similar physical problems in \cite{20}, which was introduced
and discussed in \cite{21, 22}, so we take this space as our working
space to discuss the problem \eqref{eq1}, where $p(x)$ only depends on the
space variable $x$, not  on the time variable $t$.

In this article, we will  the existence, uniqueness and localization property
of solutions for \eqref{eq1} in the space $W^{1,x}L^{p(x)}(Q)$.
Throughout this paper, unless special statement, we always suppose
that the exponent $p(x)$ is continuous on $\overline{\Omega}$
with logarithmic module of continuity
\begin{gather}
1 < p^-=\inf_{x\in\Omega}p(x) \leq p(x)\leq \sup_{x\in\Omega} p(x)
=p^+<\infty.\label{e1.4}\\
\forall x\in\Omega,\; y\in\Omega, |x-y|<1, \quad
|p(x)-p(y)|\leq\omega(|x-y|),\label{e1.5}
\end{gather}
where
$$
\limsup_{\tau\to0^+}\omega(\tau)\ln\frac{1}{\tau}=C<+\infty.
$$
First we give the definition of  (weak) solutions for problem \eqref{e1.1}.

\begin{definition} \label{def1.1} \rm
A function $u(x,t)\in W^{1,x}L^{p(x)}(Q)\cap L^\infty(0,T; L^2(\Omega))$ is
called a (weak) solution of \eqref{e1.1} if
$$
-\int_Q u\frac{\partial \varphi}{\partial t}\,dx\,dt
+\int_\Omega u\varphi dx|^T_0
+\int_Q (u^\sigma+d_0)|\nabla u|^{p(x)-2}\nabla u\nabla \varphi \,dx\,dt
=\int_Q f(x,t)\varphi \,dx\,dt
$$
for all $\varphi\in C^1(0,T;C_0^\infty(\Omega))$.
\end{definition}

Now we are in a position to give  results about the existence and uniqueness
of solutions for problem \eqref{eq1}.

\begin{theorem}\label{thm1}
 Let $p(x)$ satisfy \eqref{e1.4}--\eqref{e1.5}. If the following conditions hold
\begin{itemize}
\item[(H1)] $\max\{1, \frac{2N}{N+2}\}<p^-<N$, $2\leq\sigma<\frac{2p^+}{p^+-1}$;
\item[(H2)] $u_0\geq0$,  $f\geq0$,
$\|u_0\|_{\infty,\Omega}+\int_0^T\|f(x,t)\|_{\infty, \Omega}dt
:=K(T)<\infty$,
\end{itemize}
then  \eqref{eq1} has at least one nontrivial weak solution in $W^{1,x}L^{p(x)}(Q)$.
\end{theorem}

\begin{theorem}\label{thm2}
 Suppose that the conditions in Theorem \ref{thm1} are fulfilled and
$2<\sigma<\frac{2p^+}{p^+-1}$, $p^+\geq 2$. Then the nonnegative solution of
\eqref{eq1} is unique within the class of all nonnegative weak solutions.
\end{theorem}

Let us define
$$
\operatorname{supp}  \omega
=\overline{\big\{x\in G: \lim_{\rho\to0}\frac{\operatorname{meas}
(G\cap B_\rho(x))}{\operatorname{meas} (B_\rho(x))}>0\big\}},
$$
where $G=\{x\in\Omega: \omega>0\}$, $B_\rho(x)=\{y\in\Omega: |x-y|<\rho\}$.
Hence we can present the localization property of solutions.


\begin{theorem}\label{thm3}
 Assume that the hypotheses of Theorem \ref{thm2} are satisfied and
$2<\sigma<\frac{2(p^+-p^-)}{p^-(p^+-1)}$, $\operatorname{supp}u_0\in\Omega$.
  If $u$ is a nonnegative solution of problem \eqref{eq1} and $f\equiv 0$,
  then $\operatorname{supp}u \subset \operatorname{supp}u_0$ a.e. in $Q$.
\end{theorem}

This paper is organized as follows. In Section 2, we shall introduce the
space $W^{m,x}L^{p(x)}(Q)$ and the necessary properties, which will be needed later.
Section 3 and Section 4 are devoted to proving the  existence and uniqueness
of solutions for  problem \eqref{eq1} respectively.
In Section 5, we will discuss the localization property of solutions
 to problem \eqref{eq1}.

\section{Preliminaries}

In this section  we recall the basic knowledge of the general spaces
$L^{p(x)}(\Omega)$, $W^{m,p(x)}(\Omega)$ and  $W^{m,x}L^{p(x)}(Q)$
and the necessary results which will be useful in the sequel,
we refer to \cite{21, 22, 23, 24} for more details.
Denote
$$
E=\{\omega: \omega  \text{ is a measurable function on }
\Omega\},
$$
where $\Omega\subset \mathbb{R}^N$ is an open subset.

Let $p(x):\Omega\to[1, \infty]$ be an element in  $E$.
Denote $\Omega_\infty=\{x\in\Omega: p(x)=\infty\}$. For $u\in E$, we
define
$$
\rho(u)=\int_{\Omega\setminus \Omega_\infty}
|u(x)|^{p(x)}dx+ \operatorname{ess, sup}_{x\in\Omega_\infty}|u(x)|.
$$
The space $L^{p(x)}(\Omega)=\{u\in E: \exists\lambda>0, \rho(\lambda u)<\infty\}$
endowed with the norm
$$
\|u\|_{L^{p(x)}(\Omega)}=\inf\{\lambda>0:
\rho(\frac{u}{\lambda})\leq1\}.
$$
We define the conjugate function $p'(x)$ of $p(x)$ by
$$
p'(x)=\begin{cases}
\infty, &\text{if }  p(x)=1;\\
1,&\text{if }  p(x)=\infty;\\
\frac{p(x)}{p(x)-1}, &\text{if } 1<p(x)<\infty.
\end{cases}
$$

\begin{lemma}[\cite{21}] \label{lem2.1}
(a) The dual space of $L^{p(x)}(\Omega)$ is $L^{p'(x)}(\Omega)$, if $1\leq
p(x)<\infty$.

(b) The space $L^{p(x)}(\Omega)$ is reflexive if and
only if \eqref{e1.4} is satisfied.
\end{lemma}

\begin{lemma}[\cite{21}] \label{lem2.2}
 If   $1\leq p(x)<\infty$, $C^\infty_0(\Omega)$ is dense in the space
$L^{p(x)}(\Omega)$
and $L^{p(x)}(\Omega)$ is separable.
\end{lemma}

\begin{lemma}[\cite{21}] \label{lem2.3}
If $1\leq p(x)\leq \infty$, for every $u(x)\in L^{p(x)}(\Omega)$ and
$v(x)\in L^{p'(x)}(\Omega)$, we have
$$
\int_\Omega |u(x)v(x)|dx\leq
C\|u(x)\|_{L^{p(x)}(\Omega)}\|v(x)\|_{L^{p'(x)}(\Omega)},
$$
where $C$ is
only dependent on $p(x)$ and $\Omega$, not dependent on
$u(x),v(x)$.
\end{lemma}


\begin{lemma}[\cite{21}] \label{lem2.4}
Let $1\leq p(x)< \infty$. The following conclusions hold:
\begin{enumerate}
\item  $\|u\|_{L^{p(x)}(\Omega)}<1$ $(=1,>1)$ $\Leftrightarrow \rho(u)<1(=1,>1)$.
\item If $\|u\|_{L^{p(x)}(\Omega)}\geq1$,
  then $\|u\|_{L^{p(x)}(\Omega)}^{p^-}\leq\rho(u)\leq\|u\|_{L^{p(x)}(\Omega)}^{p^+}$.

\item If $\|u\|_{L^{p(x)}(\Omega)}\leq1$, then
$\|u\|_{L^{p(x)}(\Omega)}^{p^+}\leq\rho(u)\leq\|u\|_{L^{p(x)}(\Omega)}^{p^-}$.
\end{enumerate}
\end{lemma}

Let $m>0$ be an integer. For each
$\alpha=(\alpha_1, \alpha_2,\cdots, \alpha_n)$, $\alpha_i$ are nonnegative
integers and $|\alpha|=\Sigma^n_{i=1}\alpha_i$, and denote by $D^\alpha$ the
distributional derivative  of order $\alpha$ with respect to the
variable $x$.

We now introduce  the generalized Lebesgue-Sobolev space
$W^{m,p(x)}(\Omega)$ which is defined as
$$
W^{m,p(x)}(\Omega)=\{u\in L^{p(x)}(\Omega): D^\alpha u\in L^{p(x)}(\Omega),
|\alpha|\leq m\}.
$$
Here $W^{m,p(x)}(\Omega)$ is a Banach space  endowed with the
norm
$$
\|u\|=\sum_{|\alpha|\leq m}\|D^\alpha u\|_{L^{p(x)}(\Omega)}.
$$
The space $W^{m,p(x)}_0(\Omega)$ is defined as the closure of
$C^\infty_0(\Omega)$ in $W^{m,p(x)}(\Omega)$. The dual space
$(W^{m,p(x)}_0(\Omega))^*$ is denoted by $W^{-m,p'(x)}(\Omega)$
equipped with the norm
$$
\|f\|_{W^{-m,p'(x)}(\Omega)}=\inf\Sigma_{|\alpha|\leq
m}\|f_\alpha\|_{L^{p'(x)}(\Omega)},
$$
where infimum is taken on all
possible decompositions
$$
f=\sum_{|\alpha|\leq m}(-1)^{|\alpha|}D^\alpha f_\alpha, \quad
 f_\alpha\in L^{p'(x)}(\Omega).
$$


\begin{lemma}[\cite{21}] \label{lem2.5}
(i) $W^{m,p(x)}(\Omega)$ and $W^{m,p(x)}_0(\Omega)$ are separable if
$1\leq p(x)<\infty$.

(ii) $W^{m,p(x)}(\Omega)$ and
$W^{m,p(x)}_0(\Omega)$ are reflexive if \eqref{e1.4} holds.
\end{lemma}


We define the space
$$
W^{m,x}L^{p(x)}(Q)=\{u\in L^{p(x)}(Q): D^\alpha u\in L^{p(x)}(Q),
|\alpha|\leq m\}.
$$
It is easy to see that $W^{m,x}L^{p(x)}(Q)$ is a Banach space with the norm
$\|u\|=\sum_{|\alpha|\leq m}\|D^\alpha u\|_{L^{p(x)}(Q)}$,
where $p(x)$ is independent of $t$, see \cite{19} for further discussions.
The space $W^{m,x}_0L^{p(x)}(Q)$ is defined as the closure of
$C^\infty_0(Q)$ in $W^{m,x}L^{p(x)}(Q)$ and
$W^{m,x}_0L^{p(x)}(Q)\hookrightarrow L^{p(x)}(Q)$ is continuous
embedding.  Let $\bar{M}$ be the number of multiindexes
$\alpha$ which satisfies $0\leq|\alpha|\leq m$, then the space
$W^{m,x}_0L^{p(x)}(Q)$ can be considered as a close subspace of the
product space $\Pi_{i=1}^{\bar{M}} L^{p(x)}(Q)$. So if
$1<p(x)<\infty$, $\Pi_{i=1}^{\bar{M}} L^{p(x)}(Q)$ is reflexive and
further we can get that the space $W^{m,x}_0L^{p(x)}(Q)$ is
reflexive. The dual space $(W^{m,x}_0L^{p(x)}(Q))^*$ is denoted by
$W^{-m,x}L^{p'(x)}(Q)$ equipped with the norm
$$
\|f\|_{W^{-m,x}L^{p'(x)}(Q)}=\sup_{\|u\|_{W_0^{m,x}L^{p(x)}(Q)}\leq1}|<f,u>|
=\inf\sum_{|\alpha|\leq m}\|f_\alpha\|_{L^{p'(x)}(Q)},
$$
where the infimum is taken on all possible
decompositions
$$
f=\sum_{|\alpha|\leq m}(-1)^{|\alpha|}D_x^\alpha f_\alpha, \quad
 f_\alpha\in L^{p'(x)}(Q).
$$
In what follows, we denote
$\|u(x,t)\|_{k,\Omega}=\big(\int_\Omega |u(x,t)|^k dx\big)^{1/k}$,
$\|u(x,t)\|_{\infty,Q}=\sup_{(x,t)\in Q}|u(x,t)|$.

\section{Existence of solutions}

Let us consider the auxiliary parabolic problem
\begin{equation}
\begin{gathered}
\frac{\partial u}{\partial t}=\operatorname{div}(a_{n,H}(u)
|\nabla u|^{p(x)-2}\nabla u)+f(x,t),\quad (x,t)\in Q,\\
u(x,t)=0,\quad  (x,t)\in \Gamma,\\
u(x,0)=u_0(x),\quad  x\in \Omega,
\end{gathered} \label{e3.1}
\end{equation}
here $H$ stands for a positive parameter to be chosen later and notice that
\[
0<d_0\leq a_{n,H}(u)=\Big(\min(|u|^2,H^2)
+\frac{1}{n^2}\Big)^{\sigma/2}+d_0\leq (H^2+1)^{\sigma/2}+d_0,\]
and $1<n<\infty$

Since $C_0^\infty(\Omega)$ is dense in $W_0^{1,p(x)}(\Omega)$, we may
construct the sequence of approximate solutions
$u_n(x,t)=\sum_{k=1}^n c_k^n(t)\varphi_k(x)$, and with similar arguments
as in \cite{5}, we obtain that problem \eqref{e3.1} has a weak solution
$u_n(x,t)\in  W_0^{1,x}L^{p(x)}(Q)\cap L^2(Q)$ satisfying the identity
\begin{equation}
\int_{t_1}^{t_2}\int_\Omega[u_{nt}\xi+a_{n,H}(u_n)|\nabla u_n|^{p(x)-2}
\nabla u_n \nabla\xi-f(x,t)\xi]\,dx\,dt=0,\label{e3.2}
\end{equation}
where $t_1<t_2\in (0,T)$.
To prove the main result, we need the following a priori estimates.

\begin{lemma} \label{lem3.1} 
The solution of \eqref{e3.1} satisfies the estimate
$$
\|u_n\|_{\infty,\Omega}\leq \|u_0\|_{\infty,\Omega}
+\int_0^T\|f(x,t)\|_{\infty,\Omega}dt=K(T)<\infty.
$$
\end{lemma}

\begin{proof}
First, we introduce the function
\[
u_{nH}=\begin{cases}
H, &\text{if }  u_n>H;\\
u_n,&\text{if }  |u_n|\leq H;\\
-H,&\text{if }  u_n<-H.
\end{cases}
\]
We choose the function $u_{nH}^{2k-1}$ as a test function in
\eqref{e3.2} with $k\in N$. In \eqref{e3.2}, let $t_2=t+h$, $t_1=t$, with
$t,t+h\in(0,T)$. Then
\begin{align*}
&\int_{t}^{t+h}\int_\Omega\Big[(u_{nH})_tu_{nH}^{2k-1}
 +a_{n,H}(u_{nH})|\nabla u_{nH}|^{p(x)-2}\nabla u_{nH} \nabla u_{nH}^{2k-1}\\
&-f(x,t)u_{nH}^{2k-1}\Big]\,dx\,dt=0,
\end{align*}
i.e.
\begin{equation}
\begin{aligned}
&\frac{1}{2k}\int_{t}^{t+h}\frac{d}{dt}\Big(\int_\Omega u_{nH}^{2k}dx\Big)dt\\
&+\int_{t}^{t+h}\int_\Omega (2k-1)a_{n,H}(u_{nH})u_{nH}^{2(k-1)}
 |\nabla u_{nH}|^{p(x)}\,dx\,dt \\
&=\int_{t}^{t+h}\int_\Omega f(x,t)u_{nH}^{2k-1}\,dx\,dt.
\end{aligned} \label{e3.3}
\end{equation}
Dividing  by $h$,  letting  $h\to0$, and applying Lebesgue's dominated
convergence theorem, we have that for all $t\in (0,T)$,
\begin{equation}
\begin{aligned}
&\frac{1}{2k}\frac{d}{dt}\int_\Omega u_{nH}^{2k}dx
+\int_\Omega (2k-1)a_{n,H}(u_{nH})u_{nH}^{2(k-1)}|\nabla u_{nH}|^{p(x)}dx
\\
&= \int_\Omega f(x,t)u_{nH}^{2k-1}dx.\label{e3.4}
\end{aligned}
\end{equation}
By Lemma \ref{lem2.3}, the right-hand side of the above equality can be rewritten as
$$
\big|\int_\Omega f(x,t)u_{nH}^{2k-1}dx\big|
\leq \|u_{nH}\|^{2k-1}_{2k,\Omega}\|f\|_{2k,\Omega},\quad  k=1,2,3,\dots,
$$
whence
\begin{equation}
\begin{aligned}
&\| u_{nH}\|^{2k-1}_{2k,\Omega}\frac{d}{dt}(\| u_{nH}\|_{2k,\Omega})+(2k-1)
\int_\Omega a_{n,H}(u_{nH})u_{nH}^{2(k-1)}|\nabla u_{nH}|^{p(x)}dx \\
&\leq \|u_{nH}\|^{2k-1}_{2k,\Omega}\|f\|_{2k,\Omega},\quad k=1,2,3,\dots
\end{aligned}\label{e3.5}
\end{equation}
Integrating over $(0,t)$ for the above inequality for all $t$, we obtain
 $$
\|u_{nH}(\cdot,t)\|_{2k,\Omega}\leq \|u_{nH}(\cdot,0)\|_{2k,\Omega}
+\int_0^T\|f\|_{2k,\Omega}dt,\quad \forall k\in N.
$$
 Letting $k\to\infty$, one gets
 $$
\|u_{nH}(\cdot,t)\|_{\infty,\Omega}
\leq \|u_{nH}(\cdot,0)\|_{\infty,\Omega}
+\int_0^T\|f\|_{\infty,\Omega}dt\leq \|u_{0}\|_{\infty,\Omega}
+\int_0^T\|f\|_{\infty,\Omega}dt.
$$
If we choose $H>K(T)$, then $u_{nH}(\cdot,t)\leq \sup |u_{nH}(\cdot,t)|\leq K(T)<H$,
and hence $u_{nH}(\cdot,t)=u_n(\cdot,t)$.
\end{proof}


\begin{remark} \label{rmk3.1} \rm
According to the above arguments, we obtain $u_{nH}(\cdot,t)=u_n(\cdot,t)$, and
 $$
\min\{u^2_n,H^2\}=u_n^2,\quad
 a_{n,H}(u_{nH})=a_{n,H}(u_{n})=\Big(\frac{1}{n^2}+u_n^2\Big)^{\sigma/2}+d_0.
$$
\end{remark}

\begin{corollary}\label{coro1}
If $u_0\geq0$ and $f\geq0$, then the solution $u_n(x,t)$ is nonnegative in $Q$.
\end{corollary}

\begin{proof}
Set $u_n^-=\min\{u_n, 0\}$, then we obtain $u_n^-(\cdot,0)=0$.
By Remark \ref{rmk3.1}, and let $k=1$ in \eqref{e3.4}, we obtain
$$
\frac{1}{2}\frac{d}{dt}(\|u_n^-(x,t)\|^2_{2,\Omega})
+\int_\Omega a_{n,H}(u_{n}^-)|\nabla u_{n}^-|^{p(x)}dx\leq 0,
$$
since $\int_\Omega f(x,t)u_{n}^{-}dx\leq 0$.
Then integrating over $(0,t)$ for the above inequality for all $t$, we obtain
$$
\|u_n^-(x,t)\|_{2,\Omega}\leq \|u_n^-(\cdot,0)\|_{2,\Omega}=0.
$$
Then the required assertion follows.
\end{proof}

\begin{remark} \label{rmk3.2} \rm
 From Corollary \ref{coro1} we know that the constructed weak solution
is nonnegative. But to our best knowledge, it
still remains unknown whether any solution of problem \eqref{eq1}
is nonnegative if the given data are nonnegative.
\end{remark}

\begin{lemma} \label{lem3.2}
The solution of \eqref{e3.1} satisfies the estimates
\begin{gather}
\int_{Q}u^\sigma_n|\nabla u_n|^{p(x)}\,dx\,dt
\leq H K(T)|\Omega|^{1/2},\label{e3.6}\\
\frac{1}{n^\sigma}\int_{Q}|\nabla u_n|^{p(x)}\,dx\,dt
\leq H K(T)|\Omega|^{1/2},\label{e3.7}\\
d_0\int_{Q}|\nabla u_n|^{p(x)}\,dx\,dt\leq H K(T)|\Omega|^{1/2}.
\label{e3.8}
\end{gather}
\end{lemma}

\begin{proof}
 We proceed as in the proof of Lemma \ref{lem3.1}.
Take $k=1$ in \eqref{e3.5}, it follows
$$
\frac{d}{dt}(\|u_{n}\|_{2,\Omega})+\frac{1}{\|u_{n}\|_{2,\Omega}}
\int_\Omega a_{n,H}(u_{n})|\nabla u_{n}|^{p(x)}dx \leq
\|f\|_{2,\Omega}.
$$
Furthermore, we integrate the above equation over $(0,t)$ for all
$t\in(0,T)$,
$$
\|u_{n}(\cdot,t)\|_{2,\Omega}+\frac{1}{\|u_{n}\|_{2,\Omega}}
\int_0^t\int_\Omega a_{n,H}(u_{n})|\nabla u_{n}|^{p(x)}\,dx\,dt
\leq \|u_{n}(\cdot,0)\|_{2,\Omega}+\int_0^t\|f\|_{2,\Omega}dt,
$$
i.e.
\begin{align*}
\int_0^t\int_\Omega a_{n,H}(u_{n})|\nabla u_{n}|^{p(x)}\,dx\,dt
&\leq\|u_{n}\|_{2,\Omega} (\|u_{n}(\cdot,0)\|_{2,\Omega}
 +\int_0^t\|f\|_{2,\Omega}dt) \\
& \leq H |\Omega|^{1/2}K(T).
\end{align*}
Since $a_{n,H}(u_{n})\geq d_0$, we obtain \eqref{e3.8}; since
 $a_{n,H}(u_{n})\geq\frac{ 1}{n^\sigma}$, we obtain \eqref{e3.7};
since $H>K(T)$, one gets $a_{n,H}(u_{n})\geq u_n^\sigma$,
hence we obtain \eqref{e3.6}.
\end{proof}

\begin{lemma} \label{lem3.3}
The solution of \eqref{e3.1} satisfies the estimate
$$
\|u_{nt}\|_{W^{-1,x}L^{p(x)}(Q)}\leq C(H,\sigma, p^\pm,K(T),|\Omega|).
$$
\end{lemma}

\begin{proof}
From \eqref{e3.2}, for $\xi\in W_0^{1,x}L^{p(x)}(Q)$  we have
\begin{align*}
&\int_Qu_{nt}\xi \,dx\,dt\\
&=-\int_Q\Big[\Big(u_n^2+\frac{1}{n^2}\Big)^{\sigma/2}
 +d_0\Big]|\nabla u_n|^{p(x)-2}\nabla u_n\nabla\xi \,dx\,dt+\int_Qf\xi \,dx\,dt\\
&\leq \int_Q\Big[\Big(u_n^2+\frac{1}{n^2}\Big)^{\sigma/2}
 +d_0\Big]|\nabla u_n|^{p(x)-1}|\nabla\xi| \,dx\,dt+\int_Q|f||\xi| \,dx\,dt\\
&\leq2\|[(u_n^2+\frac{1}{n^2})^{\sigma/2}+d_0]|\nabla u_n|^{p(x)-1}
 \|_{p'(x)}\|\nabla\xi\|_{p(x)} +2\|f\|_{p'(x)}\|\xi \|_{p(x)}\\
&\leq2\max \Big\{\Big(\int_Q \Big\{\Big[(u_n^2+\frac{1}{n^2})^{\sigma/2}
 +d_0\Big]|\nabla u_n|^{p(x)-1}\Big\}^{\frac{p(x)}{p(x)-1}}\,dx\,dt\Big)^{\frac{1}{p'^+}}, \\
&\Big(\int_Q\Big\{\Big[\Big(u_n^2+\frac{1}{n^2}\Big)^{\sigma/2}
 +d_0\Big]|\nabla u_n|^{p(x)-1}\Big\}^{\frac{p(x)}{p(x)-1}}\,dx\,dt
 \Big)^{\frac{1}{p'^-}}\Big\}\|\nabla\xi\|_{p(x)}\\
&+2\max\Big\{\Big(\int_Q|f|^{p'(x)}\,dx\,dt\Big)^{\frac{1}{p'^+}},
 \Big(\int_Q|f|^{p'(x)}\,dx\,dt\Big)^{\frac{1}{p'^-}}\Big\}\|\xi\|_{p(x)}\\
&\leq (2((K^2(T)+1)^{\sigma/2}+d_0)^{\frac{1}{p^{\pm}-1}}K(T)
 |\Omega|H+2|f|_\infty|T|)\|\xi\|_{W^{1,x}L^{p(x)}(Q)},
\end{align*}
which yields the desired conclusion.
\end{proof}

From the above conclusion and the uniform estimates in $n$, we obtain a
subsequence, still denoted $\{u_n\}_n$, such that
\begin{equation}
\begin{gathered}
u_n\to u \quad\text{a.e. in }   Q;\\
\nabla u_n\rightharpoonup \nabla u \quad \text{weakly in }
L^{p(x)}(Q);\\
u_n^\sigma|\nabla u_n|^{p(x)-2}D_iu_n\rightharpoonup A_i(x,t) \quad
\text{weakly in } L^{p'(x)}(Q);\\
|\nabla u_n|^{p(x)-2}D_iu_n\rightharpoonup W_i(x,t) \quad
\text{weakly in } L^{p'(x)}(Q),
\end{gathered}
\label{e3.9}
\end{equation}
for  $u\in W^{1,x}L^{p(x)}(Q)$, $A_i(x,t)\in L^{p'(x)}(Q)$,
$W_i(x,t)\in L^{p'(x)}(Q)$.

\begin{lemma} \label{lemma3.3}
 For almost all $(x,t)\in Q$,
$$
\lim_{n\to \infty}\int_{Q}\Big(\Big(u_n^2+\frac{1}{n^2}\Big)^{\sigma/2}
-u^\sigma_n\Big)|\nabla u_n|^{p(x)-2}\nabla u_n\nabla \xi \,dx\,dt=0,
\quad \forall\xi\in W_0^{1,x}L^{p(x)}(Q).
$$
\end{lemma}

\begin{proof}
By Young's inequality, we have
\begin{align*}
I &:= \int_Q\Big(\Big(u_n^2+\frac{1}{n^2}\Big)^{\sigma/2}
-u^\sigma_n\Big)|\nabla u_n|^{p(x)-2}\nabla u_n\nabla \xi \,dx\,dt\\
&=\frac{\sigma}{2}\frac{1}{n^2}\int_Q\Big(\int_0^1
 \Big(u_n^2+s\frac{1}{n^2}\Big)^{\frac{\sigma-2}{2}}ds\Big)
 |\nabla u_n|^{p(x)-2}\nabla u_n\nabla \xi \,dx\,dt\\
&\leq \sigma \frac{1}{n^2}\Big(K^2(T)+1\Big)^{\frac{\sigma-2}{2}}\|
|\nabla u_n|^{p(x)-1}\|_{p'(x)}\|\nabla \xi\|_{p(x)}\\
&\leq C \frac{1}{n^2}\Big\{\Big(\int_Q|\nabla u_n|^{p(x)}\,dx\,dt
\Big)^{\frac{p^+-1}{p^+}},\Big(\int_Q|\nabla u_n|^{p(x)}\,dx\,dt
\Big)^{\frac{p^--1}{p^-}}\Big\}\|\nabla \xi\|_{p(x)}.
\end{align*}
By \eqref{e3.7}, we obtain
$$
I\leq C H \big(\frac{1}{n}\big)^{2-\sigma\frac{p^+-1}{p^+}}\|\nabla\xi\|_{p(x)}.
$$
Letting $n\to\infty$, we obtain the desired conclusion.
\end{proof}

\begin{lemma} \label{lemma3.4}
 For almost all $(x,t)\in Q$,
$$
A_i(x,t)=u^\sigma W_i(x,t),\quad i=1,2,\dots , N.
$$
 \end{lemma}

\begin{proof}
In \eqref{e3.9}, letting $n\to\infty$, we have
\begin{gather}
\int_Qu_n^\sigma|\nabla u_n|^{p(x)-2}\nabla u_n\nabla \xi \,dx\,dt
\to \sum_{i=1}^N \int_QA_i(x,t)D_i\xi \,dx\,dt;\label{e3.10} \\
\int_Q|\nabla u_n|^{p(x)-2}\nabla u_n\nabla \xi \,dx\,dt\to \sum_{i=1}^N
\int_QW_i(x,t)D_i\xi \,dx\,dt.\label{e3.11}
\end{gather}
By Lebesgue's dominated convergence theorem we have
\begin{equation}
\lim_{n\to\infty}\sum_{i=1}^N\int_Q(u_n^\sigma-u^\sigma)A_i(x,t)D_i\xi \,dx\,dt=0.
\label{e3.12}
\end{equation}
From \eqref{e3.9} it follows that
\begin{align*}
&\lim_{n\to\infty}\sum_{i=1}^N\int_Q[u_n^\sigma|\nabla u_n|^{p(x)-2}
D_iu_n-u^\sigma W_i(x,t)]D_i\xi \,dx\,dt \\
&=\lim_{n\to\infty}\sum_{i=1}^N\int_Q[(u_n^\sigma-u^\sigma)
 |\nabla u_n|^{p(x)-2}D_iu_n \\
&\quad +u^\sigma(|\nabla u_n|^{p(x)-2}D_iu_n-W_i(x,t))]D_i\xi \,dx\,dt =0.
\end{align*}
By \eqref{e3.10}--\eqref{e3.12} and the above equalities, we complete the
proof.
\end{proof}


\begin{lemma} \label{lem3.6}
 For almost all $(x,t)\in Q$,
$$
W_i(x,t)=|\nabla u|^{p(x)-2} D_i(u),\quad i=1,2,\dots N.
$$
\end{lemma}
\begin{proof}
In \eqref{e3.2}, choosing $\xi=(u_n-u)\Phi$ with $\Phi\in W_0^{1,x}L^{p(x)}(Q)$,
$\Phi\geq0$, we have
\begin{align*}
&\int_Q[u_{nt}(u_n-u)\Phi+\Phi(u_n^\sigma+d_0)|\nabla u_n|^{p(x)-2}
\nabla u_n\nabla(u_n-u)] \,dx\,dt  \\
+&\int_Q[(u_n-u)(u_n^\sigma+d_0)|\nabla u_n|^{p(x)-2}
\nabla u_n\nabla\Phi-f(x,t)(u_n-u)\Phi]\,dx\,dt \\
+&\int_Q ((u_n^2+\frac{1}{n^2})^{\sigma/2}-u_n^\sigma)
|\nabla u_n|^{p(x)-2}\nabla u_n\nabla\xi \,dx\,dt=0.
\end{align*}
It follows that
\begin{equation}
\int_Q\Phi(u_n^\sigma+d_0)|\nabla u_n|^{p(x)-2}\nabla u_n\nabla(u_n-u)] \,dx\,dt=0.
\label{e3.13}
\end{equation}
On the other hand, by the fact that $u_n,\ u\in L^\infty(Q)$ and
$|\nabla u|\in L^{p(x)}(Q)$, we have
\begin{gather}
\lim_{n\to\infty}\int_Q\Phi(u^\sigma+d_0)|\nabla u|^{p(x)-2}
\nabla u\nabla(u_n-u)\,dx\,dt=0.\label{e3.14} \\
\lim_{n\to\infty}\int_Q\Phi(u^\sigma_n-u^\sigma)
 |\nabla u|^{p(x)-2}\nabla u\nabla(u_n-u)\,dx\,dt=0.\label{e3.15}
\end{gather}
Note that
\begin{equation}
\begin{aligned}
0&\leq(|\nabla u|^{p(x)-2}\nabla u_n-|\nabla u|^{p(x)-2}\nabla u)\nabla(u_n-u) \\
&\leq\frac{1}{d_0}[(u_n^\sigma+d_0)|\nabla u_n|^{p(x)-2}
 \nabla u_n-(u_n^\sigma-u^\sigma)|\nabla u|^{p(x)-2}\nabla u]\nabla(u_n-u) \\
&\quad -\frac{1}{d_0}(u^\sigma+d_0)|\nabla u|^{p(x)-2}
 \nabla u\nabla (u_n-u).
\end{aligned}\label{e3.16}
\end{equation}
Bring \eqref{e3.13}--\eqref{e3.15} into \eqref{e3.16}, we obtain
$$
\lim_{n\to\infty}\int_Q \Phi(|\nabla u_n|^{p(x)-2}
\nabla u_n-|\nabla u|^{p(x)-2}\nabla u)\nabla(u_n-u)\,dx\,dt=0.
$$
The rest arguments are the same as those of \cite[Theorem 2.1]{25}.
Thus the existence of weak solutions for  problem \eqref{eq1} is obtained
by a standard limiting process.
\end{proof}

\section{Uniqueness of solutions}

In this section, we study the uniqueness of the solutions to  \eqref{e1.1}.
To obtain the main conclusion of this section, we need the following lemma.

\begin{lemma} \label{lem4.1}
Let $M(s)=|s|^{p(x)-2}s$, then for all $\xi, \eta \in \mathbb{R}^N$,
\begin{align*}
&(M(\xi)-M(\eta))(\xi-\eta)\\
&\geq \begin{cases}
2^{-p(x)}|\xi-\eta|^{p(x)}, &\text{if }  2\leq p(x)<\infty;\\
(p(x)-1)|\xi-\eta|^2(|\xi|^{p(x)}+|\eta|^{p(x)})^{\frac{p(x)-2}{p(x)}},
&\text{if }  1<p(x)<2.
\end{cases}
\end{align*}
\end{lemma}

Now we shall prove Theorem \ref{thm2} by contradiction.
Suppose $u(x,t)$ and $v(x,t)$ are two nonnegative weak solutions of problem
\eqref{e1.1} and there is a $\delta>0$ such that for some $0<\tau\leq T$,
$w=u-v>\delta$ on the set $\Omega_\delta=\Omega \cap \{x:w(x,t)>\delta\}$
and $\mu(\Omega_\delta)>0$. Let
$$
F_\varepsilon(\xi)=\begin{cases}
\frac{1}{\alpha-1}\varepsilon^{1-\alpha}-\frac{1}{\alpha-1}\xi^{1-\alpha},
&\text{if } \xi>\varepsilon;\\
0,&\text{if } \xi\leq\varepsilon.
\end{cases}
$$
where $\delta>2\varepsilon>0$ and $\alpha=\sigma/2$.

By the definition of weak solution, we take a test-function $\xi=F_\varepsilon(w)$,
\begin{equation}
\begin{aligned}
0&=\int_{Q_\tau}[w_tF_\varepsilon(w)+(v^\sigma+d_0)(|\nabla u|^{p(x)-2}
 \nabla u-|\nabla v|^{p(x)-2}\nabla v)\nabla F_\varepsilon(w)]\,dx\,dt \\
&\quad +\int_{Q_\tau}(u^\sigma-v^\sigma)|\nabla u|^{p(x)-2}
 \nabla u\nabla F_\varepsilon(w)\,dx\,dt \\
&=\int_{Q_{\varepsilon,\tau}}w_tF_\varepsilon(w)\,dx\,dt \\
&\quad +\int_{Q_{\varepsilon,\tau}}(v^\sigma+d_0)
 w^{-\alpha}(|\nabla u|^{p(x)-2}\nabla u-|\nabla v|^{p(x)-2}\nabla v)
 \nabla w\,dx\,dt \\
&\quad +\int_{Q_{\varepsilon,\tau}}(u^\sigma-v^\sigma)w^{-\alpha}
 |\nabla u|^{p(x)-2}\nabla u\nabla w\,dx\,dt \\
&=J_1+J_2+J_3,
\end{aligned} \label{e4.1}
\end{equation}
with $Q_{\varepsilon,\tau}=Q_\tau\cap\{(x,t)\in Q_\tau:w>\varepsilon\}$.

Now, let $t_0=\inf\{t\in(0,\tau]:w>\varepsilon\}$, then we estimate
 $J_1$, $J_2$, $J_3$.
\begin{equation}
\begin{aligned}
J_1&=\int_{Q_{\varepsilon,\tau}}w_tF_\varepsilon(w)\,dx\,dt\\
&=\int_{\Omega}\Big(\int_0^{t_0}w_tF_\varepsilon(w)dt
+\int_{t_0}^{\tau}w_tF_\varepsilon(w)dt\Big)dx \\
&\geq \int_\Omega\int_{\varepsilon}^{w(x,\tau)}F_\varepsilon(s)\,ds\,dx \\
&\geq \int_{\Omega_\delta}\int_\varepsilon^{w(x,\tau)}F_\varepsilon(s)\,ds\,dx\\
&\geq\int_{\Omega_\delta}
(w-2\varepsilon)F_\varepsilon(\varepsilon)dx \\
&\geq(\delta-2\varepsilon)F_\varepsilon(\varepsilon)\mu(\Omega_\delta),
\end{aligned}\label{e4.2}
\end{equation}
Let us first consider the case $p^-\geq 2$. By  the first inequality of
Lemma \ref{lem4.1}, we obtain
\begin{equation}
\begin{aligned}
J_2&=\int_{Q_{\varepsilon,\tau}}(v^\sigma+d_0)w^{-\alpha}(|\nabla u|^{p(x)-2}
\nabla u-|\nabla v|^{p(x)-2}\nabla v)\nabla w\,dx\,dt \\
&\geq \int_{Q_{\varepsilon,\tau}}(v^\sigma+d_0)w^{-\alpha}2^{-p(x)}
|\nabla w|^{p(x)}\,dx\,dt \\
&\geq 2^{-p^+}\int_{Q_{\varepsilon,\tau}}(v^\sigma+d_0)w^{-\alpha}
|\nabla w|^{p(x)}\,dx\,dt\geq 0,
\end{aligned} \label{e4.3}
\end{equation}

Noting that $\frac{p(x)}{p(x)-1}\geq \frac{p^+}{p^+-1}=\alpha>1$ and
applying Young's inequality, we estimate integrand of $J_3$ in the following way
\begin{equation}
\begin{aligned}
&|(u^\sigma-v^\sigma)w^{-\alpha}|\nabla u|^{p(x)-2}\nabla u\nabla w| \\
&=\big|\sigma w\int_0^1(\theta u+(1-\theta)v)^{\sigma-1}
d\theta w^{-\alpha}|\nabla u|^{p(x)-2}\nabla u\nabla w\big| \\
&\leq \frac{C}{W^\alpha}\Big[\frac{v^\sigma+d_0}{C}|\nabla w|^{p(x)}
+C_1(\sigma,d_0,K(T),p^\pm)|w|^{p'(x)}|\nabla u|^{p(x)}\Big] \\
&\leq \frac{v^\sigma+d_0}{2^{p^++1}w^\alpha}|\nabla w|^{p(x)}
+C_1(\sigma,d_0,K(T),p^\pm)|w|^{p'(x)-\alpha}|\nabla u|^{p(x)} \\
&\leq \frac{v^\sigma+d_0}{2^{p^++1}w^\alpha}|\nabla w|^{p(x)}
+C_1(\sigma,d_0,K(T),p^\pm)|\nabla u|^{p(x)}.
\end{aligned}\label{e4.4}
\end{equation}
Substituting \eqref{e4.4} into $J_3$, we obtain
\begin{equation}
J_3\leq \frac{1}{2}J_2+C\int_{Q_{\varepsilon,\tau}}|\nabla u|^{p(x)}\,dx\,dt.
\label{e4.5}
\end{equation}

Next we consider the case $1<p^-<p(x)<2$, $p^+>2$.
 According to the second inequality of Lemma \ref{lem4.1}, it is easy to see that
the following inequalities hold
\begin{equation}
\begin{aligned}
J_2&=\int_{Q_{\varepsilon,\tau}}(v^\sigma+d_0)w^{-\alpha}
 (|\nabla u|^{p(x)-2}\nabla u-|\nabla v|^{p(x)-2}\nabla v)\nabla w\,dx\,dt \\
&\geq  (p^--1)\int_{Q_{\varepsilon,\tau}}(v^\sigma+d_0)
 w^{-\alpha}(|\nabla u|+|\nabla v|)^{p(x)-2}|\nabla w|^{2}\,dx\,dt\geq 0.
\end{aligned}\label{e4.6}
\end{equation}
Using Young's inequality and the fact that $1<\alpha\leq\frac{p^+}{p^+-1}\leq 2$,
we evaluate integrand of $J_3$ as follows:
\begin{equation}
\begin{aligned}
&|(u^\sigma-v^\sigma)w^{-\alpha}|\nabla u|^{p(x)-2}\nabla u\nabla w| \\
&= \big|\sigma w\int_0^1(\theta u+(1-\theta)v)^{\sigma-1}
 d\theta w^{-\alpha}|\nabla u|^{p(x)-2}\nabla u\nabla w\big| \\
&\leq \frac{(v^\sigma+d_0)(p^--1)}{2w^\alpha}(|\nabla u|+|\nabla v|)^{p(x)-2}
|\nabla w|^2 \\
&\quad +C_1(\sigma,d_0,K(T),p^\pm)|w|^{2-\alpha}(|\nabla u|+|\nabla v|)^{p(x)} \\
&\leq \frac{(v^\sigma+d_0)(p^--1)}{2w^\alpha}(|\nabla u|+|\nabla v|)^{p(x)-2}
 |\nabla w|^2 \\
&\quad +C_1(\sigma,d_0,K(T),p^\pm)(|\nabla u|+|\nabla v|)^{p(x)}.
\end{aligned}\label{e4.7}
\end{equation}
Inserting \eqref{e4.7}  into $J_3$, we obtain
$$
J_3\leq \frac{1}{2}J_2+C\int_{Q_{\varepsilon,\tau}}
(|\nabla u|+|\nabla v|)^{p(x)}\,dx\,dt.
$$
Plugging the estimates \eqref{e4.2}, \eqref{e4.3}, \eqref{e4.5}
and \eqref{e4.2}, \eqref{e4.6}, \eqref{e4.7} into \eqref{e4.1} and
dropping the nonnegative terms, we arrive at the inequality
$$
(\delta-2\varepsilon)(1-2^{1-\alpha})\varepsilon^{1-\alpha}
\mu(\Omega_\delta)\leq\tilde{C},
$$
with a constant $\tilde{C}$ independent of $\varepsilon$.

Notice that
$\lim_{\varepsilon\to 0}(\delta-2\varepsilon)(1-2^{1-\alpha})
\varepsilon^{1-\alpha}\mu(\Omega_\delta)=+\infty$, we obtain a contradiction.
This means $\mu(\Omega_\delta)=0$ and $w\leq 0$, a.e. in $Q_\tau$.
Thus the proof is complete.



\section{Localization property of solutions}

In this section, we shall focus on the study of localization of solutions
to problem \eqref{e1.1}. The proof is similar to that of
\cite[Theorem 4.1]{6}, we would like to give the detailed treatment,
just for the reader's convenience.
In fact, by Definition \ref{def1.1}, it follows easily that
\begin{equation} \label{e5.1}
\int_{Q}u_\tau\xi+(u^\sigma+d_0)|\nabla u|^{p(x)-2}\nabla u\nabla \xi=0,
\end{equation}
with $\tau\in(0,T)$.
Let
$$
\Psi=\inf\{\text{dist}(x, \operatorname{supp}\ u_0\cup\partial\Omega)/\lambda, 1\},
$$
where $0<\lambda<1$, and $F_\varepsilon(\xi)$ is mentioned in Section 4
with $\alpha=\sigma /2$.
Taking $\xi=\Psi F_\varepsilon(u) (0<\varepsilon<1)$ and substituting
it into \eqref{e5.1}, we obtain
\begin{equation}
\begin{aligned}
0&=\int_{Q_{\varepsilon,\tau}}u_t\Psi F_\varepsilon(u)\,dx\,dt
+\int_{Q_{\varepsilon,\tau}}\Psi(u^\sigma+d_0)|\nabla u|^{p(x)-2}
\nabla u\nabla F_\varepsilon(u)\,dx\,dt\\ &+\int_{Q_{\varepsilon,\tau}}
F_\varepsilon(u)(u^\sigma+d_0)|\nabla u|^{p(x)-2}\nabla u\nabla\Psi \,dx\,dt
:= I_1+I_2+I_3.
\end{aligned}\label{e5.2}
\end{equation}
with $Q_{\varepsilon,\tau}=Q_\tau\cap\{(x,t)\in Q_\tau: u>\varepsilon\}$.
Denote $E=\{x\in\{\Psi=1\}:u(x,\tau)>\delta\}$ with
$\delta>2\varepsilon>0$, then
\begin{equation}
\begin{aligned}
I_1
&=\int_{Q_{\varepsilon,\tau}} u_t\Psi F_\varepsilon(u)\,dx\,dt
\geq\int_{\Omega_\varepsilon}\chi_{\operatorname{supp}\Psi}
 \Psi\int^u_\varepsilon F_\varepsilon(s)\,ds\,dx\\
&\geq \int_{\Omega_\varepsilon}\chi_{\operatorname{supp}\Psi}\Psi(u-\varepsilon)
F_\varepsilon(\delta)dx \\
&\geq \big(\delta-\frac{3}{2}\varepsilon\big)F_\varepsilon
\big(\frac{3}{2}\varepsilon\big)\operatorname{meas}(E).
\end{aligned}\label{e5.3}
\end{equation}
and
\begin{equation}
\begin{aligned}
I_2&=\int_{Q_{\varepsilon,\tau}}\Psi(u^\sigma+d_0)|\nabla u|^{p(x)-2}
\nabla u\nabla \frac{1}{\alpha-1}(-u^{1-\alpha})\,dx\,dt\\
&\geq \int_{Q_{\varepsilon,\tau}}\Psi(u^\sigma+d_0)|\nabla u|^{p(x)}
u^{-\alpha}\,dx\,dt\geq0.
\end{aligned} \label{e5.4}
\end{equation}
 Applying Young's inequality with $\eta$ and choosing
$\eta=(\varepsilon^\beta)^{1-p(x)}$, we may estimate that
\begin{equation}
\begin{aligned}
|I_3|&=\big|\int_{Q_{\varepsilon,\tau}}F_\varepsilon(u)(u^\sigma+d_0)
 |\nabla u|^{p(x)-2}\nabla u\nabla\Psi \,dx\,dt\big| \\
&\leq C \int_{Q_{\varepsilon,\tau}}\varepsilon^{1-\alpha}|\nabla u|^{p(x)-1}
 |\nabla\Psi|\,dx\,dt\\
&\leq C(\sigma,d_0,K(T),p^\pm)\varepsilon^{\beta+\frac{(1-\alpha)p^-}{p^--1}}
 \int_{Q_{\varepsilon,\tau}}|\nabla u|^{p(x)}\,dx\,dt \\
&\quad  +\varepsilon^{\beta(1-p^+)}\int_{Q_{\varepsilon,\tau}}|\nabla \Psi|^{p(x)}\,dx\,dt,
\end{aligned} \label{e5.5}
\end{equation}
where $C>0$ denote the various constants.
Choosing $\beta=\frac{\alpha p^--1}{p^--1}>0$ and putting
\eqref{e5.3}--\eqref{e5.5} into \eqref{e5.2}, we deduce
\begin{equation}
\begin{aligned}
&\frac{1}{2}[1-(3/2)^{1-\alpha}]
 \varepsilon^{2-\alpha-\beta+\frac{(\alpha-1)p^-}{p^--1}}\operatorname{meas}(E)\\
&\leq (\delta-3\varepsilon/2)[1-(3/2)^{1-\alpha}]\varepsilon^{1-\alpha-\beta
 +\frac{(\alpha-1)p^-}{p^--1}}\operatorname{meas}(E)\\
&\leq \widetilde{C}\Big(1+\varepsilon^{\frac{(\alpha-1)p^-}{p^--1}
-\beta p^+}\Big),
\end{aligned}\label{e5.6}
\end{equation}
with the positive constant $\widetilde{C}$ independent of $\varepsilon$.
Noticing that
$$
2<\sigma< \frac{2(p^+-p^-)}{p^-(p^+-1)}<\frac{2p^+}{p^+-1},
$$
we have
\begin{gather}
1<\alpha=\frac{\sigma}{2}<\frac{(p^+-p^-)}{p^-(p^+-1)},\quad
 1-\beta+\frac{(\alpha-1)p^-}{p^--1}=0;\label{e5.7} \\
\frac{(\alpha-1)p^-}{p^--1}-\beta p^+=\frac{(p^+-p^-)-\alpha p^-(p^+-1)}{p^--1}>0.
\label{e5.8}
\end{gather}
Assume that there exists the constant $\tau_0\in(0,T)$ such that
$\operatorname{meas}(E)\neq 0.$ Thus,  \eqref{e5.6}--\eqref{e5.8} yield a
contradiction. Hence, we have
\begin{equation} \label{eq5}
\operatorname{meas}\{x\in\{\Psi=1\}:u(x,\tau)>\delta\}=0, %\label{e5.9}
\end{equation}
for all $\delta\in(0,1)$ and  a.e. $\tau\in(0,T)$.
Then Theorem \ref{thm3} follows from \eqref{eq5} and the arbitrariness of $\lambda$.

\subsection*{Acknowledgments}
Ning Pan was supported by the Fundamental Research Funds for the Central
Universities (DL11BB40). Binlin Zhang was supported by Natural Science
Foundation of Heilongjiang Province of China (No. A201306)
and Research Foundation of Heilongjiang Educational Committee (No. 12541667)
and Doctoral Research Foundation of Heilongjiang Institute of
Technology (No. 2013BJ15).

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\end{thebibliography}

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