\documentclass[reqno]{amsart} \usepackage{hyperref} \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 $10$ 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} \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}