\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
$1
0$
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 $\|g_n\|_{ L^r(Q,\gamma)}\leq c, 10$ 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}
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