\documentclass[reqno]{amsart}
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\AtBeginDocument{{\noindent\small
2005-Oujda International Conference on Nonlinear Analysis.
\newline {\em Electronic Journal of Differential Equations},
Conference 14, 2006, pp. 9--20.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or
http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2006 Texas State University - San Marcos.}
\vspace{9mm}}
\setcounter{page}{9}

\begin{document}

\title[\hfilneg EJDE/Conf/14 \hfil Degenrate parabolic problems]
{Existence result for  variational degenerated parabolic problems
via pseudo-monotonicity}

\author[L. Aharouch, E. Azroul, M.  Rhoudaf \hfil EJDE/Conf/14 \hfilneg]
{Lahsen Aharouch,  Elhoussine Azroul,  Mohamed Rhoudaf}  % in alphabetical order

\address{Lahsen Aharouch \newline
D\'epartement de Math\'ematiques et Informatique\\
Facult\'e des Sciences Dhar-Mahraz\\
B.P 1796 Atlas  F\`es, Maroc}
\email{l\_aharouch@yahoo.fr}

\address{Elhoussine Azroul \newline
D\'epartement de Math\'ematiques et Informatique\\
Facult\'e des Sciences Dhar-Mahraz\\
B.P 1796 Atlas  F\`es, Maroc}
\email{azroul\_elhoussine@yahoo.fr}

\address{Mohamed  Rhoudaf \newline
D\'epartement de Math\'ematiques et Informatique\\ 
Facult\'e des Sciences Dhar-Mahraz\\
B.P 1796 Atlas  F\`es, Maroc}
\email{rhoudaf\_mohamed@yahoo.fr}

\date{}
\thanks{Published September 20, 2006.}
\subjclass[2000]{35J60}
\keywords{Weighted Sobolev spaces; boundary
value problems; truncations; \hfill\break\indent parabolic problems}

\begin{abstract}
 In this paper, we study the existence of  weak solutions for the
 initial-boundary value problems of the  nonlinear degenerated
 parabolic equation
 $$
 \frac{\partial u}{\partial t}-\mathop{\rm div}a(x,t,u,\nabla u)
 +a_0(x,t,u,\nabla u) = f ,
 $$
 where $Au = -\mathop{\rm div}a(x,t,u,\nabla u)$ is a classical
  divergence operator of Leray-lions  acting from
 $L^p(0,T,W_0^{1,p}(\Omega,w))$ to its dual.
 The source term $f$ is assumed to belong to
 $L^{p'}(0,T,W^{-1,p'}(\Omega,w^*))$.
\end{abstract}

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

 \section{Introduction}

Let $\Omega$ be a bounded open subset of $\mathbb{R}^N$ and let $Q$ be the
cylinder $\Omega \times (0,T)$ with some given $T>0$. Consider the
 parabolic initial-boundary value problem
\begin{equation}\label{1.0}
\begin{gathered}
  \frac{\partial u}{\partial t}+A(u) = f \quad
   \mbox {in } Q \\
 u(x,t) = 0 \quad  \mbox{on }  \partial \Omega \times (0,T)
 \\ u(x,0) =u_0(x)  \quad \mbox{in } \Omega,
\end{gathered}
  \end{equation}
  where $Au = -\mathop{\rm div}a(x,t,u,\nabla u)$ is a classical
  divergence operator of Leray-lions form with respect to the
  Sobolev space $L^p(0,T,W_0^{1,p}(\Omega))$ for some
  $1<p<\infty$. The right-hand side $f$ is
  supposed lying in $L^{p'}(0,T,W_0^{-1,p'}(\Omega))$.

  We consider, first, the case where $A$ satisfies the classical Leray-lions
  conditions, in particular the  classical coercivity
  \begin{equation}
a(x,t,s,\xi )\xi \geq \alpha|\xi|^p. \label{1.1}
\end{equation}
  Then $A$ is a bounded pseudo-monotone and coercive operator from
  the space $L^p(0,T,W_0^{1,p}(\Omega))$ into its dual
  $L^{p'}(0,T,W_0^{-1,p'}(\Omega))$. In this setting, problems of
  the form \eqref{1.0} were solved by  Lions \cite{lions}
and Breszis-Browder
  \cite{brbr} in the case $p\geq 2$ and by Landes
  \cite{land0} and Landes-Mustonen \cite{lamu} when $1<p<2$
  (see also \cite{bomu},\cite{bomu1},\cite{alor}
  for related topics ).
When the classical coercivity \eqref{1.1} is replaced by the
  more general condition
  \begin{equation}
  a(x,t,s,\xi )\xi \geq c  \sum_{i=1}^Nw_i(x)  |\xi_i|^p,
  \label{1.2}
\end{equation}
where now $w(x) = \{w_i(x)$,  $1\leq i \leq N \}$ is a family of
weight functions on $\Omega$, the problem \eqref{1.0} can not
be solved
in the classical Sobolev settings $L^p(0,T,W_0^{1,p}(\Omega))$.
However, to do this, we must to change this classical setting by
the general one $L^p(0,T,W_0^{1,p}(\Omega,w))$ related to the
so-called weighted Sobolev space $W_0^{1,p}(\Omega,w)$. In this
direction, we list in particular the work \cite{Li} where the
authors have studied the existence of weak solution of the
variational parabolic boundary-value problems
\begin{equation} \label{P}
\begin{gathered}
  \frac{\partial u}{\partial t}+A(u)+A_0(x,t,u,\nabla u) = f
   \quad \mbox{in }  Q \\
u(x,t) = u_0(x) \quad  \mbox{on } \partial \Omega \times (0,T)    \\
u(x,t) = 0 \quad \mbox{in } \Omega,
\end{gathered}
\end{equation}
 but under more restrictions on the weight family $w$
(compare with Remark \ref{rmk2.1}).

Note that, little information is known for the degenerate
parabolic. Similar problems for degenerate  nonlinear elliptic
equations have been studied in \cite{drabekk}
and \cite{Akdim1}.
Our aim of this paper is to study the same variational degenerate
parabolic problems \eqref{1.0} in some general case of weight.
For that some important lemmas is firstly proved and the approach
of pseudo-monotonicity  is used.
A simple model of our problem is as follows
\begin{gather*}
 \frac{\partial u}{\partial t} - \mathop{\rm div}(|x|^s |Du|^{p-2}
Du) + \sigma(x) |u|^{p-2}u = f
   \quad \mbox {in }  Q \\
 u(x,t) = u_0(x) \quad \mbox{on } \partial \Omega \times (0,T)
    \\ u(x,t) = 0 \quad \mbox{in }  \Omega.
\end{gather*}

The present paper is organized as follows:
 We start with the introduction of a basic
 assumptions and main result in section 2,
 which is proved in section 3. Finally, we give an example in section
 4.

 \section{Assumptions and Main results}

\subsection*{Hypotheses}
Let $\Omega$ be a bounded open set of $\mathbb{R}^N$, $p$ be a
real number such that $2<p<\infty$ and $w= \{w_i(x): 1\leq i\leq
N\}$ be a vector of weight functions, i.e., every component
$w_i(x)$ is a measurable function which is strictly positive a.e.
in $\Omega$. Further, we suppose in all our considerations that,
\begin{gather}
w_i \in L^1_{\rm loc}(\Omega),\label{2.1} \\
w_i^{\frac{-1}{p-1}} \in L^1_{\rm loc}(\Omega),\label{2.2}
\end{gather}
for any $0\leq i\leq N$.
 We denote by $W^{1,p}(\Omega,w)$ the
space of all real-valued functions $u \in L^p(\Omega,w_0)$ such
that the derivatives in the sense of distributions fulfill
$$
\frac{\partial u}{\partial x_i} \in L^p(\Omega, w_i)\quad
\mbox{for } i=1,\dots, N.
$$
This is a Banach space under the norm
\begin{equation}
\|u\|_{1,p,w}=\Big[ \int_{\Omega}|u(x)|^pw_0\,dx +
\sum_{i=1}^N\int_\Omega |\frac{\partial u(x)}{\partial x_i}|^p
w_i(x)\,dx \Big]^{1/p}.\label {2.3}
\end{equation} The
condition \eqref{2.1} implies that $C_0^{\infty}(\Omega)$ is a
subset of $W^{1,p}(\Omega,w)$ and consequently, we can introduce
the subspace $W_0^{1,p}(\Omega,w)$ of $W^{1,p}(\Omega,w)$ as the
closure of $C_0^{\infty}(\Omega)$ with respect to the norm
\eqref{2.3}. Moreover, the condition \eqref{2.2} implies
that $W^{1,p}(\Omega,w)$ as well as $W_0^{1,p}(\Omega,w)$ are
reflexive Banach spaces.
We recall that the dual space of weighted Sobolev spaces $W_0^{1,p}(\Omega,w)$
is equivalent to $W^{-1, p'}(\Omega, w^{*})$, where
$w^{*} =\{w_i^{*} = w_i^{1-p'}$, $i = 0, \dots, N \}$ and where $p'$ is
the conjugate of $p$ i.e. $p' = \frac{p}{p-1}$. For more details,
we refer the reader to \cite{drabek}.

Now we state the some assumptions.
\begin{itemize}
\item[(H1)] For $2 \leq p < \infty$, the expression
\begin{equation}
\||u\|| = \Big({\sum_{i=1}^N}{ \int_{\Omega} |\frac{\partial
u}{\partial x_i}|^p w_i(x)\,dx}\Big)^{1/p} \label{2.5'}
\end{equation}
is a norm on $W_0^{1,p}(\Omega,w)$ and it's
equivalent to \eqref{2.3}. There exists a weight function
$\sigma $ on $\Omega$  such that
\begin{equation}
\sigma \in
L^1(\Omega) \quad \mbox{and} \quad  \sigma ^\frac{-1}{p- 1}\in
L^1_{\rm loc}(\Omega). \label {2'.5}
\end{equation}
The Hardy inequality
 \begin{equation}\Big({ \int_{\Omega}|u(x)|^p\sigma\,dx}\Big)^{1/p} \leq c
 \Big({\sum_{i=1}^N}{ \int_{\Omega} |\frac{\partial
u}{\partial x_i}|^p w_i(x)\,dx}\Big)^{1/p},\label{2''.5}
\end{equation}
holds for every $ u \in W_0^{1,p}(\Omega,w)$ with a constant $ c>0$
independent of $u$.
Moreover, the imbedding
\begin{equation}
W_0^{1,p}(\Omega,w)\hookrightarrow L^p(\Omega,\sigma) \label {2'''.5}
\end{equation}
expressed by the inequality \eqref{2''.5} is compact.
\end{itemize}
Note that $(W_0^{1,p}(\Omega,w),\||.\||)$ is a uniformly convex
(and thus reflexive) Banach space.

 \begin{remark} \label{rmk2.1} \rm
 Assume that $w_0(x)\equiv 1 $ and there exists
$ \nu \in ]\frac{N}{P},{+\infty}[\cap [\frac{1}{P-1},
{+\infty}[$ such that
\begin{equation}
w_i^{-\nu} \in L^1(\Omega) \quad \mbox {for all } i=1,\dots,N.
\label{2.4}
\end{equation}
 Note that the assumptions
\eqref{2.1} and \eqref{2.4} imply that,
\begin{equation}
\||u\|| = \Big({  \sum_{i=1}^N}{ \int_{\Omega}
|\frac{\partial u}{\partial x_i}|^p w_i(x)\,dx}\Big)^{1/p} \label {2.5}
\end{equation}
is a norm defined on $W_0^{1,p}(\Omega,w)$ and it's equivalent
to \eqref{2.3} and that, the imbedding
  \begin{equation}
W_0^{1,p}(\Omega,w) \hookrightarrow  L^p(\Omega)    \label {2.6}
\end{equation}
is compact  \cite[pp 46]{drabek}. Thus the hypothesis
$(H_1)$ is satisfied for $ \sigma \equiv 1$.
\end{remark}


\begin{itemize}
\item[(H2)] For $ i = 1,\dots ,N $,
\begin{gather}
|a_0(x,t,s, \xi)|\leq \beta \sigma^{1/p}(x)
\ \ [ c_0(x,t) + \sigma^\frac{1}{p'}|s|^{p-1} +  {
\sum_{j=1}^N}w_j^{\frac{1}{p'}}(x)|\xi_j|^{p-1}], \label{2'.7}
\\
|a_i(x,t,s, \xi)|\leq \beta w_i^{1/p}(x)
\ \ [ c_1(x,t) + \sigma^\frac{1}{p'}|s|^{p-1} +  {
\sum_{j=1}^N}w_j^{\frac{1}{p'}}(x)|\xi_j|^{p-1}], \label{2.7}
\\
 \sum_{i=1}^N [a_i(x,t,s,\xi) - a_i(x,t,s,\eta)]
 (\xi_i-\eta_i)  > 0 \quad  \forall
 \xi\neq \eta \in \mathbb{R}^N ,\label {2.8}
\\
a_0(x,t,s, \xi).s + {  \sum_{i=1}^N}a_i(x,t,s, \xi).\xi_i
\geq \alpha {  \sum_{i=1}^N}w_i |\xi_i|^p, \label{2.9}
\end{gather}
where $c_0(x,t)$ and $c_1(x,t)$ are some  positive functions in
$L^{p'}(Q)$, and $\alpha$ and $\beta$ are some  strictly positive
constants.
\end{itemize}

\subsection*{Some lemmas}
In this subsection we establish some imbedding and compactness
results in weighted Sobolev Spaces which allow in particular to
extend in the settings of weighted Sobolev spaces.\\ Let $V
=W_0^{1,p}(\Omega,w)$, $H=L^2(\Omega,\sigma)$ and let
$V^*=W^{-1,p'}(\Omega,w^*)$, with ($2\leq p<\infty$). Let $X =
L^p(0,T,V)$. The dual space of $ X $ is $X^* = L^{p'}(0,T,V^*)$
where $\frac {1}{p'}+\frac {1}{p}=1$ and denoting the space
$W_p^1(0,T,V,H)=\{ v \in X : v' \in X^*\}$ endowed with the norm
\begin{equation}
\|u\|_{w_p^1} = \|u\|_X +\|u'\|_{X^*},\label{3.1}
\end{equation}
 is a Banach space. Here $ u'$ stands for  the generalized
derivative of $u$; i.e.,
$$
\int_0^T u'(t)\varphi (t) \,dt = - \int_0^T  u(t)\varphi' (t) \,dt
\quad \mbox{for all }  \varphi \in C_0^\infty(0,T).
$$

\begin{lemma} \label{lem2.1}
The Banach space $H$ is an Hilbert space and its dual $H'$ can be
identified with him self; $ i.e., H'\simeq H $
\end{lemma}

\begin{lemma} \label{lem2.2}
 The  evolution triple $V \subseteq H \subseteq V^*$ is verified.
 \end{lemma}

\begin{lemma}\label{lem2.3}
 Let $g \in L^r(Q,\gamma)$ and let $g_n\in L^r(Q,\gamma)$,
with $\|g_n\|_{ L^r(Q,\gamma)}\leq c, 1<r<\infty$.
If $g_n(x)\to g(x)$ a.e in $Q$, then $g_n \rightharpoonup
g$ in $L^r(Q,\gamma)$, where $\rightharpoonup$ denotes weak
convergence and $\gamma$ is a weight function on $Q$.
\end{lemma}

\begin{lemma}\label{lem2.4}
Assume that (H1) and (H2) are satisfied and let $(u_n)$ be a
sequence in $L^p(0 , T, W_0^{1,p}(\Omega,w))$ such that
$u_n\rightharpoonup u$ weakly in $L^p(0 , T, W_0^{1,p}(\Omega,w))$
and
\begin{equation}
\int_Q[a(x,t,u_n,\nabla u_n) - a(x,t,u_n,\nabla u)][\nabla u_n- \nabla u]
\,dt\, dx \to 0. \label{3.20}
\end{equation}
Then $u_n \to u$ in $L^p(0 , T, W_0^{1,p}(\Omega,w))$.
\end{lemma}

Now we recall the well-known general Sobolev imbedding theorems for
 evolution equations.

 \begin{lemma}[\cite{zei}] \label{lem2.5}
Let $V \subseteq  H \subseteq V^*$ be an evolution triple. Then
the imbedding
$$
W_p^1(0,T,V,H) \subseteq C([0,T]),H)
$$
is continuous.
\end{lemma}

\begin{lemma}[\cite{zei}] \label{lem2.6}
Let $Z_1, Y, Z_2$ be real reflexive Banach spaces. Assume that the
imbeddings $Z_1\subseteq Y\subseteq Z_2 $ are continuous, and the
imbedding $Z_1\subseteq Y$ is compact, $0 < T < \infty$, $1<p$,
$q<\infty$. Then $W = \{u \in L^p(0,T,Z_1) : u' \in L^q(0,T,Z_2)\}$
equipped with the norm
$$
\|u\|_w = \|u\|_{L^p(0,T,Z_1)} + \|u'\|_{L^q(0,T,Z_2)}
$$
is a Banach space and the imbedding
$W\subseteq L^p(0,T,Y)$ is compact.
\end{lemma}


\subsection*{Existence results}
\begin{definition} \label{def2.1}\rm
A monotone map $ T:D(T)\to X^* $ is called maximal
monotone if its graph
$$
G(T) =  \{(u,T(u))\in X \times X^* {\mbox{for all }} u\in D(T) \}
$$
is not a proper subset of any monotone set in $X \times X^*$.
\end{definition}

Let us consider the operator $\frac {\partial}{\partial t}$ which
induces a linear map $L$ from the subset
 $D(L) = \{v \in X : v' \in X^*, v(0)=0\}$ of $X$ into $X^*$
 by
\begin{equation}
\langle Lu,v \rangle_X =  \int_0^T \langle u'(t),v(t)\rangle_V \,dt \quad
 u \in D(L), \; v \in X.\label{3.2}
\end{equation}


\begin{lemma}[\cite{zei}] \label{lem2.7}
$L$ is a closed linear maximal monotone map.
\end{lemma}

In our study we deal with mappings of the form $F = L+S$ where $L$
is a given linear densely defined maximal monotone map from $D(L)
\subset X$ to $X^*$ and $S$ is a bounded demicontinuous map of
monotone type from $X$ to $X^*$.

\begin{definition} \label{def2.2} \rm
A mapping  $S$ is pseudomonotone with respect to $D(L)$, if
for any sequence $\{u_n\}$ in $D(L)$ with $u_n\rightharpoonup u $,
 $Lu_n \rightharpoonup Lu$ and $ \lim_{n\to
\infty} \sup  \langle S(u_n),u_n-u \rangle \leq 0$, we have
$ \lim_{n\to \infty} \langle S(u_n),u_n-u
\rangle = 0$ and $S(u_n) \rightharpoonup S(u)$ as $n \to \infty$.
\end{definition}

  Consider the non linear parabolic problem
\begin{gather*} %\begin{equation} \label{P}
  \frac{\partial u}{\partial t}+A(u)+A_0(x,t,u,\nabla u) = f \quad
\mbox {in } Q \\
   u(x,t) = 0 \quad  \mbox{on } \partial \Omega \times (0,T) \\
u(x,0) =u_0 \quad \mbox{in } \Omega.
\end{gather*}

\begin{definition} \label{def2.3} \rm
 A function $u$ is said to be a weak solution of the initial-boundary
 value problem \eqref{P} if $u \in C([0,T],H) \cap L^p(0,T,V)$,
 $\frac {\partial u}{\partial t} \in L^{p'}(0,T,V^*)$
 and $u$ satisfies the equation
$$
\frac {\partial u}{\partial t} + Au + A_0u = f \ \ \ \ 0<t<T, u(0)=u_0,
$$
where the operator $A+A_0: X \to X^*$ is defined by
$$
\langle (A+A_0)(u), v \rangle =  \int_Q a(x,t,u,\nabla
u)\nabla v \,dx \,dt +  \int_Q a_0(x,t,u, \nabla
u)v\,dx \,dt
$$
\end{definition}



\begin{proposition} \label{prop2.1}
 The operator $A + A_0: X \to X^* $ is :
 \begin{enumerate}
\item[(a)] bounded and  demicontinuous;
 \item[(b)]  pseudomonotone  with respect to $D(L)$
\item[(c)] strongly coercive, i.e.,
$$
\frac{\langle (A+A_0)(u),u \rangle_X}{\|u\|_X} \to
+\infty ,\ \ \mbox{as} \ \ \|u\|_X \to +\infty .
$$
 \end{enumerate}
 \end{proposition}


We first consider the Zero-initial value problem,
\begin{equation} \label{P0}
\begin{gathered}
  \frac{\partial u}{\partial t}+A(u)+A_0(x,t,u,\nabla u) = f
   \quad  \mbox{in }  Q \\
u(x,t) = 0 \quad  \mbox{on }  \partial \Omega \times (0,T)
    \\ u(x,0) = 0 \quad \mbox{in } \Omega,
\end{gathered}
\end{equation}

\begin{theorem} \label{thm2.1}
Assume that the conditions (H1)-(H2) hold, then  problem
\eqref{P0}  admits a weak solution for any $f \in X^*$.
\end{theorem}


\begin{theorem} \label{thm2.2}
 \par Assume that the conditions (H1)-(H2) hold and
$u_0 \in      W_0^{1,p}(\Omega,w)$, then the initial-boundary
value problem \eqref{P} admits a weak solution for any $f \in
X^*$.
\end{theorem}

 \section{Proofs of Main results}

\begin{proof}[Proof of lemma \ref{lem2.1}]
 Let the map $F: H \times H \to \mathbb{R}$ be defined by
$$
F(f,g) =  \int_\Omega fg\sigma \,dx.
$$
Note that $F$ is a symmetric  bilinear form, which is also
continuous and defined positively, since
$$
 \int_\Omega fg\sigma \,dx
=  \int_\Omega f\sigma^{1/2}g\sigma^{1/2} \,dx \leq \Big(
\int_\Omega |f|^2\sigma\,dx\Big)^{1/2} \Big( \int_\Omega
|g|^2\sigma\,dx \Big)^{1/2}.
$$
Then the Banach  space $H$ is an  Hilbert space.
Finally by a standard argument, we can identify $H$ with its dual
$H'$; i.e., $H' \simeq H$.
\end{proof}

\begin{proof}[Proof of lemma \ref{lem2.2}]
By the imbedding \eqref{2'''.5} and the fact that
 $2\leq p<\infty$, and $\sigma\in L^1(\Omega)$, we can write
$$
W_0^{1,p}(\Omega,w)\hookrightarrow \hookrightarrow
L^p(\Omega,\sigma)\hookrightarrow H\simeq H'\hookrightarrow
W^{-1,p'}(\Omega,w^*) .
$$
\end{proof}

\begin{proof}[Proof of lemma \ref{lem2.3}]
 Since $g_n\gamma^{1/r}$ is bounded in $L^r(Q)$ and
$g_n(x)\gamma^{1/r}(x)\to g\gamma^{1/r}$,
a.e. in $Q$, then by  \cite[lemma 3.2]{Leja}, we have
$$
g_n\gamma^{1/r}\rightharpoonup g\gamma^{1/r} \quad
\mbox{in } L^r(Q).
$$
Moreover, for all $\varphi \in L^{r'}(Q,\gamma^{1-r'})$, we have
 $\varphi\gamma^{\frac{-1}{r}}\in L^{r'}(Q)$. Then
$$
\int_Qg_n\varphi \,dx \to  \int_Q g\varphi \,dx, \quad\mbox{i.e. } g_n
\rightharpoonup g \quad \mbox{in }  L^r(Q,\gamma).
$$
\end{proof}

\begin{proof}[Proof of lemma \ref{lem2.4}]
Let $D_n = [a(x,t,u_n,\nabla u_n) - a(x,t,u_n,\nabla u)][\nabla u_n- \nabla u]$.
 Then by \eqref{2.8}, $D_n$ is a positive
function and by \eqref{3.20}, $D_n \to 0$ in $L^1(Q)$.
Extracting a subsequence, still denoted by $u_n$, and using
\eqref{2'''.5} we can write
$$
u_n\to u \quad  \mbox{a.e. in }Q, \quad
D_n\to 0 \quad \mbox{a.e. in }  Q.
$$
Then, there exists a subset $B$ of $Q$, of zero  measure
such that
 for $(t,x) \in Q\setminus B$,
$|u_n(x,t)|<\infty$, $|\nabla u(x,t)|<\infty$,
$|c_1(x,t)|<\infty$, $w_i(x)>0$ and
$u_n(x,t)\to u(x,t), D_n(x,t)\to 0$.
We set $\epsilon_n = \nabla u_n(x,t)$ and
$\epsilon = \nabla u(x,t)$. Then
\begin{align*}
D_n(x,t) &= [a(x,t,u_n,\epsilon_n) -
a(x,t,u_n,\epsilon)](\epsilon_n-\epsilon)\\
&\geq \alpha  \sum_{i=1}^N
w_i|\epsilon_n^i|^p + \alpha \sum_{i=1}^N
w_i|\epsilon^i|^p\\
&\quad  -  \sum_{i=1}^N\beta w_i^{1/p}
\Big[c_1(x,t)+\sigma^{\frac{1}{p'}}|u_n|^{p-1}+ \sum_{j=1}^N
w_j^{\frac{1}{p'}}|\epsilon_n^j|^{p-1}\Big] |\epsilon^i|\\
&\quad  -  \sum_{i=1}^N\beta w_i^{1/p}
\Big[c_1(x,t)+\sigma^{\frac{1}{p'}}|u_n|^{p-1}+ \sum_{j=1}^N
w_j^{\frac{1}{p'}}|\epsilon^j|^{p-1}\Big] |\epsilon_n^i|;
\end{align*}
i.e,
\begin{equation}
D_n(x,t) \geq \alpha
 \sum_{i=1}^N w_i|\epsilon_n^i|^p-c_{x,t}
\Big[ 1 + \sum_{j=1}^N w_j^{\frac{1}{p'}}|\epsilon_n^j|^{p-1}
+  \sum_{i=1}^N w_i^{1/p}|\epsilon_n^i|\Big],\label{3.21}
\end{equation}
where $c_{x,t}$ is a constant which depends on $x$, but does not
depend on $n$.  Since $u_n(x,t)\to u(x,t)$, we have
$|u_n(x,t)|\leq M_{x,t}$, where $M_{x,t}$ is some positive
constant. Then by a standard argument $|\epsilon_n|$ is bounded
uniformly with respect to $n$. Indeed, \eqref{3.21} becomes
$$
D_n(x,t) \geq
 \sum_{i=1}^N |\epsilon_n^i|^p\Big(\alpha w_i -
\frac{c_{x,t}}{N|\epsilon_n^i|^p} -
\frac{c_{x,t}w_i^{\frac{1}{p'}}}{|\epsilon_n^i|} -
\frac{c_{x,t}w_i^{1/p}}{|\epsilon_n^i|^{p-1}}\Big).
$$
If $|\epsilon_n|\to \infty $ (for a subsequence) there
exists at least one $i_0$ such that $|\epsilon_n^{i_0}|\to
\infty $, which implies that $D_n(x,t)\to \infty $ which
gives a contradiction.

 Let now $\epsilon^*$ be a cluster point of $\epsilon_n$. We
have $|\epsilon^*|<\infty $ and by the continuity of a with
respect to the two last variables we obtain
$$
(a(x,t,u(x,t),\epsilon^*) - a(x,t,u(x,t),\epsilon
))(\epsilon^*-\epsilon) = 0.
$$
In view of \eqref{2.8} we have
$\epsilon^* = \epsilon $. The uniqueness of the cluster point
implies
$$
\nabla u_n(x,t)\to \nabla u(x,t) \quad \mbox{a.e. in } Q.
$$
Since  the sequence $a(x,t,u_n,\nabla u_n)$ is bounded
in the space $ \prod_{i=1}^NL^{p'}(Q,w_i^*)$ and
$a(x,t,u_n,\nabla u_n)\to a(x,t,u,\nabla u)$ a.e. in $Q$,
Lemma \ref{lem2.3} implies
$$
a(x,t,u_n,\nabla
u_n)\rightharpoonup a(x,t,u,\nabla u) \quad \mbox{in }
 \prod_{i=1}^NL^{p'}(Q,w_i^*) \quad \mbox{and a.e. in }Q.
$$
We set $\bar{y}_n = a(x,t,u_n,\nabla u_n)\nabla u_n$ and
$\bar{y} = a(x,t,u,\nabla u)\nabla u$. As in
 \cite[lemma lemma 5]{bomupu} we can write $\bar{y}_n\to \bar{y}$ in
$L^1(Q)$. By \eqref{2.9}, we have
$$
\alpha  \sum_{i=1}^Nw_i|\frac{\partial u_n}{\partial
x_i}| \leq a(x,t,u_n,\nabla u_n)\nabla u_n.
$$
Let
\begin{gather*}
z_n = \sum_{i=1}^Nw_i|\frac{\partial u_n}{\partial x_i}|^p, \quad
 z =  \sum_{i=1}^Nw_i|\frac{\partial u}{\partial x_i}|^p,\\
 y_n = \frac{\bar{y}_n}{\alpha}, \quad y =\frac{\bar{y}}{\alpha}.
\end{gather*}
 Then, by Fatou's lemma we obtain
$$
\int_Q 2y\,dx dt \leq  \lim_{n\to
\infty}\inf   \int_Q y+y_n-|z_n-z|\,dx\, dt ;$$
i.e.,
$0\leq  \lim_{n\to \infty}\sup   \int_Q |z_n-z|\,dx dt$, hence
$$
0\leq  \lim_{n\to \infty}\inf \int_Q |z_n-z|\,dx dt \leq
 \lim_{n\to \infty}\sup
 \int_Q |z_n-z|\,dx dt \leq 0.
$$
This implies
$$
\nabla u_n\to \nabla u \quad \mbox{in }
 \prod_{i=1}^NL^p(Q,w_i),
$$
which with \eqref{2.5'} completes the present proof.
\end{proof}


\begin{proof}[Proof of proposition \ref{prop2.1}]
 (a) We set $ B = A +  A_0  $. Using \eqref{2'.7}, \eqref{2.7} and
H\"{o}lder's inequality we can show that $B$ is bounded.
 For showing that $B$ is  demicontinuous,
 let $v_\epsilon \to v$ in X   as $\epsilon \to 0$, and prove that,
  $$
\langle  B(v_\epsilon),\varphi \rangle \to \langle
 B(v),\varphi \rangle  \quad  \mbox{for all } \varphi\in X.
$$
Since, $a_i(x,t, v_\epsilon, \nabla
v_\epsilon) \to a_i(x,t, v, \nabla v)$ as
$\epsilon \to 0$, for a.e. $x\in \Omega$,  by the growth
conditions \eqref{2.7}, \eqref{2'.7} and  lemma \ref{lem2.3} we get
$$
a_i(x,t, v_\epsilon, \nabla
v_\epsilon) \rightharpoonup a_i(x,t, v, \nabla v)  \quad \mbox{in }
L^{p'}(Q, w_i^{1-p'}) \quad \mbox{as }  \epsilon \to 0
$$
for
$ (i = 1,\dots,N)$ and
$$
a_0(x,t, v_\epsilon, \nabla v_\epsilon) \rightharpoonup a_0(x,t, v, \nabla v)
\quad \mbox{in } L^{p'}(Q, \sigma^{1-p'}) \quad \mbox{as } \epsilon
\to 0.
$$
Finally for all $\varphi \in X$,
$$
\langle B(v_\epsilon),\varphi\rangle \to \langle
B(v),\varphi \rangle \quad \mbox{as } \epsilon \to 0
$$
(since $\varphi\in L^p(Q, \sigma)$ for all $\varphi\in X $).
\smallskip

\noindent(b) Suppose that $\{u_j\}$ is a sequence in $D(L)$ with
\begin{enumerate}
\item[(i)] $u_j \rightharpoonup u$ weakly in $X$
\item[(ii)] $Lu_j
\to Lu$ weakly in $X^*$,
 \item[(iii)] $\limsup  \langle A+A_0(u_j),u_j-u \rangle _X \leq 0$.
\end{enumerate}
By the definition of the operator $L$ in \eqref{3.2},
we obtain that $\{ u_j \}$ is a bounded sequence in
$W_p^1(0,T,V,H)$. By virtue of lemma \ref{lem2.6}, we get,
$$
u_j \to u \quad \mbox{strongly in } L^p(Q,\sigma) .
$$
On the other hand,
$$
\langle A_0u_j,u_j-u \rangle =
 \int_Q  a_0(x,t,u_j,\nabla u_j)(u_j-u)\,dx \,dt
$$
Thus the H\"older's inequality and (i) imply
\begin{align*}
\langle A_0u_j,u_j-u \rangle
&\leq \Big( \int_Q |a_0|^{p'}\sigma^{1-p'} \,dx \,dt\Big)^{1/p'}
 \|u_j-u\|_{L^p(Q,\sigma)}\\
& \leq \|a_0\|_{L^{p'}(Q,\sigma*)} \|u_j-u\|_{L^p(Q,\sigma)},
\end{align*}
i.e, $\langle A_0u_j, u_j-u \rangle \to 0$ as
$j \to \infty$. Combining the last convergence with (iii), we
obtain
$$
\lim_{j \to \infty} \sup  \langle Au_j,u_j-u
\rangle \leq 0.
$$
And by the pseudo-monotonicity of $A$ (see \cite[Proposition 1]{drabekk}),
we have
$$
Au_j \rightharpoonup Au \quad \mbox{in }  X^*  \quad \mbox{and} \quad
{\lim_{j \to \infty}} \langle Au_j,u_j-u \rangle = 0.
$$
Then
$$
{\lim_{j \to \infty}} \langle Au_j+A_0(u_j),u_j-u \rangle = 0.
$$
 On the other hand, $ {\lim_{j \to \infty}} \langle Au_j,u_j-u \rangle = 0$,
which implies
\begin{align*}
0 &=  {\lim_{j \to \infty}} \int_{Q} a(x,t,u_j,\nabla u_j)\nabla
(u_j-u)\,dx \,dt  \\
&=  {\lim_{j \to \infty}} \int_{Q} [a(x,t,u_j,\nabla u_j) -
a(x,t,u_j,\nabla u)][\nabla u_j - \nabla u]\,dx \,dt \\
&\quad + {\lim_{j \to \infty}} \int_{Q} a(x,t,u_j,\nabla u)(\nabla
u_j-\nabla u)\,dx \,dt.
\end{align*}
The last integral in the right hand tends to zero, since by the
continuity of the Nemytskii operator,
$a(x,t,u_j,\nabla u)\to a(x,t,u,\nabla u)$ in
 $ {\prod_{i=1}^N L^{p'}(Q,w_i^{1-p'})}$ as
$j \to  +\infty$.
 So that
$$
  {\lim_{j \to \infty}} \int_{Q}
 [a(x,t,u_j,\nabla u_j) - a(x,t,u_j,\nabla u)][\nabla u_j
- \nabla u]\,dx \,dt = 0.
$$
By lemma \ref{lem2.4} we have
$$
\nabla u_j \to \nabla u \quad \mbox{ a.e. in } Q .
$$
Hence $a_0(x,t,u_j,\nabla u_j) \to a_0(x,t,u,\nabla u)$
a.e. in $Q$ as
  $j \to \infty$ and since
$$
a_0(x,t,u_j,\nabla u_j) \in L^{p'}(Q,\sigma^{1-p'})
$$
  by Lemma \ref{lem2.3}, we obtain
$$
a_0(x,t,u_j,\nabla u_j) \rightharpoonup  a_0(x,t,u,\nabla u) \quad
\mbox{in }  L^{p'}(Q,\sigma^{1-p'}).
$$
Finally,
$$
B(u_j) \rightharpoonup B(u) \quad \mbox{in }  X^*.
$$

\noindent(c) The strongly coercivity follows from
\eqref{2.9}
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm2.1}]
By  proposition \ref{prop2.1} the
operator $A+A_0 : X \to X^*$ is pseudomonotone with
respect to $D(L)$, and the operator $A+A_0$ satisfies the strong
coercivity condition which implies that both of the conditions (i)
and (ii) in \cite[theorem 4]{Bemu} hold.
 So all the conditions in \cite[theorem 4]{Bemu} are met.
  Therefore, there exists a solution $u\in D(L)$ of
   the evolution equation
$$
\frac {\partial u}{\partial t} + Au + A_0u = f
$$
for any $f\in X^*$. In order to prove that $u$
is also a weak solution of the problem \eqref{P0}, we have to show
that $u\in C([0,T],H)$.
 By the definition of $D(L)$ and lemma \ref{lem2.5}, we obtain
$$
D(L)\subseteq W_p^1(0,T,V,H) \subseteq C([0,T],H).
$$
Which implies that $u \in C([0,T],H)$.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm2.2}]
Now we turn to problem \eqref{P}. Assume that (H1)-(H2) hold and
$u_0 \in  W_0^{1,p}(\Omega,w)$. Let
$$
\overline{a}_i(x,t,u,\nabla u) = a_i(x,t,u + u_0,\nabla u + \nabla u_0)
$$
for all i = 0,\dots,N.
Then it is easy to see that $\overline{a}_i$ also satisfies
     the conditions (H1)-(H2). But $\beta$ , $\alpha $ and the
     function $c_0(x,t)$ and $c_1(x,t)$ in (H1)-(H2) may depend on the
     function $u_0$.
Analogously,  $\overline{A} + \overline{A}_0: X \to X^*$ is defined by
$$
\langle (\overline{A}+\overline{A}_0)(u), v \rangle
=  \int_Q \overline{a}(x,t,u,\nabla
u)\nabla v \,dx \,dt +  \int_Q \overline{a}_0(x,t,u,
\nabla u)v\,dx \,dt
$$
for $u ,v \in L^p(0, T, V)$, where
$\overline{A} = -{\mathop{\rm div}} \overline{a}(x, t , u , \nabla u) $.
Then, by Theorem \ref{thm2.1}, we have Theorem \ref{thm2.2}
\end{proof}

 \section{An example}
 Let $\Omega$ be a bounded domain of $\mathbb{R}^N (N\geq 1)$, satisfying the cone
condition. Let us consider the Carath\'{e}odory functions
\begin{gather*}
a_i(x,t,s,\epsilon ) = w_i|\epsilon_i|^{p-1}\mathop{\rm sgn}(\epsilon_i) \quad
\mbox{for } i=1,\dots,N,
\\
a_0(x,t,s,\epsilon ) = \rho \sigma(x)s|s|^{p-2}, \quad  \rho>0,
\end{gather*}
where $ \sigma $ and $ w_i(x)$ ($i=,1,\dots,N$) are given weight functions,
strictly positive almost everywhere in $\Omega$.
We shall assume that the weight functions satisfy,
$w_i(x)=w(x)$, $x\in \Omega$, for all
$i=1,\dots,N$. Then, we can consider the Hardy inequality
\eqref{2''.5} in the form
$$
\Big(  \int_Q|u(x,t)|^p\sigma (x)\,dx \Big)^{1/p}\leq c
\Big(  \int_Q|\nabla u(x,t)|^pw\,dx \Big)^{1/p}.
$$
It is easy to show that
the functions $a_i(x,t,s,\epsilon )$ are Carath\'{e}odory
functions satisfying the growth condition \eqref{2.7} and the
coercivity $\eqref{2.8}$. On the other hand, the monotonicity
condition is satisfied, in fact,
\begin{align*}
&\sum_{i=1}^N(a_i(x,t,s,\epsilon)-a_i(x,t,s,\hat{\epsilon}))
(\epsilon_i - \hat{\epsilon}_i)\\
&= w(x) \sum_{i=1}^N(|\epsilon_i|^{p-1}\mathop{\rm sgn}\epsilon_i
- |\hat{\epsilon_i}|^{p-1}\mathop{\rm sgn}\hat{\epsilon_i})
(\epsilon_i-\hat{\epsilon_i})>0
\end{align*}
for almost all $(x,t)\in Q$ and for all
$\epsilon, \hat{\epsilon}\in \mathbb{R}^N$ with
$\epsilon\neq \hat{\epsilon}$, since
$w>0$ a.e. in $\Omega$. In particular, let us use the special
weight functions $w$ and $\sigma$ expressed in terms of the
distance to the boundary $\partial \Omega$. Denote
$d(x) = \mathop{\rm dist}(x,\partial \Omega)$ and set
$$
w(x) = d^\lambda(x),\quad \sigma (x) = d^\mu (x).
$$
In this case, the Hardy inequality reads
$$
\Big( \int_Q|u(x,t)|^pd^\mu (x)\,dx\Big)^{1/p}
\leq \Big( c \int_Q|\nabla u(x,t)|^pd^\lambda(x)\,dx \Big)^{1/p}.
$$
For $ \lambda < p-1$, $\frac{\mu - \lambda }{p} + 1 > 0$; See for example
\cite{drabekk}


\begin{corollary} \label{coro4.1}
The   parabolic initial-boundary value problem
\begin{align*}
 &\int_Q \frac{\partial u(x,t)}{\partial
t}\varphi\,dx\,dt +  \int_Q d^\lambda(x)
 \sum_{i=1}^N |\frac{\partial u(x,t)}{\partial
x_i}|^{p-1}\mathop{\rm sgn}(\frac{\partial u}{\partial
x_i})\frac{\partial \varphi(x,t)}{\partial x_i}\,dx\,dt \\
&+  \int_Q \rho d^\mu(x) u(x,t)
|u(x,t)|^{p-2} \varphi(x,t)\\
&=  \int_Q f \varphi\,dx\,dt \quad \forall \varphi \in D(Q)
\end{align*}
admits at least one solution $u$ in
$L^p(0,T,W_0^{1,p}(\Omega,d^\lambda))$, for any function $f$ in
$L^{p'}(0,T,W^{-1,p'}(\Omega,d^{\lambda'}))$ where $\lambda' =
\lambda (1-p')$ and $ u_0 \in W_0^{1,p}(\Omega,d^\lambda)$.
\end{corollary}

\begin{thebibliography}{Dellio 83}

\bibitem{adams} R. Adams, \emph{Sobolev spaces},  AC, Press, New York, (1975)

\bibitem{Akdim1} Y. Akdim, E. Azroul and A. Benkirane,
\emph{Existence Results for Quasilinear Degenerated Equations Via Strong Convergence of Truncations}, Revista Matematica Complutense
17, , N.2,  (2004) pp 359-379.

\bibitem{Bemu} J. Berkovits, V. Mustonen,
\emph{Topological degree for perturbation of linear maximal
monotone mappings and applications to a class of parabolic
problems}, Rend. Mat. Roma, Ser, VII, 12 (1992), pp. 597-621.

\bibitem{bomupu} L. Boccardo, F. Murat and  J. P. Puel
\emph{Existence of bunded solutions for nonlinear elliptic unilateral probems},
Ann. Matt. Pura Appl. (4) 152  (1988), pp. 183-196. (English, with French and Italian
 summaries.

\bibitem{bomu} L. Boccardo, F. Murat,
\emph{Strongly nonlinear Cauchy problems with gradient dependt lower order nonlinearity}, Pitman  Research Notes in Mathematics, 208 (1988), pp. 347-364.

\bibitem{bomu1} L. Boccardo, F. Murat,
\emph{Almost everywhere convergence of the gradients of solutions to
elliptic and parabolic equations,} Nonlinear analysis, T.M.A., 19 (1992),
no. 6, pp. 581-597.

\bibitem{brbr} H. Brezis, and F.E. Browder,
\emph{Strongly nonlinear parabolic initial-boundary value   problems},
 Proc. Nat. Acad. Sci. U. S. A. 76 (1976). pp. 38-40.

\bibitem{alor} A. Dallaglio A. Orsina ,
\emph{Non linear parabolic  equations with natural growth condition
and $L^1$ data.} Nolinear Anal., T.M.A., 27 no. 1 (1996). pp. 59-73.

\bibitem{drabekk} P. Drabek, A. Kufner and L. Mustonen,
\emph{Pseudo-monotonicity and degenerated or singular elliptic operators},
Bull. Austral. Math. Soc. Vol. 58 (1998), 213-221.

\bibitem{drabek}  P. Drabek, A. Kufner and F. Nicolosi,
 \emph{Non linear elliptic equations, singular and degenerated cases},
University of West Bohemia, (1996).

\bibitem{Kufner} A. Kufner,
\emph{Weighted Sobolev Spaces}, John Wiley and Sons, (1985).

\bibitem{land0} R. Landes, \emph{On the existence of weak
solutions for quasilinear parabolic initial-boundary value
problems, } Proc. Roy. Soc. Edinburgh sect. A. 89 (1981), 217-137.

\bibitem{lamu} R. Landes, V. Mustonen, \emph{A strongly
nonlinear  parabolic initial-boundary value problems}, Ark. f.
Math. 25. (1987).

\bibitem{land} R. Landes, V. Mustonen, \emph{On
parabolic initial-boundary value problems with critical growth for
the gradient}, Ann. Inst. H. Poincar\'{e}11(2)(1994)135-158.

\bibitem{Leja} J. Leray, J. L. Lions, \emph{Quelques resultats de V$\dot{i}\check{s}\dot{i}$k sur les probl\`{e}mes elliptiques
nonlin\'eaires par les m\'ethodes de Minty-Browder}, Bull. Soc.
Math. France 93 (1995), 97-107.

\bibitem{lions} J. L. Lions, \emph{quelques methodes de
r\'{e}solution des probl\`{e}mes aux limites non lin\'{e}aires},
Dunod et Gauthiers-Villars, 1969.

\bibitem{Li} Zh. Liu, \emph{nonlinear degenerate parabolic
equations}, Acta Math Hungar,77 (1-2), 1997,  147-157.

\bibitem{zei} E. Zeidler, \emph{nonlinear functional analysis
and its applications, II A and II B}, Springer-Verlag (New
York-Heidlberg, 1990).

\end{thebibliography}

\end{document}