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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2011 (2011), No. 18, pp. 1--9.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2011 Texas State University - San Marcos.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2011/18\hfil Existence of solutions]
{Existence of solutions for quasilinear parabolic equations
with nonlocal boundary conditions}

\author[B. Chen\hfil EJDE-2011/18\hfilneg]
{Baili Chen}

\address{Baili Chen \newline
Department of Mathematics and Computer Science\\
Gustavus Adolphus College\\
Saint Peter, MN 56082, USA}
 \email{bchen@gustavus.edu}

\thanks{Submitted April 22, 2010. Published February 3, 2011.}
\subjclass[2000]{35K20, 35K59, 35B45, 35D30}
\keywords{Faedo-Galerkin method; nonlocal boundary conditions;
a priori \hfill\break\indent estimates;
quasilinear parabolic equations; generalized solution}

\begin{abstract}
  We prove the existence of a generalized solution a quasilinear
  parabolic equation with nonlocal boundary conditions,
  using the Faedo-Galerkin approximation.
\end{abstract}

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

\section{Introduction}


In this paper, we are concerned with the existence of  a
generalized solution of the following quasilinear parabolic
equation with nonlocal boundary conditions:
\begin{gather}
\frac{\partial{u}}{\partial{t}}
- \sum_{i=1}^{n}\frac{\partial}{\partial x_i}(|u|^{p-2}
\frac{\partial{u}}{\partial{x_i}})  +  |u|^{p-2}u  =  f(x,t),\quad
 x\in\Omega,\; t\in [0,T] \label{e1}\\
u(x,t) = \int_\Omega k(x,y)u(y,t)dy, \quad x\in \Gamma \label{e2}\\
u(x,0) = u_0(x). \label{e3}
\end{gather}
As a physical motivation, problem \eqref{e1}--\eqref{e3} arises
from the study of quasi-static thermoelasticity.  The main
difficulty of this problem is related to the presence of both
quasilinear term in \eqref{e1} and nonlocal boundary condition
\eqref{e2}. Literatures to this type of problem are very limited.
We only found [4] in which the authors study a quasilinear
parabolic equation with  nonlocal boundary conditions different
from \eqref{e2}.


The quasilinear term in \eqref{e1} makes it difficult to apply
classical methods like semi-group method or method of upper and
lower solutions.  However, we found that  Faedo-Galerkin method
serves as a convenient tool for this type of problem.  We proved
the existence of a generalized solution of problem
\eqref{e1}--\eqref{e3} by constructing approximate solution using
Faedo-Galerkin method and applying weak convergence and
compactness arguments.

It is well known that Faedo-Galerkin method is used to prove the
existence of solutions for linear parabolic equations in
\cite{Lady}.  In \cite{Gerbi}, Faedo-Galerkin method is coupled
with contraction mapping theorems to prove the existence of weak
solutions of semilinear wave equations with dynamic boundary
conditions.  Bouziani et al. use Faedo-Galerkin method to show the
existence of a unique weak solution for a linear parabolic
equation with nonlocal boundary conditions.   Lion's book
\cite[Chapter 1]{lions}, collects the work of Dubinskii and
Raviart, in which they use Faedo-Galerkin method to prove the
existence and uniqueness of weak solution for a quasilinear
parabolic equation with homogeneous boundary condition.

Problem \eqref{e1}--\eqref{e3} is the extension of the problem in
\cite[p. 140]{lions} in which the boundary conditions are
homegeneous.

This article is organized as follows: in section 2, we give the
definition of the generalized solution of problem
\eqref{e1}--\eqref{e3} and introduce the function spaces related
to the  generalized solution.  In section 3, we demonstrate the
construction of an approximation solution by Faedo-Galerkin method
and derive a priori estimates for the approximation solution.
Section 4 is devoted to the proof of existence of the generalized
solution by compactness arguments.

\section{Preliminaries}

In this article, we use the following notation:\\
$\Omega$:   regular and bounded domain of $R^n$;
$\Gamma$:  boundary of $\Omega$;\\
$(\cdot,\cdot )$: usual inner product in $L^2(\Omega)$;\\
$W^{k,p}(\Omega)$:  Sobelev space on $\Omega$;
$H^{r}(\Omega)$:   Sobelev space $W^{r,2}(\Omega)$;\\
$L^p(\Omega)$:  $L^p$ space defined on $\Omega$;
$|\cdot |_p$:  norm in $L^p(\Omega)$;
$|\cdot |_{p,\Gamma}$:  norm in $L^p(\Gamma)$;\\
$H^{-r}(\Omega)$:  dual space of $H^{r}(\Omega)$;
$|\cdot |_{H^{-r}(\Omega)}$:  norm in $H^{-r}(\Omega)$;\\
$c$:  nonzero constant which may take different values
on each occurrence;\\
$C$:  nonnegative constant which may take different values
on each occurrence;\\
$\hookrightarrow$:  continuous embedding;\\
$K(x)$:  norm of $k(x,y)$ in $L^q(\Omega)$ with respect to $y$,\\
 i.e., $K(x)=(\int_{\Omega}|k(x,y)|^q dy)^{1/q}$;\\
$K_i (x)$:  norm of $D_i k(x,y)$ in $L^q(\Omega)$ with respect
to $y$,\\
 i.e., $K_i(x)=(\int_{\Omega}|\frac{\partial
k(x,y)}{\partial x_i}|^q dy)^{1/q}$.

In this article, we make the following assumptions:
\begin{itemize}
\item[(A1)] $n\ge 2$, $p>n$, $r>\frac{n}{2} + 2$;

\item[(A2)] $\frac{1}{p} + \frac{1}{q} = 1$;

\item[(A3)] $f\in L^q(0,T; L^{q}(\Omega))$  and
$u_0 \in L^\infty(\Omega)$;

\item[(A4)] For any $x\in \Gamma$, $K(x)<\infty$, $K_i(x)<\infty$;

\item[(A5)] $\sum_{i=1}^n \int_\Gamma K(x)^{p-1} K_i(x) d\Gamma
< 1-\frac{1}{p}$.
\end{itemize}

Here we give an example of a function $k(x,y)$ which satisfies
assumptions (A4) and (A5): When $n=2$, $p=3$ and $\Omega$ is an
unit square, let $k(x,y)=x_1x_2(y_1y_2)^{2/3}$. It is easy to
verify that $K(x)=(\int_{\Omega}|k(x,y)|^q dy)^{1/q}$ and
$K_i(x)=(\int_{\Omega}|\frac{\partial k(x,y)}{\partial x_i}|^q
dy)^{1/q}$ satisfy assumptions (A4) and (A5).

With assumption (A1), using Sobelev embedding theorems, see
\cite{Adam}, we have
$$
H^r(\Omega) \hookrightarrow W^{2,p}(\Omega)
\hookrightarrow W^{1,p}(\Omega) \hookrightarrow L^p(\Omega)
\hookrightarrow L^2(\Omega).
$$
Define a space $V$:
\begin{equation} \label{e4}
V=\{v\in H^r(\Omega):\ v(x)=\int_\Omega k(x,y)v(y)dy, \ for\ x\in \Gamma\}
\end{equation}
It is easy to see that $V$ is a subspace of $H^r(\Omega)$.


\begin{definition}\label{def2.1} \rm
Define a generalized solution of problem \eqref{e1}--\eqref{e3} as
a function $u$, such that
\begin{itemize}

\item[(i)] $u\in L^\infty (0,T;L^2(\Omega))\cap
C([0,T],H^{-r}(\Omega))$;

\item[(ii)] $\frac{du}{dt}\in L^q(0,T;H^{-r}(\Omega))$;

\item[(iii)] $u(x,0)=u_0(x)$;

\item[(iv)] for all $v\in V$ and a.e. $t\in [0,T]$,
\begin{equation} \label{e5}
(\frac{du}{dt}, v) - (\sum_{i=1}^{n}\frac{\partial}{\partial x_i}
(|u|^{p-2}\frac{\partial u}{\partial x_i}) , v) + (|u|^{p-2}u, v)
 = (f,v)\,.
\end{equation}
\end{itemize}
\end{definition}


\begin{remark} \label{rmk2.2} \rm
From the proof of existence theorem in section 4, we will see that
each inner product in the identity \eqref{e5} is a function of $t$
in $L^q(0,T)$, hence the identity holds for a.e. $t\in [0,T]$. On
the other hand, since $u(t)\in V$, the boundary condition
\eqref{e2} is satisfied.
\end{remark}


\section{Construction of an approximate solution and a priori estimates}

Since $V$ is a subspace of $H^{r}(\Omega)$, which is separable. We
can choose a countable set of distinct basis elements ${w_j},\
j=1,2,\cdots, $ which generate $V$ and are orthonormal in $L^2
(\Omega)$.  Let $V_m$ be the subspace of $V$ generated by the
first $m$ elements: ${w_1, w_2,\cdots,w_m}$. We construct the
approximate solution of the form:
\begin{equation} \label{e6}
u_m(x,t) = \sum_{j=1}^{m} g_{jm}(t) w_j(x),\quad
(x,t)\in\Omega\times[0,T].
\end{equation}
where $(g_{jm}(t))_{j=1}^m$ remains to be determined.

Denote the orthogonal projection of $u_0$ on $V_m$ as $u_m^0 =
P_{V_m} u_0$, then $u_m^0 \to u_0$ in $V$, as $m\to\infty$. Let
$(g_{jm}^0)_{j=1}^m$ be the coordinate of $u_m^0$ in the basis
$(w_j)_{j=1}^m$ of $V_m$; i.e., $u_m^0 = \sum_{j=1}^m g_{jm}^0
w_j$, let $g_{jm}(0) = g_{jm}^0$.


We need to determine $(g_{jm}(t))_{j=1}^m$ to satisfy
\begin{equation*}
({u_m}', w_j) - (\sum_{i=1}^{n}\frac{\partial}{\partial x_i}(|u_m|^{p-2}\frac{\partial u_m}{\partial x_i}) , w_j) + (|u_m|^{p-2}u_m, w_j) = (f,w_j),\quad 1\le j\le m.
\end{equation*}


Integrating by parts on the second term of left-hand side, we have
\begin{equation} \label{e7}
\begin{aligned}
&({u_m}', w_j) + \sum_{i=1}^{n}\int_{\Omega} (|u_m|^{p-2} D_i
u_m)(D_i w_j) \,dx \\
& -  \sum_{i=1}^{n}\int_{\Gamma} (|u_m|^{p-2} D_i u_m) w_j d\Gamma
 + (|u_m|^{p-2}u_m, w_j) = (f,w_j),\quad 1\le j\le m.
\end{aligned}
\end{equation}
 The above system is a system of ordinary differential
equations in $(g_{jm}(t))_{j=1}^m$.  By Caratheodory theorem
\cite{Codd}, there exists solution $(g_{jm}(t))_{j=1}^m,\ t\in
[0,t_m)$.

We need a priori estimates that permit us to extend the solution
to the whole domain $[0,T]$.

We derive a priori estimates for the approximate solution as
follows: Multiply \eqref{e7} by $g_{jm}(t)$, then sum over $j$
from $1$ to $m$, we have
\begin{align*}
&({u_m}', u_m) + \sum_{i=1}^{n}\int_{\Omega} (|u_m|^{p-2} D_i
u_m)(D_i u_m) dx\\
&-  \sum_{i=1}^{n}\int_{\Gamma} (|u_m|^{p-2} D_i u_m) u_m d\Gamma
 + (|u_m|^{p-2}u_m, u_m) = (f,u_m),\quad 1\le j\le m.
\end{align*}
which gives
 \begin{equation} \label{e8}
\begin{aligned}
&\frac{1}{2}\frac{d}{dt} |u_m(t)|_2^2 + \frac{4}{p^2} \sum_{i=1}^n
\int_\Omega (D_i(|u_m|^\frac{p-2}{2} u_m))^2 dx +  |u_m(t)|_p^p \\
&= (f, u_m) + \sum_{i=1}^n\int_\Gamma (|u_m|^{p-2} D_i u_m) u_m
d\Gamma.
\end{aligned}
\end{equation}

Integrating with respect to $t$ from $0$ to $T$ on both sides, we obtain
\begin{equation} \label{e9}
\begin{aligned}
&\frac{1}{2} |u_m(T)|_2^2 + \int_0^T \frac{4}{p^2} \sum_{i=1}^n
\int_\Omega (D_i(|u_m|^\frac{p-2}{2} u_m))^2 dx   dt   + \int_0^T
|u_m(t)|_p^p  dt \\
&= \int_0^T (f, u_m)  dt +    \int_0^T
\sum_{i=1}^n\int_\Gamma (|u_m|^{p-2} D_i u_m) u_m d\Gamma  dt
 +   \frac{1}{2} |u_m(0)|_2^2.
\end{aligned}
\end{equation}
This gives
\begin{equation} \label{e10}
\begin{aligned}
&\frac{1}{2} |u_m(T)|_2^2 + \int_0^T \frac{4}{p^2} \sum_{i=1}^n
\int_\Omega (D_i(|u_m|^\frac{p-2}{2} u_m))^2 dx   dt   + \int_0^T
|u_m(t)|_p^p  dt\\
&\le     \int_0^T |(f, u_m)|  dt +    \int_0^T
\sum_{i=1}^n\int_\Gamma |(|u_m|^{p-2} D_i u_m) u_m | d\Gamma  dt
 +   \frac{1}{2} |u_m(0)|_2^2.
\end{aligned}
\end{equation}
The first term in the right-hand side of \eqref{e10}
 can be estimated as follows:
\begin{equation} \label{e11}
\begin{aligned}
\int_0^T |(f,u_m)| dt
 &= \int_0^T  \int_\Omega |fu_m| dx dt\\
 &\le \int_0^T |f|_q |u_m|_p dt    \quad
\text{(h\"older's\ inequality)} \\
&\le \int_0^T (\frac{1}{p}
|u_m|_p^p  +  \frac{p-1}{p}|f|_q^\frac{p}{p-1}) dt. \quad
\text{(Young's inequality)}
\end{aligned}
\end{equation}
Next, we estimate second term in the right-hand side of \eqref{e10}:
For $x\in\Gamma$, we have
\[
|u_m(x,t)| = \int_\Omega |k(x,y) u_m(y,t)|dy \le |k(x,y)|_q |u_m|_p.
\end{equation*}
Then we have $|u_m(x,t)| \le K(x) |u_m|_p$\ for\ $x\in \Gamma$.
Similarly, we have $|D_i u_m(x,t)| \le K_i(x) |u_m|_p$ for
$x\in \Gamma$.

Then using h\"older's inequality and assumptions (A4) and (A5), we have
\begin{equation} \label{e12}
\begin{aligned}
&\int_0^T \big|\sum_{i=1}^n \int_\Gamma  (|u_m|^{p-2} D_i u_m
) u_m d\Gamma \big| dt\\
&\le \int_0^T \sum_{i=1}^n \int_\Gamma
K(x)^{p-1} |u_m|_p^{p-1}  K_i(x) |u_m|_p d\Gamma dt\\
&\le \int_0^T \Big(\sum_{i=1}^n\int_\Gamma K(x)^{p-1} K_i(x)
d\Gamma\Big)|u_m|_p^p dt\\
&=C\int_0^T |u_m|_p^p dt
\end{aligned}
\end{equation}
where
\[
C=\sum_{i=1}^n\int_\Gamma K(x)^{p-1} K_i(x) d\Gamma <
1-\frac{1}{p}\,.
\]
With the above estimates and \eqref{e10}, we have
\begin{align*}
&\frac{1}{2} |u_m(T)|_2^2 + \int_0^T \frac{4}{p^2} \sum_{i=1}^n
\int_\Omega (D_i(|u_m|^\frac{p-2}{2} u_m))^2 dx   dt
+ \int_0^T (1-\frac{1}{p} - C)    |u_m(t)|_p^p  dt \\
&\le \int_0^T (\frac{p-1}{p}|f|_q^\frac{p}{p-1}) dt
+  \frac{1}{2}|u_m(0)|_2^2.
\end{align*}
which holds for any finite $T>0$.

Under assumption (A1)-(A5), we have the following a priori estimates:
\begin{itemize}
\item[(B)]  $u_m$ is bounded in $L^\infty(0, T;\ L^2(\Omega))$;

\item[(C)]  $|u_m|^\frac{p-2}{2}|u_m|$ is bounded in $L^2(0,T;
H^1(\Omega))$;

\item[(D)]  $u_m$ is bounded in $L^p (0,T; L^p(\Omega))$.
\end{itemize}
Since $T$ is an arbitrary positive number, we have
\[
 \ |u_m|_p^p < \infty \quad \text{a.e. } t
\]

\section{Existence of a generalized solution}

To prove the existence of a generalized solution, we first prove
the following lemma:

\begin{lemma} \label{lem4.1}
Let $u_m$, constructed  in \eqref{e6}, be the approximate solution of
\eqref{e1}--\eqref{e3} in the sense of Definition \ref{def2.1}.
Then $u_m'$ is bounded in $L^q(0,T;\ H^{-r}(\Omega))$.
\end{lemma}

\begin{proof}
For $v\in V\subset H^r$, from \eqref{e7}, we have
\begin{equation} \label{e13}
\begin{aligned}
&({u_m}', v) + (\sum_{i=1}^{n}\int_\Omega (|u_m|^{p-2}D_i u_m) (D_i
v) dx \\
&-  \sum_{i=1}^n \int_\Gamma (|u_m|^{p-2}D_i u_m) v d\Gamma
 + (|u_m|^{p-2}u_m,v) = (f,v).
\end{aligned}
\end{equation}
The last term in the left-hand side can be estimated as
in \cite{lions}:
\begin{align*}
|(|u_m|^{p-2}u_m,v) |
&\le  |\ |u_m|^{p-1}|_q |v|_p\\
&\le  (|u_m|_p^p)^{1/q} |v|_p \\
&\le   (|u_m|_p^p)^{1/q} C|v|_{H^r},
\end{align*}
since $H^r\hookrightarrow L^p$.
Hence $|\,|u_m|^{p-2} u_m|_{H^{-r}(\Omega)} \le
C(|u_m|_p^p)^{1/q} < \infty$.
The norm of $|u_m|^{p-2} u_m$ in $L^q(0,T;H^{-r}(\Omega))$
is bounded by
\[
\Big(\int_0^T (C(|u_m|_p^p)^{1/q})^q dt\Big)^{1/q} =
\Big(\int_0^T C^q |u_m|_p^p dt\Big)^{1/q} < \infty.
\]
Therefore, $|u_m|^{p-2} u_m$ is bounded in $L^q(0,T;H^{-r}(\Omega))$.


Next, we consider the term
$\sum_{i=1}^n \int_\Gamma (|u_m|^{p-2}D_i u_m) v d\Gamma $
in the left-hand side of \eqref{e13}:
\[
v \to \sum_{i=1}^n \int_\Gamma (|u_m|^{p-2}D_i u_m)
v d\Gamma = (a(u_m), v).
\]
We have
\begin{align*}
&\sum_{i=1}^n \int_\Gamma (|u_m|^{p-2}D_i u_m) v d\Gamma\\
&\le \sum_{i=1}^n |\ (|u_m|^{p-2}D_i u_m)|_{q,\Gamma} |v|_{p,\Gamma} \\
&= \sum_{i=1}^n \Big|\Big(|\int_\Omega k(x,y) u_m(y,t) dy|^{p-2}
\int_\Omega D_i k(x,y) u_m(y,t)dy\Big)\Big|_{q,\Gamma} \\
&\quad\times \Big|\int_\Omega k(x,y)v(y,t)dy\Big|_{p,\Gamma}\\
&\le  \sum_{i=1}^n |(K(x)^{p-2} K_i (x)|u_m|_p^{p-1})|_{q,\Gamma}
 |(K(x)|v|_p)|_{p,\Gamma}\\
&\le \sum_{i=1}^n |K(x)^{p-2} K_i (x)|_{q,\Gamma} |K(x)|_{p,\Gamma}
 |u_m|_p^{p-1} |v|_p\\
&\le  \sum_{i=1}^n |K(x)^{p-2} K_i (x)|_{q,\Gamma} |K(x)|_{p,\Gamma}
 |u_m|_p^{p-1} C|v|_{H^r}.
\end{align*}
Therefore,
\[
|a(u_m)|_{H^{-r}(\Omega)} \le \sum_{i=1}^n |K(x)^{p-2}
K_i (x)|_{q,\Gamma} |K(x)|_{p,\Gamma} |u_m|_p^{p-1} C < \infty.
\]
Then the norm of $a(u_m)$ in $L^q(0,T; H^{-r}(\Omega))$ is bounded by
\[
\Big(\int_0^T \sum_{i=1}^n  (|K(x)^{p-2} K_i (x)|_{q,\Gamma}
|K(x)|_{p,\Gamma}C)^q |u_m|_p^p dt\Big)^{1/q} < \infty.
\]
Hence, $a(u_m)$ is bounded in $L^q(0,T; H^{-r}(\Omega))$.

Next, we consider the second term in the left-hand side of \eqref{e13}.  Integrating by parts gives
\begin{equation} \label{e14}
\begin{aligned}
&\sum_{i=1}^n \int_\Omega (|u_m|^{p-2} D_i u_m) (D_i v) dx \\
&= \frac{1}{c} (\sum_{i=1}^n \int_\Gamma |u_m|^{p-2}u_m D_i v
d\Gamma  -  \int_\Omega |u_m|^{p-2}u_m \Delta v dx).
\end{aligned}
\end{equation}
Consider $ v \to \sum_{i=1}^n \int_\Gamma
|u|^{p-2} u D_i v d\Gamma= (I_1(u),v)$, we have:
\begin{align*}
|(I_1(u), v)|
&\le \sum_{i=1}^n |\ |u|^{p-2}u|_{q,\Gamma} |D_i v|_{p,\Gamma}\\
&= \sum_{i=1}^n \Big| \Big(\int_\Omega k(x,y) u(y,t) dy \Big)^{p-1}
\Big|_{q,\Gamma}  \Big|\int_\Omega D_i k(x,y) v(y,t) dy \Big|_{p,\Gamma}\\
&\le \sum_{i=1}^n |(K(x)^{p-1} |u|_p^{p-1})|_{q,\Gamma}
 |(K_i(x)|v|_p)|_{p,\Gamma}\\
&= \sum_{i=1}^n |K(x)^{p-1}|_{q,\Gamma} |K_i(x)|_{p,\Gamma}
  |u|_p^{p-1} |v|_p\\
&\le \sum_{i=1}^n |K(x)^{p-1}|_{q,\Gamma} |K_i(x)|_{p,\Gamma}
|u|_p^{p-1}C |v|_{H^r}
\end{align*}
So we have
\[
|I_1(u_m)|_{H^{-r}(\Omega)} \le \sum_{i=1}^n
|K(x)^{p-1}|_{q,\Gamma} |K_i(x)|_{p,\Gamma} |u_m|_p^{p-1}C
<\infty.
\]
With this, it is easy to see that norm of $I_1(u_m)$ in
$L^q(0,T; H^{-r}(\Omega))$ is bounded.

Next, consider $ v \to  \int_\Omega |u|^{p-2}u \Delta
v dx = (I_2(u), v)$.
From the proof of \cite[Theorem 12.2]{lions}, we know $I_2(u_m)$
is bounded in $L^q(0,T; H^{-r}(\Omega))$.
Since $f\in L^q(0,T; L^{q}(\Omega))\subset  L^q(0,T;
H^{-r}(\Omega))$, from \eqref{e13} and the above discussion, we have
$u_m'$ is bounded in $L^q(0,T; H^{-r}(\Omega))$.
\end{proof}

With Lemma \ref{lem4.1}, we can  use  \cite[Theorem 12.1]{lions}.
We quote the theorem here.

\begin{theorem} \label{thm4.2}
Let $B,B_1$ be Banach spaces, and $S$ be a set. Define
\[
M(v) = (\sum_{i=1}^n\int_\Omega |v|^{p-2}(\frac{\partial v}
{\partial x_i})^2 dx)^{1/p}
\]
on $S$ with:
\begin{itemize}
\item[(a)]  $S\subset B\subset B_1$, and $M(v)\ge 0$ on $S$,
$M(\lambda v)=|\lambda| M(v)$;

\item[(b)]  the set $\{v|v\in S, M(v)\le 1\}$ is relatively compact
in $B$.
\end{itemize}
Define the set
$F=\{v: v$ is locally summable on $[0,T]$ with value in $B_1$,\\
$\int_0^T (M(v(t)))^{p_0} dt \le C$, $v'$ bounded in
$L^{p_1}(0,T; B_1)\}$.
Where $1<p_i<\infty$, $i=0,1$.

Then $F\subset L^{p_0}(0,T;B)$ and $F$ is relatively compact in
$L^{p_0}(0,T;B)$.
\end{theorem}

We need Theorem \ref{thm4.2} to prove the following lemma:

\begin{lemma} \label{lem4.3}
Let $u_m$, constructed as in \eqref{e6}, be the approximate solution of
\eqref{e1}--\eqref{e3} in the sense of Definition \ref{def2.1}, then
$ u_m \to u $ in $L^p(0,T;  L^p(\Omega))$ strongly and almost
everywhere.
\end{lemma}

\begin{proof}
Let $S=\{v: |v|^\frac{p-2}{2} v\in H^1(\Omega)\}$. Since
$ H^1(\Omega)$ is also compactly embedded in $L^2(\Omega)$, the
proof of \cite[Proposition 12.1,p. 143]{lions} also works for
$|v|^\frac{p-2}{2} v\in H^1(\Omega)$, then (b) holds.

Let $B=L^p(\Omega)$, $B_1=H^{-r}(\Omega)$, $p_0=p$, $p_1=q$, we have
\begin{align*}
\int_0^T (M(u_m))^{p_0} dt
&= \int_0^T (\sum_{i=1}^n \int_\Omega |u_m|^{p-2}(\frac{\partial u_m}{\partial x_i})^2 dx) dt\\
&= C\int_0^T \sum_{i=1}^n\int_\Omega (D_i(|u_m|^{\frac{p-2}{2}} u_m))^2
dx dt
< \infty
\end{align*}
Now with Lemma \ref{lem4.1} and a priori estimates, conclusion follows
easily from application of Theorem \ref{thm4.2}.
\end{proof}

Next, we prove that we can pass the limit in \eqref{e13}.
Lemmas \ref{lem4.4}--\ref{lem4.7}, below,
 show that we can pass the limit in each
term in the left-hand side of \eqref{e13}.

\begin{lemma} \label{lem4.4}
Let $u_m$, as constructed in \eqref{e6}, be the approximate solution of
\eqref{e1}--\eqref{e3} in the sense of Definition \ref{def2.1}, then
$(|u_m|^{p-2} u_m, v) \to (|u|^{p-2} u, v)$ as $m
\to \infty$.
\end{lemma}

\begin{proof}
We need to show that  $|u_m|^{p-2} u_m \rightharpoonup |u|^{p-2} u$
in $L^q(\Omega)$ weakly, this is a consequence of
\cite[Lemma 1.3]{lions}.
\end{proof}


\begin{lemma} \label{lem4.5}
Let $u_m$, constructed as in \eqref{e6}, be the approximate solution of
\eqref{e1}--\eqref{e3} in the sense of Definition \ref{def2.1}, then
$\int_\Gamma (|u_m|^{p-2} D_i u_m)v d\Gamma \to
\int_\Gamma (|u|^{p-2} D_i u)v d\Gamma$ as $m \to \infty$.
\end{lemma}

\begin{proof}
By a priori estimates, $u_m$ is bounded in $L^p(\Omega)$ for
almost every $t$, then there exists subsequence of $u_m$, still
denoted as $u_m$, converges to $u$ weak star in $L^p(\Omega)$
(Alaoglu's Theorem) for almost every $t\in [0,T]$.

Under the assumption that for fixed $x$,
$|k(x,y)|_q = (\int|k(x,y)|^q dy)^{1/q} < \infty$; i.e.,
$k(x,y)\in L^q(\Omega)$ for fixed $x\in \Gamma$,
we have
\[
\int_\Omega k(x,y)u_m(y,t)dy \to \int_\Omega
k(x,y)u(y,t)dy\quad\text{ as }m\to \infty.
\]
Similarly,
\[
\int_\Omega D_i k(x,y)u_m(y,t)dy \to \int_\Omega D_i
k(x,y)u(y,t)dy\quad\text{as }m\to \infty.
\]
Therefore, for $x\in \Gamma$, we have
$|u_m(x,t)|^{p-2} D_i u_m(x,t)  \to  |u(x,t)|^{p-2} D_i u(x,t)$ a.e.


Next, we prove that $|(|u_m(x,t)|^{p-2} D_i u_m(x,t))|_{q,\Gamma}
< \infty$.
For $x\in\Gamma$, we have
\begin{gather*}
u_m(x,t) = \int_\Omega k(x,y) u_m(y,t) dy,\\
|u_m(x,t)| < |k(x,y)|_q |u_m|_p
 \le K(x)C
\end{gather*}
Since $K(x)\in L^p(\Gamma)$, we have $|u_m|_{p,\Gamma}<\infty $.
Similarly, we have $|D_i u_m|_{p,\Gamma} <\infty$ and
$|v|_{p,\Gamma}<\infty$.
Then
\begin{align*}
&\left|\ |u_m|^{p-2} D_i u_m \right|_{q,\Gamma}\\
&\le\left|\ |u_m|^{p-2}\right|_{\frac{p}{p-2},\Gamma}
|D_i u_m|_{p,\Gamma}  \quad
\text{(since $\frac{1}{q}=\frac{p-2}{p}+\frac{1}{p}$, \cite[p25]{Adam})}\\
&= (|u_m|_{p,\Gamma})^{p-2} |D_i u_m|_{p,\Gamma} < \infty
\end{align*}
By Lemma \cite[Lemma 1.3]{lions}, we have:
$|u_m|^{p-2} D_i u_m \rightharpoonup  |u|^{p-2} D_i u$ weakly in
$L^q(\Gamma)$ for a.e. $t\in [0,T]$.
Since $|v|_{p,\Gamma}<\infty$, the proof is complete.
\end{proof}


\begin{lemma} \label{lem4.6}
Let $u_m$, as constructed in \eqref{e6}, be the approximate solution of
\eqref{e1}--\eqref{e3} in the sense of Definition \ref{def2.1}, then
\[
\int_\Omega (|u_m|^{p-2} D_i u_m)(D_i v) dx \to
\int_\Omega (|u|^{p-2} D_i u)(D_i v) dx .
\]
\end{lemma}


\begin{proof}
From \eqref{e14} we know, we need to prove:
\begin{itemize}
\item[(i)] $\int_\Gamma |u_m|^{p-2}u_m D_i v d\Gamma \to
\int_\Gamma |u|^{p-2}u D_i v d\Gamma $;
and

\item[(ii)] $\int_\Omega |u_m|^{p-2}u_m \Delta v dx \to
\int_\Omega |u|^{p-2}u \Delta v dx $.
\end{itemize}
(i) From the proof of Lemma \ref{lem4.5}, we have, for $x\in \Gamma$,
$|u_m(x,t)|^{p-2} u_m(x,t) \to |u(x,t)|^{p-2} u(x,t)$
almost everywhere, and
\[
|\ |u_m|^{p-2} u_m|_{q,\Gamma} = |u_m|_{p,\Gamma}^{p-1} < \infty.
\]
Therefore, we can apply \cite[Lemma 1.3]{lions} to conclude that
$|u_m(x,t)|^{p-2} u_m(x,t) \rightharpoonup  |u(x,t)|^{p-2} u(x,t)
$ weakly in $L^q(\Gamma)$.
Since $D_i v \in L^p(\Gamma)$, (i) is proved.

(ii) From Lemma \ref{lem4.3},  we have
$|u_m(x,t)|^{p-2} u_m(x,t) \to |u(x,t)|^{p-2}u(x,t)$ almost everywhere,
for $x\in \Omega$.
Since $|\ |u_m|^{p-2} u_m|_q = |u_m|_p^{p-1} < \infty$,
by \cite[Lemma 1.3]{lions}, we have:
$|u_m|^{p-2} u_m \rightharpoonup  |u|^{p-2} u $ weakly in
$L^q(\Omega)$.
Since $\Delta v\in L^p(\Omega)$, we complete the proof of (ii).
\end{proof}

\begin{lemma} \label{lem4.7}
Let $u_m$, as constructed in \eqref{e6}, be the approximate solution of
\eqref{e1}--\eqref{e3} in the sense of Definition \ref{def2.1}, then
$(u_m', v) \to (u', v)$\ and\ $u(t)$ is continuous on
$[0,T]$.
\end{lemma}

\begin{proof}
Since $u_m'$ is bounded in $L^q(0,T; H^{-r}(\Omega))$, by
Alaoglu's theorem, there exists a subsequence, still
denoted by  $u_m'$, converging to $\chi$ weak star in $L^q(0,T;
H^{-r}(\Omega))$.  By slightly modifying the proof of
\cite[Theorem 1]{Bouz} (with the space $L^q(0,T; H^{-r}(\Omega))$,
instead of $L^2(0,T; B_2^1(0,1)).$), we have $\chi=u'$ and
$u(t)$ is continuous on $[0,T]$.
\end{proof}

Based on the above discussion, we summarize the existence theorem
as follows.

\begin{theorem} \label{thm4.8}
Under  assumptions (A1)-(A5), there exists a generalized
solution $u$ of problem \eqref{e1}--\eqref{e3}, such that
\begin{itemize}
\item[(1)] $u\in L^\infty (0,T;L^2(\Omega))\cap C([0,T],
H^{-r}(\Omega))$;

\item[(2)] $|u|^\frac{p-2}{2}u$ is bounded in $L^2(0,T; H^1(\Omega))$.


\item[(3)] $\frac{du}{dt}\in L^q(0,T;H^{-r}(\Omega))$.

\item[(4)] $u(x,0)=u_0(x)$.

\item[(5)] for all $v\in V$ and a.e. $t\in [0,T]$,
\[
(\frac{du}{dt}, v) - (\sum_{i=1}^{n}\frac{\partial}
{\partial x_i}(|u|^{p-2}\frac{\partial u}{\partial x_i}) , v)
+ (|u|^{p-2}u, v) = (f,v)\,.
\]
\end{itemize}
\end{theorem}


\subsection*{Acknowledgements}
The author would like to thank the anonymous referee for the helpful
comments and suggestions.

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\end{document}