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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 46, pp. 1--21.\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{9mm}}

\begin{document}
\title[\hfilneg EJDE-2015/46\hfil Existence of solutions]
{Existence of solutions to a parabolic $p(x)$-Laplace equation
with convection term via $L^\infty$ estimates}

\author[Z. Li, B. Yan, W. Gao \hfil EJDE-2015/46\hfilneg]
{Zhongqing Li, Baisheng Yan, Wenjie Gao}

\address{Zhongqing Li (corresponding author)\newline
College of Mathematics, Jilin University, Changchun 130012, China}
\email{zqli\_jlu@163.com}

\address{Baisheng Yan \newline
College of Mathematics, Jilin University, Changchun 130012,  China.\newline
Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA}
\email{yan@math.msu.edu}

\address{Wenjie Gao \newline
College of Mathematics, Jilin University, Changchun 130012, China}
\email{wjgao@jlu.edu.cn}

\thanks{Submitted November 28, 2014. Published February 17, 2015.}
\subjclass[2000]{35K65, 35K55, 46E35}
\keywords{Parabolic $p(x)$-Laplace equation; convection term;
\hfill\break\indent De Giorgi iteration; $L^\infty$ estimates}

\begin{abstract}
 This article is devoted to the study of the existence of weak solutions
 to an initial and boundary value problem for a parabolic $p(x)$-Laplace
 equation with convection term. Using the De Giorgi iteration technique,
 the authors establish the critical a priori $L^\infty$-estimates and thus
 prove the existence of weak solutions.
\end{abstract}

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

\section{Introduction}

In this article, we consider the  initial and boundary value problem for
parabolic $p(x)$-Laplace equation
\begin{equation}\label{1}
\begin{gathered}
\frac{\partial{u}}{\partial{t}}-
\operatorname{div}\big(|\nabla u|^{p(x)-2}\nabla u\big)
=B(x,t)|\nabla u|^{p(x)}-\operatorname{div}\overrightarrow{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)\in L^\infty(\Omega), \quad x\in\Omega.
\end{gathered}
\end{equation}
Here, $\Omega\subset \mathbb{R}^N$ is a bounded domain
with smooth boundary $\partial\Omega$,  $Q_T=\Omega\times(0,T)$,
$\Gamma_T=\partial\Omega\times(0,T)$,\, $T>0$ is finite, and
$p(x)$, $B(x,t)$, $\overrightarrow{F}(x,t)$ are given quantities satisfying
conditions to be specified later.

Recently, partial differential equations involving variable exponents,
such as the $p(x)$-Laplace equation in \eqref{1}, have been extensively
investigated, owing to their physical importance and powerful application.
The mathematical model of Problem \eqref{1} originates from heat and mass
transfer in nonhomogeneous media and non-Newtonian fluids with thermo-convective
effects \cite{MR2246902}. Equations of this type also appear in the study
of digital image recovery \cite{MR2246061} and electrorheological fluids
\cite{MR1810360}. It  describes the evolution diffusion and filtration process.
In particular, the models  like \eqref{1} with variable exponent provide
a good mathematical interpretation for the mechanical properties of certain
viscous electrorheological fluids characterized by their abilities to undergo
significant changes when an electric field is applied.

We focus on  mathematical analysis concerning the existence of solutions
to Problem \eqref{1}. Similar problems with constant exponents or $L^1$
data have been studied  by many authors;
see, e.g., \cite{MR2677803, MR1378470,  MR924524,  MR1191957, MR1747629,
MR1307456, MR2593046, MR1776929}.
To study our problem, we encounter several difficulties arising from the
variable exponents.
To deal with  \eqref{1}, one must face the typical difficulty of how
to define the solution space to \eqref{1}.
When $p(x)=p$ is a constant, it is well known that $L^p(0,T;W^{1,p}_0(\Omega))$
 can be taken as the solution space.
However, in the nonconstant case and $p^-=\inf p(x) >1$,
if the solution space is defined to be $L^{p(x)}(0,T;W^{1,p(x)}_0(\Omega))$,
or $L^{p^-}(0,T;W^{1,p(x)}_0(\Omega))$, etc.,
then it leads to an unfavorable fact that the $p(x)$-Laplace operator
is not bounded and not continuous from this space into its dual.
To conquer this difficulty, we adopt the appropriate solution space $V$
as defined below,
which helps us to define a weak solution to \eqref{1}. However,
other difficulties arise from it at the same time.
On one hand, one must verify the chain rule in the variable exponent space,
 as given in Lemma \ref{lemma 1.3} with its proof in the Appendix,
even if this is an obvious fact in the case when $p$ is a constant
\cite{MR1378470,MR924524}.
On the other hand, we will get the existence result for Problem \eqref{1}
through a limit process in which Simon's compactness theorem \cite{MR916688}
plays a crucial role.
Nevertheless, the solution space $V$ prevents from directly employing the theorem.
We take into account the properties associated with $V$ and surmount this difficulty.
There are other differences between the variable exponent case and the constant
exponent case.
Some important properties and inequalities are no longer valid.
For example, the variable exponent spaces are not translation invariant,
Young's inequality with convolution
$\|f\ast g\|_{p(\cdot)}\leq C\|f\|_{p(\cdot)}\|g\|_1$
holds if and only if $p$ is constant,
and for $u\in W_0^{1,p(x)}(\Omega)$,
$\int_\Omega|u|^{p(x)}dx\leq C\int_\Omega|\nabla u|^{p(x)}dx$
is not valid for the variable exponent $p$, etc.; we refer to
monograph \cite{MR2790542} for details and more references.

To define an appropriate solution space for Problem \eqref{1}, we make
the following hypotheses on the quantities appearing in \eqref{1}.
\begin{itemize}
\item[(H1)]  $p\in C(\overline{\Omega})$,
and $p^+:=\max_{\overline{\Omega}}p(x)$,
$p^-:=\min_{\overline{\Omega}}p(x)$ satisfy  $1<p^-\leq p^+<+\infty$; furthermore, there exists a positive constant $C$ such that the following log-H\"{o}lder continuous condition holds:
\begin{equation}\label{log-H}
|p(x)-p(y)|\leq\frac{-C}{\log|x-y|} \quad \text{for every $x,y\in\Omega$
satisfying $|x-y|\leq\frac{1}{2}$.}
\end{equation}


\item[(H2)] $B\in L^\infty(Q_T)$ satisfies $0\le B(x,t)\le b$, where
$b>0$ is a constant, and $\overrightarrow{F}$ is a vector field satisfying
$|\overrightarrow{F}|^{(p^-)'}\in L^r(Q_T)$, where $(p^-)'=\frac{p^-}{p^--1}$
 and $r>\frac{N+p^-}{p^-}$.
Hence, $\overrightarrow{F}\in \big(L^{p'(x)}(Q_T)\big)^N$ as
$|\overrightarrow{F}|\in L^{(p^-)'}(Q_T)\hookrightarrow L^{p'(x)}(Q_T)$;
see the relevant definitions below.
\end{itemize}
We remark that, when $p$ is a constant, it is well known that
$W^{1,p}_0(\Omega)$ (the closure of $C_0^\infty(\Omega)$ in $W^{1,p}(\Omega)$)
is identical to
$H_0^{1,p}(\Omega):=\{f\in L^p(\Omega):|\nabla f|\in L^{p}(\Omega) \text{ with }
 f|_{\partial\Omega}=0\}$.
However, when $p$ is a function, there exists an interesting Lavrentiev
phenomenon \cite{MR1486765},
which shows that the above two space are not equivalent.
The log-H\"{o}lder continuous condition \eqref{log-H} above guarantees an
important fact that $C_0^\infty(\Omega)$ is dense in
$W^{1,p(x)}(\Omega)$ \cite{MR2120185}. Under this condition, one can define
variable Sobolev spaces with homogeneous boundary values, $W_0^{1,p(x)}(\Omega)$,
as the closure of $C_0^\infty(\Omega)$ in $W^{1,p(x)}(\Omega)$; moreover,
the condition makes $p(x)$-Poincar\'{e}'s inequality hold
\cite{MR2737220, MR1134951, MR2593046}.


We introduce the function space
\[
V=\{v\in L^{p^-}(0,T;W^{1,p(x)}_0(\Omega)):  |\nabla v|\in L^{p(x)}(Q_T)\},
\]
endowed with the norm $\|u\|_V=|\nabla u|_{L^{p(x)}(Q_T)}$,
or the equivalent norm
$\|u\|_V=|u|_{L^{p^-}(0,T;W^{1,p(x)}_0(\Omega))}+|\nabla u|_{L^{p(x)}(Q_T)}$;
the equivalence follows from $p(x)$-Poincar\'{e}'s inequality.
Then $V$ is a separable and reflexive Banach space
(see \cite{MR2677803,MR2593046}).

We now give the definition of weak solutions to Problem \eqref{1}.

\begin{definition} \label{def1.1} \rm
We say that $u\in V\cap L^\infty(Q_T)$ is a weak solution to  \eqref{1},
provided that $u_t\in V^\ast+L^1(Q_T)$, $u(x,0)=u_0(x)$ in $L^{p^-}(\Omega)$,
and
\begin{equation}\label{definition 1.1}
\begin{aligned}
&\int_0^T\langle u_t,\phi\rangle dt
 +\int_0^T\int_\Omega|\nabla u|^{p(x)-2}\nabla u\cdot\nabla\phi \,dx\,dt\\
&= \int_0^T\int_\Omega B  |\nabla u|^{p(x)}\phi \,dx\,dt
 +\int_0^T\int_\Omega\nabla\phi\cdot\overrightarrow{F}\,dx\,dt
\end{aligned}
 \end{equation}
holds for every  $\phi(x,t)\in V \cap L^\infty(Q_T)$.
Here, with $u_t=\alpha^{(1)}+\alpha^{(2)}\in V^\ast+L^1(Q_T)$,
it is understood  that
\[
\int_0^T\langle u_t,\phi\rangle dt
:=\langle u_t,\phi\rangle_{V^\ast+L^1(Q_T),V \cap L^\infty(Q_T)}
=\langle \alpha^{(1)},\phi\rangle_{V^\ast,V}
+\int_0^T\int_\Omega\alpha^{(2)}\phi \,dx\,dt.
\]
\end{definition}

When $p(x) = p$ is a constant, sup-/sub-solution method is powerful and direct
to the existence results
(see \cite{MR924524}).
Nevertheless, it is not suitable to our problem because, due to the complicated
nonlinearities of $p(x)$-Laplace,
it may be quite difficult to construct a supsolution $\overline{u}$ and
a subsolution $\underline{u}$ in $V$
which simultaneously satisfy $\underline{u}\leq\overline{u}$.
Roughly speaking, in Equation \eqref{1}, the growth power of
$|\nabla u|^{p(x)-2}\nabla u$
at the left-hand side of \eqref{1} is less than that of the convection term
$|\nabla u|^{p(x)}$ at the right-hand side,
which leads us not to directly utilizing pseudo-monotone operator method
\cite{MR0259693}.
Instead, to obtain the existence of weak solutions to Problem \eqref{1},
we will employ the $L^\infty$ estimate method and get the solution through a
limit process to the approximate equations.
We carry out the De Giorgi iteration, different from the classical constant
exponent case
(see \cite{MR1776929,MR1378470,MR1191957} and the excellent and elegant
argument therein), in the setting of variable exponent.
We first give a general form of \cite[Theorem 5.1]{MR1378470} or
\cite[Lemma 1]{MR1776929}, as stated in \eqref{lemma 1.2-2},
by  which we obtain the $L^\infty$ regularity under the classification
when $p^-\geq2$ and when $1<p^-<2$,
other than the classification appeared in \cite{MR1776929}.
It should be remarked that, we employ the infimum of $p(x)$,
which facilitates this iteration, however, on the other side of the coin,
it makes the iteration process more technical and complexity. By the way,
 our result in Theorem \ref{thm2.1} shows an interesting phenomenon:
the uniformly $L^\infty$ bound of $u$ can depend on $p^-$ other than $p(x)$
itself as in the constant exponent case \cite{MR1776929}.
In the limit process, the properties of solution space $V$ and its related
variable exponent space will be frequently used,
which is one of the features in the equation with variable exponent.

The plan of this paper is as follows.
In section 2, we apply the De Giorgi iteration to Problem \eqref{1}
to obtain a uniform bound for the {\em bounded} weak solution $u\in V$;
this a priori $L^\infty$-assumption   is crucial for such a uniform bound,
 as  in \cite{MR1378470,MR1191957}.
In section 3, we construct an approximation equation to Problem \eqref{1}.
 Based on the uniform bound of $u_n$, we obtain the strong convergence of
$u_n$ in the solution space $V$, by virtue of which we establish
the existence of solutions. Section 4 is an Appendix in which we give
some brief proofs to some lemmas in the paper.

To conclude this section, we recall some preliminary results on
the Lebesgue and Sobolev spaces with variable exponents;
for more details, see  \cite{MR1866056,MR1134951} or monograph
\cite{MR2790542,MR1810360}.
Let $p$ be a continuous function defined in $\overline{\Omega}$,
$p(x)>1$, for any $x\in\overline{\Omega}$.

 1. The space
\[
L^{p(x)}(\Omega):=\big\{u: \text{$u$ is measurable in $\Omega$ and
$\int_\Omega|u(x)|^{p(x)}dx<\infty$}\big\}.
\]
This  space  is equipped with the  Luxemburg's norm
\[
|u|_{L^{p(x)}(\Omega)}:=\inf\big\{\lambda>0:\int_\Omega|\frac{u(x)}{\lambda}
|^{p(x)}dx\leq1\big\}.
\]
The space $\left(L^{p(x)}(\Omega),|\cdot|_{L^{p(x)}(\Omega)}\right)$
is a separable, uniformly convex Banach space.

2. The space
\[
W^{1,p(x)}(\Omega):=\left\{u\in L^{p(x)}(\Omega):|\nabla u|\in L^{p(x)}(\Omega)\right\},
\]
endowed with the  norm
\[
|u|_{W^{1,p(x)}({\Omega})}:=|\nabla u|_{L^{p(x)}(\Omega)}+|u|_{L^{p(x)}(\Omega)}.
\]
We denote by $W^{1,p(x)}_0(\Omega)$ the closure of $C^\infty_0(\Omega)$
in $W^{1,p(x)}(\Omega)$.
In fact, the norm $|\nabla u|_{L^{p(x)}(\Omega)}$ and
$|u|_{W^{1,p(x)}({\Omega})}$ are equivalent norms in $W^{1,p(x)}_0(\Omega)$.
$W^{1,p(x)}({\Omega})$ and $W^{1,p(x)}_0(\Omega)$ are separable and reflexive
Banach spaces.


 3. Frequently used relationships for the estimates.
\[
 \min\big\{|u|_{L^{p(x)}(\Omega)} ^{p^-}, \; |u|_{L^{p(x)}(\Omega)} ^{p^+} \big\}
\leq  \int_\Omega|u(x)|^{p(x)}dx
\leq  \max\big \{|u|_{L^{p(x)}(\Omega)} ^{p^-}, \;
 |u|_{L^{p(x)}(\Omega)} ^{p^+} \big\}.
\]
Consequently,
\[
|u_k-u|_{L^{p(x)}(\Omega)}\to 0\Longleftrightarrow
\int_\Omega|u_k-u|^{p(x)}dx\to 0.
\]

 4. $p(x)$-H\"{o}lder's inequality:
 For any $u\in L^{p(x)}(\Omega)$ and $v\in L^{p'(x)}(\Omega)$,
with $\frac{1}{p(x)}+\frac{1}{p'(x)}=1$, we have
\[
\Big|\int_\Omega uv\,dx\Big|
\leq\big(\frac{1}{p^-}+\frac{1}{({p'})^-}\big)|u|_{L^{p(x)}(\Omega)}
 |v|_{L^{p'(x)}(\Omega)}
\leq 2|u|_{L^{p(x)}(\Omega)}|v|_{L^{p'(x)}(\Omega)}.
\]

5. Embedding relationships:
If $p_1$ and $p_2$ are in $C(\overline{\Omega})$, and $1\leq p_1(x)\leq p_2(x)$,
for any $x\in\overline{\Omega}$, then there exists a positive constant
$C_{p_1(x),p_2(x)}$ such that
\[
|u|_{L^{p_1(x)}(\Omega)}\leq C_{p_1(x),p_2(x)}|u|_{L^{p_2(x)}(\Omega)}.
\]
i.e. the embedding $L^{p_2(x)}(\Omega)\hookrightarrow L^{p_1(x)}(\Omega)$
is continuous.
If $q\in C(\overline{\Omega})$ and $1\leq q(x)<p^*(x)$, for any
$x\in\overline{\Omega}$, then the embedding
$W_0^{1,p(x)}(\Omega)\hookrightarrow L^{q(x)}(\Omega)$ is continuous
and compact, where
\[
p^*(x):=\begin{cases}
  \frac{Np(x)}{N-p(x)},&p(x)<N,\\
  +\infty,             &p(x)\geq N.
\end{cases}
\]

 6. $p(x)$-Poincar\'{e}'s inequality:
Under the condition \eqref{log-H}, there exists a positive constant $C_p$ such that
\[
|u|_{L^{p(x)}(\Omega)}\leq C_p|\nabla u|_{L^{p(x)}(\Omega)},\quad
\text{for all } u\in W_0^{1,p(x)}(\Omega).
\]

\section{A priori bounds}

First of all, we give some technical lemmas frequently used in the process of
De Giorgi iteration. In particular, \eqref{lemma 1.2-2} can be seen as
a general form of \cite[Theorem 5.1]{MR1378470} or \cite[Lemma 1]{MR1776929}.
Their proofs will be given in the Appendix for the convenience of the readers.

\begin{lemma}\label{lemma 1.1}
Assume that $a, b, \lambda$ are positive constants, with
$\lambda\geq\frac{1}{2}+\frac{b}{a}$.
Define
\begin{equation}\label{lemma 1.1-0}
  \varphi(s)=
  \begin{cases}
e^{\lambda s}-1, &s\geq0,\\
-e^{-\lambda s}+1, &s\leq0.
  \end{cases}
\end{equation}
Then the following properties hold:
\begin{enumerate}
\item   For all $s\in\mathbb{R}$,
\begin{equation}\label{lemma 1.1-1}
 |\varphi(s)|\geq \lambda |s|,\quad
a\varphi'(s)-b|\varphi(s)|\geq\frac{a}{2}e^{\lambda|s|}.
\end{equation}

\item There exist constants $d\geq0$ and $M>1$ such that, for all $s\geq d$,
\begin{equation}\label{lemma 1.1-2}
\varphi'(s)\leq\lambda M\big[\varphi\big(\frac{s}{p^-}\big)\big]^{p^-},\quad
\varphi(s)\leq M\big[\varphi\big(\frac{s}{p^-}\big)\big]^{p^-}.
\end{equation}

\item Let $\Phi(s)=\int_0^s\varphi(\sigma)d\sigma$. If $p^-\geq2$,
then there exists a positive constant $c^\ast$ such that
\begin{equation}\label{lemma 1.2-1}
 \Phi(s)\geq c^\ast\big[\varphi\big(\frac{s}{p^-}\big)\big]^{p^-}, \quad \forall
 s\ge 0;
\end{equation}
if $1<p^-<2$, then there exist $d\geq0$ and $c^\ast=c^\ast(p^-,d)$ such that
\begin{equation}\label{lemma 1.2-2}
 \begin{gathered}
 \Phi(s)\geq c^\ast\big[\varphi\big(\frac{s}{p^-}\big)\big]^{p^-}, \quad\forall  s\geq d,\\
  \Phi(s)\geq c^\ast\big[\varphi\big(\frac{s}{p^-}\big)\big]^2, \quad\forall
 0\leq s\leq d.
\end{gathered}
\end{equation}
\end{enumerate}
\end{lemma}


\begin{lemma}\label{lemma 1.3}
Assume that function $\pi:\mathbb{R}\to  \mathbb{R}$ is piecewise $C^1$ with
$\pi(0)=0$ and $\pi'=0$ outside a compact set.
Let  $\Pi(s)=\int_0^s\pi(\sigma)d\sigma$.
If $u\in V$ with $u_t\in V^\ast+L^1(Q_T)$, then
\begin{equation}\label{lemma 1.3-1}
\int_0^T  \langle u_t,\pi(u)\rangle dt
= \langle u_t,\pi(u)\rangle_{V^\ast+L^1(Q_T),V \cap L^\infty(Q_T)}
=\int_\Omega\Pi(u(T))dx-\int_\Omega\Pi(u(0))dx.
\end{equation}
\end{lemma}

Using the lemmas above, we begin the De Giorgi iteration to get the a
priori $L^\infty$ estimate.

\begin{theorem}\label{thm2.1}
Let $u\in L^{\infty}(Q_T)\cap V$ be a weak solution to Problem \eqref{1}. Then
\[
\|u\|_{L^{\infty}(Q_T)}\leq \|u_0\|_{L^\infty(\Omega)}+C,
\]
where $C$ is a constant depending on
$p^-, N, T, r, b, \Omega, |||\overrightarrow{F}|^{(p^-)'}\|_{L^r(Q_T)}$,
but independent of $u$.
\end{theorem}

\begin{proof}
Let $k$ be a real number such that $k>\|u_0\|_{L^\infty(\Omega)}$ and
let $\varphi$ be the function  defined  in \eqref{lemma 1.1-0} with
 constant $\lambda\ge \frac12 +2b$, where $b>0$ is the constant in
Hypothesis (H2). (We shall use \eqref{lemma 1.1-1} with $a=1$ and
$a=1/2$ below.) Define
\[
G_k(u)= \begin{cases}
u-k, &\text{if }  u>k,\\
u+k, &\text{if }  u<-k,\\
0, &\text{if }  |u|\leq k.
\end{cases}
\]
Note that $u\in L^{\infty}(Q_T)\cap V$; so does $\varphi(G_k(u))$.
Then, for each $\tau\in[0,T]$,
one may choose $v=\varphi(G_k(u))\chi_{[0,\tau]}$ as a test function in
\eqref{definition 1.1}
(where $\chi_A$ is the characteristic function on the set $A$).
Noting  that $\nabla v=\chi_{[0,\tau]}\chi\{|u|>k\}\varphi'(G_k(u))\nabla u$,
we have
\begin{equation}\label{thm2.1-1}
\begin{split}
&\int_0^\tau\langle u_t,\varphi(G_k(u))\rangle dt
 +\int_0^\tau\int_\Omega|\nabla u|^{p(x)}\varphi'(G_k(u))\chi\{|u|>k\}\,dx\,dt\\
&= \int_0^\tau\int_\Omega B |\nabla u|^{p(x)}\varphi(G_k(u))\,dx\,dt
 +\int_0^\tau\int_\Omega\chi\{|u|>k\}\varphi'(G_k(u))
 \nabla u\cdot\overrightarrow{F}\,dx\,dt.
\end{split}
\end{equation}
Denote $A_k(t)=\left\{x\in\Omega:|u(x,t)|>k\right\}$.
In what follows, we write $\varphi=\varphi(G_k(u))$  and
$\varphi'=\varphi'(G_k(u))$ for simplicity.
 Thanks to the choice of $k$, one has
\begin{equation} \label{thm2.1-2}
\begin{aligned}
\int_0^\tau\langle u_t,\varphi(G_k(u))\rangle dt
&=\int_\Omega\Phi(G_k(u))(\tau)dx-\int_\Omega\Phi(G_k(u_0))dx\\
&=\int_{A_k(\tau)}\Phi(G_k(u))(\tau)dx-\int_{A_k(0)}\Phi(G_k(u_0))dx\\
&=\int_{A_k(\tau)}\Phi(G_k(u))(\tau)dx.
\end{aligned}
\end{equation}
From Young's inequality with $\epsilon$, it follows that
\begin{equation}\label{thm2.1-3}
\begin{aligned}
&\int_0^\tau\int_{A_k(t)}\varphi' \nabla u\cdot\overrightarrow{F}\,dx\,dt\\
&\leq\epsilon\int_0^\tau\int_{A_k(t)}|\nabla u|^{p^-}\varphi' \,dx\,dt
+C(\epsilon)\int_0^\tau\int_{A_k(t)}|\overrightarrow{F}|^{(p^-)'}\varphi' \,dx\,dt.
\end{aligned}
\end{equation}
Substituting \eqref{thm2.1-2} and \eqref{thm2.1-3}
in \eqref{thm2.1-1} yields
\begin{equation}\label{thm2.1-4}
\begin{split}
 &\int_{A_k(\tau)}\Phi(G_k(u))(\tau)dx
 +\int_0^\tau\int_{A_k(t)}|\nabla u|^{p(x)}\left(\varphi'-B|\varphi|\right)\,dx\,dt\\
&\leq\epsilon\int_0^\tau\int_{A_k(t)}|\nabla u|^{p^-}\varphi' \,dx\,dt
+C(\epsilon)\int_0^\tau\int_{A_k(t)}|\overrightarrow{F}|^{(p^-)'}\varphi' \,dx\,dt.
\end{split}
 \end{equation}
Note that $\varphi'-B|\varphi|\geq\varphi'-b|\varphi|
\geq\frac{1}{2}e^{\lambda|G_k(u)|}>0$
by \eqref{lemma 1.1-1} (with $a=1$). By utilizing
$|\nabla u|^{p(x)}\geq|\nabla u|^{p^-}-1$ and choosing $\epsilon=\frac{1}{2}$,
 we get from \eqref{thm2.1-4} that
\begin{equation} \label{thm2.1-5}
\begin{aligned}
&\int_{A_k(\tau)}\Phi(G_k(u))(\tau)dx
 +\int_0^\tau\int_{A_k(t)}|\nabla u|^{p^-}\big(\frac{1}{2}\varphi'-B|\varphi|\big)\,dx\,dt \\
&\leq C\int_0^\tau\int_{A_k(t)}|\overrightarrow{F}|^{(p^-)'}\varphi' \,dx\,dt
+\int_0^\tau\int_{A_k(t)}\left(\varphi'-B|\varphi|\right)\,dx\,dt \\
&\leq\int_0^\tau\int_{A_k(t)}\left(C|\overrightarrow{F}|^{(p^-)'}+1\right)\varphi' \,dx\,dt.
\end{aligned}
\end{equation}
Using \eqref{lemma 1.1-1} with $a=\frac12$, we have
$\frac12 \varphi'-B|\varphi|\geq\frac12 \varphi'-b|\varphi|
\geq\frac{1}{4}e^{\lambda|G_k(u)|}>0$.
Denoting $w_k=\varphi\left(\frac{|G_k(u)|}{p^-}\right)$,
we proceed to estimate \eqref{thm2.1-5},
\begin{equation}\label{thm2.1-6}
\begin{split}
 \int_0^\tau\int_{A_k(t)}|\nabla u|^{p^-}
\Big(\frac{1}{2}\varphi'-B|\varphi|\Big)\,dx\,dt
 &\geq\frac{1}{4}\int_0^\tau\int_{A_k(t)}|e^{\lambda\frac{|G_k(u)|}{p^-}}\nabla u|^{p^-}\,dx\,dt\\
 &\geq\frac{1}{4}\big(\frac{1}{\lambda}\big)^{p^-}
 \int_0^\tau\int_{A_k(t)}|\nabla w_k|^{p^-}\,dx\,dt.
\end{split}
\end{equation}
By definition,
$A_k(t)\setminus A_{k+d}(t)=t\{x\in\Omega:k<|u(x,t)|\leq k+d\}$;
hence $0<|G_k(u)|\leq d$
and $\varphi'(G_k(u))=\lambda e^{\lambda|G_k(u)|}\leq\lambda e^{\lambda d}$  on $A_k(t)\setminus A_{k+d}(t)$.
So, from \eqref{lemma 1.1-2}, it follows that
\begin{equation} \label{thm2.1-7}
\begin{aligned}
&\int_0^\tau\int_{A_k(t)}\left(C|\overrightarrow{F}|^{(p^-)'}+1\right)\varphi'
\,dx\,dt \\
&\leq\lambda M\int_0^\tau\int_{A_{k+d}(t)}
 \left(C|\overrightarrow{F}|^{(p^-)'}+1\right)|w_k|^{p^-}\,dx\,dt \\
&\quad +\int_0^\tau\int_{A_k(t)\setminus A_{k+d}(t)}
 \left(C|\overrightarrow{F}|^{(p^-)'}+1\right)\varphi' \,dx\,dt \\
&\leq\lambda M\int_0^\tau\int_{A_{k+d}(t)}h|w_k|^{p^-}\,dx\,dt
+\lambda e^{\lambda d}\int_0^\tau\int_{A_k(t)\setminus A_{k+d}(t)}h \,dx\,dt,
\end{aligned}
\end{equation}
where $h=C|\overrightarrow{F}|^{(p^-)'}+1$.
Putting \eqref{thm2.1-5}, \eqref{thm2.1-6} and \eqref{thm2.1-7}
together, we deduce
\begin{equation}\label{thm2.1-8}
\begin{split}
&\int_{A_k(\tau)}\Phi(G_k(u))(\tau)dx
+\frac{1}{4}\big(\frac{1}{\lambda}\big)^{p^-}
  \int_0^\tau\int_{A_k(t)}|\nabla w_k|^{p^-}\,dx\,dt\\
&\leq\lambda M\int_0^\tau\int_{A_{k+d}(t)}h|w_k|^{p^-}\,dx\,dt
+\lambda e^{\lambda d}\int_0^\tau\int_{A_k(t)\setminus A_{k+d}(t)}h \,dx\,dt.
     \end{split}
\end{equation}
\smallskip

\noindent\textbf{Case 1.  $p^-\geq2$.}
In this case, by \eqref{lemma 1.2-1}, one has
\begin{equation}\label{thm2.1.1-1}
 \int_{A_k(\tau)}\Phi(G_k(u))(\tau)dx
 \geq c^\ast\int_{A_k(\tau)}|w_k|^{p^-}dx.
\end{equation}
Substituting  \eqref{thm2.1.1-1} in \eqref{thm2.1-8}
and taking the supremum for $\tau\in[0,t_1]$, with $t_1\leq T$
to be determined later,  we have
\begin{equation}\label{thm2.1.1-2}
\begin{split}
&c^\ast\sup_{\tau\in[0,t_1]}\int_{A_k(\tau)}|w_k|^{p^-}dx
+\frac{1}{4}\big(\frac{1}{\lambda}\big)^{p^-}
  \int_0^{t_1}\int_{A_k(t)}|\nabla w_k|^{p^-}\,dx\,dt\\
&\leq\lambda M\int_0^{t_1}\int_{A_k(t)}h|w_k|^{p^-}\,dx\,dt
+\lambda e^{\lambda d}\int_0^{t_1}\int_{A_k(t)\setminus A_{k+d}(t)}h \,dx\,dt.
     \end{split}
\end{equation}
By the embedding inequality (see \cite{MR1230384,MR0241822}), we have
\begin{equation}\label{thm2.1.1-3}
\begin{aligned}
&\Big(\int_0^{t_1}\int_{A_k(t)}|w_k|^{p^-\frac{N+p^-}{N}}\,dx\,dt
 \Big)^\frac{N}{N+p^-}\\
&\leq\gamma  \Big(\sup_{\tau\in[0,t_1]}\int_{A_k(\tau)}|w_k|^{p^-}dx
+\int_0^{t_1}\int_{A_k(t)}|\nabla w_k|^{p^-}\,dx\,dt\Big),
\end{aligned}
\end{equation}
where $\gamma$ is a constant depending on $N,p^-$, but independent of
$t_1\le T$. Hence, from (\ref{thm2.1.1-2}), it follows  that
\begin{align*}
&J_{k_{t_1}}:=\Big(\int_0^{t_1}\int_{A_k(t)}|w_k|^{p^-\frac{N+p^-}{N}}\,dx\,dt
\Big)^\frac{N}{N+p^-}\\
&\leq C\Big( \int_0^{t_1}\int_{A_k(t)}h|w_k|^{p^-}\,dx\,dt
+ \int_0^{t_1}\int_{A_k(t)\setminus A_{k+d}(t)}h \,dx\,dt\Big),
\end{align*}
where $C$ is a constant independent of $t_1$. Consequently,
 by H\"{o}lder's inequality (thanks to the assumption
$|\overrightarrow{F}|^{(p^-)'}\in L^r(Q_T)$  with $r>\frac{N+p^-}{p^-}$), we deduce
\begin{flalign*}
J_{k_{t_1}}
& \leq C\Big(\int_0^{t_1}\int_{A_k(t)}|w_k|^{p^-\frac{N+p^-}{N}}\,dx\,dt\Big)^\frac{N}{N+p^-}
      \Big(\int_0^{t_1}\int_{A_k(t)}h^\frac{N+p^-}{p^-}\,dx\,dt\Big)^\frac{p^-}{N+p^-} \\
&\quad+C\Big(\int_0^{t_1}\int_{A_k(t)}h^r\,dx\,dt\Big)^{1/r}
   \Big(\int_0^{t_1}\mu(A_k(t))dt\Big)^{1-\frac{1}{r}} \\
&\leq C\Big(\int_0^{t_1}\int_{A_k(t)}|w_k|^{p^-\frac{N+p^-}{N}}\,dx\,dt
 \Big)^\frac{N}{N+p^-}  \|h\|_{L^r(Q_{t_1})}
  \big(t_1\mu(\Omega)\big)^{\frac{p^-}{N+p^-}-\frac{1}{r}} \\
&\quad+C\|h\|_{L^r(Q_{t_1})}
   \Big(\int_0^{t_1}\mu(A_k(t))dt\Big)^{1-\frac{1}{r}},
\end{flalign*}
where $\mu(\Omega)$ represents the Lebesgue measure of $\Omega$.
Choosing $t_1$ small enough such that
\begin{equation}\label{thm2.1.1-(1)}
 C\|h\|_{L^r(Q_{t_1})}\left(t_1\mu(\Omega)\right)^{\frac{p^-}{N+p^-}
 -\frac{1}{r}}\leq\frac{1}{2}
\end{equation}
and we obtain
\begin{equation}\label{thm2.1.1-5}
J_{k_{t_1}}\leq C\|h\|_{L^r(Q_T)}
\Big(\int_0^{t_1}\mu(A_k(t))dt\Big)^{1-\frac{1}{r}}.
\end{equation}

For any $l>k\geq\|u_0\|_{L^\infty(\Omega)}$, using \eqref{lemma 1.1-1},
we conclude that
\begin{equation} \label{thm2.1.1-6}
\begin{aligned}
  J_{k_{t_1}}
 &\geq\Big(\int_0^{t_1}\int_{A_k(t)}|
 \frac{\lambda G_k(u)}{p^-}|^{p^-\frac{N+p^-}{N}}\,dx\,dt\Big)^\frac{N}{N+p^-} \\
 &\geq\big(\frac{\lambda}{p^-}\big)^{p^-}
  \Big(\int_0^{t_1}\int_{A_k(t)}\big(|u|-k\big)^{p^-\frac{N+p^-}{N}}\,dx\,dt
 \Big)^\frac{N}{N+p^-} \\
&\geq\big(\frac{\lambda}{p^-}\big)^{p^-}
 (l-k)^{p^-}\Big(\int_0^{t_1}\mu({A_l(t)})dt\Big)^\frac{N}{N+p^-}.
\end{aligned}
\end{equation}
Let $\psi_k=\int_0^{t_1}\mu({A_k(t)})dt$.
It follows from \eqref{thm2.1.1-5} and \eqref{thm2.1.1-6} that
\begin{equation}\label{thm2.1.1-7}
\psi_l\leq\frac{C}{(l-k)^{\frac{p^-(N+p^-)}{N}}}
\psi_k^{(1-\frac{1}{r})\frac{N+p^-}{N}}.
\end{equation}
\smallskip

\noindent\textbf{Case 2.  $1<p^-<2$.} In this case,
from \eqref{lemma 1.2-2} (it should be remarked that the constant $d$
in \eqref{lemma 1.1-2} and \eqref{lemma 1.2-2} could be the same if we
choose $d$ suitably large), we have
\begin{equation}\label{thm2.1.2-1}
\int_{A_k(\tau)}\Phi(G_k(u))(\tau)dx
 \geq c^\ast\int_{A_{k+d}(\tau)}|w_k|^{p^-}dx
 +c^\ast\int_{A_k(\tau)\backslash A_{k+d}(\tau)}|w_k|^2dx.
\end{equation}
Substituting \eqref{thm2.1.2-1} into \eqref{thm2.1-8}
and taking the supremum for $\tau\in[0,t_1]$,
 where $t_1\leq T$ to be chosen later,  we derive
\begin{equation} \label{thm2.1.2-2}
\begin{aligned}
&c^\ast\sup_{\tau\in[0,t_1]}\int_{A_{k+d}(\tau)}|w_k|^{p^-}dx
 +\frac{1}{4}\big(\frac{1}{\lambda}\big)^{p^-}
   \int_0^{t_1}\int_{A_{k+d}(t)}|\nabla w_k|^{p^-}\,dx\,dt \\
&\quad +c^\ast\sup_{\tau\in[0,t_1]}\int_{A_k(\tau)\backslash A_{k+d}(\tau)}|w_k|^2dx
    +\frac{1}{4}\big(\frac{1}{\lambda}\big)^{p^-}
  \int_0^{t_1}\int_{A_k(t)\backslash A_{k+d}(t)}|\nabla w_k|^{p^-}\,dx\,dt \\
&\leq\lambda M\int_0^{t_1}\int_{A_{k+d}(t)}h|w_k|^{p^-}\,dx\,dt
+\lambda e^{\lambda d}\int_0^{t_1}\int_{A_k(t)\setminus A_{k+d}(t)}h \,dx\,dt.
\end{aligned}
\end{equation}
Again, recall the following embedding estimates \cite{MR1230384, MR0241822}:
\begin{gather}\label{thm2.1.2-3}
\begin{aligned}
&\int_0^{t_1}\int_{A_{k+d}(t)}|w_k|^{p^-\frac{N+p^-}{N}}\,dx\,dt\\
&\leq\gamma^{p^-\frac{N+p^-}{N}}
 \Big(\sup_{\tau\in[0,t_1]}\int_{A_{k+d}(\tau)}|w_k|^{p^-}dx
+\int_0^{t_1}\int_{A_{k+d}(t)}|\nabla w_k|^{p^-}\,dx\,dt\Big)^{1+\frac{p^-}{N}},
\end{aligned} \\
\label{thm2.1.2-4}
\begin{aligned}
&\int_0^{t_1}\int_{A_k(t)\backslash A_{k+d}(t)}|w_k|^{p^-\frac{N+2}{N}}\,dx\,dt\\
&\leq\gamma^{p^-\frac{N+2}{N}}
 \Big(\sup_{\tau\in[0,t_1]}\int_{A_k(\tau)\backslash A_{k+d}(\tau)}|w_k|^2dx\\
&\quad +\int_0^{t_1}\int_{A_k(t)\backslash A_{k+d}(t)}|\nabla w_k|^{p^-}\,dx\,dt
\Big)^{1+\frac{p^-}{N}}.
\end{aligned}
\end{gather}
Combining \eqref{thm2.1.2-3}, \eqref{thm2.1.2-4} with
\eqref{thm2.1.2-2}, we obtain
\begin{flalign*}
J^{(1)}_{k_{t_1}}
&:=\Big(\int_0^{t_1}\int_{A_{k+d}(t)}|w_k|^{p^-\frac{N+p^-}{N}}\,dx\,dt
 \Big)^{\frac{N}{N+p^-}} \\
&\quad +\Big(\int_0^{t_1}\int_{A_k(t)\backslash
 A_{k+d}(t)}|w_k|^{p^-\frac{N+2}{N}}\,dx\,dt\Big)^{\frac{N}{N+p^-}}\\
&\leq C\int_0^{t_1}\int_{A_{k+d}(t)}h|w_k|^{p^-}\,dx\,dt
   +C\int_0^{t_1}\int_{A_k(t)\backslash A_{k+d}(t)}h|w_k|^{p^-}\,dx\,dt\\
&\quad +C\int_0^{t_1}\int_{A_k(t)}h \,dx\,dt
:=(E1)+(E2)+(E3).
\end{flalign*}
We estimate $(E1)$ as follows.
\begin{align*}
&(E1)\\
&\leq C\Big(\int_0^{t_1}\int_{A_{k+d}(t)}|w_k|^{p^-\frac{N+p^-}{N}}\,dx\,dt
 \Big)^{\frac{N}{N+p^-}}
\Big(\int_0^{t_1}\int_{A_{k+d}(t)}h^{\frac{N+p^-}{p^-}}\,dx\,dt\Big)
 ^{\frac{p^-}{N+p^-}}\\
&\leq C\Big(\int_0^{t_1}\int_{A_{k+d}(t)}|w_k|^{p^-\frac{N+p^-}{N}}\,dx\,dt
 \Big)^{\frac{N}{N+p^-}}
\|h\|_{L^r(Q_{t_1})}\left(t_1\mu(\Omega)\right)^{\frac{p^-}{N+p^-}-\frac{1}{r}}.
\end{align*}
Using H\"{o}lder's inequality and Young's inequality with $\epsilon$, we have
\begin{align*}
&(E2)\\
&\leq C\Big(\int_0^{t_1}\int_{A_k(t)\backslash A_{k+d}(t)}
 |w_k|^{p^-\frac{N+2}{N}}\,dx\,dt\Big)^{\frac{N}{N+2}}
  \Big(\int_0^{t_1}\int_{A_k(t)\backslash A_{k+d}(t)}h^{\frac{N+2}{2}}\,dx\,dt
 \Big)^{\frac{2}{N+2}}\\
&\leq\frac{1}{2}\Big(\int_0^{t_1}\int_{A_k(t)\backslash A_{k+d}(t)}
 |w_k|^{p^-\frac{N+2}{N}}\,dx\,dt\Big)^{\frac{N}{N+p^-}}\\
&\quad +C\Big(\int_0^{t_1}\int_{A_k(t)\backslash A_{k+d}(t)}h^{\frac{N+2}{2}}\,dx\,dt
 \Big)^{\frac{2}{2-p^-}}\\
&\leq\frac{1}{2}\Big(\int_0^{t_1}\int_{A_k(t)\backslash A_{k+d}(t)}
 |w_k|^{p^-\frac{N+2}{N}}\,dx\,dt\Big)^{\frac{N}{N+p^-}}\\
&\quad +C\|h\|_{L^r(Q_{t_1})}^{\frac{N+2}{2-p^-}}
\Big(\int_0^{t_1}\mu(A_k(t))dt\Big)^{\frac{2}{2-p^-}(1-\frac{N+2}{2r})}.
\end{align*}
For $(E3)$, we have
\[
(E3)\leq C\|h\|_{L^r(Q_{t_1})} \Big(\int_0^{t_1}\mu(A_k(t))dt\Big)^{1-\frac{1}{r}}.
\]
Now select $t_1\in(0,(\mu(\Omega))^{-1}]$ sufficiently small so that
\begin{equation}\label{thm2.1.2-(1)}
 C\|h\|_{L^r(Q_{t_1})}\left(t_1\mu(\Omega)\right)^{\frac{p^-}{N+p^-}
 -\frac{1}{r}}\leq\frac{1}{2}.
\end{equation}
From the above estimates, we have
\begin{equation}\label{thm2.1.2-5}
\begin{aligned}
J^{(1)}_{k_{t_1}}
&\leq C\|h\|_{L^r(Q_{t_1})} \Big(\int_0^{t_1}\mu(A_k(t))dt\Big)^{1-\frac{1}{r}}\\
&\quad +C\|h\|_{L^r(Q_{t_1})}^{\frac{N+2}{2-p^-}}
 \Big(\int_0^{t_1}\mu(A_k(t))dt\Big)^{\frac{2}{2-p^-}(1-\frac{N+2}{2r})}.
\end{aligned}
\end{equation}
Noticing that $r>\frac{N+p^-}{p^-}$, after a straightforward computation,
 we have $\frac{2}{2-p^-}(1-\frac{N+2}{2r})>1-\frac{1}{r}$.
 Meanwhile, the choice of $t_1$ ensures $\psi_k\leq t_1\mu(\Omega)\leq1$.
As a result, \eqref{thm2.1.2-5} becomes
\begin{equation}\label{thm2.1.2-5'}
J^{(1)}_{k_{t_1}}
\leq C\Big(\int_0^{t_1}\mu(A_k(t))dt\Big)^{1-\frac{1}{r}}.
\end{equation}

For any $l>k\geq\|u_0\|_{L^\infty(\Omega)}$, using \eqref{lemma 1.1-1},
we deduce that
\begin{align*}
 J^{(1)}_{k_{t_1}}
&\geq\Big(\int_0^{t_1}\int_{A_{k+d}(t)}|\frac{\lambda G_k(u)}{p^-}
|^{p^-\frac{N+p^-}{N}}\,dx\,dt\Big)^{\frac{N}{N+p^-}}\\
&\quad +\Big(\int_0^{t_1}\int_{A_k(t)\backslash A_{k+d}(t)}
   |\frac{\lambda G_k(u)}{p^-}|^{p^-\frac{N+2}{N}}\,dx\,dt\Big)^{\frac{N}{N+p^-}}\\
&\geq\big(\frac{\lambda}{p^-}\big)^{p^-}
  \Big(\int_0^{t_1}\int_{A_{k+d}(t)}\big(|u|-k\big)^{p^-\frac{N+p^-}{N}}
 \,dx\,dt\Big)^\frac{N}{N+p^-}\\
&\quad +\big(\frac{\lambda}{p^-}\big)^{{p^-}\frac{N+2}{N+p^-}}
 \Big(\int_0^{t_1}\int_{A_k(t)\backslash A_{k+d}(t)}
   \big(|u|-k\big)^{p^-\frac{N+2}{N}}\,dx\,dt\Big)^{\frac{N}{N+p^-}} \\
&\geq\big(\frac{\lambda}{p^-}\big)^{p^-}(l-k)^{p^-}
  \Big(\int_0^{t_1}\mu\left({A_l(t)}\cap{A_{k+d}(t)}\right)dt\Big)^\frac{N}{N+p^-}\\
&\quad+\big(\frac{\lambda}{p^-}\big)^{{p^-}\frac{N+2}{N+p^-}}(l-k)^{{p^-}
 \frac{N+2}{N+p^-}}
 \Big(\int_0^{t_1}\mu\left({A_l(t)}\backslash{A_{k+d}(t)}\right)dt\Big)^{\frac{N}{N+p^-}}.
\end{align*}
In fact, we have
\begin{equation}\label{thm2.1.2-5''}
\begin{split}
 \left( J^{(1)}_{k_{t_1}}\right)^{\frac{N+p^-}{N}}
&\geq\big(\frac{\lambda}{p^-}\big)^{p^-\frac{N+p^-}{N}}(l-k)^{p^-\frac{N+p^-}{N}}
 \int_0^{t_1}\mu\left({A_l(t)}\cap{A_{k+d}(t)}\right)dt\\
&\quad+\big(\frac{\lambda}{p^-}\big)^{{p^-}\frac{N+2}{N}}(l-k)^{{p^-\frac{N+2}{N}}}
\int_0^{t_1}\mu\left({A_l(t)}\backslash{A_{k+d}(t)}\right)dt.
\end{split}
\end{equation}
Consequently, combining \eqref{thm2.1.2-5''} and \eqref{thm2.1.2-5'},
 with  $\psi_k=\int_0^{t_1}\mu({A_k(t)})dt$, we  have again
\begin{equation}\label{thm2.1.2-6}
\psi_l\leq\frac{C}{\min\big\{(l-k)^{\frac{p^-(N+p^-)}{N}},(l-k)
^{\frac{p^-(N+2)}{N}}\big\}}
\psi_k^{(1-\frac{1}{r})\frac{N+p^-}{N}}.
\end{equation}

Now we have proved \eqref{thm2.1.2-6} and \eqref{thm2.1.1-7}. Our hypothesis $r>\frac{N+p^-}{p^-}$ guarantees $\left(1-\frac{1}{r}\right)\frac{N+p^-}{N}>1$. Therefore, thanks to the iteration lemma in \cite{MR1776929}, we eventually obtain that
$\psi_{(\|u_0\|_{L^\infty(\Omega)}+D)}=0,$ where $D>0$ is a constant depending only on $p^-, N, t_1, r, b, \Omega,\||\overrightarrow{F}|^{(p^-)'}\|_{L^r(Q_{t_1})}$.
This proves that, for a fixed $\lambda$ validating  Lemma \ref{lemma 1.1},
\begin{equation}\label{thm2.1.1-9}
\|u(x,t)\|_{L^\infty(Q_{t_1})}\leq\|u_0\|_{L^\infty(\Omega)}+D.
\end{equation}


Finally, partition the time interval $[0,T]$ into finite subintervals
$[0,t_1]$, $[t_1,t_2]$ $\cdot\cdot\cdot$ $[t_{n-1},T]$
such that the conditions similar to those in \eqref{thm2.1.1-(1)}
and \eqref{thm2.1.2-(1)} are available for each subinterval $[t_i,t_{i+1}]$;
then, using the same method, we  deduce an inequality of the form
\eqref{thm2.1.1-9}.
Eventually, we conclude that
$\|u(x,t)\|_{L^\infty(Q_T)}\leq \|u_0\|_{L^\infty(\Omega)}+C$,
where the constant  $C$
depends only on $p^-, N, T, r, b, \Omega,
\||\overrightarrow{F}|^{(p^-)'}\|_{L^r(Q_T)}$.
\end{proof}

\section{Application to the existence of solutions to \eqref{1}}

With the $L^\infty$-estimate obtained above,
we can prove the existence of solutions to Problem \eqref{1}.
First, we recall a lemma from \cite{MR924524},
which plays an important role in our estimates.

\begin{lemma}\label{lemma 3.1}
Let $\theta(s)=se^{\eta s^2}$, $s\in\mathbb{R}$, where
$\eta\geq\frac{b^2}{4a^2}$ is fixed, and let
 $\Theta(s)=\int_0^s\theta(\tau)d\tau$. Then $\theta(0)=0$ and
\begin{equation} \label{lemma 3.1-2}
 \Theta(s)\geq0, \quad a\theta'(s)-b|\theta(s)|\geq\frac{a}{2}, \quad\forall \, s\in\mathbb{R}.
\end{equation}
\end{lemma}

We are now in a position to prove the existence of solutions to \eqref{1}
based on the $L^\infty$ estimate.

\begin{theorem}\label{thm3.1}
Under the hypotheses {\rm (H1)} and {\rm (H2)}, there exists a solution
 $u\in L^\infty(Q_T)\cap V$ to  \eqref{1}.
\end{theorem}

\begin{proof}
\textbf{Step 1: The approximation equation.}
We introduce the following approximation equation of Problem \eqref{1}.
\begin{equation}\label{thm3.1-0}
\begin{gathered}
\frac{\partial{u_n}}{\partial{t}}-
\operatorname{div}\left(|\nabla u_n|^{p(x)-2}\nabla u_n\right)
=B(x,t)\min\{|\nabla u_n|^{p(x)},n\}
 -\operatorname{div}\overrightarrow{F}(x,t), \\
 (x,t)\in Q_T,\\
u_n(x,t)=0, \quad (x,t)\in\Gamma_T,\\
u_n(x,0)=u_0(x)\in L^\infty(\Omega), \quad x\in\Omega.
\end{gathered}
\end{equation}
For each fixed $n\in \mathbb{N}$, since $\min\left\{|\nabla u_n|^{p(x)},n\right\}$
is bounded,  the existence of a weak solution $u_n\in L^{\infty}\cap V $
to \eqref{thm3.1-0} follows from the standard methods (for instance, the
pseudo-monotonicity operator theory in \cite{MR0259693,MR1134951,MR1033498},
or the difference and variation methods in \cite{MR2593046}).

We  write the term $B(x,t)\min\{|\nabla u_n|^{p(x)},n\}$
in \eqref{thm3.1-0} as $B_n(x,t)|\nabla u_n|^{p(x)}$, with
 $B_n(x,t)$  defined by
\[
B_n(x,t)=\begin{cases}
0, & \text{if } |\nabla u_n(x,t)|=0,\\[4pt]
B(x,t)\frac{\min\{|\nabla u_n(x,t)|^{p(x)},n\}}{|\nabla u_n(x,t)|^{p(x)}},
&\text{if }|\nabla u_n(x,t)|\ne 0.
\end{cases}
\]
Then $B_n\in L^\infty(Q_T)$ satisfies $0\le B_n(x,t)\le B(x,t)\le b$.
Hence, by  Theorem \ref{thm2.1}, we have the  uniform bound
\begin{equation}\label{thm3.1-1}
\|u_n(x,t)\|_{L^\infty(Q_T)}
\leq\|u_0\|_{L^\infty(\Omega)}+C,
\end{equation}
where $C$ depends only on
 $p^-, N, T, r, b, \Omega, \||\overrightarrow{F}|^{(p^-)'}\|_{L^r(Q_T)}$
and it is independent of $n$.
Our goal is to show that a subsequence of the approximate solution
sequence $\{u_n\}$ converges to a measurable function $u$,
which coincides with a weak solution of Problem \eqref{1}.
\smallskip

\noindent\textbf{Step 2: The weak convergence  $u_n\rightharpoonup u$ in
$L^{p^-}(0,T;W^{1,p(x)}_0(\Omega))$.}
Choosing $\theta(u_n)$ as a testing function in \eqref{thm3.1-0}, we have
\begin{equation}\label{thm3.1-2}
\begin{split}
&\int_0^T\langle \frac{\partial{u_n}}{\partial t},\theta(u_n)\rangle dt
 +\iint_{Q_T}|\nabla u_n|^{p(x)}\theta'(u_n)\,dx\,dt\\
&= \iint_{Q_T} B\min\{|\nabla u_n|^{p(x)},n\}\theta(u_n)\,dx\,dt
 +\iint_{Q_T}\theta'(u_n)\nabla{u_n}\cdot\overrightarrow{F}\,dx\,dt.
\end{split}
\end{equation}
Lemma \ref{lemma 1.3} yields
$\int_0^T\langle \frac{\partial{u_n}}{\partial t},\theta(u_n)\rangle dt
=\int_\Omega\left[\Theta(u_n(T))-\Theta(u_0)\right]dx$.
Using Young's inequality with $\epsilon$ in the last term of the right-hand
side, \eqref{thm3.1-2} becomes
\begin{align*}
&\int_\Omega\Theta(u_n(T))dx
+\iint_{Q_T}|\nabla u_n|^{p(x)}\theta'(u_n)\,dx\,dt\\
&\leq \int_\Omega\Theta(u_0)dx+ \iint_{Q_T} B |\nabla u_n|^{p(x)}|\theta(u_n)|\,dx\,dt\\
&\quad+\epsilon\iint_{Q_T}|\nabla u_n|^{p(x)}\theta'(u_n)\,dx\,dt
+\iint_{Q_T}\epsilon^{-\frac{1}{p(x)-1}}|\overrightarrow{F}|^{p'(x)}\theta'(u_n)\,dx\,dt.
\end{align*}
Taking $\epsilon=1/2$, we rewrite the above inequality as
\begin{equation}\label{thm3.1-3}
\begin{split}
&\int_\Omega\Theta(u_n(T))dx
+\iint_{Q_T}\left[\frac{1}{2}\theta'(u_n)-B|\theta(u_n)|\right]|\nabla u_n|^{p(x)}\,dx\,dt\\
&\leq \int_\Omega\Theta(u_0)dx
+\left(\frac{1}{2}\right)^{-\frac{1}{p^--1}}
\iint_{Q_T}|\overrightarrow{F}|^{p'(x)}\theta'(u_n)\,dx\,dt.
\end{split}
\end{equation}
With the aid of \eqref{lemma 3.1-2} in Lemma \ref{lemma 3.1} (with $a=\frac{1}{2}$,
and $\frac{1}{2}\theta'(u_n)-B|\theta(u_n)|
\geq\frac{1}{2}\theta'(u_n)-b|\theta(u_n)|\geq\frac{1}{4}$), we deduce that
\begin{equation}\label{thm3.1-4}
\frac{1}{4}\iint_{Q_T}|\nabla u_n|^{p(x)}\,dx\,dt
\leq \int_\Omega\Theta(u_0)dx
+\left(\frac{1}{2}\right)^{-\frac{1}{p^--1}}
\iint_{Q_T}|\overrightarrow{F}|^{p'(x)}\theta'(u_n)\,dx\,dt.
\end{equation}
Since $u_n$ is uniformly bounded with respect to $n$ and
$u_0\in L^\infty(\Omega)$, it follows that
\begin{equation}\label{thm3.1-5}
\iint_{Q_T}|\nabla u_n|^{p(x)}\,dx\,dt
\leq C\Big(|\overrightarrow{F}|_{L^{p'(x)}(Q_T)},
\|u_0\|_{L^\infty(\Omega)},\sup_n\|u_n\|_{L^\infty(Q_T)}\Big).
\end{equation}
This implies that $u_n$ is uniformly bounded in $V$. By the way, obviously,
the following inequality holds
\begin{align*}
&|u_n|^{p^-}_{L^{p^-}(0,T;W^{1,p(x)}_0(\Omega))}\\
&=\int_0^T|\nabla u_n|^{p^-}_{L^{p(x)}(\Omega)}dt\\
&\leq\max\Big\{\Big(\iint_{Q_T}|\nabla u_n|^{p(x)}\,dx\,dt
 \Big)^{\frac{p^-}{p^+}}T^{1-\frac{p^-}{p^+}},
\iint_{Q_T}|\nabla u_n|^{p(x)}\,dx\,dt\Big\},
\end{align*}
which implies
\begin{equation}\label{thm3.1-6}
|u_n|_{L^{p^-}(0,T;W^{1,p(x)}_0(\Omega))}
\leq C\Big(|\overrightarrow{F}|_{L^{p'(x)}(Q_T)},
 \|u_0\|_{L^\infty(\Omega)},\sup_n\|u_n\|_{L^\infty(Q_T)},p^-,p^+,T\Big).
\end{equation}
Therefore, $u_n$ is bounded in the space 
$L^\infty(Q_T)\cap L^{p^-}(0,T;W^{1,p(x)}_0(\Omega))$.
 We can extract a subsequence of $u_n$, still denoted by $u_n$,
such that $u_n\rightharpoonup u$, weakly in
$L^{p^-}(0,T;W^{1,p(x)}_0(\Omega))$. Simultaneously,
$u_n\rightharpoonup u$, weakly* in $L^\infty(Q_T)$.
\smallskip

\noindent \textbf{Step 3: The strong convergence  $u_n\to  u$ in
$L^{p^-}(0,T;L^{p(x)}(\Omega))$.}
From \eqref{thm3.1-0}, we deduce that
\begin{equation}\label{thm3.2-0}
\begin{split}
 \frac{\partial{u_n}}{\partial{t}}
=\operatorname{div}\left(|\nabla u_n|^{p(x)-2}\nabla u_n-\overrightarrow{F}\right)
+B\min\{|\nabla u_n|^{p(x)},n\}\in V^\ast+L^1(Q_T).
\end{split}
\end{equation}
For each $v\in V$, by the definition of the norm on $V$ and
$p(x)$-H\"{o}lder's inequality, we have
\begin{align*}
&\sup_{\|v\|_V\leq1}
|\langle\operatorname{div}\left(|\nabla u_n|^{p(x)-2}\nabla u_n-\overrightarrow{F}\right),v\rangle_{V^\ast,V}|\\
&=\sup_{\|v\|_V\leq1}
\big|\iint_{Q_T}\left(-|\nabla u_n|^{p(x)-2}\nabla u_n\cdot\nabla v
+\overrightarrow{F}\cdot\nabla v\right)\,dx\,dt\big|\\
&\leq \sup_{\|v\|_V\leq1}\big[
2||\nabla u_n|^{p(x)-2}\nabla u_n|_{L^{p'(x)}(Q_T)}
|\nabla v|_{L^{p(x)}(Q_T)}
+2|\overrightarrow{F}|_{L^{p'(x)}(Q_T)}|\nabla v|_{L^{p(x)}(Q_T)}\big]\\
&\leq 2\max\Big\{\Big(\iint_{Q_T}|\nabla u_n|^{p(x)}\,dx\,dt
 \Big)^{\frac{1}{(p')^+}},
\Big(\iint_{Q_T}|\nabla u_n|^{p(x)}\,dx\,dt\Big)^{\frac{1}{(p')^-}}\Big\}\\
&\quad +2|\overrightarrow{F}|_{L^{p'(x)}(Q_T)}.
\end{align*}
It follows from \eqref{thm3.1-5} that
\begin{equation}\label{thm3.2-1}
\big\|\operatorname{div}\big(|\nabla u_n|^{p(x)-2}\nabla u_n-\overrightarrow{F}\big)
\big\|_{V^\ast}\leq C,
\end{equation}
where $C$ is independent of $n$. Thanks to the embedding relationship
\begin{equation}\label{thm3.2-2}
\begin{aligned}
&L^{(p^-)'}(0,T;W^{-1,p'(x)}(\Omega))
\hookrightarrow V^\ast\\
&\hookrightarrow L^{(p^+)'}(0,T;W^{-1,p'(x)}(\Omega))
=L^{(p')^-}(0,T;W^{-1,p'(x)}(\Omega)),
\end{aligned}
\end{equation}
from \eqref{thm3.2-1}, \eqref{thm3.1-5} and \eqref{thm3.2-0},
 we conclude that $\frac{\partial{u_n}}{\partial{t}}$
is bounded in the space \\
$L^{(p')^-}(0,T;W^{-1,p'(x)}(\Omega))+L^1(Q_T)$.

For a fixed $s$  such that $s>\frac{N}{2}+1$, the following embedding
relationships hold $1^\diamond$ $s>\frac{N}{2}$, we have
$H_0^s(\Omega)\hookrightarrow L^\infty(\Omega)$,
   and then $L^1(\Omega)\hookrightarrow H^{-s}(\Omega)$;
$2^\diamond$ $s-1>\frac{N}{2}$, one has
$H_0^s(\Omega)\hookrightarrow W^{1,p(x)}(\Omega)$,
consequently, $W^{-1,p'(x)}(\Omega)\hookrightarrow H^{-s}(\Omega)$.
As a result, we have
\begin{equation}\label{thm3.2-3}
\|\frac{\partial{u_n}}{\partial{t}}\|_{L^1\left(0,T;H^{-s}(\Omega)\right)}\leq C,
\end{equation}
where $C$ is independent of $n$.
Noticing that
$W_0^{1,p(x)}(\Omega)\overset{\text{compact}}{\hookrightarrow}
L^{p(x)}(\Omega)\hookrightarrow H^{-s}(\Omega)$ and by
 \eqref{thm3.1-6},
we employ Simon's compactness theorem in \cite{MR916688}
to obtain that $u_n\to  u$, strongly in $L^{p^-}(0,T;L^{p(x)}(\Omega))$.
\smallskip

\noindent \textbf{Step 4: The convergence $\nabla u_n\to \nabla u$ a.e. in $Q_T$.}
From  the strong convergence obtained in Step 3,
one may choose a subsequence of $u_n$, still denoted by $u_n$ for simplicity,
such that $u_n\to  u$, a.e. in $Q_T$.
We now use  Egoroff's theorem (recalling $\mu(Q_T)<+\infty$) to obtain,
for fixed $\delta>0$, there exists a measurable closed subset
$E_\delta\subset Q_T$ such that
\begin{enumerate}
\item   $\mu(Q_T-E_\delta)\leq\delta$;
\item $u_n\rightrightarrows u$ uniformly on $E_\delta$.
 It follows that $|u_n-u_m|<k$, for fixed $k>0$, and sufficiently large $m,n$.
\end{enumerate}
Let $\zeta$ be a cut-off function satisfying
$\zeta\in C_0^\infty(Q_T)$; $\zeta=1$ on $E_\delta$; $0\leq\zeta\leq1$ on $Q_T$.
Introduce the following truncation function
\[
T_k(s)=  \begin{cases}
 s, &\text{if } |s|<k,\\
 k, &\text{if } s\geq k,\\
 -k,&\text{if } s\leq-k.
 \end{cases}
 \]
Subtracting Equations \eqref{thm3.1-0} for different parameters
 $n$ and $m$, we have
\begin{equation}\label{thm3.3-0}
\begin{gathered}
\begin{aligned}
&\frac{\partial({u_n-u_m})}{\partial{t}}-
\operatorname{div}\left(|\nabla u_n|^{p(x)-2}\nabla u_n
 -|\nabla u_m|^{p(x)-2}\nabla u_m\right)\\
&=B\left(\min\{|\nabla u_n|^{p(x)},n\}
 -\min\{|\nabla u_m|^{p(x)},m\}\right),    \quad &(x,t)\in Q_T,
\end{aligned}\\
(u_n-u_m)(x,t)=0, \quad (x,t)\in\Gamma_T,\\
(u_n-u_m)(x,0)=0, \quad x\in\Omega.
\end{gathered}
\end{equation}
Since $T_k$ is Lipschitz continuous, one may take $\zeta T_k(u_n-u_m)$
as a test function in \eqref{thm3.3-0}; hence we have
\begin{equation} \label{thm3.3-1}
\begin{aligned}
&\int_0^T\langle\frac{\partial({u_n-u_m})}{\partial{t}},\zeta T_k(u_n-u_m)\rangle dt
  \\
&+\iint_{Q_T}\big(|\nabla u_n|^{p(x)-2}\nabla u_n-|\nabla u_m|^{p(x)-2}\nabla u_m
\big)\cdot
\big(\nabla u_n-\nabla u_m\big)\zeta T_k'(u_n-u_m)\,dx\,dt \\
&+\iint_{Q_T}\big(|\nabla u_n|^{p(x)-2}\nabla u_n-|\nabla u_m|^{p(x)-2}\nabla u_m\big)\cdot
\nabla\zeta T_k(u_n-u_m)\,dx\,dt \\
&=\iint_{Q_T} B\left(\min\{|\nabla u_n|^{p(x)},n\}-\min\left\{|\nabla u_m|^{p(x)},m\right\}\right)\zeta T_k(u_n-u_m)\,dx\,dt.
\end{aligned}
\end{equation}
Since $\zeta(x,0)=\zeta(x,T)$, by   Lemma \ref{lemma 1.3},
we handle the first term on the left-hand side of \eqref{thm3.3-1} as follows,
\[
\int_0^T\langle\frac{\partial({u_n-u_m})}{\partial{t}},\zeta T_k(u_n-u_m)\rangle dt
=-\int_\Omega\int_0^T\zeta_t\int_0^{u_n-u_m}T_k(s)dsdtdx.
\]
Noticing that $T_k$ is an odd function, $|T_k(s)|\leq k$, we get
\begin{align*}
&\iint_{Q_T}\left(|\nabla u_n|^{p(x)-2}\nabla u_n-|\nabla u_m|^{p(x)-2}\nabla u_m\right)\cdot
\left(\nabla u_n-\nabla u_m\right)\zeta T_k'(u_n-u_m)\,dx\,dt\\
&\leq k\iint_{Q_T}|\zeta_t||u_n-u_m|\,dx\,dt\\
&\quad +k\iint_{Q_T}||\nabla u_n|^{p(x)-2}\nabla u_n-|\nabla u_m|^{p(x)-2}\nabla u_m|
|\nabla\zeta|\,dx\,dt\\
&\quad +bk\iint_{Q_T}|\min\{|\nabla u_n|^{p(x)},n\}
 -\min\{|\nabla u_m|^{p(x)},m\}|\zeta \,dx\,dt
\leq kC(\delta).
\end{align*}
Noting that $T_k'\geq0$, $T_k'(s)=1$ on $|s|<k$ and that $u_n$ converges
uniformly on $E_\delta$, we obtain
\begin{align*}
&\iint_{E_\delta}\big(|\nabla u_n|^{p(x)-2}\nabla u_n-|\nabla u_m|^{p(x)-2}
 \nabla u_m\big)\cdot
\left(\nabla u_n-\nabla u_m\right)\,dx\,dt\\
&=\iint_{E_\delta}\big(|\nabla u_n|^{p(x)-2}\nabla u_n-|\nabla u_m|^{p(x)-2}
\nabla u_m\big)\cdot
\left(\nabla u_n-\nabla u_m\right)T_k'(u_n-u_m)\,dx\,dt\\
&\leq \iint_{Q_T}\!\big(|\nabla u_n|^{p(x)-2}\nabla u_n-|\nabla u_m|^{p(x)-2}
 \nabla u_m\big)\cdot
\left(\nabla u_n-\nabla u_m\right)\zeta T_k'(u_n-u_m)\,dx\,dt.
\end{align*}
Hence, based on the above estimates, by \eqref{thm3.1-1},
\eqref{thm3.1-5} and the arbitrariness of $k$, we have
\begin{equation}\label{thm3.3-2}
\limsup_{n,m\to +\infty}\iint_{E_\delta}
\left(|\nabla u_n|^{p(x)-2}\nabla u_n-|\nabla u_m|^{p(x)-2}\nabla u_m\right)\cdot
\left(\nabla u_n-\nabla u_m\right)\,dx\,dt=0.
\end{equation}
From  \eqref{thm3.3-2} and using the method in \cite{MR1398379,MR1776929}
(or the method to be used in Step 5 below), we may obtain that
$\iint_{E_\delta}|\nabla u_n-\nabla u_m|^{p(x)}\,dx\,dt\to 0$
(it is equivalent to $|\nabla u_n-\nabla u_m|_{L^{p(x)}(E_\delta)}\to 0$),
which shows that $\{\nabla u_n\}_{n=1}^\infty$ is a Cauchy sequence in
 $(L^{p(x)}(E_\delta))^N$.
Thus, we can extract a subsequence of $u_n$, still denoted by itself,
such that $\nabla u_n\to  \alpha$, strongly in $(L^{p^-}(E_\delta))^N$.
In step 3, we know that $u_n\to  u$,
strongly in $L^{p^-}(0,T;L^{p(x)}(\Omega))$, it is easy to say $u_n\to  u$,
strongly in $L^{p^-}(E_\delta)$.
It follows from above analysis that $\alpha=\nabla u$, i.e.
$\nabla u_n\to \nabla u$ a.e. in $E_\delta$.
The arbitrariness of $\delta$ indicates that
$\nabla u_n\to \nabla u$ a.e. in $Q_T$.
\smallskip

\noindent\textbf{Step 5: The convergence
$\iint_{Q_T}|\nabla u_n-\nabla u|^{p(x)}\,dx\,dt\to 0$.}
For  the function $\theta$ defined in Lemma \ref{lemma 3.1}, it follows that
$\theta(u_n-u_m)\in L^\infty(Q_T)\cap V$ since
$u_n, u_m\in L^\infty(Q_T)\cap V$. Therefore, $\theta(u_n-u_m)$ can be taken as a test function in \eqref{thm3.3-0}  to yield that
\begin{equation} \label{thm3.4-1}
\begin{aligned}
&\int_0^T\langle\frac{\partial({u_n-u_m})}{\partial{t}},\theta(u_n-u_m)\rangle dt \\
&+\iint_{Q_T}\left(|\nabla u_n|^{p(x)-2}\nabla u_n-|\nabla u_m|^{p(x)-2}\nabla u_m\right)\cdot
\left(\nabla u_n-\nabla u_m\right)\theta'(u_n-u_m)\,dx\,dt \\
&=\iint_{Q_T}B\left(\min\left\{|\nabla u_n|^{p(x)},n\right\}
-\min\left\{|\nabla u_m|^{p(x)},m\right\}\right)\theta(u_n-u_m)\,dx\,dt.
\end{aligned}
\end{equation}
Use \eqref{lemma 3.1-2} in Lemma \ref{lemma 3.1} to estimate the first
term on the left-hand side of \eqref{thm3.4-1} to obtain
\[
\int_0^T\langle\frac{\partial({u_n-u_m})}{\partial{t}},\theta(u_n-u_m)\rangle dt
=\int_\Omega\Theta(u_n-u_m)(T)dx\geq0.
\]
After a simple computation,  the right-hand side of \eqref{thm3.4-1}
can be estimated as follows.
\begin{align*}
& \iint_{Q_T}B\left(\min\left\{|\nabla u_n|^{p(x)},n\right\}
-\min\left\{|\nabla u_m|^{p(x)},m\right\}\right)\theta(u_n-u_m)\,dx\,dt\\
&\leq b\iint_{Q_T}\left(|\nabla u_n|^{p(x)}+|\nabla u_m|^{p(x)}\right)
 |\theta(u_n-u_m)|\,dx\,dt\\
&=b\iint_{Q_T}\left(|\nabla u_n|^{p(x)-2}\nabla u_n\cdot\nabla u_m
 +|\nabla u_m|^{p(x)-2}\nabla u_m\cdot\nabla u_n\right)
|\theta(u_n-u_m)|\,dx\,dt\\
&\quad +b\iint_{Q_T}\left(|\nabla u_n|^{p(x)-2}\nabla u_n
 -|\nabla u_m|^{p(x)-2}\nabla u_m\right)\\
&\quad \cdot\left(\nabla u_n-\nabla u_m\right)
|\theta(u_n-u_m)|\,dx\,dt.
\end{align*}
Consequently, \eqref{thm3.4-1} can be estimated as
\begin{equation} \label{thm3.4-2}
\begin{aligned}
&\iint_{Q_T}\left(|\nabla u_n|^{p(x)-2}\nabla u_n-|\nabla u_m|^{p(x)-2}
 \nabla u_m\right)\\
&\quad \cdot\left(\nabla u_n-\nabla u_m\right)
\left[\theta'(u_n-u_m)-b|\theta(u_n-u_m)|\right]\,dx\,dt \\
&\leq b\iint_{Q_T}\left(|\nabla u_n|^{p(x)-2}\nabla u_n\cdot\nabla u_m
 +|\nabla u_m|^{p(x)-2}\nabla u_m\cdot\nabla u_n\right)
 |\theta(u_n-u_m)|\,dx\,dt.
\end{aligned}
\end{equation}
With the help of \eqref{lemma 3.1-2} in Lemma \ref{lemma 3.1} (with $a=1$),
since $\nabla u_n\to \nabla u$ a.e. in $Q_T$
(Step 4),  we may utilize Fatou's Lemma in \eqref{thm3.4-2} as
 $m\to +\infty$ to obtain that
\begin{align*}
E(n)&:=\iint_{Q_T}\left(|\nabla u_n|^{p(x)-2}\nabla u_n
 -|\nabla u|^{p(x)-2}\nabla u\right)\cdot
\left(\nabla u_n-\nabla u\right)\,dx\,dt\\
&\leq 2b\iint_{Q_T}\left(|\nabla u_n|^{p(x)-2}\nabla u_n\cdot\nabla u+|\nabla u|^{p(x)-2}\nabla u\cdot\nabla u_n\right)
|\theta(u_n-u)|\,dx\,dt\\
&\leq 4b||\nabla u_n|^{p(x)-2}\nabla u_n|_{L^{p'(x)}(Q_T)}
|\theta(u_n-u)\nabla u|_{L^{p(x)}(Q_T)}\\
&\quad +4b||\nabla u|^{p(x)-2}\nabla u\theta(u_n-u)|_{L^{p'(x)}(Q_T)}
|\nabla u_n|_{L^{p(x)}(Q_T)}\\
&\leq C\max\Big\{\Big(\iint_{Q_T}|\nabla u_n|^{p(x)}\,dx\,dt
 \Big)^{\frac{1}{(p')^\pm}}\Big\}\\
&\quad \times \max\Big\{\Big(\iint_{Q_T}|\theta(u_n-u)|^{p(x)}|\nabla u|^{p(x)}\,dx\,dt
 \Big)^{\frac{1}{p^\pm}}\Big\}\\
&\quad +C\max\Big\{\Big(\iint_{Q_T}|\theta(u_n-u)|^{p'(x)}|\nabla u|^{p(x)}\,dx\,dt
 \Big)^{\frac{1}{(p')^\pm}}\Big\}\\
&\quad \times \max\Big\{\Big(\iint_{Q_T}|\nabla u_n|^{p(x)}\,dx\,dt
 \Big)^{\frac{1}{p^\pm}}\Big\}\\
&\leq C\max\Big\{\Big(\iint_{Q_T}|\theta(u_n-u)|^{p(x)}|\nabla u|^{p(x)}
  \,dx\,dt\Big)^{\frac{1}{p^\pm}}\Big\}\\
&+C\max\Big\{\Big(\iint_{Q_T}|\theta(u_n-u)|^{p'(x)}|\nabla u|^{p(x)}\,dx\,dt
 \Big)^{\frac{1}{(p')^\pm}}\Big\}.
\end{align*}
In view of  \eqref{thm3.1-5} and \eqref{thm3.1-1}, $\theta(u_n-u)$
is uniformly bounded.
The Lebesgue dominated convergence theorem yields
\begin{equation}\label{thm3.4-3}
E(n):=\iint_{Q_T}\left(|\nabla u_n|^{p(x)-2}\nabla u_n-|\nabla u|^{p(x)-2}
\nabla u\right)\cdot
\left(\nabla u_n-\nabla u\right)\,dx\,dt\to 0.
\end{equation}
We now estimate
\begin{equation}\label{thm3.4-4}
\begin{aligned}
&\iint_{Q_T}|\nabla u_n-\nabla u|^{p(x)}\,dx\,dt \\
&=\int_0^T\int_{\{x\in\Omega;p(x)\geq2\}}|\nabla u_n-\nabla u|^{p(x)}\,dx\,dt\\
&\quad +\int_0^T\int_{\{x\in\Omega;1< p(x)<2\}}|\nabla u_n-\nabla u|^{p(x)}\,dx\,dt 
=I^{(1)}+I^{(2)}.
\end{aligned}
\end{equation}
Applying the following basic inequality, for any $y,z\in \mathbb{R}^N$,
\[
 \left(|y|^{p(x)-2}y-|z|^{p(x)-2}z\right)\cdot(y-z)\geq
\begin{cases}
2^{2-p^+}|y-z|^{p(x)}, &\text{if }\ p(x)\geq2,\\
(p^--1)\frac{|y-z|^2}{(|y|+|z|)^{2-p(x)}}, &\text{if } 1<p(x)<2.
\end{cases}
\]
we compute the  two parts in \eqref{thm3.4-4}:
\begin{equation}\label{thm3.4-5}
\begin{aligned}
I^{(1)}
&\leq\frac{1}{2^{2-p^+}}
\int_0^T\int_{\{x\in\Omega;p(x)\geq2\}}
\Big(|\nabla u_n|^{p(x)-2}\nabla u_n\\
&\quad -|\nabla u|^{p(x)-2}\nabla u\Big)\cdot
\left(\nabla u_n-\nabla u\right)\,dx\,dt\\
&\leq2^{p^+-2}E(n)\to 0.
\end{aligned}
\end{equation}
Using $p(x)$-H\"{o}lder's inequality, for $I^{(2)}$, by
\eqref{thm3.1-5} and \eqref{thm3.4-3}, we have
\begin{align}
I^{(2)}
&=\int_0^T\int_{\{x\in\Omega;1< p(x)<2\}}
\frac{|\nabla u_n-\nabla u|^{p(x)}}{\left(|\nabla u_n|+|\nabla u|\right)
^{\frac{p(x)}{2}(2-p(x))}}
\Big(|\nabla u_n| \nonumber \\
&\quad +|\nabla u|\Big)^{\frac{p(x)}{2}(2-p(x))}\,dx\,dt
 \nonumber\\
&\leq 2\Big|\frac{|\nabla u_n-\nabla u|^{p(x)}}
 {\left(|\nabla u_n|+|\nabla u|\right)^{\frac{p(x)}{2}(2-p(x))}}\Big|
_{L^{\frac{2}{p(x)}}(Q_T)}\Big| \Big(|\nabla u_n|  \nonumber\\
&\quad +|\nabla u|\Big)^{\frac{p(x)}{2}(2-p(x))}\Big|
_{L^{\frac{2}{2-p(x)}}(Q_T)}
 \nonumber\\
&\leq 2\max\Big\{\Big(\iint_{Q_T}\frac{|\nabla u_n-\nabla u|^{2}}
{\left(|\nabla u_n|+|\nabla u|\right)^{2-p(x)}}\,dx\,dt\Big)
 ^{\frac{p^\pm}{2}}\Big\}  \nonumber\\
&\quad \times\max\Big\{\Big(\iint_{Q_T}\left(|\nabla u_n|+|\nabla u|\right)^{p(x)}
 \,dx\,dt\Big)^{\frac{2-p^\pm}{2}}\Big\}  \nonumber\\
&\leq C\max\Big\{\big(\frac{1}{p^--1}\big)^{\frac{p^\pm}{2}}
 \left(E(n)\right)^{\frac{p^\pm}{2}}\Big\}  \nonumber\\
&\quad \times\max\Big\{\Big(\iint_{Q_T}\left(|\nabla u_n|^{p(x)}
+|\nabla u|^{p(x)}\right)\,dx\,dt\Big)^{\frac{2-p^\pm}{2}}\Big\}\to 0.
 \label{thm3.4-6}
\end{align}
Combining \eqref{thm3.4-4}, \eqref{thm3.4-5} and \eqref{thm3.4-6},
we arrive at
\begin{equation}\label{thm3.4-7}
\iint_{Q_T}|\nabla u_n-\nabla u|^{p(x)}\,dx\,dt\to 0,
\end{equation}
which implies
\begin{equation}\label{thm3.4-8}
|\nabla u_n-\nabla u|_{L^{p(x)}({Q_T})}\to 0;
\end{equation}
that is,  $u_n\to  u$ strongly in the solution space $V$
(simultaneously, $u_n\to  u$, strongly in $L^{p^-}(0,T;W^{1,p(x)}_0(\Omega))$).
\smallskip

\noindent\textbf{Step 6: Passing to the limit.}
It follows from \eqref{thm3.4-8}, the property of Nemytskii
operator (\cite{MR1134951,MR1033498}) and generalized Lebesgue dominated
convergence theorem that
\begin{align*}
&|\nabla u_n|^{p(x)-2}\nabla u_n\to |\nabla u|^{p(x)-2}\nabla u, \,\text{strongly in}\, \left(L^{p'(x)}({Q_T})\right)^N,\\
&\min\big\{|\nabla u_n|^{p(x)},n\big\}
\to |\nabla u|^{p(x)},\,\text{strongly in}\, L^1({Q_T}).
\end{align*}
For every $v\in V$,
\begin{align*}
&\big|\langle-\operatorname{div}\left(|\nabla u_n|^{p(x)-2}
 \nabla u_n-|\nabla u|^{p(x)-2}\nabla u\right),v\rangle_{V^\ast,V}\big|\\
&=\big|\iint_{Q_T}\left(|\nabla u_n|^{p(x)-2}\nabla u_n
 -|\nabla u|^{p(x)-2}\nabla u\right)\cdot\nabla v\,dx\,dt\big|\\
&\leq 2\big||\nabla u_n|^{p(x)-2}\nabla u_n-|\nabla u|^{p(x)-2}
\nabla u\big|_{L^{p'(x)}(Q_T)} |\nabla v|_{L^{p(x)}(Q_T)}.
\end{align*}
It follows that
\begin{align*}
&\|-\operatorname{div}\left(|\nabla u_n|^{p(x)-2}\nabla u_n
 -|\nabla u|^{p(x)-2}\nabla u\right)\|_{V^\ast}\\
&\leq 2||\nabla u_n|^{p(x)-2}\nabla u_n-|\nabla u|^{p(x)-2}
 \nabla u|_{L^{p'(x)}(Q_T)}\to 0.
\end{align*}
Therefore, for the principal term in the approximate equation
\eqref{thm3.1-0}, we have
\[
-\operatorname{div}(|\nabla u_n|^{p(x)-2}\nabla u_n)
\to -\operatorname{div}(|\nabla u|^{p(x)-2}\nabla u),\quad
\text{strongly in } V^\ast.
\]
As a consequence, one has
$u_{nt}\to  u_t$, \,strongly in $V^\ast+L^1({Q_T})$.

On the other hand, as stated in Step 3,
 $V^\ast+L^1({Q_T})\hookrightarrow L^1(0,T;H^{-s}(\Omega))$ for $s$
sufficiently large.
Therefore, from \eqref{thm3.1-6} and \eqref{thm3.2-3},
we deduce  (according to
$W^{1,1}(0,T;H^{-s}(\Omega))\hookrightarrow C([0,T];H^{-s}(\Omega))$
in \cite{MR2597943}) that $u_n\to  u$, strongly in $C([0,T];H^{-s}(\Omega) )$,
from which  $u_n(x,0)=u_0(x)$ makes a perfect sense.

Finally,
since  $u_n(x,0)\to  u(x,0)$, strongly in $H^{-s}(\Omega)$, it follows
that $u(x,0)=u_0(x)$. This  proves that $u\in V\cap L^\infty(Q_T)$
 is a weak solution to Problem \eqref{1}.
\end{proof}


\section{Appendix}

\begin{proof}[Proof of Lemma \ref{lemma 1.1}]
Note that
\[
\varphi'(s)= \begin{cases}
\lambda e^{\lambda s}, &s\geq0,\\
\lambda e^{-\lambda s}, &s\leq0.
  \end{cases}
\]
(1) Obviously, $|\varphi(s)|=e^{\lambda|s|}-1\geq \lambda |s|$.
Remember that $\lambda\geq\frac{1}{2}+\frac{b}{a}$.
If $s\geq0$, then
\[
a\lambda e^{\lambda s}-b(e^{\lambda s}-1)\geq(a\lambda-b)
e^{\lambda s}\geq\frac{a}{2}e^{\lambda s}.
\]
If $s\leq0$, then
\[
a\lambda e^{-\lambda s}-b(e^{-\lambda s}-1)\geq(a\lambda-b)
 e^{-\lambda s}\geq\frac{a}{2}e^{-\lambda s}.
\]

(2) The inequality $\lambda e^{\lambda s}
\leq \lambda M [e^{\lambda\frac{s}{p^-}}-1]^{p^-}$ is equivalent to
$[\frac{\exp(\lambda\frac{s}{p^-})}{\exp({\lambda\frac{s}{p^-}}-1)}]^{p^-}\leq M$,
which, for $s\ge d$,  is guaranteed by
\[
  \lim_{s\to +\infty}\frac{\exp(\lambda
\frac{s}{p^-})}{\exp({\lambda\frac{s}{p^-}}-1)}=1.
\]
 Likewise, the inequality
$e^{\lambda s}-1\leq M\left[e^{\lambda\frac{s}{p^-}}-1\right]^{p^-}$
for $s\ge d$ is  ensured by the limit
\[
\lim_{s\to +\infty}\frac{\exp(\lambda s)}{\exp({\lambda\frac{s}{p^-}}-1)^{p^-}}=1.
\]

(3) We prove the case $1<p^-<2$ only; the proof of the case $p^-\ge 2$
is entirely similar. The desired inequalities follow easily from the
following limits:
\[
\lim_{s\to +\infty}\frac{\frac{1}{\lambda}(e^{\lambda s}-1)-s}
{(e^{\lambda\frac{s}{p^-}}-1)^{p^-}}=\frac{1}{\lambda};\quad
\lim_{s\to +\infty}\frac{\frac{1}{\lambda}(e^{\lambda s}-1)-s}
{(e^{\lambda\frac{s}{p^-}}-1)^2}=2\lambda.
\]
\end{proof}


\begin{proof}[Proof of Lemma \ref{lemma 1.3}]
Since $\pi\in C^1$ with $\pi(0)=0$ and $\pi$, $\pi'$ are bounded,
 it follows that $\pi(u)\in V \cap L^\infty(Q_T)$.
The left-hand side of \eqref{lemma 1.3-1} exists.
By Lemma 3.2 in \cite{MR2677803} or \cite{MR1747629},
it follows from $u\in V$ with $u_t\in V^\ast+L^1(Q_T)$ that
$u\in C([0,T];L^1(\Omega))$ and hence $\Pi(u)\in C([0,T];L^1(\Omega))$.
So, the right-hand side of \eqref{lemma 1.3-1} does exist.
For the decomposition of the time derivative
$u_t=\alpha^{(1)}+\alpha^{(2)}\in V^\ast+L^1(Q_T)$, noting the
embedding relationship
$$
L^{p^+}(0,T;W^{1,p(x)}_0(\Omega))\hookrightarrow V\hookrightarrow
L^{p^-}(0,T;W^{1,p(x)}_0(\Omega)),
$$
by  standard mollification method in \cite{MR1307456},
there exist $u_n\in C^\infty([0,T];W^{1,p(x)}_0(\Omega))$,
$u_{nt}=\alpha^{(1)}_n+\alpha^{(2)}_n$,
$\alpha^{(1)}_n\in C^\infty([0,T];W^{-1,p'(x)}(\Omega))$,
$\alpha^{(2)}_n\in C^\infty([0,T];L^1(\Omega))$
such that
$u_n\to  u$,  strongly in $V$; $\alpha^{(1)}_n\to  \alpha^{(1)}$,
strongly in $V^\ast$;
$\alpha^{(2)}_n\to  \alpha^{(2)}$, strongly in $L^1(0,T;L^1(\Omega))$. Because
$\Pi(u_n)\in C^1([0,T];L^1(\Omega))$ and $\pi(u_n)\in V \cap L^\infty(Q_T)$, we have
\begin{equation}\label{lemma 1.3-2}
\begin{aligned}
\Pi(u_n(T))-\Pi(u_n(0))
&=\int_0^T[\Pi(u_n)]_tdt\\
&= \langle\alpha^{(1)}_n,\pi(u_n)\rangle_{V^\ast,V}
+\iint_{Q_T}\alpha^{(2)}_n\pi(u_n)\,dx\,dt.
\end{aligned}
\end{equation}
Since $u_n\to  u$, strongly in $C([0,T];L^1(\Omega))$, we have $u_n\to  u$,
a.e. in $Q_T$. (If necessary, by a further subsequence to be denoted by the same
$u_n$.)
Furthermore, the sequence $\pi(u_n)\to \pi(u)$, a.e. in $Q_T$ and remains bounded;
hence $\pi(u_n)\to \pi(u)$, weakly* in $L^\infty(Q_T)$.
Combing with $\alpha^{(2)}_n\to \alpha^{(2)}$, strongly in $L^1(Q_T)$, one has
$\iint_{Q_T}\alpha^{(2)}_n\pi(u_n)\,dx\,dt\to \iint_{Q_T}\alpha^{(2)}\pi(u)\,dx\,dt$.
 Moreover,
from $u_n\to  u$, strongly in $V$ and the properties of $\pi$, one has
 $\pi(u_n)\to \pi(u)$, strongly in $V$. Together with
$\alpha^{(1)}_n\to  \alpha^{(1)}$, strongly in $V^\ast$,
it yields $\langle\alpha^{(1)}_n,\pi(u_n)\rangle_{V^\ast,V}\to
\langle\alpha^{(1)},\pi(u)\rangle_{V^\ast,V}$.
Finally, $\Pi(u_n)\to \Pi(u)$ in
$C([0,T];L^1(\Omega))\hookrightarrow L^1(Q_T)$.
Meanwhile $\Pi(u_n(T))\to \Pi(u(T))$ and
$\Pi(u_n(0))\to \Pi(u(0))$, strongly in $L^1(Q_T)$.
Consequently,
$\int_\Omega\Pi(u_n(T))dx-\int_\Omega\Pi(u_n(0))dx\to
\int_\Omega\Pi(u(T))dx-\int_\Omega\Pi(u(0))dx$.
Hence \eqref{lemma 1.3-1} follows from \eqref{lemma 1.3-2}
 by passing to the limit as $n\to\infty$.
\end{proof}


\subsection*{Acknowledgments}
This research was supported by the National Science Foundation of China 
(11271154, 11401252),
by the Key Lab of Symbolic Computation and
Knowledge Engineering of Ministry of Education,
and by the 985 Program and the Tang Ao-Qing Professorship of Jilin University.
The first author is also supported by the Graduate Innovation Fund
of Jilin University (2014084).

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\end{document}