\documentclass[twoside]{article} \usepackage{amsfonts, amsmath} % used for R in Real numbers \pagestyle{myheadings} \markboth{Strongly nonlinear parabolic initial-boundary value problems} {Abdelhak Elmahi} \begin{document} \setcounter{page}{203} \title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent 2002-Fez conference on Partial Differential Equations,\newline Electronic Journal of Differential Equations, Conference 09, 2002, pp 203--220. \newline http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp ejde.math.swt.edu (login: ftp)} \vspace{\bigskipamount} \\ % Strongly nonlinear parabolic initial-boundary value problems in Orlicz spaces % \thanks{ {\em Mathematics Subject Classifications:} 35K15, 35K20, 35K60. \hfil\break\indent {\em Key words:} Orlicz-Sobolev spaces, compactness, parabolic equations. \hfil\break\indent \copyright 2002 Southwest Texas State University. \hfil\break\indent Published December 28, 2002. } } \date{} \author{Abdelhak Elmahi} \maketitle \begin{abstract} We prove existence and convergence theorems for nonlinear parabolic problems. We also prove some compactness results in inhomogeneous Orlicz-Sobolev spaces. \end{abstract} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \newtheorem{definition}[theorem]{Definition} \newtheorem{corollary}[theorem]{Corollary} \numberwithin{equation}{section} \section{Introduction} Let $\Omega$ be a bounded domain in $\mathbb{R}^N,T>0$ and let $A(u)=\sum_{| \alpha | \leq 1}(-1)^{| \alpha |}D^\alpha A_\alpha (x,t,u,\nabla u)$ be a Leray-Lions operator defined on $L^p(0,T;W^{1,p}(\Omega ))$, $10$ for $t>0$, $\frac{M(t)}t\to 0$ as $t\to 0$ and $\frac{M(t)}t\to \infty$ as $t\to \infty$. Equivalently, $M$ admits the representation: $M(t)=\int_0^ta(\tau )d\tau$ where $a:\mathbb{R}^{+}\to \mathbb{R}^{+}$ is non-decreasing, right continuous, with $a(0)=0$, $a(t)>0$ for $t>0$ and $a(t)\to \infty$ as $t\to \infty$. The N-function $\overline{M}$ conjugate to $M$ is defined by $\overline{M}% (t)=\int_0^t\overline{a}(\tau )d\tau$, where $\overline{a}:\mathbb{R}% ^{+}\to \mathbb{R}^{+}$ is given by $\overline{a}(t)=\sup \{s:a(s)\leq t\}$ \cite{1,10,11}. The N-function $M$ is said to satisfy the $\Delta _2$ condition if, for some $k>0$: $$\label{2.1} M(2t)\leq k\,M(t)\quad \text{for all }t\geq 0,$$ when this inequality holds only for $t\geq t_0>0$, $M$ is said to satisfy the $\Delta _2$ condition near infinity. Let $P$ and $Q$ be two N-functions. $P\ll Q$ means that $P$ grows essentially less rapidly than $Q$; i.e., for each $\varepsilon>0$, $\frac{P(t)}{Q(\varepsilon \,t)}\to 0\quad \text{as }t\to \infty .$ This is the case if and only if \thinspace \thinspace \thinspace \thinspace \thinspace \thinspace \thinspace $\lim _{t\to \infty }\,\frac{Q^{-1}(t)}{P^{-1}(t)}=0.$ An N-function is said to satisfy the $\triangle '$-condition if, for some $k_0>0$ and some $t_0\geq 0$: $$\label{2.2} M(k_0tt')\leq M(t)M(t'),\quad \text{for all }t,t'\geq t_0.$$ It is easy to see that the $\triangle '$-condition is stronger than the $\triangle _2$-condition. The following N-functions satisfy the $\triangle'$-condition: $M(t)=t^p(\mathop{\rm Log}^qt) ^s$, where $10$}). Note that L_M(\Omega ) is a Banach space under the norm $\| u\| _{M,\Omega }=\inf \Big\{ \lambda >0: \int_\Omega M(\frac{u(x)}\lambda )dx\leq 1\Big\}$ and \mathcal{L}_M(\Omega ) is a convex subset of L_M(\Omega ). The closure in L_M(\Omega ) of the set of bounded measurable functions with compact support in \overline{\Omega } is denoted by E_M(\Omega ). The equality E_M(\Omega )=L_M(\Omega ) holds if and only if M satisfies the \Delta _2 condition, for all t or for t large according to whether \Omega has infinite measure or not. The dual of E_M(\Omega ) can be identified with L_{\overline{M}}(\Omega ) by means of the pairing \int_\Omega u(x)v(x)dx, and the dual norm on L_{\overline{M}}(\Omega ) is equivalent to \| .\| _{\overline{M},\Omega }. The space L_M(\Omega ) is reflexive if and only if M and \overline{M} satisfy the \Delta _2 condition, for all t or for t large, according to whether \Omega  has infinite measure or not. We now turn to the Orlicz-Sobolev space. W^1L_M(\Omega ) (resp. W^1E_M(\Omega )) is the space of all functions u such that u and its distributional derivatives up to order 1 lie in L_M(\Omega ) (resp. E_M(\Omega )). This is a Banach space under the norm $\| u\| _{1,M,\Omega }=\sum_{| \alpha | \leq 1}\| D^\alpha u\| _{M,\Omega }.$ Thus W^1L_M(\Omega ) and W^1E_M(\Omega ) can be identified with subspaces of the product of N+1 copies of L_M(\Omega ). Denoting this product by \Pi L_M, we will use the weak topologies \sigma(\Pi L_M,\Pi E_{\overline{M}}) and \sigma (\Pi L_M,\Pi L\overline{_M}). The space W_0^1E_M(\Omega ) is defined as the (norm) closure of the Schwartz space \mathcal{D}(\Omega ) in W^1E_M(\Omega ) and the space W_0^1L_M(\Omega ) as the \sigma (\Pi L_M,\Pi E_{\overline{M}}) closure of \mathcal{D}(\Omega ) in W^1L_M(\Omega ). We say that u_n converges to u for the modular convergence in  W^1L_M(\Omega ) if for some \lambda >0, \int_\Omega M(\frac{D^\alpha u_n-D^\alpha u}\lambda )dx\to 0 for all | \alpha | \leq 1. This implies convergence for \sigma (\Pi L_M,\Pi L\overline{_M}). If M satisfies the \Delta _2 condition on \mathbb{R}^{+}(near infinity only when \Omega  has finite measure), then modular convergence coincides with norm convergence. Let W^{-1}L_{\overline{M}}(\Omega ) (resp. W^{-1}E_{\overline{M}}(\Omega )) denote the space of distributions on \Omega  which can be written as sums of derivatives of order \leq 1 of functions in L_{\overline{M}}(\Omega ) (resp. E_{\overline{M}}(\Omega )). It is a Banach space under the usual quotient norm. If the open set \Omega  has the segment property, then the space \mathcal{D}(\Omega ) is dense in W_0^1L_M(\Omega ) for the modular convergence and for the topology \sigma (\Pi L_M,\Pi L\overline{_M}) (cf. \cite{8,9}). Consequently, the action of a distribution in W^{-1}L_{\overline{M}}(\Omega ) on an element of W_0^{1}L_M(\Omega ) is well defined. For k>0, we define the truncation at height k,T_k:\mathbb{R}\to\mathbb{R} by $$\label{2.3} T_k(s)=\begin{cases} s\quad &\text{if }| s| \leq k\\ k s/| s| &\text{if }| s| >k. \end{cases}$$ The following abstract lemmas will be applied to the truncation operators. \begin{lemma} \label{lemma 2.1} Let F:\mathbb{R}\to \mathbb{R} be uniformly lipschitzian, with F(0)=0. Let M be an N-function and let u\in W^{1}L_{M}(\Omega ) (resp. W^{1}E_{M}(\Omega )). Then F(u)\in W^{1}L_{M}(\Omega ) (resp. W^{1}E_{M}(\Omega )). Moreover, if the set of discontinuity points of F' is finite, then $\frac{\partial }{\partial x_{i}}F(u)= \begin{cases} F'(u)\frac{\partial u}{\partial x_{i}} & \text{a.e. in } \{ x\in \Omega :u(x)\notin D\} \\ 0 &\text{a.e. in }\{ x\in \Omega :u(x)\in D\} . \end{cases}$ \end{lemma} \begin{lemma} \label{lemma 2.2} Let F:\mathbb{R}\to \mathbb{R} be uniformly lipschitzian, with F(0)=0. We suppose that the set of discontinuity points of F' is finite. Let M be an N-function, then the mapping F:W^{1}L_{M}(\Omega)\to W^{1}L_{M}(\Omega ) is sequentially continuous with respect to the weak* topology \sigma (\Pi L_{M},\Pi E_{\overline{M}}). \end{lemma} \paragraph{Proof} By the previous lemma, F(u)\in W^1L_M(\Omega ) for all u\in W^1L_M(\Omega ) and $\| F(u)\| _{1,M,\Omega }\leq C\,\| u\| _{1,M,\Omega },$ which gives easily the result. \hfill\square Let \Omega  be a bounded open subset of \mathbb{R}^N, T>0 and set Q=\Omega \times ] 0,T[. Let m\geq 1 be an integer and let M be an N-function. For each \alpha \in \mathbf{N}^N, denote by D_x^\alpha  the distributional derivative on Q of order \alpha  with respect to the variable x\in \mathbb{R}^N. The inhomogeneous Orlicz-Sobolev spaces are defined as follows \begin{gather*} W^{m,x}L_M(Q)=\{ u\in L_M(Q):D_x^\alpha u\in L_M(Q)\;\forall | \alpha | \leq m\}\\ W^{m,x}E_M(Q)=\{ u\in E_M(Q):D_x^\alpha u\in E_M(Q)\; \forall | \alpha | \leq m\} \end{gather*} The last space is a subspace of the first one, and both are Banach spaces under the norm $\| u\| =\sum_{| \alpha | \leq m}\| D_x^\alpha u\| _{M,Q}.$ We can easily show that they form a complementary system when \Omega  satisfies the segment property. These spaces are considered as subspaces of the product space \Pi L_M(Q) which have as many copies as there is \alpha -order derivatives, | \alpha | \leq m. We shall also consider the weak topologies \sigma ( \Pi L_M,\Pi E_{\overline{M}})  and \sigma ( \Pi L_M,\Pi L_{\overline{M}}). If u\in W^{m,x}L_M(Q) then the function :t\longmapsto u(t)=u(t,.) is defined on [ 0,T] with values in W^mL_M(\Omega ). If, further, u\in W^{m,x}E_M(Q) then the concerned function is a W^mE_M(\Omega )-valued and is strongly measurable. Furthermore the following imbedding holds: W^{m,x}E_M(Q)\subset L^1(0,T;W^mE_M(\Omega )). The space W^{m,x}L_M(Q) is not in general separable, if u\in W^{m,x}L_M(Q), we can not conclude that the function u(t) is measurable on [ 0,T]. However, the scalar function t\mapsto \|u(t)\| _{M,\Omega } is in L^1( 0,T) . The space W_0^{m,x}E_M(Q) is defined as the (norm) closure in W^{m,x}E_M(Q) of \mathcal{D}(Q). We can easily show as in \cite{9} that when \Omega  has the segment property then each element u of the closure of \mathcal{D}(Q) with respect of the weak * topology \sigma ( \Pi L_M,\Pi E_{\overline{M}})  is limit, in W^{m,x}L_M(Q), of some subsequence ( u_i)  \subset  \mathcal{D}(Q) for the modular convergence; i.e., there exists \lambda >0 such that for all | \alpha | \leq m, $\int_QM( \frac{D_x^\alpha u_i-D_x^\alpha u}\lambda ) \,dx\,dt\to 0\text{ as }i\to \infty ,$ this implies that ( u_i)  converges to u in W^{m,x}L_M(Q) for the weak topology \sigma ( \Pi L_M,\Pi L_{\overline{M}}) . Consequently $\overline{\mathcal{D}(Q)}^{\sigma ( \Pi L_M,\Pi E_{\overline{M}}) }=\overline{\mathcal{D}(Q)}^{\sigma ( \Pi L_M,\Pi L_{\overline{M}}) },$ this space will be denoted by W_0^{m,x}L_M(Q). Furthermore, W_0^{m,x}E_M(Q)=W_0^{m,x}L_M(Q)\cap \Pi E_M. Poincar\'e's inequality also holds in W_0^{m,x}L_M(Q) i.e. there is a constant C>0 such that for all u\in W_0^{m,x}L_M(Q) one has $\sum_{| \alpha | \leq m}\| D_x^\alpha u\| _{M,Q}\leq C\sum_{| \alpha | =m}\| D_x^\alpha u\| _{M,Q}.$ Thus both sides of the last inequality are equivalent norms on W_0^{m,x}L_M(Q). We have then the following complementary system $\begin{pmatrix} W_0^{m,x}L_M(Q) & F \\ W_0^{m,x}E_M(Q) & F_0 \end{pmatrix},$ F being the dual space of W_0^{m,x}E_M(Q). It is also, except for an isomorphism, the quotient of \Pi L_{\overline{M}} by the polar set % W_0^{m,x}E_M(Q)^{\bot }, and will be denoted by F=W^{-m,x}L_{\overline{M}% }(Q) and it is shown that $W^{-m,x}L_{\overline{M}}(Q)=\Big\{ f=\sum_{| \alpha | \leq m}D_x^\alpha f_\alpha :f_\alpha \in L_{\overline{M}}(Q)\Big\} .$ This space will be equipped with the usual quotient norm $\| f\| =\inf \sum_{| \alpha | \leq m}\| f_\alpha \| _{\overline{M},Q}$ where the infimum is taken on all possible decompositions $f=\sum_{| \alpha | \leq m}D_x^\alpha f_\alpha ,\quad f_\alpha \in L_{\overline{M}}(Q).$ The space F_0 is then given by $F_0=\Big\{ f=\sum_{| \alpha | \leq m}D_x^\alpha f_\alpha :f_\alpha \in E_{\overline{M}}(Q)\Big\}$ and is denoted by F_0=W^{-m,x}E_{\overline{M}}(Q). \begin{remark} \label{remark 2.1}\rm We can easily check, using \cite[lemma 4.4]{9}, that each uniformly lipschitzian mapping F, with F(0)=0, acts in inhomogeneous Orlicz-Sobolev spaces of order 1: W^{1,x}L_{M}(Q) and W_{0}^{1,x}L_{M}(Q). \end{remark} \section{Galerkin solutions}\label{sec 4} In this section we shall define and state existence theorems of Galerkin solutions for some parabolic initial-boundary problem. Let \Omega  be a bounded subset of \mathbb{R}^N, T>0 and set Q=\Omega \times ] 0,T[ . Let $A(u)=\sum_{| \alpha | \leq m}(-1)^{| \alpha | }D_x^\alpha (A_\alpha (u))$ be an operator such that \label{4.1} \begin{aligned} &A_\alpha (x,t,\xi ):\Omega \times [ 0,T] \times \mathbb{R} ^{N_0}\to \mathbb{R}\text{ is continuous in (t,\xi ), for a.e. x\in \Omega} \\ &\text{and measurable in }x,\text{ for all }(t,\xi )\in [ 0,T] \times \mathbb{R}^{N_0}, \\ &\text{where, N_0 is the number of all \alpha-order's derivative, |\alpha | \leq m.} \end{aligned} $$\label{4.2} | A_\alpha (x,s,\xi )| \leq \chi ( x) \Phi (| \xi | ) \text{ with \chi (x)\in L^1( \Omega ) and \Phi :\mathbb{R}^{+}\to \mathbb{R}^{+} increasing.}$$ $$\label{4.3} \sum_{| \alpha | \leq m}A_\alpha (x,t,\xi )\xi _\alpha \geq -d(x,t)\text{ with }d(x,t)\in L^1(Q),\text{ }d\geq 0.$$ Consider a function \psi \in L^2(Q) and a function \overline{u}\in L^2( \Omega ) \cap W_0^{m,1}( \Omega ) . We choose an orthonormal sequence ( \omega _i) \subset \mathcal{D% }( \Omega )  with respect to the Hilbert space L^2( \Omega )  such that the closure of ( \omega _i)  in C^m(\overline{\Omega })  contains \mathcal{D}( \Omega ) . C^m( \overline{\Omega })  being the space of functions which are m times continuously differentiable on \overline{\Omega }. For V_n=\mathop{\rm span}\langle \omega _1,\dots,\omega _n\rangle  and $\| u\| _{C^{1,m}( Q) }=\sup \big\{ | D_x^\alpha u(x,t)| ,| \frac{\partial u}{\partial t}( x,t) | :| \alpha | \leq m,(x,t)\in Q\big\}$ we have $\mathcal{D}(Q)\subset \overline{\left\{ \cup _{n=1}^\infty C^1( [ 0,T] ,V_n) \right\} }^{C^{1,m}(Q)}$ this implies that for \psi  and \overline{u}, there exist two sequences % (\psi _n) and (\overline{u}_n) such that \begin{gather} \label{4.4} \psi _n\in C^1([ 0,T] ,V_n),\quad \psi _n\to \psi \text{ in }L^2(Q). \\ \label{4.5} \overline{u}_n\in V_n,\quad \overline{u}_n\to \overline{u} \text{ in }L^2( \Omega ) \cap W_0^{m,1}( \Omega ) . \end{gather} Consider the parabolic initial-boundary value problem $$\label{4.6} \begin{gathered} \frac{\partial u}{\partial t}+A(u)=\psi \;\text{in }Q, \\ D_x^\alpha u=0 \text{ on }\partial \Omega \times ] 0,T[ ,\text{ for all\textit{\ }}| \alpha | \leq m-1, \\ u(0)=\overline{u}\text{ in }\Omega . \end{gathered}$$ In the sequel we denote A_\alpha (x,t,u,\nabla u,\dots ,\nabla ^mu) by A_\alpha (x,t,u) or simply by A_\alpha (u). \begin{definition} \label{definition 1} \rm A function u_{n}\in C^{1}( [ 0,T] ,V_{n}) \ is called Galerkin solution of (\ref{4.6}) if $\int_{\Omega }\frac{\partial u_{n}}{\partial t}\varphi dx+\int_{\Omega }\sum_{| \alpha | \leq m}A_{\alpha }(u_{n}).D_{x}^{\alpha }\varphi dx=\int_{\Omega }\psi _{n}(t)\varphi dx$ for all \varphi \in V_{n}\ and all t\in [ 0,T] ;\;u_{n}(0)=% \overline{u}_{n}. \end{definition} We have the following existence theorem. \begin{theorem}[\cite{12}] \label{theorem 4.1} Under conditions (\ref{4.1})-(\ref{4.3}), there exists at least one Galerkin solution of (\ref{4.6}). \end{theorem} Consider now the case of a more general operator $A(u)=\sum_{| \alpha | \leq m}(-1)^{| \alpha | }D_x^\alpha (A_\alpha (u))$ where instead of (\ref{4.1}) and (\ref{4.2}) we only assume that \begin{gather} A_\alpha (x,t,\xi ):\Omega \times [ 0,T] \times \mathbb{R}% ^{N_0}\to \mathbb{R}\text{ is continuous in }\xi ,\text{ for a.e. }% (x,t)\in Q \nonumber \\ \text{and measurable in (x,t) for all }\xi \in \mathbb{R}^{N_0}. \label{4.7} \\ \label{4.8} | A_\alpha (x,s,\xi )| \leq C( x,t) \Phi ( |\xi | ) \text{ with }C(x,t)\in L^1( Q) . \end{gather} We have also the following existence theorem \begin{theorem}[\cite{13}] \label{thm4.2} There exists a function u_{n} in C( [ 0,T] ,V_{n}) such that \frac{\partial u_{n}}{\partial t} is in L^{1}( 0,T;V_{n}) and $\int_{Q_{\tau }}\frac{\partial u_{n}}{\partial t}\varphi \,dx\,dt +\int_{Q_{\tau }}\sum_{| \alpha | \leq m}A_{\alpha }(x,t,u_{n}).D_{x}^{\alpha }\varphi \,dx\,dt =\int_{Q_{\tau }}\psi _{n}\varphi \,dx\,dt$ for all \tau \in [ 0,T]  and all \varphi \in C([ 0,T] ,V_{n}), where Q_{\tau }=\Omega \times [ 0,\tau ] ;\;u_{n}(0)=\overline{u}_{n}. \end{theorem} \section{Strong convergence of truncations} In this section we shall prove a convergence theorem for parabolic problems which allows us to deal with approximate equations of some parabolic initial-boundary problem in Orlicz spaces (see section \ref{sec 7}). Let \Omega, be a bounded subset of \mathbb{R}^N with the segment property and let T>0, Q=\Omega \times ] 0,T[ . Let M be an N-function satisfying a \Delta ' condition and the growth condition $M( t) \ll | t| ^{\frac N{N-1}}$ and let P be an N-function such that P\ll M. Let A:W^{1,x}L_M(Q)\to W^{-1,x}L_{\overline{M}}(Q) be a mapping given by $A(u)=-\mathop{\rm div} a(x,t,u,\nabla u)$ where a(x,t,s,\xi ):\Omega \times [ 0,T] \times \mathbb{R}\times \mathbb{R}^N\to \mathbb{R}^N is a Carath\'eodory function satisfying for a.e. (x,t)\in \Omega \times ]0,T[  and for all s\in \mathbb{R} and all \xi ,\xi ^{*}\in \mathbb{R}^N: \begin{gather} \label{20} | a(x,t,s,\xi )| \leq c(x,t)+k_1\overline{P}^{-1}M(k_2| s| )+k_3\overline{M}^{-1}M(k_4| \xi | ) \\ \label{21} [ a(x,t,s,\xi )-a(x,t,s,\xi ^{*})] [ \xi -\xi ^{*}] >0\quad \text{if } \xi \neq \xi ^{*} \\ \label{22} \alpha M(\frac{| \xi | }\lambda )-d(x,t)\leq a(x,t,s,\xi )\xi \, \end{gather} where c(x,t)\in E_{\overline{M}}(Q), c\geq 0, d(x,t)\in L^1(Q), k_1,k_2,k_3,k_4\in \mathbb{R}^{+} and \alpha ,\lambda \in\mathbb{R}_{*}^{+}. Consider the nonlinear parabolic equations $$\label{23} \frac{\partial u_n}{\partial t}-\mathop{\rm div }a(x,t,u_n,\nabla u_n)=f_n+g_n \quad \text{in }\mathcal{D}'(Q)$$ and assume that: \begin{gather} \label{24} u_n\rightharpoonup u\quad \text{weakly in }W^{1,x}L_M(Q)\text{for } \sigma (\Pi L_M,\Pi E_{\overline{M}}), \\ \label{25} f_n\to f\quad \text{strongly in } W^{-1,x}E_{\overline{M}}(Q), \\ \label{26} g_n\rightharpoonup g\quad \text{weakly in }L^1(Q). \end{gather} We shall prove the following convergence theorem. \begin{theorem} \label{thm5.1} Assume that (\ref{20})-(\ref{26}) hold. Then, for any k>0, the truncation of u_{n}\ at height k (see (\ref{2.3}) for the definition of the truncation) satisfies $$\nabla T_{k}(u_{n})\to \nabla T_{k}(u)\quad \text{strongly in }(L_{M}^{\rm loc}(Q))^{N}. \label{27}$$ \end{theorem} \begin{remark}\rm An elliptic analogous theorem is proved in Benkirane-Elmahi \cite{2}. \end{remark} \begin{remark} \rm Convergence (\ref{27}) allows, in particular, to extract a subsequence % n' such that: $\nabla u_{n'}\to \nabla u\quad \text{a.e. in }Q.$ Then by lemma 4.4 of \cite{8}, we deduce that $a(x,t,u_{n'},\nabla u_{n'})\rightharpoonup a(x,t,u,\nabla u)\quad\text{weakly in L_{\overline{M}}(Q))^{N} for }\sigma (\Pi L_{\overline{M}},\Pi E_{M}).$ \end{remark} \paragraph{Proof of Theorem \ref{thm5.1}} \textbf{Step 1:} For each k>0, define S_k(s)=\int_0^sT_k(\tau )d\tau . Since T_k is continuous, for all w\in W^{1,x}L_M(Q) we have S_k(w)\in W^{1,x}L_M(Q) and \nabla S_k(w)=T_k(w)\nabla w. So that, by mollifying as in \cite{6}, it is easy to see that for all \varphi \in \mathcal{D}(Q) and all v\in W^{1,x}L_M(Q) with \frac{\partial v}{\partial t}\in W^{-1,x}L_{\overline{M}}(Q)+L^1(Q), we have $$\label{28} \langle\langle \frac{\partial v}{\partial t},\varphi T_k(v)\rangle\rangle =-\int_Q\frac{\partial \varphi }{\partial t}S_k(v)\,dx\,dt.$$ where \langle\langle ,\rangle\rangle means for the duality pairing between W_0^{1,x}L_M(Q)+L^1(Q) and W^{-1,x}L_{\overline{M}}(Q)\cap L^\infty (Q). Fix now a compact set K with K\subset Q and a function \varphi _K\in \mathcal{D}(Q) such that 0\leq \varphi _K\leq 1 in Q and \varphi _K=1 on K. Using in (\ref{23}) v_n=\varphi _K( T_k(u_n)-T_k(u)) \in W^{1,x}L_M(Q)\cap L^\infty (Q) as test function yields \label{29} \begin{aligned} &\langle\langle \frac{\partial u_n}{\partial t},\varphi _KT_k(u_n)\rangle\rangle -\langle\langle \frac{\partial u_n}{\partial t},\varphi _KT_k(u)\rangle\rangle \\ &+\int_Q\varphi _Ka(x,t,u_n,\nabla u_n)[ \nabla T_k(u_n)-\nabla T_k(u)] dx\,dt &\\ &+\int_Q( T_k(u_n)-T_k(u)) a(x,t,u_n,\nabla u_n)\nabla \varphi _K\,dx\,dt\\ &=\langle\langle f_n,v_n\rangle\rangle +\langle\langle g_n,v_n\rangle\rangle . \end{aligned} Since u_n\in W^{1,x}L_M(Q) and \frac{\partial u_n}{\partial t}\in W^{-1,x}L_{\overline{M}}(Q)+L^1(Q) then by (\ref{28}), $\langle\langle \frac{\partial u_n}{\partial t}, \varphi _KT_k(u_n)\rangle\rangle =-\int_Q\frac{\partial \varphi _K}{\partial t}S_k(u_n)\,dx\,dt.$ On the other hand since ( u_n)  is bounded in W^{1,x}L_M(Q) and \frac{\partial u_n}{\partial t}=h_n+g_n while g_n is bounded in L^1(Q) and so in \mathcal{M}(Q) and h_n=\mathop{\rm div} a(x,t,u_n,\nabla u_n)+f_n is bounded in W^{-1,x}L_{\overline{M}}(Q), then by \cite[Corollary 1]{Cpt}, u_n\to u strongly in L_M^{\rm loc}(Q). Consequently, T_k(u_n)\to T_k(u) and S_k(u_n)\to S_k(u) in L_M^{\rm loc}(Q). So that $\int_Q\frac{\partial \varphi _K}{\partial t}S_k(u_n)\,dx\,dt\to \int_Q \frac{\partial \varphi _K}{\partial t}S_k(u)\,dx\,dt$ and also \int_Q( T_k(u_n)-T_k(u)) a(x,t,u_n,\nabla u_n)\nabla \varphi _K\,dx\,dt\to 0 as n\to \infty . Furthermore \langle\langle f_n,v_n\rangle\rangle \to 0, by (\ref{25}). Since g_n\in L^1(Q) and T_k(u_n)-T_k(u)\in L^\infty (Q), $\langle\langle g_n,\varphi _K( T_k(u_n)-T_k(u)) \rangle\rangle =\int_Qg_n\varphi _K( T_k(u_n)-T_k(u)) \,dx\,dt$ which tends to 0 by Egorov's theorem. Since \varphi _KT_k(u) belongs to W_0^{1,x}L_M(Q)\cap L^\infty (Q) while \frac{\partial u_n}{\partial t} is the sum of a bounded term in % W^{-1,x}L_{\overline{M}}(Q) and of g_n which weakly converges in L^1(Q) one has $\langle\langle \frac{\partial u_n}{\partial t},\varphi _KT_k(u)\rangle\rangle \to \langle\langle \frac{\partial u}{\partial t},\varphi _KT_k(u)\rangle\rangle =-\int_Q\frac{\partial \varphi }{\partial t}S_k(u)\,dx\,dt.$ We have thus proved that $$\label{30} \int_Q\varphi _Ka(x,t,u_n,\nabla u_n)[ \nabla T_k(u_n)-\nabla T_k(u)] \,dx\,dt\to 0\quad \text{as } n\to \infty .$$ \noindent\textbf{Step 2:} Fix a real number r>0 and set Q_{(r)}=\{ x\in Q:| \nabla T_k(u)| \leq r\} and denote by \chi _r the characteristic function of Q_{(r)}. Taking s\geq r one has: \begin{aligned} 0\leq& \int_{Q_{(r)}}\varphi _K\big[ a(x,t,u_n,\nabla T_k(u_n))-a(x,t,u_n,\nabla T_k(u))\big]\\ &\times\big[ \nabla T_k(u_n)-\nabla T_k(u)\big] \,dx\,dt \\ &\leq \int_{Q_{(s)}}\varphi _K\big[ a(x,t,u_n,\nabla T_k(u_n))-a(x,t,u_n,\nabla T_k(u))\big] \\ &\times\big[ \nabla T_k(u_n)-\nabla T_k(u)] \,dx\,dt \\ =&\int_{Q_{(s)}}\varphi _K\big[ a(x,t,u_n,\nabla T_k(u_n))-a(x,t,u_n,\nabla T_k(u)\chi _s)\big] \\ &\times\big[ \nabla T_k(u_n)-\nabla T_k(u)\chi _s] \,dx\,dt \\ \leq& \int_Q\varphi _K\big[ a(x,t,u_n,\nabla T_k(u_n))-a(x,t,u_n,\nabla T_k(u)\chi _s)\big]\\ &\times\big[\nabla T_k(u_n)-\nabla T_k(u)\chi _s] \,dx\,dt \\ =&\int_Q\varphi _Ka(x,t,u_n,\nabla u_n)\big[ \nabla T_k(u_n)-\nabla T_k(u)] \,dx\,dt \\ &-\int_Q\varphi _K\big[ a(x,t,u_n,\nabla u_n)-a(x,t,u_n,\nabla T_k(u_n))\big]\\ &\times\big[ \nabla T_k(u_n)-\nabla T_k(u)\chi _s] \,dx\,dt \\ &+\int_Q\varphi _Ka(x,t,u_n,\nabla u_n)[ \nabla T_k(u)-\nabla T_k(u)\chi _s] \,dx\,dt \\ &-\int_Q\varphi _Ka(x,t,u_n,\nabla T_k(u)\chi _s)[ \nabla T_k(u_n)-\nabla T_k(u)\chi _s] \,dx\,dt. \end{aligned}\label{31} Now pass to the limit in all terms of the right-hand side of the above equation. By (\ref{30}), the first one tends to 0. Denoting by \chi _{G_n} the characteristic function of G_n=\{(x,t)\in Q:| u_n(x,t)| >k\} , the second term reads $$\label{32} \int_Q\varphi _K[ a(x,t,u_n,\nabla u_n)-a(x,t,u_n,0)] \chi _{G_n}\nabla T_k(u)\chi _s\,dx\,dt$$ which tends to 0 since [ a(x,t,u_n,\nabla u_n)-a(x,t,u_n,0)]  is bounded in (L_{\overline{M}}(Q))^N, by (\ref{20}) and (\ref{24}) while \chi _{G_n}\nabla T_k(u)\chi _s converges strongly in (E_M(Q))^N to 0 by Lebesgue's theorem. The fourth term of (\ref{31}) tends to \label{33} \begin{aligned} -\int_Q&\varphi _Ka(x,t,u,\nabla T_k(u)\chi _s)[ \nabla T_k(u)-\nabla T_k(u)\chi _s] \,dx\,dt\\ &=\int_{Q\setminus Q_{(s)}}\varphi_Ka(x,t,u,0)\nabla T_k(u)\,dx\,dt \end{aligned} since a(x,t,u_n,\nabla T_k(u)\chi _s) tends strongly to a(x,t,u,\nabla T_k(u)\chi _s) in (E_{\overline{M}}(Q))^N while \nabla T_k(u_n)-\nabla T_k(u)\chi _s converges weakly to \nabla T_k(u)-\nabla T_k(u)\chi _s in (L_M(Q))^{N} for \sigma (\Pi L_M,\Pi E_{\overline{M}}). Since a(x,t,u_n,\nabla u_n) is bounded in (L_{\overline{M}}(Q))^N one has (for a subsequence still denoted by u_n) $$\label{34} a(x,t,u_n,\nabla u_n)\rightharpoonup h\quad \text{weakly in }(L_{\overline{M}% }(Q))^N\text{ for }\sigma (\Pi L_{\overline{M}},\Pi E_M).$$ Finally, the third term of the right-hand side of (\ref{31}) tends to $$\label{35} \int_{Q\setminus Q_{(s)}}\varphi _Kh\nabla T_k(u)\,dx\,dt.$$ We have, then, proved that \label{36} \begin{aligned} 0\leq &\lim \sup _{n\to \infty }\int_{Q_{(r)}}\varphi _K\big[ a(x,t,u_n,\nabla T_k(u_n))-a(x,t,u_n,\nabla T_k(u))\big]\\ &\times\big[\nabla T_k(u_n)-\nabla T_k(u)\big] \,dx\,dt \\ \leq &\int_{Q\setminus Q_{(s)}}\varphi _K[ h-a(x,t,u,0)] \nabla T_k(u)\,dx\,dt. \end{aligned} Using the fact that [ h-a(x,t,u,0)] \nabla T_k(u)\in L^1(\Omega) and letting s\to +\infty  we get, since | Q\setminus Q_{(s)}| \to 0, $$\label{37} \int_{Q_{(r)}}\varphi _K[ a(x,t,u_n,\nabla T_k(u_n))-a(x,t,u_n,\nabla T_k(u))] [ \nabla T_k(u_n)-\nabla T_k(u)] \,dx\,dt$$ which approaches 0 as n\to \infty. Consequently $\int_{Q_{(r)}\cap K}[ a(x,t,u_n,\nabla T_k(u_n))-a(x,t,u_n,\nabla T_k(u))] [ \nabla T_k(u_n)-\nabla T_k(u)] \,dx\,dt\to 0\,$ as n\to \infty . As in \cite{2}, we deduce that for some subsequence \nabla T_k(u_n)\to \nabla T_k(u) \thinspace a.e. in\ Q_{(r)}\cap K. Since\ r,\ k and K are arbitrary, we can construct a subsequence (diagonal in r, in k and in j, where ( K_j)  is an increasing sequence of compacts sets covering Q), such that $$\label{38} \nabla u_n\to \nabla u\quad \quad \text{a.e. in } Q.$$ \noindent\textbf{Step 3:} As in \cite{2} we deduce that $\int_Q\varphi _Ka(x,t,u_n,\nabla u_n)\nabla T_k(u_n)\,dx\,dt\to \int_Q\varphi _Ka(x,t,u,\nabla u)\nabla T_k(u)\,dx\,dt$ as n\to \infty, and that $$\label{39} \,a(x,t,u_n,\nabla T_k(u_n))\nabla T_k(u_n)\to a(x,t,u,\nabla T_k(u))\nabla T_k(u)\text{ strongly in } L^1( K) .$$ This implies that (see \cite{2} if necessary): \nabla T_k(u_n)\to \nabla T_k(u) in ( L_M(K)) ^N for the modular convergence and so strongly and convergence (\ref{27}) follows. Note that in convergence (\ref{27}) the whole sequence (and not only for a subsequence) converges since the limit \nabla T_k(u) does not depend on the subsequence. \section{Nonlinear parabolic problems} \label{sec 6} Now, we are able to establish an existence theorem for a nonlinear parabolic initial-boundary value problems. This result which specially applies in Orlicz spaces generalizes analogous results in of Landes-Mustonen \cite{13}. We start by giving the statement of the result. Let \Omega  be a bounded subset of \mathbb{R}^N with the segment property, T>0, and Q=\Omega \times ] 0,T[ . Let M be an N-function satisfying the growth condition $M(t)\ll | t| ^{\frac N{N-1}},$ and the \triangle '-condition. Let P be an N-function such that P\ll M. Consider an operator A:W_0^{1,x}L_M(Q)\to W^{-1,x}L_{\overline{M}% }(Q) of the form $$\label{6.1} \ A(u)=-\mathop{\rm div} a(x,t,u,\nabla u)+a_0(x,t,u,\nabla u)$$ where a:\Omega \times [ 0,T] \times \mathbb{R}\times \mathbb{R} ^N\to \mathbb{R}^N and a_0:\Omega \times [ 0,T] \times \mathbb{R}\times \mathbb{R}^N\to \mathbb{R} are Carath\'eodory functions satisfying the following conditions, for a.e. (x,t)\in \Omega \times [ 0,T]  for all s\in \mathbb{R} and \xi \neq \xi ^{*}\in \mathbb{R}^N: \begin{gather} \label{6.2} \begin{gathered} | a(x,t,s,\xi )| \leq c(x,t)+k_1 \overline{P}^{-1}M(k_2| s| )+k_3\overline{M}^{-1}M(k_4| \xi | ), \\ | a_0(x,t,s,\xi )| \leq c(x,t)+k_1\overline{M}^{-1}M(k_2| s| )+k_3\overline{M}^{-1}P(k_4| \xi | ), \end{gathered} \\ \label{6.3} [ a(x,t,s,\xi )-a(x,t,s,\xi ^{*})] [ \xi -\xi ^{*}] >0, \\ \label{6.4} a(x,t,s,\xi )\xi \,+a_0(x,t,s,\xi )s\geq \alpha M(\frac{| \xi | }\lambda ) -d(x,t) \end{gather} where c(x,t)\in E_{\overline{M}}(Q), c\geq 0, d(x,t)\in L^1( Q), k_1,k_2,k_3,k_4\in \mathbb{R}^{+} and \alpha ,\lambda \in \mathbf{R}_{*}^{+}. Furthermore let $$\label{6.5} f\in W^{-1,x}E_{\overline{M}}( Q)$$ We shall use notations of section \ref{sec 4}. Consider, then, the parabolic initial-boundary value problem $$\label{6.6} \begin{gathered} \frac{\partial u}{\partial t}+A(u)=f\ \ \ \text{in }Q \\ u(x,t)=0 \text{ on }\partial \Omega \times ] 0,T[ \\ u(x,0)=\psi (x)\ \text{in }\Omega . \end{gathered}$$ where \psi  is a given function in L^2( \Omega ) . We shall prove the following existence theorem. \begin{theorem} \label{thm6.1} Assume that (\ref{6.2})-(\ref{6.5}) hold. Then there exists at least one weak solution u\in W_{0}^{1,x}L_{M}(Q)\cap L^{2}(Q)\cap C( [ 0,T] ,L^{2}(\Omega )) of (\ref{6.6}), in the following sense: $$\begin{gathered} -\int_{Q}u\frac{\partial \varphi }{\partial t}\,dx\,dt+[ \int_{\Omega }u(t)\varphi (t)dx] _{0}^{T}+\int_{Q}a(x,t,u,\nabla u).\nabla \varphi \,dx\,dt \\ \;+\int_{Q}a_{0}(x,t,u,\nabla u).\varphi \,dx\,dt=\left\langle f,\varphi \right\rangle \end{gathered} \label{6.7}$$ for all \varphi \in C^{1}( [ 0,T] ,L^{2}(\Omega )) . \end{theorem} \begin{remark} \rm In (\ref{6.6}), we have u\in W_{0}^{1,x}L_{M}(Q)\subset L^{1}(0,T;W^{-1,1}(\Omega )) and \frac{\partial u}{\partial t}\in W^{-1,x}L_{\overline{M}}(Q)\subset L^{1}(0,T;W^{-1,1}(\Omega )). Then u\in W^{1,1}(0,T;W^{-1,1}(\Omega ))\subset C([ 0,T] ,W^{-1,1}(\Omega ))  with continuity of the imbedding. Consequently u is, possibly after modification on a set of zero measure, continuous from [ 0,T]  into W^{-1,1}(\Omega ) in such a way that the third component of (\ref{6.6}% ), which is the initial condition, has a sense. \end{remark} \paragraph{Proof of Theorem \ref{thm5.1} } It is easily adapted from the proof given in \cite{13}. For convenience we suppose that \psi =0. For each n, there exists at least one solution u_n of the following problem (see Theorem \ref{thm4.2} for the existence of u_n): $$\label{6.8} \begin{gathered} u_n\in C( [ 0,T] ,V_n) , \quad \frac{\partial u_n}{\partial t}\in L^1(0,T;V_n), \quad u_n(0)=\psi _n\equiv 0 \quad \text{and, }\\ \text{for all }\tau \in [ 0,T],\quad \int_{Q_\tau }\frac{\partial u_n}{\partial t}\varphi \,dx\,dt +\int_{Q_\varepsilon }a(x,t,u_n,\nabla u_n).\nabla \varphi \,dx\,dt \\ +\int_{Q_\varepsilon }a_0(x,t,u_n,\nabla u_n).\varphi\,dx\,dt =\int_{Q_\varepsilon }f_n\varphi \,dx\,dt,\quad \forall \varphi \in C([ 0,T] ,V_n) . \end{gathered}$$ where f_k\subset \cup _{n=1}^\infty C( [ 0,T] ,V_n)  with f_k\to f in W^{-1,x}E_{\overline{M}}(Q). Putting \varphi =u_n in (\ref{6.8}), and using (\ref{6.2}) and (\ref{6.4}) yields $$\label{6.9} \begin{gathered} \| u_n\| _{W_0^{1,x}L_M(Q)}\leq C,\quad \| u_n\| _{L^\infty (0,T;L^2(\Omega ))}\leq C \\ \| a_0(x,t,u_n,\nabla u_n)\| _{L_{\overline{M}}(Q)} \leq C\quad \text{and}\quad \| a(x,t,u_n,\nabla u_n)\| _{L_{\overline{M}}(Q)}\leq C. \end{gathered}$$ Hence, for a subsequence $$\label{6.10} \begin{gathered} u_n\rightharpoonup u \text{ weakly in }W_0^{1,x}L_M(Q)\text{ for }\sigma ( \Pi L_M,\Pi E_{\overline{M}}) \text{and weakly in }L^2(Q), \\ a_0(x,t,u_n,\nabla u_n)\rightharpoonup h_0,\;a(x,t,u_n,\nabla u_n)\rightharpoonup h\text{ in }L_{\overline{M}}(Q)\text{ for }\sigma ( \Pi L_{\overline{M}},\Pi E_M) \end{gathered}$$ where h_0\in L_{\overline{M}}(Q) and h\in ( L_{\overline{M}}(Q)) ^N. As in \cite{13}, we get that for some subsequence u_n(x,t)\to u(x,t)  a.e. in Q (it suffices to apply Theorem 3.9 instead of Proposition 1 of \cite{13}). Also we obtain $-\int_Qu\frac{\partial \varphi }{\partial t}\,dx\,dt+[ \int_\Omega u(t)\varphi (t)dx] _0^T+\int_Qh\nabla \varphi \,dx\,dt+\int_Qh_0\varphi \,dx\,dt=\langle f,\varphi \rangle ,$ for all \varphi \in C^1( [ 0,T] ;\mathcal{D}(\Omega )) . The proof will be completed, if we can show that $$\label{6.11} \int_Q( h\nabla \varphi +h_0\varphi ) \,dx\,dt=\int_Q( a(x,t,u,\nabla u)\nabla \varphi +a_0(x,t,u,\nabla u)\varphi ) \,dx\,dt$$ for all \varphi \in C^1([ 0,T] ;\mathcal{D}(\Omega )) and that % u\in C( [ 0,T] ,L^2(\Omega )) . For that, it suffices to show that $$\label{6.12} \lim _{n\to \infty }\int_Q( a(x,t,u_n,\nabla u_n)[ \nabla u_n-\nabla u] +a_0(x,t,u_n\nabla u_n)[ u_n-u] ) \,dx\,dt\leq 0.$$ Indeed, suppose that (\ref{6.12}) holds and let s>r>0 and set Q^r=\{ (x,t)\in Q:| \nabla u(x,t)| \leq r\} . Denoting by \chi _s the characteristic function of Q^s, one has \label{6.13} \begin{aligned} 0\leq &\int_{Q^r}\big[ a(x,t,u_n,\nabla u_n)-a(x,t,u_n,\nabla u)\big] \big[ \nabla u_n-\nabla u\big] \,dx\,dt \\ \leq& \int_{Q^s}\big[ a(x,t,u_n,\nabla u_n)-a(x,t,u_n,\nabla u)\big] \big[ \nabla u_n-\nabla u\big] \,dx\,dt \\ =&\int_{Q^s}\big[ a(x,t,u_n,\nabla u_n)-a(x,t,u_n,\nabla u.\chi _s)\big] \big[ \nabla u_n-\nabla u.\chi _s\big] \,dx\,dt \\ \leq& \int_Q\big[ a(x,t,u_n,\nabla u_n)-a(x,t,u_n,\nabla u.\chi _s)\big] \big[ \nabla u_n-\nabla u.\chi _s\big] \,dx\,dt \\ =&\int_Qa_0(x,t,u_n,\nabla u_n)(u_n-u)-\int_Qa(x,t,u_n,\nabla u_n.\chi _s)[ \nabla u_n-\nabla u.\chi _s] \,dx\,dt \\ &+\int_Q\big[ a(x,t,u_n,\nabla u_n)( \nabla u_n-\nabla u) +a_0(x,t,u_n,\nabla u_n)( u_n-u) \big] \,dx\,dt \\ &+\int_{Q\setminus Q^s}a(x,t,u_n,\nabla u_n)\nabla u\,dx\,dt. \end{aligned} The first term of the right-hand side tends to 0 since ( a_0(x,t,u_n,\nabla u_n))  is bounded in L_{\overline{M}}(Q) by (\ref{6.2}) and u_n\to u strongly in L_M(Q). The second term tends to \int_{Q\setminus Q^s}a(x,t,u_n,0)\nabla u\,dx\,dt since a(x,t,u_n,\nabla u_n.\chi _s) tends strongly in ( E_{\overline{M}}(Q)) ^N to a(x,t,u,\nabla u.\chi _s) and \nabla u_n\rightharpoonup \nabla u weakly in ( L_M(Q)) ^N for \sigma ( \Pi L_M,\Pi E_{\overline{M}}) . The third term satisfies (\ref{6.12}) while the fourth term tends to \int_{Q\setminus Q^s}h\nabla u\,dx\,dt since a(x,t,u_n,\nabla u_n)\rightharpoonup h weakly in ( L_{\overline{M}}(Q)) ^N for \break \sigma ( \Pi L_{\overline{M}},\Pi E_M)  and M satisfies the \triangle _2-condition. We deduce then that \begin{align*} 0\leq &\limsup _{n\to \infty }\int_{Q^s}[ a(x,t,u_n,\nabla u_n)-a(x,t,u_n,\nabla u)] [ \nabla u_n-\nabla u] \,dx\,dt \\ \leq &\int_{Q\setminus Q^s}[ h-a(x,t,u,0)] \nabla u\,dx\,dt\to 0\quad \text{ as }s\to \infty . \end{align*} and so, by (\ref{6.3}), we can construct as in \cite{2} a subsequence such that \nabla u_n\to \nabla u a.e. in Q. This implies that a(x,t,u_n,\nabla u_n)\to a(x,t,u,\nabla u) and that \break a_0(x,t,u_n,\nabla u_n)\to a_0(x,t,u,\nabla u) a.e. in Q. Lemma 4.4 of \cite{8} shows that h=a(x,t,u,\nabla u) and h_0=a_0(x,t,u,\nabla u) and (\ref{6.11}) follows. The remaining of the proof is exactly the same as in \cite{13}. \hfill\square \begin{corollary} \label{cor 6.2} The function u can be used as a testing function in (\ref{6.6}) i.e. $\frac{1}{2}\big[ \int_{\Omega }( u(t)) ^{2}dx] _{0}^{\tau }+\int_{Q_{\tau }}[ a(x,t,u,\nabla u).\nabla u+a_{0}(x,t,u,\nabla u)u\big] \,dx\,dt=\int_{Q_{\tau }}fu\,dx\,dt$ for all \tau \in [ 0,T] . \end{corollary} The proof of this corollary is exactly the same as in \cite{13}. \section{Strongly nonlinear parabolic problems\label{sec 7}} In this last section we shall state and prove an existence theorem for strongly nonlinear parabolic initial-boundary problems with a nonlinearity g(x,t,s,\xi ) having growth less than M(| \xi | ). This result generalizes Theorem 2.1 in Boccardo-Murat \cite{5}. The analogous elliptic one is proved in Benkirane-Elmahi \cite{2}. The notation is the same as in section \ref{sec 6}. Consider also assumptions (\ref{6.2})-(\ref{6.5}) to which we will annex a Carath\'eodory function g:\Omega \times [ 0,T] \times \mathbb{R} \times \mathbb{R}^N\to \mathbb{R}^N satisfying, for a.e. (x,t)\in \Omega \times [ 0,T]  and for all s\in \mathbb{R} and all \xi \in \mathbb{R}^N: \begin{gather} \label{7.1} \ g(x,t,s,\xi )s\geq 0 \\ \label{7.2} | g(x,t,s,\xi )| \leq b(| s| )( c'(x,t)+R( | \xi | ) ) \end{gather} where c'\in L^1(Q) and b:\mathbb{R}^{+}\to \mathbb{R}% ^{+}  and where R is a given N-function such that R\ll M. Consider the following nonlinear parabolic problem $$\label{7.3} \begin{gathered} \frac{\partial u}{\partial t}+A(u)+g(x,t,u,\nabla u)=f\quad \text{in }Q,\\ u(x,t)=0 \quad \text{on }\partial \Omega \times ( 0,T) , \\ u(x,0)=\psi (x)\quad \text{in }\Omega . \end{gathered}$$ We shall prove the following existence theorem. \begin{theorem} \label{thm7.1} Assume that (\ref{6.1})-(\ref{6.5}), (\ref{7.1}) and (\ref{7.2}) hold. Then, there exists at least one distributional solution of (\ref{7.3}). \end{theorem} \paragraph{Proof} It is easily adapted from the proof of theorem 3.2 in \cite{2} Consider first g_n(x,t,s,\xi )=\frac{g(x,t,s,\xi )}{1+\frac 1ng(x,t,s,\xi)}  and put $A_n(u)=A(u)+g_n(x,t,u,\nabla u)$, we see that $A_n$ satisfies conditions (\ref{6.2})-(\ref{6.4}) so that, by Theorem \ref{thm6.1}, there exists at least one solution $u_n\in W_0^{1,x}L_M(Q)$ of the approximate problem $$\label{7.4} \begin{gathered} \frac{\partial u_n}{\partial t}+A(u_n)+g_n(x,t,u_n,\nabla u_n)=f \quad \text{in }Q \\ u_n(x,t)=0\quad \text{on }\partial \Omega \times ] 0,T[ \\ u_n(x,0)=\psi (x)\quad \text{in }\Omega \end{gathered}$$ and, by Corollary \ref{cor 6.2}, we can use $u_n$ as testing function in (% \ref{7.4}). This gives $\int_Q[ a(x,t,u_n,\nabla u_n).\nabla u_n+a_0(x,t,u_n,\nabla u_n).u_n] \,dx\,dt\leq \langle f,u_n\rangle$ and thus $( u_n)$ is a bounded sequence in $W_0^{1,x}L_M(Q)$. Passing to a subsequence if necessary, we assume that $$\label{7.5} \ u_n\rightharpoonup u\quad \text{weakly in }W_0^{1,x}L_M(Q) \text{ for }\sigma ( \Pi L_M,\Pi E_{\overline{M}})$$ for some $u\in W_0^{1,x}L_M(Q)$. Going back to (\ref{7.4}), we have $\int_Qg_n(x,t,u_n,\nabla u_n)u_n\,dx\,dt\leq C.$ We shall prove that $g_n(x,t,u_n,\nabla u_n)$ are uniformly equi-integrable on $Q$. Fix $m>0$. For each measurable subset $E\subset Q$, we have \begin{align*} &\int_E| g_n(x,t,u_n,\nabla u_n)| \\ &\leq \int_{E\cap \{| u_n| \leq m\} }| g_n(x,t,u_n,\nabla u_n)| +\int_{E\cap \{ | u_n| >m\} }| g_n(x,t,u_n,\nabla u_n)| \\ &\leq b(m)\int_E[ c'(x,t)+R(| \nabla u_n| )] \,dx\,dt+\frac 1m\int_{E\cap \{ | u_n| >m\} }|g_n(x,t,u_n,\nabla u_n)| \,dx\,dt \\ &\leq b(m)\int_E[ c'(x,t)+R(| \nabla u_n| )] \,dx\,dt+\frac 1m\int_Qu_ng_n(x,t,u_n,\nabla u_n)\,dx\,dt \\ &\leq b(m)\int_Ec'(x,t)\,dx\,dt+b(m)\int_ER(\frac{| \nabla u_n| }{\lambda '})\,dx\,dt+\frac Cm \end{align*} Let $\varepsilon >0$, there is $m>0\;$such that$\;\frac Cm<\frac \varepsilon 3$. Furthermore, since $c''\in L^1( Q)$ there exists $\delta _1>0$ such that $b(m)\int_Ec^{\prime \prime }(x,t)\,dx\,dt<\frac \varepsilon 3$. On the other hand, let $\mu >0$ such that $\| \nabla u_n\|_{M,Q}\leq \mu ,\forall n$. Since $R\ll M$, there exists a constant $K_\varepsilon >0$ depending on $\varepsilon$ such that $b(m)R(s)\leq M(\frac \varepsilon 6\frac s\mu )+K_\varepsilon$ for all $s\geq 0$. Without loss of generality, we can assume that $\varepsilon <1$. By convexity we deduce that $b(m)R(s)\leq \frac \varepsilon 6M(\frac s\mu )+K_\varepsilon$ for all $s\geq 0$. Hence \begin{align*} b(m)\int_ER( \frac{| \nabla u_n| }{\lambda '})\,dx\,dt &\leq \frac \varepsilon 6\int_EM(\frac{| \nabla u_n| }\mu )\,dx\,dt+K_\varepsilon | E| \\ &\leq \frac \varepsilon 6\int_QM( \frac{| \nabla u_n| }\mu )\,dx\,dt+K_\varepsilon | E| \\ &\leq \frac \varepsilon 6+K_\varepsilon | E| . \end{align*} When $| E| \leq \varepsilon /(6K_\varepsilon)$, we have $b(m)\int_ER(\frac{| \nabla u_n| }{\lambda '})\,dx\,dt\leq \frac \varepsilon 3,\quad \forall n.$ Consequently, if $| E| <\delta =\inf ( \delta _1,\frac \varepsilon {6K_\varepsilon })$ one has $\int_E| g_n(x,t,u_n,\nabla u_n)| \,dx\,dt\leq \varepsilon ,\quad \forall n,$ this shows that the $g_n(x,t,u_n,\nabla u_n)$ are uniformly equi-integrable on $Q$. By Dunford-Pettis's theorem, there exists $h\in L^1(Q)$ such that $$\label{7.6} g_n(x,t,u_n,\nabla u_n)\rightharpoonup h\quad \text{weakly in }L^1(Q).$$ Applying then Theorem \ref{thm5.1}, we have for a subsequence, still denoted by $u_n$, $$\label{7.7} u_n\to u,\nabla u_n\to \nabla u\text{ a.e. in }Q\text{ and } u_n\to u\text{ strongly in }W_0^{1,x}L_M^{\rm loc}(Q).$$ We deduce that $a(x,t,u_n,\nabla u_n)\rightharpoonup a(x,t,u,\nabla u)$ weakly in $( L_{\overline{M}}(Q)) ^N$ for \break $\sigma ( \Pi L_{\overline{M},}\Pi L_M)$ and since $\frac{\partial u_n}{\partial t}\to \frac{\partial u}{\partial t}$ in $\mathcal{D}'(Q)$ then passing to the limit in (\ref{7.4}) as $n\to +\infty$, we obtain $\frac{\partial u}{\partial t}+A(u)+g(x,t,u,\nabla u)=f\quad \text{in }\mathcal{D}'(Q).$ This completes the proof of Theorem \ref{thm7.1}. \begin{thebibliography}{99} \frenchspacing \bibitem{1} \textsc{R. Adams}, \textit{Sobolev spaces}, Ac. Press, New York, 1975. \bibitem{2} \textsc{A. 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Appl. 146 (1987), 65-96. \end{thebibliography} \noindent\textsc{Abdelhak Elmahi}\\ Department de Mathematiques, C.P.R.\\ B.P. 49, F\`{e}s - Maroc\\ e-mail: elmahi\_abdelhak@yahoo.fr \end{document}