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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2017 (2017), No. 13, pp. 1--15.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2017 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2017/13\hfil Electrorheological fluid equations]
{Well-posedness of weak solutions to electrorheological fluid equations with
 degeneracy on the boundary}

\author[H. Zhan, J. Wen \hfil EJDE-2017/13\hfilneg]
{Huashui Zhan, Jie Wen}

\address{Huashui Zhan \newline
School of Applied Mathematics,
 Xiamen University of Technology,
 Xiamen, Fujian 361024, China}
\email{2012111007@xmut.edu.cn}

\address{Jie Wen \newline
School of Sciences,
Jimei University,
Xiamen, Fujian 361021, China}
\email{1195103523@qq.com}

\dedicatory{Communicated by Zhaosheng Feng}

\thanks{Submitted August 12, 2016. Published January 12, 2017.}
\subjclass[2010]{35L65, 35K85, 35R35}
\keywords{Electrorheological fluid  equation; boundary degeneracy; 
\hfill\break\indent H\"older's inequality; local stability}

\begin{abstract}
 In this article we study the electrorheological fluid  equation
 $$
 {u_t}= \operatorname{div} ({\rho^\alpha}{| {\nabla u} |^{p(x) - 2}}\nabla u),
 $$
 where $\rho (x) = \operatorname{dist} (x,\partial \Omega )$
 is the distance from the boundary, $p(x)\in C^{1}(\overline{\Omega})$, and
 $p^{-}=\min_{x\in \overline{\Omega}}p(x)>1$. We show how the degeneracy of
 $\rho^{\alpha}$ on the boundary affects the well-posedness of the weak solutions.
 In particular,  the local stability of the weak solutions is established without
 any boundary value condition.
\end{abstract}

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


\section{Introduction}

Let $\Omega\subset \mathbb{R}^{N}$ be a bounded domain with smooth
boundary $\partial \Omega$,  and $p(x)$ is a measurable function.
The evolutionary $p(x)$-Laplacian equation
\begin{equation}
u_{t} =\operatorname{div}(|\nabla u|^{p(x)-2}\nabla u),\quad (x,t) \in {Q_T}
= \Omega  \times (0,T),\label{e1.1}
\end{equation}
comes from a new interesting type of fluids
called electrorheological fluids \cite{AM,R}.
We consider an electromagnetic field with vector of magnetic density
$\vec{B}=(0,0,u(x,t))$, where $x=(x_1, x_2)\in\Omega\subset\mathbb{R}^2$.
Let $\vec{H}=(H_1, H_2, H_3)$ be a magnetic field intensity,
$\vec{J}=(J_1, J_2, J_3)$ be a current
density, $\vec{E}=(E_1, E_2,E_3)$ be an electrostatic field intensity and
$r$ be a resistivity. Review Maxwell's equations
\begin{gather}
\frac{\partial \vec{B}}{\partial t}+\operatorname{rot}\vec{E}=0, \label{e1.2}\\
\vec{J}\approx\operatorname{rot}\vec{H}, \label{e1.3}\\
\vec{B}=\lambda\vec{H}, \label{e1.4}\\
\vec{E}=r\vec{J}, \label{e1.5}
\end{gather}
where $\lambda>0$. By \eqref{e1.4}, we have $H_3=u/\lambda$. Therefore,
\begin{equation}
\begin{gathered}
J_1\approx\frac{\partial H_3}{\partial x_2}
 -\frac{\partial H_2}{\partial x_3}=\frac{1}{\lambda}\frac{\partial u}{\partial x_2},\\
J_2\approx\frac{\partial H_1}{\partial x_3}-\frac{\partial H_3}{\partial x_1}
=-\frac{1}{\lambda}\frac{\partial u}{\partial x_1}, \\
J_3\approx\frac{\partial H_2}{\partial x_1}-\frac{\partial H_1}{\partial x_2}=0.
\end{gathered}\label{e1.6}
\end{equation}
Now, if we suppose that ${r}=r_0|\vec{J}|^{q(x)}$, taking into account
 \eqref{e1.6} we know that
$$
|\vec{J}|=\sqrt{J_1^2+J_2^2+J_3^2}=\frac{1}{\lambda}|\nabla u|,
$$
where $r_0>0$ is a constant, $q(x)$ is a function which depends on the environment.
If
\begin{equation}
a(x)=\frac{r_0}{\lambda^{q(x)+1}}>0,\label{e1.7}
\end{equation}
then
$$
\vec{E}=r\vec{J}=a(x)|\nabla u|^{q(x)}(\frac{\partial u}{\partial x_2},
 -\frac{\partial u}{\partial x_1},0),
$$
as a generalization of Ohm's law \eqref{e1.5}.
Hence the third coordinate of the vector  $\operatorname{rot}\vec{E}$ is
\begin{align*}
(\operatorname{rot}\vec{E})_3
&= \frac{\partial E_2}{\partial x_1}-\frac{\partial E_1}{\partial x_2} \\
&= \frac{\partial}{\partial x_1}(-a(x)|\nabla
u|^{q(x)}\frac{\partial u}{\partial x_1})
-\frac{\partial}{\partial x_2}(a(x)|\nabla u|^{q(x)}\frac{\partial u}{\partial x_2}) \\
& = -\operatorname{div}(a(x)|\nabla u|^{q(x)}\nabla u).
\end{align*}
Using \eqref{e1.2}, according to Mashiyev-Buhrii \cite{MB}, letting
$p(x)=q(x)+2$, we have
\begin{equation}
u_t-\operatorname{div}(a(x)|\nabla u|^{p(x)-2}\nabla u)=0,\quad
(x,t)\in Q_T,\label{e1.8}
\end{equation}
with the initial value
\begin{equation}
 u\big|_{t=0} = u_0(x),\quad x\in\Omega,\label{e1.9}
\end{equation}
and the homogeneous boundary value
\begin{equation}
u\big|_{\Gamma_{T}} = 0,\quad (x,t)\in \Gamma_{T}=\partial \Omega \times (0,T),
\label{e1.10}
\end{equation}
which have been researched widely recently, one can refer to
\cite{AS1,FZ,KR,KWZ}.

If $r_0=r(x)$ is a function,
then $a(x)$ in \eqref{e1.7} may be degenerate on the boundary.
 For example, if $r(x)\big|_{\Sigma_p}=0$, where $\Sigma_p\subseteq \partial \Omega$,
then the equation is degenerate on $\Sigma_p$. We will study the problem by taking a
special but basic formula of the diffusion function $a(x)=\rho^{\alpha}(x)$,
where $\rho(x)=\operatorname{dist}(x,\partial \Omega)$, and $\alpha>0$.
Then equation \eqref{e1.8} becomes
\begin{equation}
{u_t} = \operatorname{div} ({\rho^\alpha }{| {\nabla u} |^{p(x) - 2}}\nabla u)  ,
\quad (x,t) \in \Omega  \times (0,T).\label{e1.11}
\end{equation}
If $p(x)\equiv p$ , the above equation becomes
\begin{equation}
{u_t} = \operatorname{div}({| \rho^{\alpha}{\nabla u} |^{p - 2}}\nabla u),
\label{e1.12}
\end{equation}
which was first studied by Yin-Wang \cite{YW}.
 They had proved the following results:

\begin{theorem}\label{thm1.1}
Let $p>1$, and
\begin{equation}
{u_0} \in {L^\infty }(\Omega),\quad {\rho^\alpha }| {\nabla {u_0}} |^{p}
\in {L^1}(\Omega).\label{e1.13}
\end{equation}
 If $\alpha<p-1$, then there exists an unique solution of equation \eqref{e1.12}
with the initial-boundary conditions \eqref{e1.9}-\eqref{e1.10}.
While, if $\alpha\geq p-1$,  there exists an unique solution of  \eqref{e1.12}
only with the initial value \eqref{e1.9}. In other words, if $\alpha\geq p-1$,
the stability of the solutions  of  \eqref{e1.12} is true without any boundary
condition.
  \end{theorem}

Inspired by \cite{YW}, we studied  \eqref{e1.11} in a similar way as
the one described in \cite{ZW}, and obtain a similar theorem.

\begin{theorem}\label{thm1.2}
Let $p>1$, and
\begin{equation}
{u_0} \in {L^\infty }(\Omega),\quad {\rho^\alpha }| {\nabla {u_0}} |^{p(x)}
\in {L^1}(\Omega).\label{e1.14}
\end{equation}
If $\alpha<p^{-}-1$, then there exists an unique solution of \eqref{e1.11}
with the initial-boundary conditions \eqref{e1.9}-\eqref{e1.10}.
While, if $\alpha\geq  p^{+}-1$, then there exists an unique solution of
equation \eqref{e1.11} with the initial value \eqref{e1.9}.
\end{theorem}

We are interested in this problem because we would like to know how the
 degeneracy of the diffusion function $\rho^\alpha$  affects equation
 \eqref{e1.11} essentially. To see that,  we suppose that $u$ and $v$ are
two classical solutions of  \eqref{e1.11} with the initial  values $u(x,0)$
 and $v(x,0)$ respectively. Then
\begin{align*}
&\int_{\Omega}(u-v)(u-v)_{t}dx+\int_{\Omega} \rho^\alpha (| \nabla u |^{p(x) - 2}\nabla u - | \nabla v |^{p(x) - 2}\nabla v) \cdot
 \nabla (u-v)\, dx \\
&=\int_{\partial \Omega}\rho^\alpha (u-v)(| \nabla u |^{p(x) - 2}\nabla u
- | \nabla v |^{p(x) - 2}\nabla v)\cdot\vec{n}d\Sigma=0,
\end{align*}
where $\vec{n}$ is the outer unit normal vector of $\Omega$. So
$$
\frac{1}{2}\frac{d}{dt}\int_{\Omega}(u-v)^2dx\leq 0,
$$
\begin{equation}
\int_{\Omega}|u(x,t)-v(x,t)|^2dx\leq \int_{\Omega}|u_0(x)-v_0(x)|^2dx.\label{e1.15}
\end{equation}
This implies that the classical solutions (if there are) of equation \eqref{e1.11}
are stable without any boundary value condition, only if that $\alpha>0$.
Certainly,  since equation \eqref{e1.11} is degenerate on the boundary and may
be degenerate or singular at points where $|\nabla u|=0$, it only has a weak
solution generally,  so whether the inequality \eqref{e1.15} is true or not
remains to be verified.

Obviously, since $p(x)$ is a function, there exists a gap if
$p^{-}-1\leq\alpha <p^{+}-1$ in Theorem \ref{thm1.2}.  In our paper,
 roughly speaking, only if  $\alpha\geq p^{-}-1$, we can establish the stability
of the weak solutions of equation \eqref{e1.11} without any boundary value condition.
The conclusions not  only make a supplementary of the results of
\cite{YW,ZW,Z2},
but also provide a new and more effective way to establish the stability of
the solutions (see Theorems \ref{thm2.6} and \ref{thm2.7} below).

\section{Basic functional spaces and main results}\noindent

Throughout this article we assume that $1< p(x)\in C^{1}(\overline{\Omega})$,
 and denote
\[
 p^+ =\max_{\bar{\Omega}}p(x), \quad 1<p^{-}=\min_{\bar{\Omega}}p(x).
\]
First of all, we introduce some basic functional spaces.
The space
\[
L^{p(x)}(\Omega) =\{u: u \text{ is a measurable
real-valued function}, \int_{\Omega}|u(x)|^{p(x)} dx <\infty\}.
\]
is equipped with the  Luxemburg norm
$$
\|u\|_{L^{p(x)}}(\Omega) = \inf\{\lambda >0:
\int_{\Omega}\bigl|\frac{u(x)}{\lambda}\bigl|^{p(x)}dx\leq 1\}.
$$
The space $(L^{p(x)}(\Omega),\|\cdot\|_{L^{p(x)}(\Omega)})$ is a separable,
uniformly convex Banach space.

The space
$$
W^{1,p(x)}(\Omega) =\{ u \in L^{p(x)}(\Omega): |\nabla u|\in L^{p(x)}(\Omega)\}.
$$
is endowed with the norm
\[
\|u\|_{W^{1,p(x)}}=\|u\|_{L^{p(x)}(\Omega)} + \|\nabla u\|_{L^{p(x)}(\Omega)},\quad
\forall u\in W^{1,p(x)}(\Omega).
\]
We use $W_{0}^{1,p(x)}(\Omega)$ to denote the closure of
$C^{\infty}_{0}(\Omega)$ in $W^{1,p(x)}$.
Some properties
of the function spaces $W^{1,p(x)}(\Omega)$ are quoted in the following lemma.

\begin{lemma}\label{lem2.1}
\begin{itemize}
\item[(i)] The spaces $(L^{p(x)}(\Omega), \|\cdot \|_{L^{p(x)}(\Omega)})$,
$(W^{1,p(x)}(\Omega), \|\cdot \|_{W^{1,p(x)}(\Omega)})$ and
$W^{1,p(x)}_{0}(\Omega)$ are reflexive Banach spaces.

\item[(ii)] $p(x)$-H\"{o}lder's inequality.
Let $q_{1}(x)$ and $q_{2}(x)$ be real functions with
$\frac{1}{q_1(x)}+\frac{1}{q_2(x)} = 1$ and $q_1(x) > 1$.
Then, the conjugate space of
$L^{q_1(x)}(\Omega)$ is $L^{q_2(x)}(\Omega)$. And for any
$u \in L^{q_1(x)}(\Omega)$ and $ v \in L^{q_2(x)}(\Omega)$, we have
$$
\bigl|\int_{\Omega}uv dx\bigl|\leq 2\|u\|_{L^{q_1(x)}(\Omega)}
\|v\|_{L^{q_2(x)}(\Omega)}.
$$

\item[(iii)]
\begin{gather*}
 \|u\|_{L^{p(x)}(\Omega)} = 1 \implies \int_{\Omega}|u|^{p(x)} dx = 1,\\
 \|u\|_{L^{p(x)}}(\Omega) > 1 \implies
  |u|^{p^{-}}_{L^{p(x)}}\leq\int_{\Omega}|u|^{p(x)} dx\leq|u|^{p^{+}}_{L^{p(x)}},\\
\|u\|_{L^{p(x)}}(\Omega) < 1 \implies  \ |u|^{p^{+}}_{L^{p(x)}}
\leq\int_{\Omega}|u|^{p(x)} dx\leq|u|^{p^{-}}_{L^{p(x)}}.
\end{gather*}

\item[(iv)] If $p_{1}(x)\leq p_{2}(x)$, then
$L^{p_{1}(x)}(\Omega)\supset L^{p_{2}(x)}(\Omega)$.

\item[(v)] If $p_{1}(x)\leq p_{2}(x)$, then
\[
W^{1,p_{2}(x)}(\Omega)\hookrightarrow W^{1,p_{1}(x)}(\Omega).
\]

\item[(vi)] $p(x)$-Poincar\'{e}s inequality. If $p(x)\in C(\Omega)$,
then there is a constant $C >0$, such that
$$
\|u\|_{L^{p(x)}}(\Omega) \leq C\|\nabla u\|_{L^{p(x)}(\Omega)},\quad
\forall u\in W^{1,p(x)}_{0}(\Omega).
$$
This implies that $\|\nabla u\|_{L^{p(x)}}(\Omega)$  and
$\|u\|_{W^{1,p(x)}(\Omega)}$ are equivalent norms of $W^{1,p(x)}_{0}$.
\end{itemize}
\end{lemma}

 Zhikov \cite{Z1} showed that
\[
W^{1,p(x)}_{0} (\Omega)
\neq \{v\in W^{1,p(x)}_{0} (\Omega)|v|_{\partial \Omega} = 0\}
={\mathaccent"7017 W}^{1,p(x)}(\Omega).
\]
Hence, the property of the space is
different from the case when $p$ is a constant. This fact gives
a general idea used in studying the well-posedness of the solutions to
the evolutionary $p$-Laplacian
equation which can not be used directly.

 If the exponent $p(x)$ is required to satisfy logarithmic H\"{o}lder
continuity condition
$$
|p(x)-p(y)|\leq \omega(|x-y|), \quad \forall x,y\in \Omega,\; |x-y|<\frac{1}{2},
$$
with
$$
\limsup_{s \to {0^+ }} \omega (s)\ln (\frac{1} {s}) = C < \infty,
$$
then
$W_{0}^{1,p(x)}(\Omega)= \mathring{W} ^{1,p(x)}(\Omega)$.
In fact Antontsev-Shmarev \cite{AS1} established the well-posedness of equation \eqref{e1.8}.

Now, we introduce some other Banach spaces used to define the weak solution
of the equation. For every fixed $t\in[0, T]$, we define
\begin{gather*}
V_t(\Omega) =\{u(x) : u(x)\in L^2(\Omega)\cap W^{1,1}_0(\Omega),
|\nabla u(x)|^{p(x)}\in L^1 (\Omega)\}, \\
\|u\|_{V_t(\Omega)} = \|u\|_{2,\Omega} + \|\nabla u\|_{p(x),\Omega} ,
\end{gather*}
and denote by $V'_t(\Omega)$ its dual, where
$\|u\|_{2,\Omega}=\|u\|_{L^2(\Omega)}$,
$\|\nabla u\|_{p(x),\Omega} =\|\nabla u\|_{L^{p(x)}(\Omega)}$.
Also we use the Banach space
 \begin{gather*}
W(Q_T) = \big\{u : [0,T]\to V_t(\Omega)|u\in L^2(Q_T),
|\nabla u|^{p(x)} \in L^1(Q_T), u = 0\text{ on } \Gamma_T\big\}, \\
\|u\|_{W(Q_T)} = \|\nabla u\|_{p(x),Q_T} + \|u\|_{2,Q_T} .
\end{gather*}
The space $W'(Q_T)$ is the dual of $W(Q_T)$
(the space of linear functionals over $W(Q_T)$):
$ w\in W'(Q_T)$ if and only if
\begin{gather*}
w=w_0+\sum_{i=1}^{n}D_iw_i,\;  w_0\in L^2(Q_T),\;  w_i\in L^{p'(x,t)}(Q_T),\\
\forall\phi\in W(Q_T),\langle\langle w,\phi
\rangle\rangle =\iint_{Q_T}\Big(w_0\phi+\sum_{i}w_iD_i\phi\Big)\,dx\,dt.
\end{gather*}
The norm in $W'(Q_T)$ is defined by
$$
\|v\|_{W'(Q_T)}= \sup\{\langle\langle v,\phi\rangle\rangle :
\phi\in  W(Q_T),\|\phi\|_{W(Q_T)}\leq 1\}.
$$

\begin{definition}\label{def2.2}\rm
 A function $u(x,t)$ is said to be a weak solution of \eqref{e1.11} with the initial
value \eqref{e1.9}, if
\begin{equation}
u \in L^{\infty}(Q_T),\ u_{t}\in W'(Q_T),\quad
 {\rho ^\alpha }{| {\nabla u} |^{p(x)}} \in {L^1}({Q_T}),\label{e2.1}
\end{equation}
and for any function $\varphi  \in L^\infty (0,T; W_0^{1,p(x)}(\Omega))
\cap W(Q_T)$,  it holds
\begin{equation}
\langle\langle u_t,\varphi\rangle\rangle
+\iint_{{Q_T}} ( \rho^\alpha| {\nabla u} |^{p(x) - 2}
\nabla u \cdot \nabla \varphi )\,dx\,dt= 0.\label{e2.2}
\end{equation}
The initial value, as usual,  is satisfied in the sense of that
\begin{equation}
\lim_{t\to 0}\int_{\Omega}u(x,t)\phi(x) dx
=\int_{\Omega}u_0(x)\phi(x) dx,\forall \phi(x)\in C_0^{\infty}(\Omega).\label{e2.3}
\end{equation}
\end{definition}

The main result of this article is stated as follows.

\begin{theorem}\label{thm2.3}
Let $1<p^{-}$, $0<\alpha$. If
\begin{equation}
 u_0(x)\in L^{\infty}(\Omega),\quad
\rho^{\alpha}|\nabla u_0|^{p(x)} \in L^1(\Omega),\label{e2.4}
\end{equation}
then  \eqref{e1.11} with  initial value \eqref{e1.9} has
a weak solution  $u$ in the sense of Definition \ref{def2.2}.
If $\alpha<p^{-}-1$, then  \eqref{e1.11} with initial-boundary values
\eqref{e1.9}-\eqref{e1.10} has a weak solution $u$.
The boundary value condition \eqref{e1.10} is satisfied in the sense of  trace.
 \end{theorem}


\begin{theorem}\label{thm2.4} 
 Let $u$ and $v$ be two weak solutions of equation \eqref{e1.11}
with initial values $u(x,0)$ and $v(x,0)$ respectively. If
$p^{-}-1>\alpha>0$, 
\begin{gather}
%u(x,t)=v(x,t)=0,\quad (x,t)\in \partial \Omega\times (0,T), \nonumber \label{e2.5} \\
\int_{\Omega}\rho^{\alpha-1}|\nabla u|^{p(x)-1}dx<\infty, \quad
\int_{\Omega}\rho^{\alpha-1}|\nabla u|^{p(x)-1}dx<\infty,\label{e2.6}
\end{gather}
 then
\begin{equation}
\int_\Omega^{} | u(x,t) - v(x,t) |dx  
\leq \int_\Omega^{} | u_0(x) - v_0(x) |dx.\label{e2.7}
\end{equation}
 \end{theorem}

The above theorem is a weaker version of
\cite[Theorem 1.4]{ZW} when $\alpha<p^{-}-1$. We can use it to prove the 
following Theorems.


  \begin{theorem}\label{thm2.5}
Let $u$ and $v$ be two weak solutions of equation \eqref{e1.11} with the
different initial values $u(x,0), v(x,0)$ respectively, and the exponent
$p(x)$ be required to satisfy logarithmic H\"{o}lder continuity condition.
If  $\alpha\geq p^{-}-1$, $u$ and $v$ satisfy \eqref{e2.6}, 
$u_t\in L^2(Q_T)$ and $v_t\in L^2(Q_T)$, then then the stability
 \eqref{e2.7} is still true.
\end{theorem}


 \begin{theorem}\label{thm2.6}
Let  $p>1$ and $0<\alpha<p^{-}-1$. If  $u$ and $v$ are two solutions of
equation \eqref{e1.11} with the differential initial values $u_0(x)$ and
$v_0(x)$ respectively, then there exists a positive constant
$\beta\geq\max\{\frac{p^{+}-\alpha}{p^{-}-1}, 2\}$ such that
\begin{equation}
\int_{\Omega}\rho^{\beta}|u(x,t)-v(x,t)|^2dx
\leq  c\int_{\Omega}\rho^{\beta}|u_{0}(x)-v_{0}(x)|^2dx. \label{e2.8}
\end{equation}
In particular, for any small enough constant  $\delta>0$, there holds
\begin{equation}
\int_{\Omega_{\delta}}|u(x,t)-v(x,t)|^2dx
\leq  c\delta^{-\beta}\int_{\Omega}|u_{0}(x)-v_{0}(x)|^2dx. \label{e2.9}
\end{equation}
 \end{theorem}

Here,  $\Omega_{\delta}=\{x\in\Omega: \operatorname{dist}(x,\partial \Omega)>\delta\}$, by the arbitrary of $\delta$, we have the uniqueness of the solution. The inequality \eqref{e2.8} shows the local stability of the solutions.


\begin{theorem}\label{thm2.7}   
Let  $p>1$, $\alpha\geq p^{-}-1$,
$b_{i}(s)$ be a Lipschitz function, and the exponent $p(x)$ be required to
satisfy logarithmic H\"{o}lder continuity condition. If  $u$, $v$ are two
solutions of equation \eqref{e1.11} with the different initial values
$u_0(x), v_0(x)$ respectively, then the inequality \eqref{e2.8} is true,
which implies the uniqueness of the solution.
 \end{theorem}

The proof of the existence (Theorem \ref{thm2.3}) is quite different from that
shown in \cite{YW,ZW,Z2}. There  to prove the stability of solutions,
the authors used two ways to deal with the cases  $\alpha<p^{-}-1$
and $\alpha\geq p^{+}-1$. In this article, we  adopt a similar method
to prove Theorems \ref{thm2.4} and \ref{thm2.5},  and then 
develop it to prove Theorems \ref{thm2.6} and \ref{thm2.7}.
The methods used here seem to be more effective, and can be extended to
the degenerate parabolic equation related to the $p(x)$-Laplacian directly.

\section{Proof of Theorem \ref{thm2.3}}

Following \cite{AS2}, we have the following lemma.

\begin{lemma}\label{lem3.1}
Let $q\geq 1$. If $u_{\varepsilon}\in L^{\infty}(0,T;L^2(\Omega))\cap W(Q_T)$,
 $\| u_{\varepsilon t}\|_{W'(Q_T)}\leq c$,
and  $\|\nabla(|u_{\varepsilon}|^{q-1}u_{\varepsilon})\|_{p^{-},Q_T}\leq c$,
then there is a subsequence of $\{u_{\varepsilon}\}$ which is relatively compactness
 in $L^{s}(Q_T)$ with $s\in(1,\infty)$.
\end{lemma}

To study  \eqref{e1.11}, let us consider the associated regularized problem
\begin{gather}
u_{\varepsilon t} - \operatorname{div}
(\rho^\alpha_\varepsilon (| \nabla u_\varepsilon |^2
+\varepsilon)^{\frac{p(x) - 2}{2}}\nabla {u_\varepsilon}) = 0,\quad
(x,t)\in {Q_T},\label{e3.1} \\
{u_\varepsilon }(x,t) = 0,\quad (x,t) \in \partial \Omega  \times (0,T),\label{e3.2} \\
{u_\varepsilon }(x,0) = {u_{0\varepsilon}}(x),\quad x\in\Omega.\label{e3.3}
\end{gather}
where ${\rho _\varepsilon } = \rho\ast \delta_\varepsilon+ \varepsilon$,
$\varepsilon  > 0$, $\delta_\varepsilon$ is the usual mollifier,
${u_{\varepsilon ,0}} \in {C^\infty_{0} }(\Omega)$ and 
$\rho_\varepsilon ^\alpha {| {\nabla {u_{\varepsilon,0}}} |^{p(x)}}\in
{L^1}(\Omega)$ is uniformly bounded, and ${u_{\varepsilon,0}}$ converges to
$u_0$ in $W_0^{1,p(x)}(\Omega)$. It is well-known that the above problem has
a unique classical solution \cite{G,TM}.

\begin{lemma}\label{lem3.2}
  There is a subsequence of ${u_\varepsilon}$ (we still denote it as
${u_\varepsilon}$), which converges to a weak solution $u$ of equation
\eqref{e1.11} with the initial value \eqref{e1.9}.
\end{lemma}

 \begin{proof}
By the maximum principle, there is a constant $c$  dependent on
${\| {{u_{0}}} \|_{{L^\infty }(\Omega )}}$  and independent on $\varepsilon$,
such that
\begin{equation}
{\| {{u_\varepsilon }} \|_{{L^\infty }({Q_T})}} \leqslant c. \label{e3.4}
\end{equation}
Multiplying \eqref{e2.1} by $u_\varepsilon$  and integrating it over $Q_T$,
we have
\begin{equation}
\frac{1}{2}\int_\Omega u_\varepsilon^2dx
+ \iint_{{Q_T}} \rho_\varepsilon^\alpha (|\nabla u_\varepsilon|^2
+\varepsilon)^{\frac{p(x)-2}{2}}|\nabla u_\varepsilon|^2\,dx\,dt
 = \frac{1}{2}\int_\Omega u_{0}^2dx \leq c.\label{e3.5}
\end{equation}
For small enough $\lambda>0$, let
 $\Omega_{\lambda}=\{x\in \Omega: \operatorname{dist}(x,\partial \Omega)>\lambda\}$.
Since $p^{-}>1$, by \eqref{e3.5} we have
\begin{equation}
\int_{0}^{T}\int_{\Omega_{\lambda}}|\nabla u_{\varepsilon}|\,dx\,dt
\leq c \Big(\int_{0}^{T}\int_{\Omega_{\lambda}}|\nabla u_{\varepsilon}|^{p^{-}}
\,dx\,dt\Big)^{1/p^{-}} \leq c(\lambda).\label{e3.6}
\end{equation}
Now, for any $v\in W(Q_T)$, $\|v\|_{W(Q_T)}=1$, and
$$
\langle u_{\varepsilon t}, v\rangle
=-\iint_{Q_T}\rho _\varepsilon^\alpha (|\nabla u_\varepsilon|^2
+\varepsilon)^{\frac{p(x)-2}{2}}\nabla u_\varepsilon\cdot \nabla v \,dx\,dt,
$$
by Young's inequality, we can show that
$$
|\langle u_{\varepsilon t}, v\rangle |
\leq c\Big[\iint_{{Q_T}} \rho_{\varepsilon}^\alpha |\nabla u_\varepsilon|^{p(x)}
\,dx\,dt+\iint_{{Q_T}} (|v|^{p(x)}+|\nabla v|^{p(x)})\,dx\,dt\Big]\leq c,
$$
then
\begin{equation}
 \| u_{\varepsilon t}\|_{W'(Q_T)}\leq c.\label{e3.7}
\end{equation}
 Now, let $\varphi\in C_0^1(\Omega)$, $0\leq\varphi\leq 1$ such that
$\varphi|_{\Omega_{2\lambda}}=1$ and
$\varphi|_{\Omega\setminus\Omega_{\lambda}}=0$.
Then
$$
|\langle (\varphi u_{\varepsilon})_{t}, v\rangle |
=|\langle \varphi u_{\varepsilon t}, v\rangle |\leq |
\langle  u_{\varepsilon t}, v\rangle|;
$$
so we have
\begin{equation}
 \|(\varphi(x) u)_{\varepsilon t}\|_{W'(Q_T)}
\leq \|  u_{\varepsilon t}\|_{W'(Q_T)}\leq c.\label{e3.8}
\end{equation}
By \eqref{e3.6},
\begin{equation}
\iint_{Q_T}|\nabla (\varphi u_{\varepsilon})|^{p^{-}}\,dx\,dt
\leq c(\lambda)(1+\int_{0}^{T}\int_{\Omega_{\lambda}}|\nabla u_{\varepsilon}|^{p^{-}}
\,dx\,dt)\leq c(\lambda),\label{e3.9}
\end{equation}
and so
\begin{equation}
 \|\nabla(|\varphi u_{\varepsilon})\|_{p^{-},Q_T}\leq c(\lambda).\label{e3.10}
 \end{equation}
By Lemma \ref{lem3.1}, $\varphi u_{\varepsilon}$ is
 relatively compactness in $L^{s}(Q_T)$ with $s\in(1,\infty)$.
Then $\varphi u_{\varepsilon}\to \varphi u$ a.e. in $Q_T$. In particular,
by the arbitraries of $\lambda$, it follows that  $u_{\varepsilon}\to  u$ a.e.
in $Q_T$.

Hence, by \eqref{e3.4}, \eqref{e3.5}, \eqref{e3.7}, there exists  a function $u$
and the $n$-dimensional vector function
$\overrightarrow \zeta   = ({\zeta_1}, \cdots ,{\zeta_n})$ satisfying
$$
u \in  L^{\infty}(Q_T),\ u_{t}\in W'(Q_T), \quad
| {\overrightarrow \zeta} | \in {L^{\frac{p(x)}{{p(x) - 1}}}}({Q_T}),
$$
and
\begin{gather*}
{u_{\varepsilon}} \rightharpoonup * u,\quad \text{in }  {L^{\infty}(Q_T)}, \\
{\nabla u_{\varepsilon}} \rightharpoonup \nabla u \quad \text{in }
 L_{\rm loc}^{p(x)}(Q_T), \\
\rho _\varepsilon ^\alpha {| {\nabla {u_\varepsilon }}
|^{p(x)- 2}}\nabla {u_\varepsilon } \rightharpoonup \overrightarrow \zeta \quad
\text{in } {L^{\frac{p(x)}{p(x)-1}}}({Q_T}).
\end{gather*}
To prove that $u$ satisfies  \eqref{e1.11}, we notice that for any function
 $\varphi  \in C_0^\infty ({Q_T})$, we have
\begin{equation}
\iint_{{Q_T}}[u_{\varepsilon t}\varphi
+ \rho_\varepsilon^\alpha(|\nabla u_\varepsilon|^2
+\varepsilon)^{\frac{p(x)-2}{2}}\nabla {u_\varepsilon} \cdot \nabla \varphi]\,dx\,dt
 = 0.\label{e3.11}
\end{equation}
 Then
\begin{equation}
\iint_{{Q_T}} (\frac{\partial u}{\partial t}\varphi
 + \vec \varsigma  \cdot \nabla \varphi)\,dx\,dt= 0.\label{e3.12}
\end{equation}
Now,  similar to \cite{ZW,Z2}, we can prove that
\begin{equation}
\iint_{Q_T} \rho^\alpha| \nabla u |^{p(x) - 2}\nabla u \cdot \nabla \varphi
\,dx\,dt
 = \iint_{Q_T} \overrightarrow \zeta   \cdot \nabla \varphi \,dx\,dt \label{e3.13}
\end{equation}
for any function $\varphi  \in C_0^\infty ({Q_T})$.  Thus $u$ satisfies  \eqref{e1.11}.

Similarly, we can prove \eqref{e1.9} as in \cite{AS2} in the same manner.
The proof is complete.
\end{proof}

\begin{lemma}\label{lem3.3}
If $\alpha<p^{-}-1$, and let $u$ be the solution of equation \eqref{e1.11}
with the initial value \eqref{e1.9}, then the trace of  $u$   on the boundary
 $\partial \Omega$ can be defined in the traditional way.
 \end{lemma}

The above lemma was proved in \cite{ZW,Z2}. 
Note that Theorem \ref{thm2.3} is the directly consequence
of Lemmas \ref{lem3.2} and \ref{lem3.3}.

\section{Proof of Theorem \ref{thm2.4}}

For small $\eta>0$, let
\begin{equation}
S_{\eta}(s)=\int_{0}^{s}h_{\eta}(\tau)d\tau,
\quad  h_{\eta}(s)=\frac{2}{\eta}\big(1-\frac{| s|}{\eta}\big)_{+}.\label{e4.1}
\end{equation}
Obviously $h_{\eta}(s)\in C(\mathbb{R})$, and
\begin{equation}
\begin{gathered}
h_{\eta}(s)\geq 0,\quad\ | sh_{\eta}(s)| \leq 1,\quad
| S_{\eta}(s)| \leq 1,\\
\lim_{\eta \to 0} S_{\eta}(s)=\operatorname{sgn}(s),\quad
\lim_{\eta \to0} sS_{\eta}'(s)=0.
\end{gathered} \label{e4.2}
\end{equation}

\begin{proof}
If $\alpha<p^{-}-1$, by Lemma \ref{lem3.3}, the weak solution of  \eqref{e1.11}
can be defined by the trace on the boundary $\partial \Omega$ in the
traditional way. Let $u$ and $v$ be two weak solutions of  \eqref{e1.11}
with the initial values $u(x,0)$ and $v(x,0)$ respectively.

Let $\beta>0$ and
\begin{equation}
\phi(x)=\rho^\beta(x).\label{e4.3}
\end{equation}
Then, we can choose $S_{\eta}(\phi(u - v))$ as the test function, and find
\begin{equation}
\begin{aligned}
&\int_{\Omega} S_{\eta}(\phi(u - v))\frac{\partial (u - v)}{\partial t}dx \hfill \\
&+ \int_{\Omega} \rho^\alpha(| \nabla u |^{p(x)- 2}\nabla u
 - | \nabla v |^{p(x)- 2}\nabla v) \cdot \phi \nabla (u - v)S'_{\eta}(\phi(u - v))
  dx\hfill \\
&+ \int_{\Omega} \rho^\alpha(| \nabla u |^{p(x) - 2}\nabla u
  - | \nabla v |^{p(x)- 2}\nabla v) \cdot \nabla\phi
 (u - v)S'_{\eta}(\phi(u - v))dx \hfill \\
 =0.
\end{aligned}\label{e4.4}
\end{equation}
Thus, we have
\begin{gather} \label{e4.5}
\begin{aligned}
\lim_{\eta\to 0}\int_\Omega^{} S_{\eta}(\phi(u - v))
 \frac{\partial (u - v)}{\partial t}dx 
&=\int_\Omega^{} \operatorname{sgn}(\phi(u - v))\frac{\partial
(u - v)}{\partial t}dx \\
&=\int_\Omega^{} \operatorname{sgn}(u - v)\frac{\partial (u - v)}{\partial t}dx
= \frac{d}{dt}\| u - v \|_1,
\end{aligned} \\
\int_{\Omega} \rho^\alpha (| \nabla u |^{p(x)- 2}\nabla u - | \nabla v |^{p(x) - 2}\nabla v) \cdot \nabla (u - v)S'_{\eta}(\phi(u - v))
\phi(x)dx \geq 0,\label{e4.6}
\end{gather}
and
\begin{equation}
\begin{aligned}
&\big|\int_{\Omega} \rho^\alpha (| \nabla u |^{p(x)- 2}\nabla u
 - | \nabla v |^{p(x) - 2}\nabla v) \cdot \nabla\phi
 (u - v)S'_{\eta}(\phi(u-v)) dx\big| \\
&\leq c\int_{\{x:\rho^{\beta}|u-v|<\eta\}}| \rho^{\alpha-1}(| \nabla u |^{p(x)- 2}
\nabla u - |\nabla v|^{p(x)- 2}\nabla v)| \\
&\quad\times |\phi(u - v)S'_{\eta}(\phi(u-v)) |dx,
\end{aligned}\label{e4.7}
\end{equation}
which tends to $0$ as $\eta\to 0$, because of \eqref{e2.6} and
$$
\lim_{\eta\to 0}\phi(u - v)S'_{\eta}(\phi(u-v))=0.
$$
Now,  let $\eta\to 0$ in \eqref{e4.4}.  Then
\begin{equation}
\frac{{\,d}}{{{\,d}t}}{\| {u - v} \|_1}
\leqslant c{\| {u - v} \|_1}.\label{e4.8}
\end{equation}
This implies
\begin{equation}
\int_\Omega | {u(x,t) - v(x,t)} |dx
\leqslant c(T) \int_\Omega | {{u_0} - {v_0}} |dx.\label{e4.9}
\end{equation}
The proof is complete.
\end{proof}

\section{Proof of Theorem \ref{thm2.5}}

\begin{proof}
If $\alpha\geq p^{-}-1$,  the weak solution of equation \eqref{e1.11}
lacks the regularity on the boundary, we can not define the trace
on $\partial \Omega$.
Denote
\begin{equation}
\Omega_{\lambda}=\{x\in\Omega: \operatorname{dist}(x,\partial \Omega)>\lambda\},
\label{e5.1}
\end{equation}
let $\beta>0$ and
\begin{equation}
\phi(x)=[\operatorname{dist}((x,\Omega\setminus\Omega_{\lambda})]^{\beta}
=d_{\lambda}^{\beta}.\label{e5.2}
\end{equation}

Let $u$ and $v$ be two weak solutions of equation \eqref{e1.11} with the
initial values $u(x,0)$ and $v(x,0)$ respectively.
We can choose $S_{\eta}(\phi(u_{\varepsilon} - v_{\varepsilon}))$ as
the test function, where $u_{\varepsilon}$ and $v_{\varepsilon}$ are the mollified function of the solutions $u$ and $v$ respectively. Then
\begin{equation}
\begin{aligned}
&\int_{\Omega_{\lambda}} S_{\eta}(\phi (u_{\varepsilon}
 - v_{\varepsilon}))\frac{\partial (u - v)}{\partial t}dx   \\
&+ \int_{\Omega_{\lambda}} \rho^\alpha(| \nabla u |^{p(x)- 2}\nabla u
 - | \nabla v |^{p(x)- 2}\nabla v) \cdot \phi \nabla (u_{\varepsilon}
  - v_{\varepsilon})S'_{\eta}(\phi(u_{\varepsilon} - v_{\varepsilon})) dx  \\
&+ \int_{\Omega_{\lambda}} \rho^\alpha(| \nabla u |^{p(x) - 2}\nabla u
 - | \nabla v |^{p(x)- 2}\nabla v) \cdot \nabla\phi  (u_{\varepsilon}
 - v_{\varepsilon})S'_{\eta}(\phi(u_{\varepsilon} - v_{\varepsilon}))dx \\
 &=0.
\end{aligned}\label{e5.3}
\end{equation}

For any given $\lambda>0$, denoting $Q_{T\lambda}=\Omega_{\lambda}\times(0,T)$,
by (iii) of Lemma \ref{lem2.1} and \eqref{e2.1} in Definition \ref{def2.2},
 we know that $|\nabla u|\in L^{p(x)}(Q_{T\lambda})$,
$|\nabla v|^{p(x)}\in L^{p(x)}(Q_{T\lambda})$. Thus according to the
definition of the mollified functions  $u_{\varepsilon}$ and $v_{\varepsilon}$,
the exponent $p(x)$ is required to satisfy the logarithmic H\"{o}lder
continuity condition, by \cite{FZ,Z1,KR}, we have
\begin{gather}
u_{\varepsilon}\in L^{\infty}(Q_T), \quad v_{\varepsilon}\in L^{\infty}(Q_T),\quad
u_{\varepsilon}\to u, v_{\varepsilon}\to v, \quad \text{a.e. in } Q_T,
\label{e5.4} \\
\begin{gathered}
\||\nabla u_{\varepsilon}|^{p(x)}\|_{1,\Omega_{\lambda}}
\leq \||\nabla u|^{p(x)}\|_{1,\Omega_{\lambda}}, \quad
\||\nabla v_{\varepsilon}|^{p(x)}\|_{1,\Omega_{\lambda}}
 \leq \||\nabla v|^{p(x)}\|_{1,\Omega_{\lambda}},\\
 \nabla u_{\varepsilon}\to \nabla u, \quad
\nabla v_{\varepsilon}\to \nabla v,\quad \text{in }  L^{p(x)}(\Omega_{\lambda}).
\end{gathered}\label{e5.5}
\end{gather}
Since
$0\leq S'_{\eta}(\phi(u_{\varepsilon} - v_{\varepsilon}))\leq \frac{2}{\eta}$,
it follows that
$$
|\nabla (u_{\varepsilon} - v_{\varepsilon})S'_{\eta}(\phi(u_{\varepsilon}
- v_{\varepsilon}))|_{L^{p(x)}(\Omega_{\lambda})}\leq c(\eta)
|\nabla (u_{\varepsilon} - v_{\varepsilon})|_{L^{p(x)}(\Omega_{\lambda})}
\leq c(\eta),
$$
For any $\varphi\in L^{\frac{p(x)}{p(x)-1}}(\Omega_{\lambda})$, it holds
\begin{equation}
\begin{aligned}
&\int_{\Omega_{\lambda}}\nabla (u_{\varepsilon} - v_{\varepsilon})
 S'_{\eta}(\phi(u_{\varepsilon} - v_{\varepsilon}))\varphi \,dx
-\int_{\Omega_{\lambda}}\nabla (u - v)S'_{\eta}(\phi(u- v))\varphi \,dx \\
&=\int_{\Omega_{\lambda}}\nabla (u_{\varepsilon} - v_{\varepsilon})
 [S'_{\eta}(\phi(u_{\varepsilon} - v_{\varepsilon}))-S'_{\eta}(\phi(u- v))]
 \varphi\, dx \\
&\quad +\int_{\Omega_{\lambda}}[\nabla (u_{\varepsilon} - v_{\varepsilon})
 -\nabla(u-v)]S'_{\eta}(\phi(u- v))\varphi \,dx\\
&=I_1+I_2.
\end{aligned}\label{e5.6}
\end{equation}
Since $\nabla u_{\varepsilon}\to \nabla u$ and
$\nabla v_{\varepsilon}\to \nabla v$ in $L^{p(x)}(\Omega_{\lambda})$,
it follows that
\begin{equation}
\lim_{\varepsilon\to 0}I_2=0, \label{e5.7}
\end{equation}
while
\begin{equation}
\begin{aligned}
&\lim_{\varepsilon\to 0}I_1\\
&\leq \lim_{\varepsilon\to 0}\|\nabla(u_{\varepsilon}
  - v_{\varepsilon})\|_{L^{p(x)}(\Omega_{\lambda})}
 \|[S'_{\eta}(\phi(u_{\varepsilon} - v_{\varepsilon}))
 -S'_{\eta}(\phi(u- v))]\varphi\|_{L^{\frac{p(x)}{p(x)-1}}(\Omega_{\lambda})}\\
&\leq\lim_{\varepsilon\to 0}\|\nabla(u- v)\|_{L^{p(x)}(\Omega_{\lambda})}
\|[S'_{\eta}(\phi(u_{\varepsilon} - v_{\varepsilon}))
 -S'_{\eta}(\phi(u- v))]\varphi\|_{L^{\frac{p(x)}{p(x)-1}}
 (\Omega_{\lambda})}\\
&=0,
\end{aligned} \label{e5.8}
\end{equation}
by the controlled convergent theorem. Thus we have
\begin{equation}
\nabla (u_{\varepsilon} - v_{\varepsilon})S'_{\eta}(\phi(u_{\varepsilon}
- v_{\varepsilon}))\rightharpoonup\nabla (u - v)S'_{\eta}(\phi(u- v)),
 \quad \text{in }  L^{p(x)}(\Omega_{\lambda}).
\label{e5.9}
\end{equation}
Since on $\Omega_{\lambda}$, one has
$$
|\rho^{\alpha}\phi(|\nabla u|^{p-2}\nabla u
 -|\nabla v|^{p-2}\nabla v)|\in L^{\frac{p(x)}{p(x)-1}}(\Omega_{\lambda})
$$
by the weak convergency of \eqref{e5.9} we have
\begin{equation}
\begin{aligned}
&\lim_{\varepsilon\to 0}\int_{\Omega_{\lambda}}
\rho^\alpha(| \nabla u |^{p(x)- 2}\nabla u
- | \nabla v |^{p(x)- 2}\nabla v) \cdot \phi \nabla (u_{\varepsilon}
- v_{\varepsilon})S'_{\eta}(\phi(u_{\varepsilon} - v_{\varepsilon})) \,dx \\
&=\int_{\Omega_{\lambda}} \rho^\alpha(| \nabla u |^{p(x)- 2}\nabla u
 - | \nabla v |^{p(x)- 2}\nabla v) \cdot \phi \nabla (u_{\varepsilon}
 - v_{\varepsilon})S'_{\eta}(\phi(u - v)) dx.
\end{aligned}\label{e5.10}
\end{equation}
At the same time, it is clear that
\begin{equation}
\begin{aligned}
&\lim_{\varepsilon\to 0}\int_{\Omega_{\lambda}} \rho^\alpha(| \nabla u |^{p(x) - 2}
 \nabla u - | \nabla v |^{p(x)- 2}\nabla v) \cdot \nabla\phi
 (u_{\varepsilon} - v_{\varepsilon})S'_{\eta}(\phi(u_{\varepsilon}
 - v_{\varepsilon}))\,dx \\
&=\int_{\Omega_{\lambda}} \rho^\alpha(| \nabla u |^{p(x) - 2}\nabla u
 - | \nabla v |^{p(x)- 2}\nabla v) \cdot \nabla\phi  (u_{\varepsilon}
 - v_{\varepsilon})S'_{\eta}(\phi(u- v))dx,
\end{aligned}\label{e5.11}
\end{equation}
by the controlled convergent theorem. Also, since
$u_{t}, v_{t}\in L^2(Q_T)$,
by H\"older's inequality, we have
\begin{equation}
\iint_{Q_T}|\frac{\partial u}{\partial t}|\,dx\,dt<\infty, \quad
\iint_{Q_T}|\frac{\partial v}{\partial t}|\,dx\,dt< \infty. \label{e5.12}
\end{equation}
By the controlled convergent theorem, we have
\begin{equation}
\lim_{\varepsilon\to 0} \int_{\Omega_{\lambda}} S_{\eta}(\phi (u_{\varepsilon}
 - v_{\varepsilon}))\frac{\partial (u - v)}{\partial t}dx
=\int_{\Omega_{\lambda}} S_{\eta}(\phi (u- v))
 \frac{\partial (u - v)}{\partial t}dx.\label{e5.13}
\end{equation}

Now, we let $\varepsilon\to 0$, and then let $\lambda\to 0$, at last,
let $\eta\to 0$ in \eqref{e5.3}.  As the proof of \eqref{e4.5}-\eqref{e4.9},
we arrive at the desired result.
\end{proof}


\section{Uniqueness in the case $0<\alpha< p^{-}-1$}

\begin{proof}
Let $u$ and $v$ be two solutions of equation \eqref{e1.11} with 
initial values $u_0(x)$ and $v_0(x)$ respectively.
According to the definition of $W(Q_T)$, $L^2(Q_T)\subset W(Q_T)$,
 when $\varphi\in L^2(Q_T)$, we have
\begin{equation}
\langle\langle (u-v)_{t}, \varphi\rangle\rangle
=\iint_{Q_{\tau s}}\varphi\frac{\partial (u-v)}{\partial t}\,dx\,dt.\label{e6.1}
\end{equation}
From the definition of the weak solution, we have
\begin{equation}
\iint_{Q_T}\varphi\frac{\partial (u-v)}{\partial t}\,dx\,dt
=-\iint_{Q_T}\rho^{\alpha}(|\nabla u|^{p(x)-2}\nabla u-|\nabla v|^{p(x)-2}\nabla v)
\nabla\varphi \,dx\,dt,\label{e6.2}
\end{equation}
for any $\varphi  \in L^\infty (0,T; W_0^{1,p(x)}(\Omega))\cap L^2(Q_T)$.
By Lemma \ref{lem3.3},  $\alpha<p^{-}-1$, then the trace of  $u$   on the boundary
$\partial \Omega$ can be defined in the traditional way.
For any fixed $\tau,s\in [0,T]$, $\chi_{[\tau,s]}$ is the characteristic
function on $[\tau,s]$. Since $\beta\geq 2$, and
\begin{equation}
\chi_{[\tau,s]}(u-v)\rho^{\beta}\in L^2(Q_T)
\cap L^\infty (0,T; W_0^{1,p(x)}(\Omega))\label{e6.3}
\end{equation}
we may choose it as a test function in the above equality.
Thus, by denoting $Q_{\tau s}=\Omega\times[\tau, s]$, we have
\begin{equation}
\begin{aligned}
&\iint_{Q_{\tau s}}(u-v)\rho^{\beta}\frac{\partial (u-v)}{\partial t}\,dx\,dt\\
&=-\iint_{Q_{\tau s}}\rho^{\alpha}(|\nabla u|^{p(x)-2}
 \nabla u-|\nabla v|^{p(x)-2}\nabla v)\nabla[(u-v)\rho^{\beta}] \,dx\,dt \\
&=\iint_{Q_{\tau s}}\rho^{\alpha+\beta}(|\nabla u|^{p(x)-2}\nabla u
 -|\nabla v|^{p(x)-2}\nabla v)\nabla(u-v) \,dx\,dt \\
&\quad +\iint_{Q_{\tau s}}\rho^{\alpha}(|\nabla u|^{p(x)-2}
 \nabla u-|\nabla v|^{p(x)-2}\nabla v)(u-v)\nabla \rho^{\beta} \,dx\,dt.
\end{aligned}\label{e6.4}
\end{equation}
The first term on the right hand side of \eqref{e6.4}  satisfies
\begin{equation}
\iint_{Q_{\tau s}}\rho^{\alpha+\beta}(|\nabla u|^{p(x)-2}
\nabla u-|\nabla v|^{p(x)-2}\nabla v)\nabla(u-v) \,dx\,dt\geq 0.\label{e6.5}
\end{equation}
The second term on the right hand side of \eqref{e6.4}, by (iii) of
 Lemma \ref{lem2.1},
satisfies
\begin{equation}
\begin{aligned}
&\big|\iint_{Q_{\tau s}}(u-v)\rho^{\alpha}(|\nabla u|^{p(x)-2}\nabla u
 -|\nabla v|^{p(x)-2}\nabla v)\nabla \rho^{\beta}\,dx\,dt\big| \\
&\leq \iint_{Q_{\tau s}}|u-v|\rho^{\alpha}(|\nabla u|^{p(x)-1}
 +|\nabla v|^{p(x)-1})|\nabla \rho^{\beta}|\,dx\,dt \\
&\leq c\int_{\tau}^{s}\|\rho^{\alpha\frac{p(x)-1}{p(x)}}(|\nabla u|^{p(x)-1}
 +|\nabla v|^{p(x)-1})\|_{L^{\frac{p(x)}{p(x)-1}}(\Omega)} \\
&\quad\times \|\rho^{\frac{\alpha}{p(x)}}|\nabla \rho^{\beta}(u-v)|\|_{L^{p(x)}(\Omega)}dt \\
&\leq c\int_{\tau}^{s}\|\rho^{\alpha\frac{p(x)-1}{p(x)}}(|\nabla u|^{p(x)-1}
 +|\nabla v|^{p(x)-1})\|_{L^{\frac{p(x)}{p(x)-1}}(\Omega)} \\\
&\quad\times \|\rho^{\frac{\alpha}{p(x)}+(\beta-1)}|(u-v)|\|_{L^{p(x)}(\Omega)}dt \\
&\leq c\int_{\tau}^{s}\Big(\int_{\Omega}\rho^{\alpha}(|\nabla u|^{p(x)}
 +|\nabla v|^{p(x)})dx\Big)^{1/p'_1} \\
&\quad\times \Big(\int_{\Omega}\rho^{\alpha+p(x)(\beta-1)}|u-v|^{p(x)}dx\Big)^{1/p_1}dt \\
&\leq c\int_{\tau}^{s}\Big(\int_{\Omega}\rho^{\alpha+p(x)(\beta-1)}|u-v|^{p(x)}dx
\Big)^{1/p_1}dt.
\end{aligned}\label{e6.6}
\end{equation}
Here, we  used  that $|\nabla \rho|=1$  almost everywhere,
that  $p_1=p^{+}$ or $p^{-}$, and that $p'(x)= \frac{p(x)}{p(x)-1}$, 
where $p'_1=p'^{+}$ or $p'^{-}$.

Now, from $\beta\geq\frac{p^{+}-\alpha}{p^{-}-1}$, we have
\begin{equation}
\Big(\int_{\Omega}\rho^{\alpha+p(x)(\beta-1)}|u-v|^{p(x)}dx\Big)^{1/p_1}
\leq c\Big(\int_{\Omega_1+\Omega_2}\rho^{\beta}|u-v|^{p(x)}dx\Big)^{1/p_1},\label{e6.7}
\end{equation}
where $ \Omega_1=\{x\in \Omega: p(x)\geq 2\}$, $\Omega_2=\{x\in \Omega: 1<p(x)<2\}$.
Then
\begin{equation}
\begin{aligned}
\Big(\int_{\Omega_1}\rho^{\beta}|u-v|^{p(x)}dx\Big)^{1/p_1}
&\leq c\Big(\int_{\Omega_1}\rho^{\beta}|u-v|^2dx\Big)^{1/p_1}\\
&\leq c\Big(\int_{\Omega}\rho^{\beta}|u-v|^2dx\Big)^{1/p_1}.
\end{aligned} \label{e6.8}
\end{equation}
In $\Omega_2$,  by H\"older's inequality and (iii) in Lemma 
\ref{lem2.1}, we obtain
\begin{equation}
\begin{aligned}
\Big(\int_{\Omega_2}\rho^{\beta}|u-v|^{p(x)}dx\Big)^{1/p_1}
&\leq c\Big(\|\rho^{\frac{p(x)}{2}}|u-v|^{p(x)}\|_{L^{\frac{2}{p(x)}}(\Omega_2)}
 \Big)^{1/p_1} \\
&\leq c\Big(\int_{\Omega}\rho^{\beta}|u-v|^2dx\Big)^q.\label{e6.9}
\end{aligned} \end{equation}
where $q<1$.
Also we have
\begin{equation}
\begin{aligned}
&\iint_{Q_{\tau s}}(u-v)\rho^{\beta}\frac{\partial (u-v)}{\partial t}\,dx\,dt \\
&=\int_{\Omega}\rho^{\beta}[u(x,s)-v(x,s)]^2dx
 -\int_{\Omega}\rho^{\beta}[u(x,\tau)-v(x,\tau)]^2dx.
\end{aligned} \label{e6.10}
\end{equation}
From \eqref{e6.4}--\eqref{e6.10}, it follows that
\begin{equation}
\begin{aligned}
&\int_{\Omega}\rho^{\beta}[u(x,s)-v(x,s)]^2dx
 -\int_{\Omega}\rho^{\beta}[u(x,\tau)-v(x,\tau)]^2dx \\
&\leq c\int_\tau^s\Big(\int_{\Omega}\rho^{\beta}|u(x,t)-v(x,t)|^2dx\Big)^qdt\\
&\leq c\Big(\int_\tau^s\int_{\Omega}\rho^{\beta}|u(x,t)-v(x,t)|^2\,dx\,dt\Big)^q,
\end{aligned}\label{e6.11}
\end{equation}
where $q<1$. Let $\kappa(s)=\int_{\Omega}\rho^{\beta}[u(x,s)-v(x,s)]^2dx$.
Then we deduce
$$
\frac{\kappa(s)-\kappa(\tau)}{s-\tau}
\leq c\frac{\big(\int_\tau^s\kappa (t)dt\big)^q}{s-\tau}.
$$
By the L'Hospital Rule, 
\begin{equation}
\kappa'(\tau)\leq c\lim_{s\to \tau}\frac{\kappa(s)}
{\big(\int_\tau^s\kappa (t)dt\big)^{1-q}}
=c\lim_{s\to \tau}\frac{\kappa'(s)}{\kappa(s)}{\big(\int_\tau^s\kappa (t)dt\big)^q}=0.
\label{e6.12}
\end{equation}
Thus, because $\tau$ is arbitrary, we have
\begin{equation}
\int_{\Omega}\rho^{\beta}| u(x,\tau)-v(x,\tau)|^2 dx
\leq \int_{\Omega}\rho^{\beta}| u_{0}-v_{0}|^2 dx. \label{e6.13}
\end{equation}
The proof is complete.
\end{proof}

\section{Uniqueness in the case $\alpha\geq p^{-}-1$}

When $\alpha\geq p^{-}-1$, let $u$ be a weak solution of
equation \eqref{e1.11} with the initial value \eqref{e1.9}.
Generally, we can not define the trace of $u$ on the boundary.

\begin{proof} Let the constant $\beta\geq \max\{\frac{p^{+}-\alpha}{p^{-}-1}, 2\}$.
 Denote $\Omega_{\lambda}$, $Q_{T\lambda}=\Omega_{\lambda}\times(0,T)$
as \eqref{e5.1}-\eqref{e5.4}, and let $\xi_{\lambda}=d_{\lambda}^{\beta}$.
 Let  $u$ and $v$ be two solutions of equation \eqref{e1.11} with the initial
values $u_0(x)$ and $v_0(x)$ respectively. We choose
$\chi_{[\tau,s]}(u_{\varepsilon}-v_{\varepsilon})\xi_{\lambda}$ as a test function,
where $u_{\varepsilon}$ and $v_{\varepsilon}$ are the mollified function of the
solutions $u$ and $v$ respectively.  Then
\begin{equation}
\begin{aligned}
&\langle\langle (u-v)_{t}, \chi_{[\tau,s]}(u_{\varepsilon}
-v_{\varepsilon})\xi_{\lambda}\rangle\rangle \\
&=\iint_{Q_{\tau s}}(u_{\varepsilon}-v_{\varepsilon})\xi_{\lambda}
 \frac{\partial (u-v)}{\partial t}\,dx\,dt \\
&=-\iint_{Q_{\tau s}}\rho^{\alpha}(|\nabla u|^{p-2}\nabla u
 -|\nabla v|^{p-2}\nabla v)\nabla[(u_{\varepsilon}
 -v_{\varepsilon})\xi_{\lambda}] \,dx\,dt.
\end{aligned}\label{e7.1}
\end{equation}
Now, by the weak convergence of \eqref{e5.4} and
$$
|\rho^{\alpha}(|\nabla u|^{p(x)-2}\nabla u-|\nabla v|^{p(x)-2}\nabla v)|
\in L^{\frac{p(x)}{p(x)-1}}(\Omega_{\lambda})
$$
we obtain
\begin{equation}
\begin{aligned}
&\lim_{\varepsilon\to 0}\iint_{Q_{\tau s}}\rho^{\alpha}\xi_{\lambda}
 (|\nabla u|^{p(x)-2}\nabla u-|\nabla v|^{p(x)-2}\nabla v)
 \nabla(u_{\varepsilon}-v_{\varepsilon}) \,dx\,dt \\
&=\iint_{Q_{\tau s}}\rho^{\alpha}\xi_{\lambda}(|\nabla u|^{p(x)-2}\nabla u
 -|\nabla v|^{p(x)-2}\nabla v)\nabla(u-v) \,dx\,dt.
\end{aligned}\label{e7.2}
\end{equation}
By \eqref{e5.4}-\eqref{e5.5} and the Lebesgue controlled convergence theorem,
we have
\begin{equation}
\begin{aligned}
&\lim_{\varepsilon\to 0}\iint_{Q_{\tau s}}\rho^{\alpha}
(|\nabla u|^{p(x)-2}\nabla u-|\nabla v|^{p(x)-2}\nabla v)
(u_{\varepsilon}-v_{\varepsilon})\nabla\xi_{\lambda} \,dx\,dt \\
&=\iint_{Q_{\tau s}}\rho^{\alpha}(|\nabla u|^{p(x)-2}\nabla u
 -|\nabla v|^{p(x)-2}\nabla v)(u-v)\nabla\xi_{\lambda} \,dx\,dt.
\end{aligned} \label{e7.3}
\end{equation}
So
\begin{equation}
\begin{aligned}
&\lim_{\varepsilon\to 0}\iint_{Q_{\tau s}}\rho^{\alpha}
(|\nabla u|^{p(x)-2}\nabla u-|\nabla v|^{p(x)-2}\nabla v)
\nabla[(u_{\varepsilon}-v_{\varepsilon})\xi_{\lambda}] \,dx\,dt \\
&=\iint_{Q_{\tau s}}\rho^{\alpha}\xi_{\lambda}(|\nabla u|^{p(x)-2}\nabla u
 -|\nabla v|^{p(x)-2}\nabla v)\nabla(u-v) \,dx\,dt \\
&\quad +\iint_{Q_{\tau s}}\rho^{\alpha}(|\nabla u|^{p(x)-2}\nabla u
 -|\nabla v|^{p(x)-2}\nabla v)(u-v)\nabla\xi_{\lambda} \,dx\,dt.
\end{aligned} \label{e7.4}
\end{equation}
At the same time, by H\"older's inequality, similar to \eqref{e6.6}-\eqref{e6.8},
we  have
\begin{equation}
\iint_{Q_{\tau s}}\rho^{\alpha}\xi_{\lambda}(|\nabla u|^{p(x)-2}\nabla u
-|\nabla v|^{p(x)-2}\nabla v)\nabla(u-v) \,dx\,dt\geq 0,\label{e7.5}
\end{equation}
and
\begin{equation}
\begin{aligned}
&\big|\iint_{Q_{\tau s}}(u-v)\rho^{\alpha}(|\nabla u|^{p(x)-2}\nabla u
-|\nabla v|^{p(x)-2}\nabla v)\nabla\xi_{\lambda}\,dx\,dt\big| \\
&\leq c\int_{\tau}^{s}
\Big(\int_{\Omega_{\lambda}}\rho^{\alpha}d_{\lambda}^{p(x)(\beta-1)}
|u-v|^{p(x)}dx\Big)^{1/p_1}dt \\
&\leq c\int_{\tau}^{s}\Big(\int_{\Omega}\rho^{\alpha+p(x)(\beta-1)}|u-v|^{p(x)}dx
\Big)^{1/p_1}dt \\
&\leq c\Big(\int_{\tau}^{s}\int_{\Omega}|u-v|^2\,dx\,dt\Big)^q,
\end{aligned} \label{e7.6}
\end{equation}
where $q<1$.
From \eqref{e5.12}, since
$(u_{\varepsilon}-v_{\varepsilon})\xi_{\lambda}\in L^{\infty}(Q_T)$,
  we can use the Lebesgue controlled convergence theorem to deduce that
\[
\lim_{\varepsilon\to 0}\iint_{Q_{\tau s}}(u_{\varepsilon}-v_{\varepsilon})
\xi_{\lambda}\frac{\partial (u-v)}{\partial t}\,dx\,dt
=\iint_{Q_{\tau s}}(u-v)\xi_{\lambda}\frac{\partial (u-v)}{\partial t}\,dx\,dt.
%\label{e7.7}
\]
Now, after letting $\varepsilon\to 0$ and $\lambda\to 0$ in \eqref{e7.1},
by a similar argument for \eqref{e6.10}-\eqref{e6.13}, we arrive at the desired
result.
\end{proof}

\subsection*{Acknowledgments}
This research is supported by the NSF of China 11371297 and 
 NSF of Fujian Province 2015J01592.


 \begin{thebibliography}{99}

\bibitem{AM} E. Acerbi,  G. Mingione;
 Regularity results for stationary electrorheological fluids,
\emph{Arch. Ration. Mech. Anal.}, \textbf{164} (2002),   213-259.

\bibitem{AS1} S. Antontsev, S. Shmarev;
 Anisotropic parabolic equations with variable nonlinearity,
\emph{Publ. Mat.}, \textbf{53} (2009), 355-399.

\bibitem{AS2}  S. Antontsev, S. Shmarev;
 Parabolic equations with double variable nonlinearlities,
 \emph{Math. and Compu. in Simulation}, \textbf{81} (2011), 2018-2032.

\bibitem{FZ} X. L. Fan, D. Zhao;
On the spaces ${L^{p(x)}(\Omega)}$ and ${W^{m,p(x)}}$,
\emph{J. Math. Anal. Appl.}, \textbf{263} (2001), 424-446.

\bibitem{G} L. Gu;
 Second order parabolic partial differential equations (in chinese),
\emph{The Publishing Company of Xiamen University}, 2002.

\bibitem{KWZ}  Y. Kim. L. Wang, C. Zhang;
 Global bifurcation for a class of degenerate elliptic equations with
variables exponents, \emph{J. Math. Anal. Appl. } \textbf{371} (2010), 624-637.

\bibitem{LGYC} S. Lian, W. Gao, H.Yuan, C. Cao;
 Existence of solutions to an initial Dirichlet problem of evolutional
$p(x)$-Laplace equations, \emph{ Ann. I. H. Poincare -AN.}, \textbf{29} (2012), 377-399.

\bibitem{KR} O. Kovcik, J. Rkosnk;
 On spaces ${L^{p(x)}}$ and ${W^{k,p(x)}}$,
 \emph{Czechoslovak Math.J.}, \textbf{41} (1991), 592-618.

\bibitem{MB}  R. A. Mashiyev, O. M. Buhrii;
 Existence of solutions of the parabolic variational inequality
with variable exponent of nonlinearity, \emph{J. Math. Anal.
 Appl.} \textbf{377} (2011), 450-463.

\bibitem{R}  M. Ruzicka;
 \newblock{Electrorheological Fluids:
Modeling and Mathematical Theory}, \emph{Lecture Notes in Math.}, vol.1748,
Springer, Berlin, 2000.

\bibitem{TM} E.  Taylor Michael;
 Partial differential equations III, \emph{Springer-Verlag}, 1999.

\bibitem{Y} F. Yao;
 Holder regularity for the general parabolic $p(x,t)$-Laplacian equations,
\emph{Nonlinear Differential Equations and Applications.},
 DOI 10.1007/s00030-014-0277-y, 2014.

\bibitem{YW} J. Yin, C. Wang;
 Properties of the boundary flux of a singular diffusion process,
\emph{Chinese Annals of Mathematics, Ser.B}, \textbf{25} (2004),  175-182.

\bibitem{Z2}  H. Zhan;
 The boundary value condition of an evolutionary p(x)-Laplacian equation,
\emph{Boundary value problems},  \textbf{2015} (112) (2015), DOI
 10.1186/s13661-015-0377-6.

\bibitem{ZW}  H. Zhan, J. Wen;
 Evolutionary $p(x)$-Laplacian equation free from the limitation of the boundary
value,  \emph{Electron. J. Diffrential Equ.},  \textbf{143} (2016), 1-13.

\bibitem{ZZX} C. Zhang, S. Zhuo,  X. Xue;
Global gradient estimates for the parabolic $p(x,t)$-Laplacian equation,
\emph{Nonlinear Analysis.}, \textbf{105} (2014), 86-101.

\bibitem{Z1} V. V. Zhikov;
 On the density of smooth functions in Sobolev-Orlicz spaces,
\emph{Otdel.Mat.Inst.Steklov.(POMI)}, \textbf{310} (2004), 67-81,
translation in \emph{J. Math. Sci. (NY)}, \textbf{132} (2006), 285-294.

\end{thebibliography}

\end{document}