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\title[Quasilinear parabolic equations with non trivial boundary conditions]{Quasilinear parabolic equations with $p(x)$-Laplacian diffusion terms and nonlocal boundary conditions}

\author{Abita Rahmoune}
\address{Laghouat university,\\ Faculty of Mathematics and Computer Sciences\\ El Maamoura Street, Algeria}
\email{abitarahmoune@yahoo.fr}
\author{Benyattou Benabderrahmane}
\address{Department of Mathematics, Faculty of Mathematics \& Computer Sciences,\\
University of M'Sila, Algeria.}
%
\subjclass{35K20; 35K59; 35B45; 35D30}
\keywords{Nonlocal boundary conditions; quasi-linear parabolic equations; Generalized
Lebesgue space and Sobolev spaces with variable exponents}
\email{bbenyattou@yahoo.com}

\begin{abstract}
In this study, we prove the existence of local solution for a quasi linear
generalized parabolic equation with nonlocal boundary conditions for an
elliptic operator involving the variable-exponent nonlinearities, using
Faedo-Galerkin arguments and compactness method.
\end{abstract}
\maketitle 

\section{Introduction}

Let $\Omega $ be a bounded domain in $\mathbb{R}^{n}$, $n\geq 2$ with a
smooth boundary $\Gamma =\partial \Omega .$ We consider the following quasi
linear parabolic equations with nonlocal boundary conditions:%
\begin{gather}
\frac{\partial u}{\partial t}-\sum_{i=1}^{n}\frac{\partial }{\partial x_{i}}%
\left( \left\vert u\right\vert ^{p\left( x\right) -2}\frac{\partial u}{%
\partial x_{i}}\right) +\left\vert u\right\vert ^{p\left( x\right)
-2}u=f\left( x,t\right) \text{\ in}\ Q_{T}=\Omega \times \left( 0,T\right) ,
\label{1} \\
u(x,0)=u_{0}(x)\text{,\ \ }x\in \Omega ,  \label{3} \\
u\left( x,t\right) =\int_{\Omega }K\left( x,y\right) u\left( y,t\right) dy,%
\text{ \ }x\in \Gamma ,\text{ }t\in \left( 0,T\right) ,  \label{4}
\end{gather}%
where the exponent $p(.)$ is given measurable function on $\overline{\Omega }
$ such that:%
\begin{equation}
2\leq n<p_{1}\leq p\left( x\right) \leq p_{2}\leq \infty ,  \label{01}
\end{equation}%
Where%
\begin{equation*}
p_{2}=\underset{x\in \Omega }{ess\text{ }\sup }p\left( x\right) ,\text{ \ }%
p_{1}=\underset{x\in \Omega }{ess\text{ }\inf }p\left( x\right) .
\end{equation*}%
We also assume that $p\left( .\right) $ satisfies the following Zhikov--Fan
uniform local continuity condition :%
\begin{equation}
\left\vert p\left( x\right) -p\left( y\right) \right\vert \leq \frac{M}{%
\left\vert \log \left\vert x-y\right\vert \right\vert }\text{, for all }x,%
\text{ }y\text{ in }\Omega \text{ with }\left\vert x-y\right\vert <\frac{1}{2%
},\text{ }M>0.  \label{03}
\end{equation}%
In recent years, many authors have paid attention to the study of nonlinear
hyperbolic, parabolic and elliptic equations with nonstandard growth
condition. For instance, modeling of physical phenomena such as flows of
electro-rheological fluids or fluids with temperature-dependent viscosity,
thermoelasticity, nonlinear viscoelasticity, filtration processes through a
porous media and image processing. More details on these problems can be
found in \cite{Antontsev5, ChenLevine, Aboulaich, Antontsev6, Antontsev7,
Lian, Rahmoune3, Rahmoune2} and references therein.

\textbf{Constant exponent}. In \eqref{1}, when $p\left( .\right) =p$ is
constant, local, global existence and long-time behavior have been
considered by many authors.

For instance, in the absence of the term $\left\vert u\right\vert ^{p-2}u$
and when the kernel datum function $K\left( x,y\right) =0$, using the
compactness method and Faedo-Galerkin techniques, the existence and
uniqueness of a weak solution has been proved see \cite{Lions2}.

Baili Chen in \cite{Chen} generalized the result of Lions to the situation
when the presence of $\left\vert u\right\vert ^{p-2}u$ and when $K\left(
x,y\right) \neq 0$ in problem \eqref{1}, applying exactly the same technique
introduced in \cite[Probl\`{e}me 12, page 140.]{Lions2}, the author by
constructing the approximate Galerkin solution, he proved the existence of
generalized solution, the uniqueness questions are still open.

Problem \eqref{1}-\eqref{4} is the extension of the problems in Lion's book
\cite[p.140]{Lions2} in which the boundary conditions are homogeneous and in
\cite{Chen} in which the variable-exponent is constant. The uniqueness
questions in problem \eqref{1}-\eqref{4} are more complicated than in \cite%
{Chen} and are still open.

The main difficulty of this problem concern the weak converging approximate
solution is related to the presence of the quasilinear terms in \eqref{1} in
the variable-exponent.

In this paper a class of quasi linear generalized parabolic equation with
nonlocal boundary conditions for an elliptic operator involving the
variable-exponent nonlinearities was considered. Hence by using
Faedo-Galerkin arguments and compactness method as in \cite{Lions2}, we will
show the local existence of problem \eqref{1}-\eqref{4}.

\section{Preliminaries}

In this section we list and recall some well-known results and facts from
the theory of the Sobolev spaces with variable exponent. (For the details
see \cite{Diening3, Diening1, Diening2, Fan, Fu, Kovacik}).

Throughout the rest of the paper we assume that $\Omega $ is a bounded
domain of $\mathbb{R}^{n}$, $n\geq 2$ with smooth boundary $\Gamma ,$ Let $p:%
\overline{\Omega }\rightarrow \lbrack 1,\infty ]$ be a measurable function.
We denote by $L^{p(.)}(\Omega )$ the set of measurable functions $u$ on $%
\Omega $ such that%
\begin{equation*}
A_{p(.)}\left( u\right) =\int_{\Omega }\left\vert u\left( x\right)
\right\vert ^{p\left( x\right) }dx<\infty .
\end{equation*}%
The variable-exponent space $L^{p(.)}\left( \Omega \right) $ equipped with
the Luxemburg norm%
\begin{equation*}
\left\Vert u\right\Vert _{p(.),\Omega }=\left\Vert u\right\Vert
_{p(.)}=\left\Vert u\right\Vert _{L^{p\left( .\right) }({\Omega )}}=\inf
\left\{ \lambda >0,\text{\ }A_{p(.)}\left( \frac{u}{\lambda }\right) \leq
1\right\}
\end{equation*}%
is a Banach space.

In general, variable-exponent Lebesgue spaces are similar to classical
Lebesgue spaces in many aspects, see the first discussed the $L^{p(x)}$
spaces and $W^{k,p\left( x\right) }$ spaces by Kov\`{a}cik and R\'{a}kosnik
in \cite{Kovacik}.

Let us list some properties of the spaces $L^{p(.)}(\Omega )$ which will be
used in the study of the problem \eqref{1}-\eqref{4}.

\begin{itemize}
\item It follows directly from the definition of the norm that (see \cite%
{Diening3}),
\begin{equation*}
\min \left( \left\Vert u\right\Vert _{p(.)}^{p_{1}},\left\Vert u\right\Vert
_{p(.)}^{p_{2}}\right) \leq A_{p(.)}\left( u\right) \leq \max \left(
\left\Vert u\right\Vert _{p(.)}^{p_{1}},\left\Vert u\right\Vert
_{p(.)}^{p_{2}}\right) .
\end{equation*}

\item Let $p,$ $q,$ $s\geq 1$ be measurable functions defined on $\overline{%
\Omega }$ such that%
\begin{equation*}
\frac{1}{s\left( x\right) }=\frac{1}{p\left( x\right) }+\frac{1}{q\left(
x\right) },\text{ for a.e.\ \ }x\in \Omega .
\end{equation*}%
if $u\in $ $L^{p(.)}(\Omega )$, $v\in $ $L^{q(.)}(\Omega )$\ then $u.v\in
L^{s(.)}(\Omega )$ and the following generalized H\"{o}lder inequality%
\begin{equation*}
\left\Vert uv\right\Vert _{s\left( .\right) }\leq 2\left\Vert u\right\Vert
_{p(.)}\left\Vert v\right\Vert _{q(.)}.
\end{equation*}%
holds.
\end{itemize}

We us consider the following variable-exponent Lebesgue Sobolev space (see
\cite{Diening3}),%
\begin{equation*}
W^{1,p\left( .\right) }(\Omega )=\left\{ v\in L^{p(.)}(\Omega ):\text{such
that }\left\vert \nabla v\right\vert \text{ exists and }\left\vert \nabla
v\right\vert \in L^{p\left( .\right) }(\Omega )\right\} .
\end{equation*}%
This space is a Banach space with respect to the norm%
\begin{equation*}
\left\Vert u\right\Vert _{W_{0}^{1,p\left( .\right) }(\Omega )}=\left\Vert
u\right\Vert _{p(.)}+\sum_{i}\left\Vert \nabla u_{i}\right\Vert _{p(.)};%
\text{ }i=1,2.
\end{equation*}%
Furthermore, we set $W_{0}^{1,p\left( .\right) }(\Omega ),$ to be the
closure of $C_{0}^{\infty }\left( \Omega \right) $ in $W^{1,p\left( .\right)
}(\Omega ).$ Here we note that the space $W_{0}^{1,p\left( .\right) }(\Omega
)$ is usually defined in a different way for the variable exponent case.
However (see Diening et al \cite{Diening3}), both definitions are equivalent
under \eqref{03}. The $\left( W_{0}^{1,p\left( .\right) }(\Omega )\right)
^{\prime }$ is the dual space of $W_{0}^{1,p\left( .\right) }(\Omega )$ with
respect to the inner product in $L^{2}(\Omega )$ and is defined as $%
W^{-1,p^{\prime }\left( .\right) }(\Omega ),$ in the same way as the
classical Sobolev spaces, where $\frac{1}{p\left( _{.}\right) }+\frac{1}{%
p\prime \left( .\right) }=1,$ the function $p^{\prime }\left( .\right) $ is
called the dual variable exponent of $p\left( .\right) $.

\begin{itemize}
\item Let $p,$ $q:\Omega \rightarrow \left[ 1,+\infty \right) $ be
measurable functions satisfying condition \eqref{03}. If $p(x)\leq q(x)$
almost everywhere in $\Omega $, then the embedding $L^{q(.)}(\Omega
)\hookrightarrow L^{p(.)}(\Omega )$ is continuous.\ \
\end{itemize}

\begin{lemma}
(\cite{Diening3})Let $\Omega $ be a bounded domain in $\mathbb{R}^{n}$, $%
n\geq 1$ with a smooth boundary $\Gamma =\partial \Omega $, $p(.)$ is given
measurable function on $\overline{\Omega }$ satisfy conditions \eqref{03}
and $q=const\geq 1$. If $q\leq p(x)$ a.e. in $\Omega $, then%
\begin{equation}
\left\Vert v\right\Vert _{q}\leq C_{q,\Omega }\left\Vert v\right\Vert
_{p\left( .\right) }\text{ \ with the constant\ \ }C_{q,\Omega }=\left(
1+\left\vert \Omega \right\vert \right) ^{\frac{1}{q}}\text{.}  \label{04}
\end{equation}
\end{lemma}

\section{Notations and Preliminaries}

In this article, on $f$, $u_{0}$ and $K\left( x,y\right) $ we make the
following assumptions%
\begin{equation}
f\in L^{p_{2}^{\prime }}(0,T;L^{p_{2}^{\prime }}\left( \Omega \right) ),%
\text{ }\frac{1}{p_{2}}+\frac{1}{p_{2}^{\prime }}=1,  \label{7}
\end{equation}%
\begin{equation}
u_{0}\in L^{\infty }(\Omega ),  \label{8}
\end{equation}%
\begin{equation}
\text{For any }x\in \Gamma ,\text{\ \ }K\left( x\right) <\infty \text{, \ }%
K_{i}\left( x\right) <\infty ,  \label{010}
\end{equation}%
\begin{equation}
\sum_{i=1}^{n}\int_{\Gamma }K\left( x\right) ^{p_{2}-1}K_{i}\left( x\right)
d\Gamma <\infty ,\text{ }\sum_{i=1}^{n}\int_{\Gamma }K\left( x\right)
^{p_{1}-1}K_{i}\left( x\right) d\Gamma <\infty ,  \label{011}
\end{equation}%
\begin{equation}
\gamma =\max \left(
\begin{array}{c}
C_{p_{1},\Omega }^{p_{2}}\left( \sum_{i=1}^{n}\int_{\Gamma }K\left( x\right)
^{p_{2}-1}K_{i}\left( x\right) d\Gamma \right) , \\
\left( C_{p_{1},\Omega }^{p_{1}}\sum_{i=1}^{n}\int_{\Gamma }K\left( x\right)
^{p_{1}-1}K_{i}\left( x\right) d\Gamma \right)%
\end{array}%
\right) \leq \frac{p_{1}-1}{p_{2}},  \label{9}
\end{equation}%
for any $x\in \Gamma $, where%
\begin{gather*}
K\left( x\right) =\left( \int_{\Omega }\left\vert K\left( x,y\right)
\right\vert ^{p_{1}^{\prime }}dy\right) ^{\frac{1}{p_{1}^{\prime }}}\text{:}
\\
\text{Norm of }k(x,y)\text{ in }L^{p_{1}^{\prime }}\left( \Omega \right)
\text{ with respect to }y;\text{ }\frac{1}{p_{1}}+\frac{1}{p_{1}^{\prime }}=1
\end{gather*}%
\begin{gather*}
K_{i}\left( x\right) =\left( \sum_{i=1}^{n}\int_{\Omega }\left\vert \frac{%
\partial }{\partial x_{i}}K\left( x,y\right) \right\vert ^{p_{1}^{\prime
}}dy\right) ^{\frac{1}{p_{1}^{\prime }}}\text{:} \\
\text{Norm of }\frac{\partial }{\partial x_{i}}k(x,y)\text{ in }%
L^{p_{1}^{\prime }}\left( \Omega \right) \text{ with respect to }y,\text{ }%
\frac{1}{p_{1}}+\frac{1}{p_{1}^{\prime }}=1
\end{gather*}%
and $C_{p_{1},\Omega }$ defined in \eqref{04}.

Moreover, we assume that%
\begin{equation}
r>\frac{n}{2}+2.  \label{02}
\end{equation}%
Let
\begin{equation}
\ \alpha _{1}=\left( \frac{p_{2}}{p_{1}}\left( \gamma +\frac{1}{p_{2}}%
\right) \right) ^{\frac{p_{2}}{p_{1}-p_{2}}}.  \label{H1}
\end{equation}%
We define the polynomial $Q$ by%
\begin{equation*}
Q\left( \alpha \right) =\min \left( \alpha ,\alpha ^{\frac{p_{1}}{p_{2}}%
}\right) -\left( \gamma +\frac{1}{p_{2}}\right) \max \left( \alpha ,\alpha ^{%
\frac{p_{1}}{p_{2}}}\right) \text{ \ }\forall \alpha \in \left[ 0,+\infty %
\right] .
\end{equation*}%
Let%
\begin{equation*}
h\left( \alpha \right) =\alpha ^{\frac{p_{1}}{p_{2}}}-\left( \gamma +\frac{1%
}{p_{2}}\right) \alpha .
\end{equation*}%
Notice that$\ h\left( \alpha \right) =Q\left( \alpha \right) ,$ for $1\leq
\alpha \leq \infty .$ It is easy to check that the function $h(\alpha )$ is
increasing for $1\leq \alpha <\alpha _{1}$ and decreasing for $\alpha _{1}$ $%
<\alpha \leq +\infty $.

where $\alpha _{1}$ is its unique local maximum defined by \eqref{H1}. We
will assume that:%
\begin{equation}
1\leq \left\Vert u_{0}\right\Vert _{p(.)}^{p_{2}}=\alpha _{0}<\alpha _{1}
\label{A3}
\end{equation}%
and%
\begin{equation}
\frac{1}{2}\left\vert u_{0}\right\vert ^{2}+C_{2,\Omega }^{\frac{p_{2}}{%
p_{2}-1}}\frac{p_{2}-1}{p_{2}}\left\vert \Omega \right\vert ^{\frac{1}{2}%
}\int_{0}^{T}\left\vert f\right\vert _{p_{2}}^{\frac{p_{2}}{p_{2}-1}%
}dt<\int_{0}^{T}Q\left( \alpha _{1}\right) dt.  \label{A4}
\end{equation}%
The classical formulation of the problem is as follows. Find a displacement
field $u:\Omega \times \left( 0,T\right) \rightarrow \mathbb{R},$ such that:%
\begin{gather}
\left( u^{\prime },v\right) -\left( \sum_{i=1}^{n}\frac{\partial }{\partial
x_{i}}\left( \left\vert u\right\vert ^{p\left( x\right) -2}\frac{\partial u}{%
\partial x_{i}}\right) ,v\right) +\left( \left\vert u\right\vert ^{p\left(
x\right) -2}u,v\right) =\left( f,v\right) ,\text{ }\forall v\in V  \label{10}
\\
u(x,0)=u_{0}(x),\text{\ \ }x\in \Omega \text{.}  \notag
\end{gather}%
Where
\begin{equation*}
V=\left\{ v\in H^{r}\left( \Omega \right) :v\left( x\right) =\int_{\Omega
}K\left( x,y\right) v\left( y\right) dy\text{ \ for }x\in \Gamma \right\} ,
\end{equation*}%
With assumption \eqref{01}-\eqref{02}, using Sobelev embedding theorems, see
\cite{Adams}, we have%
\begin{equation*}
H^{r}\left( \Omega \right) \hookrightarrow W^{2,p_{2}}\left( \Omega \right)
\hookrightarrow W^{1,p_{2}}\left( \Omega \right) \hookrightarrow
L^{p_{2}}\left( \Omega \right) \hookrightarrow L^{2}\left( \Omega \right)
\end{equation*}%
It is easy to see that $V$ is a subspace of $H^{r}\left( \Omega \right) .$

Whenever it doesn't cause a confusion, we use the following shorthand
notations:

$L^{q}(\Omega )$: $L^{q}$ space defined on $\Omega $;\ $|.|_{q}=|.|_{q,%
\Omega }$: norm in $L^{q}(\Omega )$;\ $|.|_{q,\Gamma }$: norm in $%
L^{q}(\Gamma );$\ $H^{-r}\left( \Omega \right) $: dual space of $H^{r}\left(
\Omega \right) ;$\ $\left\vert .\right\vert _{H^{-r}\left( \Omega \right) }$
norm in $H^{-r}\left( \Omega \right) ;$ $C$: nonnegative constant which may
take different values on each occurrence.

\section{Local Existence}

\begin{theorem}
\label{Theorem1} Under hypothesis \eqref{01}-\eqref{A4}, for any finite $%
T>0, $ the problem \eqref{1}-\eqref{4} admits a weak solution $u$ such that%
\begin{equation}
u\in L^{\infty }\left( 0,T;L^{2}(\Omega )\right) \cap C\left( \left[ 0,T%
\right] ;H^{-r}\left( \Omega \right) \right) \cap L^{p\left( .\right)
}\left( \Omega \times \left( 0,T\right) \right) ,  \label{12}
\end{equation}%
\begin{equation}
\frac{\partial u}{\partial t}\in L^{p_{2}^{\prime }}\left( 0,T;H^{-r}\left(
\Omega \right) \right) ,  \label{13}
\end{equation}%
\begin{equation}
\left\vert u\right\vert ^{\frac{p\left( .\right) -2}{2}}u\in \ L^{2}\left(
0,T;H^{1}\left( \Omega \right) \right) ,  \label{14}
\end{equation}%
for all $v\in V$ and a.e. $t\in \left[ 0,T\right] ,$%
\begin{gather}
\left( u^{\prime },v\right) -\left( \sum_{i=1}^{n}\frac{\partial }{\partial
x_{i}}\left( \left\vert u\right\vert ^{p\left( x\right) -2}\frac{\partial u}{%
\partial x_{i}}\right) ,v\right) +\left( \left\vert u\right\vert ^{p\left(
x\right) -2}u,v\right) =\left( f,v\right) ,  \label{14B} \\
u(x,0)=u_{0}(x),\text{ }x\in \Omega \text{.}  \notag
\end{gather}
\end{theorem}

\begin{proof}
Since $V$ is a subspace of $H^{r}\left( \Omega \right) $ which is separable.
We can choose a countable set of distinct basis elements $w_{j}$ $\left(
j=1,2,...\right) $ which generate $V$ and are orthonormal in $L^{2}\left(
\Omega \right) .$ Let $V_{m}$ be the subspace of $V$ generated by the first $%
m$ elements: $w_{1},w_{2},...,w_{m}$. We search $u$ of the form:%
\begin{equation}
u_{m}(x,t)=\underset{i=1}{\overset{m}{\sum }}K_{im}(t)w_{i}\left( x\right) ,
\label{15}
\end{equation}%
satisfying:
\begin{equation}
\left\{
\begin{array}{l}
\left( u_{m}^{\prime },w_{j}\right) -\left( \sum_{i=1}^{n}\frac{\partial }{%
\partial x_{i}}\left( \left\vert u_{m}\right\vert ^{p\left( x\right) -2}%
\frac{\partial u_{m}}{\partial x_{i}}\right) ,w_{j}\right) \\
+\left( \left\vert u_{m}\right\vert ^{p\left( x\right) -2}u_{m},w_{j}\right)
=\left( f\left( t\right) ,w_{j}\right) ,\text{ \ }1\leq j\leq m, \\
\multicolumn{1}{c}{u_{m}(0)=u_{0m}.}%
\end{array}%
\right.  \label{16}
\end{equation}%
with%
\begin{equation}
u_{0m}=\sum_{i=1}^{m}\alpha _{im}w_{i}\longrightarrow u_{0}\ \text{\ when\ }%
m\longrightarrow \infty \ \text{in }L^{p\left( .\right) }\left( \Omega
\right) .  \label{17}
\end{equation}%
Integrating by parts on the second term of left-hand side of \eqref{16}, we
have%
\begin{equation}
\left\{
\begin{array}{c}
\left( u_{m}^{\prime },w_{j}\right) +\left( \sum_{i=1}^{n}\left( \left\vert
u_{m}\right\vert ^{p\left( x\right) -2}\frac{\partial u_{m}}{\partial x_{i}}%
\right) ,\frac{\partial }{\partial x_{i}}w_{j}\right) +\left( \left\vert
u_{m}\right\vert ^{p\left( x\right) -2}u_{m},w_{j}\right) \\
=\int_{\Gamma }\left\vert u_{m}\right\vert ^{p\left( x\right) -2}\frac{%
\partial u_{m}}{\partial x_{i}}w_{j}d\Gamma +\left( f\left( t\right)
,w_{j}\right) ,\text{ \ }1\leq j\leq m, \\
u_{m}(0)=u_{0m}.%
\end{array}%
\right.  \label{18b}
\end{equation}%
By Peano's Theorem, for every finite $m$ the problem \eqref{16}, \eqref{18b}
has a solution on $(0,T_{m})$ for each $m$. The following estimates permit
us to confirm that $T_{m}$ is independent of $m$.

a) A priori estimates

Multiplying the equation \eqref{18b} by $K_{jm}(t)$, summing over $j=1,...,$
$m$, we obtain%
\begin{gather}
\frac{1}{2}\frac{d}{dt}\left\vert u_{m}\left( t\right) \right\vert
^{2}+\sum_{i=1}^{n}\int_{\Omega }\frac{4}{p^{2}\left( x\right) }\left( \frac{%
\partial }{\partial x_{i}}\left( \left\vert u_{m}\right\vert ^{\frac{p\left(
x\right) -2}{2}}u_{m}\right) \right) ^{2}dx+\int_{\Omega }\left\vert
u_{m}\right\vert ^{p\left( x\right) }dx  \label{18} \\
=\int_{\Gamma }\left\vert u_{m}\right\vert ^{p\left( x\right) -2}\frac{%
\partial u_{m}}{\partial x_{i}}u_{m}(t)d\Gamma +\left( f\left( t\right)
,u_{m}\right)  \notag
\end{gather}%
\ Integrating on $\left( 0,T\right) $ on both sides of \eqref{18}, we get%
\begin{gather}
\frac{1}{2}\left\vert u_{m}\left( T\right) \right\vert
^{2}+\int_{0}^{T}\sum_{i=1}^{n}\int_{\Omega }\frac{4}{p^{2}\left( x\right) }%
\left( \frac{\partial }{\partial x_{i}}\left( \left\vert u_{m}\right\vert ^{%
\frac{p\left( x\right) -2}{2}}u_{m}\right) \right) ^{2}dxdt  \notag \\
+\int_{0}^{T}\min \left( \left\vert \left\vert u_{m}\right\vert \right\vert
_{p\left( .\right) }^{p_{2}},\left\vert \left\vert u_{m}\right\vert
\right\vert _{p\left( .\right) }^{p_{1}}\right) dt  \notag \\
\leq \int_{0}^{T}\int_{\Gamma }\left\vert \left\vert u_{m}\right\vert
^{p\left( x\right) -2}\frac{\partial u_{m}}{\partial x_{i}}%
u_{m}(t)\right\vert d\Gamma dt+\int_{0}^{T}\left\vert \left( f\left(
t\right) ,u_{m}\right) \right\vert dt+\frac{1}{2}\left\vert
u_{0m}\right\vert ^{2}.  \label{20}
\end{gather}%
The second term in the right-hand side of \eqref{20} can be estimated as
follows%
\begin{gather*}
\left\vert \left( f\left( t\right) ,u_{m}\right) \right\vert \leq \left\vert
f\right\vert _{2}\left\vert u_{m}\right\vert _{2}\leq C_{2,\Omega
}\left\vert f\right\vert _{2}\left\vert \left\vert u_{m}\right\vert
\right\vert _{p\left( .\right) }\text{ (holder's inequality) and \eqref{04}}
\\
\leq C_{2,\Omega }^{\frac{p_{2}}{p_{2}-1}}\frac{p_{2}-1}{p_{2}}\left\vert
f\right\vert _{2}^{\frac{p_{2}}{p_{2}-1}}+\frac{1}{p_{2}}\left\vert
\left\vert u_{m}\right\vert \right\vert _{p\left( .\right) }^{p_{2}}\text{ \
(Young's inequality)} \\
\leq C_{2,\Omega }^{\frac{p_{2}}{p_{2}-1}}\frac{p_{2}-1}{p_{2}}\left\vert
\Omega \right\vert ^{\frac{1}{2}}\left\vert f\right\vert _{p_{2}}^{\frac{%
p_{2}}{p_{2}-1}}+\frac{1}{p_{2}}\max \left( \left\Vert u_{m}\right\Vert
_{p\left( .\right) }^{p_{2}},\left\Vert u_{m}\right\Vert _{p\left( .\right)
}^{p_{1}}\right) .
\end{gather*}%
Next, we estimate first term in the right-hand side of \eqref{20} using %
\eqref{04}: For $x\in \Gamma $, we have%
\begin{equation}
\left\vert u_{m}\left( x,t\right) \right\vert \leq K\left( x\right)
\left\vert u_{m}\right\vert _{p_{1}}\leq C_{p_{1},\Omega }K\left( x\right)
\left\Vert u_{m}\right\Vert _{p\left( .\right) }.  \label{20b}
\end{equation}%
Similarly, for $x\in \Gamma $ we have%
\begin{equation}
\left\vert \sum_{i=1}^{n}\frac{\partial }{\partial x_{i}}u_{m}\left(
x,t\right) \right\vert \leq K_{i}\left( x\right) \left\vert u_{m}\right\vert
_{p_{1}}\leq C_{p_{1},\Omega }K_{i}\left( x\right) \left\Vert
u_{m}\right\Vert _{p\left( .\right) }  \label{20B}
\end{equation}%
Then using holder's inequality and assumptions \eqref{010} and \eqref{9}, we
have:%
\begin{gather*}
\sum_{i=1}^{n}\int_{\Gamma }\left\vert \left\vert u_{m}\right\vert ^{p\left(
x\right) -2}\frac{\partial u_{m}}{\partial x_{i}}u_{m}(t)\right\vert d\Gamma
\leq \sum_{i=1}^{n}\int_{\Gamma }\left\vert u_{m}\right\vert ^{p\left(
x\right) -1}\left\vert \frac{\partial u_{m}}{\partial x_{i}}\right\vert
d\Gamma \\
\leq \max \left( \sum_{i=1}^{n}\int_{\Gamma }\left\vert u_{m}\right\vert
^{p_{2}-1}\left\vert \frac{\partial u_{m}}{\partial x_{i}}\right\vert
d\Gamma ,\sum_{i=1}^{n}\int_{\Gamma }\left\vert u_{m}\right\vert
^{p_{1}-1}\left\vert \frac{\partial u_{m}}{\partial x_{i}}\right\vert
d\Gamma \right) \\
\leq \max \left(
\begin{array}{c}
C_{p_{1},\Omega }^{p_{2}}\sum_{i=1}^{n}\int_{\Gamma }K\left( x\right)
^{p_{2}-1}\left\Vert u_{m}\right\Vert _{p\left( .\right)
}^{p_{2}-1}K_{i}\left( x\right) \left\Vert u_{m}\right\Vert _{p\left(
.\right) }d\Gamma , \\
C_{p_{1},\Omega }^{p_{1}}\sum_{i=1}^{n}\int_{\Gamma }K\left( x\right)
^{p_{1}-1}\left\Vert u_{m}\right\Vert _{p\left( .\right)
}^{p_{1}-1}K_{i}\left( x\right) \left\Vert u_{m}\right\Vert _{p\left(
.\right) }d\Gamma%
\end{array}%
\right) \\
=\max \left(
\begin{array}{c}
C_{p_{1},\Omega }^{p_{2}}\left( \sum_{i=1}^{n}\int_{\Gamma }K\left( x\right)
^{p_{2}-1}K_{i}\left( x\right) d\Gamma \right) \left\Vert u_{m}\right\Vert
_{p\left( .\right) }^{p_{2}}, \\
C_{p_{1},\Omega }^{p_{1}}\left( \sum_{i=1}^{n}\int_{\Gamma }K\left( x\right)
^{p_{1}-1}K_{i}\left( x\right) d\Gamma \right) \left\Vert u_{m}\right\Vert
_{p\left( .\right) }^{p_{1}}%
\end{array}%
\right) \\
\leq \max \left(
\begin{array}{c}
C_{p_{1},\Omega }^{p_{2}}\left( \sum_{i=1}^{n}\int_{\Gamma }K\left( x\right)
^{p_{2}-1}K_{i}\left( x\right) d\Gamma \right) , \\
\left( C_{p_{1},\Omega }^{p_{1}}\sum_{i=1}^{n}\int_{\Gamma }K\left( x\right)
^{p_{1}-1}K_{i}\left( x\right) d\Gamma \right)%
\end{array}%
\right) \\
\times \max \left( \left\Vert u_{m}\right\Vert _{p\left( .\right)
}^{p_{2}},\left\Vert u_{m}\right\Vert _{p\left( .\right) }^{p_{1}}\right)
\end{gather*}

this implies that%
\begin{gather}
\frac{1}{2}\left\vert u_{m}\left( t\right) \right\vert
^{2}+\int_{0}^{T}\sum_{i=1}^{n}\int_{\Omega }\frac{4}{p^{2}\left( x\right) }%
\left( \frac{\partial }{\partial x_{i}}\left( \left\vert u_{m}\right\vert ^{%
\frac{p\left( x\right) -2}{2}}u_{m}\right) \right)
^{2}dxdt+\int_{0}^{T}Q\left( \left\vert \left\vert u_{m}\right\vert
\right\vert _{p(.)}^{p_{2}}\right) dt  \notag \\
\leq \frac{1}{2}\left\vert u_{0m}\right\vert ^{2}+C_{2,\Omega }^{\frac{p_{2}%
}{p_{2}-1}}\frac{p_{2}-1}{p_{2}}\left\vert \Omega \right\vert ^{\frac{1}{2}%
}\int_{0}^{T}\left\vert f\right\vert _{p_{2}}^{\frac{p_{2}}{p_{2}-1}}dt,
\label{025}
\end{gather}%
at this step we will assume that $Q\left( \left\vert \left\vert
u_{m}\right\vert \right\vert _{p(.)}^{p_{2}}\right) \geq 0$, so from %
\eqref{A4} and \eqref{025}, we have the following a priori estimates:
\begin{equation}
\left\vert u_{m}\right\vert \leq C\ (C\text{ is independent\ of}\ m);
\label{029}
\end{equation}%
\begin{equation}
\int_{0}^{T}\sum_{i=1}^{n}\int_{\Omega }\frac{4}{p^{2}\left( x\right) }%
\left( \frac{\partial }{\partial x_{i}}\left( \left\vert u_{m}\right\vert ^{%
\frac{p\left( x\right) -2}{2}}u_{m}\right) \right) ^{2}dxdt\leq C\ (C\text{
independent\ of}\ m).  \label{030}
\end{equation}%
So the solution $u_{m}\left( t\right) $ of problem \eqref{1}-\eqref{4}
exists on $\left[ 0,T\right] $ for each $m$, and
\begin{gather}
u_{m}\text{ \ }\ \text{is bounded in}\ L^{\infty }\left( 0,T;L^{2}(\Omega
)\right) ;  \label{031} \\
\left\vert u_{m}\right\vert ^{\frac{p\left( .\right) -2}{2}}u_{m}\text{ \ is
bounded in}\ L^{2}\left( 0,T;H^{1}\left( \Omega \right) \right)  \notag
\end{gather}

\begin{claim}
There exists an integer $N$ such that
\begin{equation}
\left\vert \left\vert u_{m}\right\vert \right\vert _{p(.)}^{p_{2}}<\alpha
_{1}\text{ \ \ }\forall t\in \left[ 0,T_{m}\right) \text{ \ \ \ }m>N.
\label{027}
\end{equation}
\end{claim}

\begin{proof}[Proof of the Claim]
Suppose \eqref{027} false. Then for each $m>N$, there exists $t\in \left[
0,T_{m}\right) $ such that $\left\vert \left\vert u_{m}\left( t\right)
\right\vert \right\vert _{p(.)}^{p_{2}}\geq \alpha _{1}.$ We note that from %
\eqref{A3} and \eqref{17} there exists $N_{0}$ such that%
\begin{equation*}
1\leq \left\vert \left\vert u_{m}\left( 0\right) \right\vert \right\vert
_{p(.)}^{p_{2}}<\alpha _{1}\text{ \ \ }\forall m>N_{0}
\end{equation*}%
Then by continuity there exists a first $T_{m}^{\ast }\in \left(
0,T_{m}\right) $ such that%
\begin{equation}
\left\vert \left\vert u_{m}\left( T_{m}^{\ast }\right) \right\vert
\right\vert _{p(.)}^{p_{2}}=\alpha _{1},  \label{028}
\end{equation}%
from where%
\begin{equation*}
Q\left( \left\vert \left\vert u_{m}\right\vert \right\vert
_{p(.)}^{p_{2}}\right) =h\left( \left\vert \left\vert u_{m}\left( t\right)
\right\vert \right\vert _{p\left( .\right) }^{p_{2}}\right) \geq 0\text{\ \
\ }\forall t\in \left[ 0,T_{m}^{\ast }\right] .
\end{equation*}%
Now from \eqref{A4} and \eqref{025}, there exist $N>N_{0}$ and $\beta \in $ $%
(1;\alpha _{1})$ such that%
\begin{gather*}
0\leq \frac{1}{2}\left\vert u_{m}\left( t\right) \right\vert
^{2}+\int_{0}^{t}\sum_{i=1}^{n}\int_{\Omega }\frac{4}{p^{2}\left( x\right) }%
\left( \frac{\partial }{\partial x_{i}}\left( \left\vert u_{m}\right\vert ^{%
\frac{p\left( x\right) -2}{2}}u_{m}\right) \right) ^{2}dxds \\
+\int_{0}^{t}Q\left( \left\vert \left\vert u_{m}\right\vert \right\vert
_{p(.)}^{p_{2}}\right) ds\leq \int_{0}^{t}Q\left( \beta \right) ds\text{ \ \
}\forall t\in \left[ 0,T_{m}^{\ast }\right] ,\text{ \ }\forall m>N
\end{gather*}%
Then the monotonicity of $Q$ implies that%
\begin{equation*}
\left\vert \left\vert u_{m}\left( t\right) \right\vert \right\vert
_{p(.)}^{p_{2}}\leq \beta <\alpha _{1}\text{ \ }\forall t\in \left[
0,T_{m}^{\ast }\right]
\end{equation*}%
and in particular, $\left\vert \left\vert u_{m}\left( T_{m}^{\ast }\right)
\right\vert \right\vert _{p(.)}^{p_{2}}<\alpha _{1}$, which is a
contradiction to \eqref{028}. And then the supposition $Q\left( \left\vert
\left\vert u_{m}\right\vert \right\vert _{p(.)}^{p_{2}}\right) \geq 0$ is
true$.$
\end{proof}

From \eqref{027} the solution $u_{m}\left( t\right) $ of problem \eqref{1}-%
\eqref{4} satisfies other of \eqref{031},
\begin{equation}
u_{m}\text{ is bounded in}\ L^{p\left( .\right) }\left( \Omega \times \left(
0,T\right) \right) .  \label{026}
\end{equation}

\begin{lemma}
\label{Lemma1}Let $u_{m}$, constructed in \eqref{15}, be the approximate
solution of \eqref{1}-\eqref{4}. Then
\begin{equation}
\frac{\partial }{\partial t}u_{m}\left( t\right) \text{ is bounded in }%
L^{p_{2}^{\prime }}(0,T;H^{-r}\left( \Omega \right) ).  \label{lema1}
\end{equation}
\end{lemma}

\begin{proof}
Let $v\in H^{r}\left( \Omega \right) $, from \eqref{16} we have%
\begin{gather}
\left( \frac{\partial u_{m}(t)}{\partial t},v\right) +\left(
\sum_{i=1}^{n}\left( \left\vert u_{m}\right\vert ^{p\left( x\right) -2}\frac{%
\partial u_{m}}{\partial x_{i}}\right) ,\frac{\partial }{\partial x_{i}}%
v\right) +\left( \left\vert u_{m}\right\vert ^{p\left( x\right)
-2}u_{m},v\right)  \label{30} \\
=\sum_{i=1}^{n}\int_{\Gamma }\left\vert u_{m}\right\vert ^{p\left( x\right)
-2}\frac{\partial u_{m}}{\partial x_{i}}vd\Gamma +\left( f\left( t\right)
,v\right) ,  \notag
\end{gather}%
The last term in the left-hand side can be estimated as follows:%
\begin{gather*}
\left\vert \left( \left\vert u_{m}\right\vert ^{p\left( x\right)
-2}u_{m},v\right) \right\vert \leq \left\vert \left\vert u_{m}\right\vert
^{p\left( x\right) -1}\right\vert _{p_{2}^{\prime }}\left\vert v\right\vert
_{p_{2}}\leq C\left\vert \left\vert u_{m}\right\vert ^{p\left( x\right)
-1}\right\vert _{p^{\prime }\left( .\right) }\left\vert v\right\vert _{p_{2}}%
\text{ \ (}p_{2}^{\prime }\leq p^{\prime }\left( .\right) \leq p_{1}^{\prime
}\text{)} \\
\leq C\max \left( \left( \int_{\Omega }\left\vert u_{m}\right\vert ^{p\left(
x\right) }dx\right) ^{\frac{1}{p_{1}^{\prime }}},\left( \int_{\Omega
}\left\vert u_{m}\right\vert ^{p\left( x\right) }dx\right) ^{\frac{1}{%
p_{2}^{\prime }}}\right) \left\vert v\right\vert _{p_{2}} \\
\leq C\max \left( \left( \int_{\Omega }\left\vert u_{m}\right\vert ^{p\left(
x\right) }dx\right) ^{\frac{1}{p_{1}^{\prime }}},\left( \int_{\Omega
}\left\vert u_{m}\right\vert ^{p\left( x\right) }dx\right) ^{\frac{1}{%
p_{2}^{\prime }}}\right) \left\vert v\right\vert _{H^{r}}
\end{gather*}%
Hence,
\begin{equation*}
\left\vert \left\vert u_{m}\right\vert ^{p\left( .\right)
-2}u_{m}\right\vert _{H^{-r}\left( \Omega \right) }\leq C\max \left( \left(
\int_{\Omega }\left\vert u_{m}\right\vert ^{p\left( x\right) }dx\right) ^{%
\frac{1}{p_{1}^{\prime }}},\left( \int_{\Omega }\left\vert u_{m}\right\vert
^{p\left( x\right) }dx\right) ^{\frac{1}{p_{2}^{\prime }}}\right) <\infty .
\end{equation*}%
The norm of $\left\vert u_{m}\right\vert ^{p\left( .\right) -2}u_{m}$ in $%
L^{p_{2}^{\prime }}(0,T;H^{-r}\left( \Omega \right) )$ is bounded by%
\begin{equation*}
C\left( \int_{0}^{T}\max \left( \left( \int_{\Omega }\left\vert
u_{m}\right\vert ^{p\left( x\right) }dx\right) ^{\frac{p_{2}^{\prime }}{%
p_{1}^{\prime }}},\int_{\Omega }\left\vert u_{m}\right\vert ^{p\left(
x\right) }dx\right) \right) ^{\frac{1}{p_{2}^{\prime }}}<\infty
\end{equation*}%
Therefore, $\left\vert u_{m}\right\vert ^{p\left( .\right) -2}u_{m}$ is
bounded in $L^{p_{2}^{\prime }}(0,T;H^{-r}\left( \Omega \right) ).$

Next, we consider the term $\sum_{i=1}^{n}\int_{\Gamma }\left\vert
u_{m}\right\vert ^{p\left( x\right) -2}\frac{\partial u_{m}}{\partial x_{i}}%
vd\Gamma $ in the left-hand side of \eqref{30}:%
\begin{gather*}
\left\vert \sum_{i=1}^{n}\int_{\Gamma }\left\vert u_{m}\right\vert ^{p\left(
x\right) -2}\frac{\partial u_{m}}{\partial x_{i}}vd\Gamma \right\vert \leq
\left( \sum_{i=1}^{n}\left\vert \left\vert u_{m}\right\vert ^{p\left(
x\right) -2}\frac{\partial u_{m}}{\partial x_{i}}\right\vert _{p_{1}^{\prime
},\Gamma }\right) \left\vert v(t)\right\vert _{p_{1},\Gamma } \\
=\sum_{i=1}^{n}\left\vert \left\vert \int_{\Omega }K\left( x,y\right)
u_{m}\left( y\right) dy\right\vert ^{p\left( x\right) -2}\int_{\Omega }\frac{%
\partial }{\partial x_{i}}K_{i}\left( x,y\right) u_{m}\left( y\right)
dy\right\vert _{p_{1}^{\prime },\Gamma } \\
\times \left\vert \int_{\Omega }K\left( x,y\right) v(y)dy\right\vert
_{p_{1},\Gamma } \\
\leq C\sum_{i=1}^{n}\left\vert K\left( x\right) ^{p\left( x\right)
-2}K_{i}\left( x\right) \left\vert u_{m}\left( y\right) \right\vert
_{p_{1}}^{p\left( x\right) -1}\right\vert _{p_{1}^{\prime },\Gamma
}\left\vert K\left( x\right) \left\vert v(y)\right\vert _{p_{1}}\right\vert
_{p_{1},\Gamma } \\
\leq C\sum_{i=1}^{n}\left\vert K\left( x\right) ^{p\left( x\right)
-2}K_{i}\left( x\right) \right\vert _{p_{1}^{\prime },\Gamma }\left\vert
K\left( x\right) \right\vert _{p_{1},\Gamma }\left\vert u_{m}\left( y\right)
\right\vert _{p_{1}}^{p\left( x\right) -1}\left\vert v(y)\right\vert _{p_{1}}
\\
\leq C\max \left( \sum_{i=1}^{n}\left\vert K\left( x\right)
^{p_{1}-2}K_{i}\left( x\right) \right\vert _{p_{1}^{\prime },\Gamma
},\sum_{i=1}^{n}\left\vert K\left( x\right) ^{p_{2}-2}K_{i}\left( x\right)
\right\vert _{p_{1}^{\prime },\Gamma }\right) \left\vert K\left( x\right)
\right\vert _{p_{1},\Gamma } \\
\times \max \left( \left\vert u_{m}\left( y\right) \right\vert
_{p_{1}}^{p_{1}-1},\left\vert u_{m}\left( y\right) \right\vert
_{p_{1}}^{p_{2}-1}\right) \left\vert v(y)\right\vert _{H^{r}\left( \Omega
\right) } \\
\leq C\max \left( \sum_{i=1}^{n}\left\vert K\left( x\right)
^{p_{1}-2}K_{i}\left( x\right) \right\vert _{p_{1}^{\prime },\Gamma
},\sum_{i=1}^{n}\left\vert K\left( x\right) ^{p_{2}-2}K_{i}\left( x\right)
\right\vert _{p_{1}^{\prime },\Gamma }\right) \left\vert K\left( x\right)
\right\vert _{p_{1},\Gamma } \\
\times \max \left( \left\vert u_{m}\left( y\right) \right\vert _{p\left(
.\right) }^{p_{1}-1},\left\vert u_{m}\left( y\right) \right\vert _{p_{\left(
.\right) }}^{p_{2}-1}\right) \left\vert v(y)\right\vert _{H^{r}}
\end{gather*}%
Therefore,%
\begin{gather*}
\left\vert \sum_{i=1}^{n}\int_{\Gamma }\left\vert u_{m}\right\vert ^{p\left(
x\right) -2}\frac{\partial u_{m}}{\partial x_{i}}d\Gamma \right\vert
_{H^{-r}\left( \Omega \right) } \\
\leq C\max \left( \sum_{i=1}^{n}\left\vert K\left( x\right)
^{p_{1}-2}K_{i}\left( x\right) \right\vert _{p_{1}^{\prime },\Gamma
},\sum_{i=1}^{n}\left\vert K\left( x\right) ^{p_{2}-2}K_{i}\left( x\right)
\right\vert _{p_{1}^{\prime },\Gamma }\right) \\
\times \max \left( \left\vert u_{m}\left( y\right) \right\vert _{p\left(
.\right) }^{p_{1}-1},\left\vert u_{m}\left( y\right) \right\vert _{p_{\left(
.\right) }}^{p_{2}-1}\right) \left\vert K\left( x\right) \right\vert
_{p_{1},\Gamma }<\infty
\end{gather*}%
Then the norm of\ $\sum_{i=1}^{n}\int_{\Gamma }\left\vert u_{m}\right\vert
^{p\left( x\right) -2}\frac{\partial u_{m}}{\partial x_{i}}d\Gamma $ in $%
L^{p_{2}^{\prime }}(0,T;H^{-r}\left( \Omega \right) )$ is bounded by%
\begin{equation*}
C\left(
\begin{array}{c}
\int_{0}^{T}\max \left( \sum_{i=1}^{n}\left\vert K\left( x\right)
^{p_{1}-2}K_{i}\left( x\right) \right\vert _{p_{1}^{\prime },\Gamma
}^{p_{2}^{\prime }},\sum_{i=1}^{n}\left\vert K\left( x\right)
^{p_{2}-2}K_{i}\left( x\right) \right\vert _{p_{1}^{\prime },\Gamma
}^{p_{2}^{\prime }}\right) \\
\times \max \left( \left\vert u_{m}\left( y\right) \right\vert _{p\left(
.\right) }^{\left( p_{1}-1\right) p_{2}^{\prime }},\left\vert u_{m}\left(
y\right) \right\vert _{p_{\left( .\right) }}^{\left( p_{2}-1\right)
p_{2}^{\prime }}\right) \left\vert K\left( x\right) \right\vert
_{p_{1},\Gamma }^{p_{2}^{\prime }}dt%
\end{array}%
\right) ^{\frac{1}{p_{2}^{\prime }}}<\infty
\end{equation*}%
Hence $\sum_{i=1}^{n}\int_{\Gamma }\left\vert u_{m}\right\vert ^{p\left(
x\right) -2}\frac{\partial u_{m}}{\partial x_{i}}d\Gamma $ is bounded in $%
L^{p_{2}^{\prime }}(0,T;H^{-r}\left( \Omega \right) ).$

Next, we consider the second term in the left-hand side of \eqref{30}.
Integrating by parts gives%
\begin{gather}
\left( \sum_{i=1}^{n}\left( \left\vert u_{m}\right\vert ^{p\left( x\right)
-2}\frac{\partial u_{m}}{\partial x_{i}}\right) ,\frac{\partial v}{\partial
x_{i}}\right) =\int_{\Omega }\sum_{i=1}^{n}\frac{1}{p\left( x\right) -1}%
\left( \frac{\partial }{\partial x_{i}}\left( \left\vert u_{m}\right\vert
^{p\left( x\right) -2}u_{m}\right) \right) \frac{\partial v}{\partial x_{i}}%
dx  \label{31} \\
=\int_{\Gamma }\sum_{i=1}^{n}\frac{1}{p\left( x\right) -1}\left\vert
u_{m}\right\vert ^{p\left( x\right) -2}u_{m}\frac{\partial v}{\partial x_{i}}%
d\Gamma -\int_{\Omega }\frac{1}{p\left( x\right) -1}\left\vert
u_{m}\right\vert ^{p\left( x\right) -2}u_{m}\Delta vdx.  \notag
\end{gather}%
First, we have%
\begin{gather*}
\left\vert \int_{\Gamma }\sum_{i=1}^{n}\frac{1}{p\left( x\right) -1}%
\left\vert u_{m}\right\vert ^{p\left( x\right) -2}u_{m}\frac{\partial v}{%
\partial x_{i}}d\Gamma \right\vert \leq \frac{1}{p_{2}-1}\sum_{i=1}^{n}\left\vert \left\vert u_{m}\right\vert ^{p\left( x\right) -2}u_{m}\right\vert
_{p_{1}^{\prime },\Gamma }\left\vert \frac{\partial v}{\partial x_{i}}%
\right\vert _{p_{1},\Gamma } \\
=\frac{1}{p_{2}-1}\sum_{i=1}^{n}\left\vert \left( \int_{\Omega }K\left(
x,y\right) u_{m}\left( y\right) dy\right) ^{p\left( x\right) -1}\right\vert
_{p_{1}^{\prime },\Gamma }\left\vert \int_{\Omega }\frac{\partial }{\partial
x_{i}}K\left( x,y\right) v\left( y\right) dy\right\vert _{p_{1},\Gamma } \\
\leq C\sum_{i=1}^{n}\left\vert K\left( x\right) ^{p\left( x\right)
-1}\left\vert u_{m}\right\vert _{p_{1}}^{p\left( x\right) -1}\right\vert
_{p_{1}^{\prime },\Gamma }\left\vert K_{i}\left( x\right) \left\vert
v\right\vert _{p_{1}}\right\vert _{p_{1},\Gamma } \\
\leq C\sum_{i=1}^{n}\left\vert K\left( x\right) ^{p\left( x\right)
-1}\right\vert _{p_{1}^{\prime },\Gamma }\left\vert K_{i}\left( x\right)
\right\vert _{p_{1},\Gamma }\left\vert u_{m}\right\vert _{p_{1}}^{p\left(
x\right) -1}\left\vert v\right\vert _{p_{1}} \\
\leq C\max \left( \sum_{i=1}^{n}\left\vert K\left( x\right)
^{p_{1}-1}\right\vert _{p_{1}^{\prime },\Gamma },\sum_{i=1}^{n}\left\vert
K\left( x\right) ^{p_{2}-1}\right\vert _{p_{1}^{\prime },\Gamma }\right) \\
\times \max \left( \left\vert u_{m}\right\vert _{p_{1}}^{p_{1}-1},\left\vert
u_{m}\right\vert _{p_{1}}^{p_{2}-1}\right) \left\vert K_{i}\left( x\right)
\right\vert _{p_{1},\Gamma }\left\vert v(y)\right\vert _{H^{r}}
\end{gather*}%
So we have%
\begin{gather*}
\left\vert \int_{\Gamma }\sum_{i=1}^{n}\frac{1}{p\left( x\right) -1}%
\left\vert u_{m}\right\vert ^{p\left( x\right) -2}u_{m}d\Gamma \right\vert
_{H^{-r}\left( \Omega \right) } \\
\leq C\max \left( \sum_{i=1}^{n}\left\vert K\left( x\right)
^{p_{1}-1}\right\vert _{p_{1}^{\prime },\Gamma },\sum_{i=1}^{n}\left\vert
K\left( x\right) ^{p_{2}-1}\right\vert _{p_{1}^{\prime },\Gamma }\right) \\
\times \max \left( \left\vert u_{m}\right\vert _{p_{1}}^{p_{1}-1},\left\vert
u_{m}\right\vert _{p_{1}}^{p_{2}-1}\right) \left\vert K_{i}\left( x\right)
\right\vert _{p_{1},\Gamma }<\infty
\end{gather*}%
consequently, $\int_{\Gamma }\sum_{i=1}^{n}\frac{1}{p\left( x\right) -1}%
\left\vert u_{m}\right\vert ^{p\left( x\right) -2}u_{m}d\Gamma $ is bounded
in $L^{p_{2}^{\prime }}(0,T;H^{-r}\left( \Omega \right) ).$

Next, consider $\int_{\Omega }\frac{1}{p\left( x\right) -1}\left\vert
u_{m}\right\vert ^{p\left( x\right) -2}u_{m}\Delta vdx$, by the same manner,
we have%
\begin{gather*}
\left\vert \int_{\Omega }\frac{1}{p\left( x\right) -1}\left\vert
u_{m}\right\vert ^{p\left( x\right) -2}u_{m}\Delta vdx\right\vert \leq \frac{%
1}{p_{1}-1}\left\vert \left\vert u_{m}\right\vert ^{p\left( x\right)
-1}\right\vert _{p_{2}^{\prime }}\left\vert \Delta v\right\vert _{p_{2}} \\
\leq C\left\vert \left\vert u_{m}\right\vert ^{p\left( x\right)
-1}\right\vert _{p^{\prime }\left( .\right) }\left\vert \Delta v\right\vert
_{p_{2}} \\
\leq C\max \left( \left( \int_{\Omega }\left\vert u_{m}\right\vert ^{p\left(
x\right) }dx\right) ^{\frac{1}{p_{1}^{\prime }}},\left( \int_{\Omega
}\left\vert u_{m}\right\vert ^{p\left( x\right) }dx\right) ^{\frac{1}{%
p_{2}^{\prime }}}\right) \left\vert v\right\vert _{H^{r}}
\end{gather*}%
therefore, $\int_{\Omega }\frac{1}{p\left( x\right) -1}\left\vert
u_{m}\right\vert ^{p\left( x\right) -2}u_{m}\Delta vdx\Gamma $ is bounded in
$L^{p_{2}^{\prime }}(0,T;H^{-r}\left( \Omega \right) ).$ Since $f\in
L^{p_{2}^{\prime }}(0,T;L^{p_{2}^{\prime }}\left( \Omega \right) )\subset
L^{p_{2}^{\prime }}(0,T;H^{-r}\left( \Omega \right) )$, from this discussion
and \eqref{30} it yields that $\frac{\partial }{\partial t}u_{m}$ is bounded
in $L^{p_{2}^{\prime }}(0,T;H^{-r}\left( \Omega \right) )$.
\end{proof}

\begin{theorem}
\label{Theorem3} Let $B$, $B_{1}$ be Banach spaces, and $S$ be a set. Define%
\begin{equation*}
M\left( v\right) =\max \left( \left( \sum_{i=1}^{n}\int_{\Omega }\left\vert
v\right\vert ^{p_{1}-2}\left( \frac{\partial v}{\partial x_{i}}\right)
^{2}dx\right) ^{\frac{1}{p_{1}}},\left( \sum_{i=1}^{n}\int_{\Omega
}\left\vert v\right\vert ^{p_{2}-2}\left( \frac{\partial v}{\partial x_{i}}%
\right) ^{2}dx\right) ^{\frac{1}{p_{2}}}\right)
\end{equation*}%
on $S$ with:

\begin{description}
\item[a)] $S\subset B\subset B_{1},$ and $M\left( v\right) \geq 0$ on $S$,\
\begin{gather*}
M\left( \lambda v\right) =\max \left( \left( \sum_{i=1}^{n}\int_{\Omega
}\left\vert v\right\vert ^{p_{1}-2}\left( \frac{\partial v}{\partial x_{i}}%
\right) ^{2}dx\right) ^{\frac{1}{p_{1}}},\left( \sum_{i=1}^{n}\int_{\Omega
}\left\vert v\right\vert ^{p_{2}-2}\left( \frac{\partial v}{\partial x_{i}}%
\right) ^{2}dx\right) ^{\frac{1}{p_{2}}}\right) \\
=\left\vert \lambda \right\vert M\left( v\right)
\end{gather*}

\item[b)] the set $\left\{ v\mid v\in S,\text{ }M\left( v\right) \leq
1\right\} $ is relatively compact in $B$.
\end{description}

Define the set
\begin{equation*}
F=\left\{
\begin{array}{c}
v:v\text{ }\ \text{is locally summable on }[0,T]\text{ with value in }B_{1}%
\text{;} \\
\text{ }\int_{0}^{T}\left( M\left( v\left( t\right) \right) \right)
^{q_{0}}dt\leq C,\text{ }v^{\prime }\text{ bounded in }L^{q_{1}}(0,T;B_{1})%
\text{,}%
\end{array}%
\right\}
\end{equation*}%
where $1<q_{i}<\infty ,$ $i=0,1$. Then $F\subset L^{q_{0}}(0,T;B)$, and $F$
is relatively compact in $L^{q_{0}}(0,T;B)$.
\end{theorem}

We need Theorem \eqref{Theorem3} to prove the following lemma \eqref{Lemma3}.

\begin{lemma}
\label{Lemma3}Let $u_{m}$, constructed as in \eqref{15}, be the approximate
solution of \eqref{1}-\eqref{4}, then $u_{m}\rightarrow u$ in $%
L^{p_{2}}(0,T;L^{p_{2}}\left( \Omega \right) )$ strongly and almost
everywhere.
\end{lemma}

\begin{proof}
Let
\begin{equation*}
S=\left\{ v:\max \left( \left\vert v\right\vert ^{\frac{p_{1}-2}{2}%
}v,\left\vert v\right\vert ^{\frac{p_{2}-2}{2}}v\right) \in H^{1}\left(
\Omega \right) \right\}
\end{equation*}%
Since $H^{1}\left( \Omega \right) $ is compactly embedded in $L^{2}\left(
\Omega \right) $, the proof of \ \cite[ Proposition 12.1,p. 143]{Lions2}
also works for both $\left\vert v\right\vert ^{\frac{p_{1}-2}{2}}v\ $\ and $%
\left\vert v\right\vert ^{\frac{p_{2}-2}{2}}v,$ then (b) holds.

Let $B=L^{p_{2}}\left( \Omega \right) ,$ $B_{1}=H^{-r}\left( \Omega \right)
, $ $q_{0}=p_{2},$ $q_{1}=p_{2}^{\prime },$ we have
\begin{gather*}
\int_{0}^{T}\left( M\left( v\left( t\right) \right) \right) ^{q_{0}}dt\leq
C\int_{0}^{T}\max \left(
\begin{array}{c}
\left( \sum_{i=1}^{n}\int_{\Omega }\left\vert v\right\vert ^{p_{1}-2}\left(
\frac{\partial v}{\partial x_{i}}\right) ^{2}dx\right) ^{\frac{p_{2}}{p_{1}}%
}, \\
\left( \sum_{i=1}^{n}\int_{\Omega }\left\vert v\right\vert ^{p_{2}-2}\left(
\frac{\partial v}{\partial x_{i}}\right) ^{2}dx\right)%
\end{array}%
\right) dt \\
\leq C\int_{0}^{T}\max \left(
\begin{array}{c}
\left( \sum_{i=1}^{n}\int_{\Omega }\left( \frac{\partial }{\partial x_{i}}%
\left( \left\vert v\right\vert ^{\frac{p_{1}-2}{2}}v\right) \right)
^{2}dx\right) ^{\frac{p_{2}}{p_{1}}}, \\
\left( \sum_{i=1}^{n}\int_{\Omega }\left( \frac{\partial }{\partial x_{i}}%
\left( \left\vert v\right\vert ^{\frac{p_{2}-2}{2}}v\right) \right)
^{2}dx\right)%
\end{array}%
\right) dt<\infty
\end{gather*}
\end{proof}

Now with Lemma \eqref{Lemma1} and a priori estimates, conclusion follows
easily from application of Theorem \eqref{Theorem3}.

Next, we prove that we can pass the limit in \eqref{30}. Lemmas %
\eqref{Lemma4}-\eqref{Lemma6}, below, show that we can pass the limit in
each term in the left-hand side of \eqref{30}

\begin{lemma}
\label{Lemma4}Let $u_{m}$, constructed as in \eqref{15}, be the approximate
solution of \eqref{1}-\eqref{4}, then $\left( \left\vert u_{m}\right\vert
^{p\left( x\right) -2}u_{m},v\right) \rightarrow \left( \left\vert
u\right\vert ^{p\left( x\right) -2}u,v\right) $ as $m\rightarrow \infty .$
\end{lemma}

\begin{proof}
Since $u_{m}\ $is bounded in$\ L^{p\left( .\right) }\left( \Omega \times
\left( 0,T\right) \right) $ then $\left\vert u_{m}\right\vert ^{p\left(
.\right) -2}u_{m}$ is bounded in $L^{\frac{p\left( .\right) }{p\left(
.\right) -1}}\left( \Omega \times \left( 0,T\right) \right) $; hence, using
same arguments as in \cite[Lemma 1.3]{Lions2}, we have%
\begin{equation*}
\left\vert u_{m}\right\vert ^{p\left( .\right) -2}u_{m}\rightarrow
\left\vert u\right\vert ^{p\left( .\right) -2}u\text{ weakly in }L^{\frac{%
p\left( .\right) }{p\left( .\right) -1}}\left( \Omega \times \left(
0,T\right) \right) .
\end{equation*}
\end{proof}

\begin{lemma}
\label{Lemma05}Let $u_{m}$, constructed as in \eqref{15}, be the approximate
solution of \eqref{1}-\eqref{4}, then $\int_{\Gamma }\sum_{i=1}^{n}\frac{1}{%
p\left( x\right) -1}\left( \left\vert u_{m}\right\vert ^{p\left( x\right) -2}%
\frac{\partial }{\partial x_{i}}u_{m}\right) vd\Gamma \rightarrow
\int_{\Gamma }\sum_{i=1}^{n}\frac{1}{p\left( x\right) -1}\left( \left\vert
u\right\vert ^{p\left( x\right) -2}\frac{\partial }{\partial x_{i}}u\right)
vd\Gamma $ as $m\rightarrow \infty .$
\end{lemma}

\begin{proof}
By a priori estimates, $u_{m}$ is bounded in $L^{p\left( .\right) }$ for
almost every $t$, then there exists subsequence of $u_{m}$, still denoted as
$u_{m}$, converges to $u_{m}$ \ weak star in $L^{p\left( .\right) }$
(Alaoglu's Theorem) for almost every $t\in \lbrack 0,T]$. Under the
assumption that for fixed $x\in \Gamma ,$ we have
\begin{equation*}
\int_{\Omega }K\left( x,y\right) u_{m}\left( y\right) dy\rightarrow
\int_{\Omega }K\left( x,y\right) u\left( y\right) dy\text{ \ as }%
m\rightarrow \infty
\end{equation*}%
Similarly%
\begin{equation*}
\int_{\Omega }\frac{\partial }{\partial x_{i}}K\left( x,y\right) u_{m}\left(
y\right) dy\rightarrow \int_{\Omega }\frac{\partial }{\partial x_{i}}K\left(
x,y\right) u\left( y\right) dy\text{ \ as }m\rightarrow \infty
\end{equation*}%
Therefore, for $x\in \Gamma $, we have%
\begin{equation*}
\left\vert u_{m}\right\vert ^{p\left( .\right) -2}\frac{\partial }{\partial
x_{i}}u_{m}\rightarrow \left\vert u\right\vert ^{p\left( .\right) -2}\frac{%
\partial }{\partial x_{i}}u\text{ \ a.e.}
\end{equation*}%
Since
\begin{gather*}
\max \left( \int_{\Gamma }K^{p_{1}}\left( x\right) d\Gamma ,\int_{\Gamma
}K^{p_{2}}\left( x\right) d\Gamma \right) <\infty \text{,} \\
\text{\ and }\max \left( \int_{\Gamma }K_{i}^{p_{1}}\left( x\right) d\Gamma
,\int_{\Gamma }K_{i}^{p_{2}}\left( x\right) d\Gamma \right) <\infty ,
\end{gather*}%
we have%
\begin{equation*}
\left\vert u_{m}\right\vert _{p\left( .\right) ,\Gamma }\leq C\max \left(
\int_{\Gamma }K^{p_{1}}\left( x\right) d\Gamma ,\int_{\Gamma
}K^{p_{2}}\left( x\right) d\Gamma \right) \max \left( \left\Vert
u_{m}\right\Vert _{p\left( .\right) }^{p_{1}},\left\Vert u_{m}\right\Vert
_{p\left( .\right) }^{p_{2}}\right) <\infty
\end{equation*}%
and%
\begin{equation*}
\left\vert \frac{\partial }{\partial x_{i}}u_{m}\right\vert _{p\left(
.\right) ,\Gamma }\leq C\max \left( \int_{\Gamma }K_{i}^{p_{1}}\left(
x\right) d\Gamma ,\int_{\Gamma }K_{i}^{p_{2}}\left( x\right) d\Gamma \right)
\max \left( \left\Vert u_{m}\right\Vert _{p\left( .\right)
}^{p_{1}},\left\Vert u_{m}\right\Vert _{p\left( .\right) }^{p_{2}}\right)
<\infty .
\end{equation*}%
Then%
\begin{gather*}
\left\vert \left\vert u_{m}\right\vert ^{p\left( .\right) -2}\frac{\partial
}{\partial x_{i}}u_{m}\right\vert _{p_{2}^{\prime },\Gamma }\leq C\left\vert
\left\vert u_{m}\right\vert ^{p\left( .\right) -2}\frac{\partial }{\partial
x_{i}}u_{m}\right\vert _{p^{\prime }\left( .\right) ,\Gamma }\text{ \ \
since (}p_{2}^{\prime }\leq p^{\prime }\left( .\right) \leq p_{1}^{\prime }%
\text{)} \\
\leq \left\vert \left\vert u_{m}\right\vert ^{p\left( .\right)
-2}\right\vert _{\frac{p\left( .\right) }{p\left( .\right) -2},\Gamma
}\left\vert \frac{\partial }{\partial x_{i}}u_{m}\right\vert _{p\left(
.\right) ,\Gamma }\text{ \ since (}\frac{1}{p^{\prime }\left( .\right) }=%
\frac{p\left( .\right) -2}{p\left( .\right) }+\frac{1}{p\left( .\right) }%
\text{)} \\
\leq \max \left( \left( \int_{\Gamma }\left\vert u_{m}\right\vert ^{p\left(
x\right) }d\Gamma \right) ^{\frac{1}{p_{1}}},\left( \int_{\Omega }\left\vert
u_{m}\right\vert ^{p\left( x\right) }d\Gamma \right) ^{\frac{1}{p_{2}}%
}\right) \\
\times \max \left( \left( \int_{\Omega }\left\vert \frac{\partial }{\partial
x_{i}}u_{m}\right\vert ^{p\left( x\right) }d\Gamma \right) ^{\frac{1}{p_{1}}%
},\left( \int_{\Gamma }\left\vert \frac{\partial }{\partial x_{i}}%
u_{m}\right\vert ^{p\left( x\right) }d\Gamma \right) ^{\frac{1}{p_{2}}%
}\right) <\infty .
\end{gather*}%
So, applying the same arguments as in \cite[Lemma 1.3]{Lions2} to conclude
that%
\begin{equation*}
\left\vert u_{m}\right\vert ^{p\left( .\right) -2}\frac{\partial }{\partial
x_{i}}u_{m}\rightarrow \left\vert u\right\vert ^{p\left( .\right) -2}\frac{%
\partial }{\partial x_{i}}u\text{\ \ weakly in }L^{p_{2}^{\prime }}\left(
\Gamma \right) \text{.}
\end{equation*}%
for a.e. $t\in \lbrack 0,T]$. Since,
\begin{equation*}
\max \left( \left( \int_{\Omega }\left\vert \frac{\partial }{\partial x_{i}}%
v\right\vert ^{p\left( x\right) }d\Gamma \right) ^{\frac{1}{p_{1}}},\left(
\int_{\Omega }\left\vert \frac{\partial }{\partial x_{i}}v\right\vert
^{p\left( x\right) }d\Gamma \right) ^{\frac{1}{p_{2}}}\right) <\infty ,
\end{equation*}%
the proof is complete.
\end{proof}

\begin{lemma}
\label{Lemma5b}Let $u_{m}$, constructed as in \eqref{15}, be the approximate
solution of \eqref{1}-\eqref{4}, then $\int_{\Gamma }\sum_{i=1}^{n}\frac{1}{%
p\left( x\right) -1}\left\vert u_{m}\right\vert ^{p\left( x\right) -2}u_{m}%
\frac{\partial v}{\partial x_{i}}d\Gamma \rightarrow \int_{\Gamma
}\sum_{i=1}^{n}\frac{1}{p\left( x\right) -1}\left\vert u\right\vert
^{p\left( x\right) -2}u\frac{\partial v}{\partial x_{i}}d\Gamma $ as $%
m\rightarrow \infty .$
\end{lemma}
\end{proof}

\begin{proof}
From the proof of Lemma \eqref{Lemma05}, we have, for $x\in \Gamma ,$ $%
\left\vert u_{m}\right\vert ^{p\left( .\right) -2}u_{m}\rightarrow
\left\vert u\right\vert ^{p\left( .\right) -2}u$ almost everywhere, and
\begin{gather*}
\left\vert \left\vert u_{m}\right\vert ^{p\left( .\right)
-2}u_{m}\right\vert _{p_{2}^{\prime },\Gamma }\leq C\left\vert \left\vert
u_{m}\right\vert ^{p\left( .\right) -2}u_{m}\right\vert _{p^{\prime }\left(
.\right) ,\Gamma } \\
\leq \max \left( \left( \int_{\Gamma }\left\vert u_{m}\right\vert ^{p\left(
x\right) }d\Gamma \right) ^{\frac{1}{p_{1}}},\left( \int_{\Gamma }\left\vert
u_{m}\right\vert ^{p\left( x\right) }d\Gamma \right) ^{\frac{1}{p_{2}}%
}\right) <\infty .
\end{gather*}%
Therefore, by applying \cite[Lemma 1.3]{Lions2} we conclude that%
\begin{equation*}
\left\vert u_{m}\right\vert ^{p\left( .\right) -2}u_{m}\rightarrow
\left\vert u\right\vert ^{p\left( .\right) -2}u\text{\ \ weakly in }%
L^{p_{2}^{\prime }}\left( \Gamma \right) \text{.}
\end{equation*}%
Since $\frac{\partial v}{\partial x_{i}}\in L^{p_{2}^{\prime }}\left( \Gamma
\right) $, the proof is complete.

\begin{lemma}
\label{Lemma5}Let $u_{m}$, constructed as in \eqref{15}, be the approximate
solution of \eqref{1}-\eqref{4}, then $\int_{\Omega }\frac{1}{p\left(
x\right) -1}\left( \left\vert u_{m}\right\vert ^{p\left( x\right)
-2}u_{m}\right) \Delta vdx\rightarrow \int_{\Omega }\frac{1}{p\left(
x\right) -1}\left( \left\vert u\right\vert ^{p\left( x\right) -2}u\right)
\Delta vdx$ as $m\rightarrow \infty .$
\end{lemma}

\begin{proof}
From lemma (\eqref{Lemma3}), we have $\left\vert u_{m}\right\vert ^{p\left(
.\right) -2}u_{m}\rightarrow \left\vert u\right\vert ^{p\left( .\right) -2}u$
almost everywhere, for $x\in \Omega $, since
\begin{gather*}
\left\vert \left\vert u_{m}\right\vert ^{p\left( .\right)
-2}u_{m}\right\vert _{p_{2}^{\prime },\Omega }\leq C\left\vert \left\vert
u_{m}\right\vert ^{p\left( .\right) -2}u_{m}\right\vert _{p^{\prime }\left(
.\right) ,\Omega } \\
\leq \max \left( \left( \int_{\Omega }\left\vert u_{m}\right\vert ^{p\left(
x\right) }dx\right) ^{\frac{1}{p_{1}}},\left( \int_{\Omega }\left\vert
u_{m}\right\vert ^{p\left( x\right) }dx\right) ^{\frac{1}{p_{2}}}\right)
<\infty .
\end{gather*}%
by \cite[Lemma 1.3]{Lions2}, we have $\left\vert u_{m}\right\vert ^{p\left(
.\right) -2}u_{m}\rightarrow \left\vert u\right\vert ^{p\left( .\right) -2}u$
weakly in $L^{p_{2}^{\prime }}\left( \Omega \right) .$ Since $\Delta v\in $ $%
L^{p_{2}}\left( \Omega \right) $, the proof is complete.
\end{proof}

\begin{lemma}
\label{Lemma6}Let $u_{m}$, constructed as in \eqref{15}, be the approximate
solution of \eqref{1}-\eqref{4}, then $\left( \sum_{i=1}^{n}\left(
\left\vert u_{m}\right\vert ^{p\left( x\right) -2}\frac{\partial u_{m}}{%
\partial x_{i}}\right) ,\frac{\partial }{\partial x_{i}}v\right) \rightarrow
\left( \sum_{i=1}^{n}\left( \left\vert u\right\vert ^{p\left( x\right) -2}%
\frac{\partial u}{\partial x_{i}}\right) ,\frac{\partial }{\partial x_{i}}%
v\right) $ as $m\rightarrow \infty $.
\end{lemma}

\begin{proof}
Replacing the results of \eqref{Lemma5b} and \eqref{Lemma5} in \eqref{31},
the proof is complete.
\end{proof}

\begin{lemma}
\label{LemaaP}Let $u_{m}$, constructed as in \eqref{15}. be the approximate
solution of \eqref{1}-\eqref{4}, then $\left( \frac{\partial }{\partial t}%
u_{m},v\right) \rightarrow \left( \frac{\partial }{\partial t}u,v\right) $
and $u(t)$ is continuous on $[0,T]$.
\end{lemma}

\begin{proof}
Since $\frac{\partial }{\partial t}u_{m}\left( t\right) $ is bounded in $%
L^{p_{2}^{\prime }}(0,T;H^{-r}\left( \Omega \right) ),$ by Alaoglu's
theorem, there exists a subsequence, still denoted by $\frac{\partial }{%
\partial t}u_{m}\left( t\right) $, converging to $\chi $ weak star in $%
L^{p_{2}^{\prime }}(0,T;H^{-r}\left( \Omega \right) ).$ By slightly
modifying the proof of \cite[Theorem 1]{Abedelfatah} (with the space $%
L^{p_{2}^{\prime }}(0,T;H^{-r}\left( \Omega \right) )$ instead of $%
L^{2}(0,T;B_{2}^{1}\left( 0,1\right) ).$)$,$ we have $\chi =u^{\prime }$ and
$u$ is continuous on $\left[ 0,T\right] .$
\end{proof}

This ends the proof of Lemma \eqref{LemaaP}.

Combining all above results, the existence theorem \eqref{Theorem1} follows.
\end{proof}

\section*{Acknowledgements}

The authors would like to thank the anonymous referees and the handling
editor for their careful reading and for relevant remarks/suggestions which
helped him to improve the paper.

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