\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 $k>0$: \begin{equation} \label{2.1} M(2t)\leq k\,M(t)\quad \text{for all }t\geq 0, \end{equation} 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 $t_0\geq 0$: \begin{equation} \label{2.2} M(k_0tt')\leq M(t)M(t'),\quad \text{for all }t,t'\geq t_0. \end{equation} 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 \begin{equation} \label{2.3} T_k(s)=\begin{cases} s\quad &\text{if }| s| \leq k\\ k s/| s| &\text{if }| s| >k. \end{cases} \end{equation} 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 \begin{equation} \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} \end{equation} \begin{equation} \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.} \end{equation} \begin{equation} \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. \end{equation} 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 \begin{equation} \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} \end{equation} 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 \begin{equation} \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) \end{equation} 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 \begin{equation} \nabla T_{k}(u_{n})\to \nabla T_{k}(u)\quad \text{strongly in }(L_{M}^{\rm loc}(Q))^{N}. \label{27} \end{equation} \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 \begin{equation} \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. \end{equation} 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 \begin{equation} \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} \end{equation} 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 \begin{equation} \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 . \end{equation} \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{equation} \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} \end{equation} 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 \begin{equation} \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 \end{equation} 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 \begin{equation} \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} \end{equation} 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 $u_n$) \begin{equation} \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). \end{equation} Finally, the third term of the right-hand side of (\ref{31}) tends to \begin{equation} \label{35} \int_{Q\setminus Q_{(s)}}\varphi _Kh\nabla T_k(u)\,dx\,dt. \end{equation} We have, then, proved that \begin{equation} \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} \end{equation} 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 $| Q\setminus Q_{(s)}| \to 0$, \begin{equation} \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 \end{equation} 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 \begin{equation} \label{38} \nabla u_n\to \nabla u\quad \quad \text{a.e. in } Q. \end{equation} \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 \begin{equation} \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) . \end{equation} 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 \begin{equation} \label{6.1} \ A(u)=-\mathop{\rm div} a(x,t,u,\nabla u)+a_0(x,t,u,\nabla u) \end{equation} 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 \begin{equation} \label{6.5} f\in W^{-1,x}E_{\overline{M}}( Q) \end{equation} We shall use notations of section \ref{sec 4}. Consider, then, the parabolic initial-boundary value problem \begin{equation} \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} \end{equation} 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{equation} \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} \end{equation} 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 $u_n$): \begin{equation} \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} \end{equation} 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 \begin{equation} \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} \end{equation} Hence, for a subsequence \begin{equation} \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} \end{equation} 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 \begin{equation} \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 \end{equation} 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 \begin{equation} \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. \end{equation} 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 \begin{equation} \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} \end{equation} 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 \begin{equation} \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} \end{equation} 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 \begin{equation} \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} \end{equation} 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 \begin{equation} \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}}) \end{equation} 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 \begin{equation} \label{7.6} g_n(x,t,u_n,\nabla u_n)\rightharpoonup h\quad \text{weakly in }L^1(Q). \end{equation} Applying then Theorem \ref{thm5.1}, we have for a subsequence, still denoted by $u_n$, \begin{equation} \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). \end{equation} 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. Benkirane and A. Elmahi}, \textit{Almost everywhere convergence of the gradients to elliptic equations in Orlicz spaces and application, }Nonlinear Anal., T. M. A., 28 n$^{\circ }$ 11 (1997), pp. 1769-1784. \bibitem{3} \textsc{A. Benkirane and A. Elmahi}, \textit{An existence theorem for a strongly nonlinear elliptic problem in Orlicz spaces, }% Nonlinear Analysis T.M.A., (to appear). \bibitem{4} \textsc{A. Bensoussan, L. Boccardo and F. Murat}, \textit{On a nonlinear partial differential equation having natural growth terms and unbounded solution}, Ann. Inst. Henri Poincar\'{e}, 5 n$^{\circ }$ 4 (1988), pp. 347-364. \bibitem{5} \textsc{L. Boccardo and F. Murat}, \textit{Strongly nonlinear Cauchy problems with gradient dependent lower order nonlinearity}, Pitman Research Notes in Mathematics, 208 (1989), pp. 247-254. \bibitem{6} \textsc{L. Boccardo and F. Murat}, \textit{Almost everywhere convergence of the gradients}, Nonlinear analysis, T.M.A., 19 (1992), n$^{\circ }$\thinspace 6, pp. 581-597. \bibitem{7} \textsc{T. Donaldson}, \textit{Inhomogeneous Orlicz-Sobolev spaces and nonlinear para\-bolic initial-boundary value problems}, J. Diff. Equations, 16 (1974), 201-256. \bibitem{Cpt} \textsc{A. Elmahi},\textit{\ Compactness results in inhomogeneous Orlicz-Sobolev spaces,} Lecture notes in Pure and Applied Mathematics, Marcel Dekker, Inc, 229 (2002), 207-221. \bibitem{8} \textsc{J.-P. Gossez}, \textit{Nonlinear elliptic boundary value problems for equations with rapidly (or slowly) increasing coefficients% }, Trans. Amer. Math. Soc. 190 (1974), pp.163-205. \bibitem{9} \textsc{J.-P. Gossez}, \textit{Some approximation properties in Orlicz-Sobolev spaces},\ Studia Math., 74 (1982), pp.17-24. \bibitem{10} \textsc{M. Krasnosel'skii and Ya. Rutickii}, \textit{convex functions and Orlicz spaces}, Noordhoff Groningen, 1969. \bibitem{11} \textsc{A. Kufner}, O. John and S. Fuc\'{\i }k, \textit{% Function spaces}, Academia, Prague, 1977. \bibitem{12} \textsc{R. Landes}, \textit{On the existence of weak solutions for quasilinear parabolic initial-boundary value problems}, Proc. Roy. Soc. Edinburgh sect. A. 89 (1981), 217-137. \bibitem{13} \textsc{R. Landes and V. Mustonen}, \textit{A strongly nonlinear parabolic initial-boundary value problem}, Ark. f. Mat. 25 (1987), 29-40. \bibitem{14} \textsc{Morrey C. B. Jr.}, \textit{Multiple integrals in the calculus of variation}, Springer-Verlag, Berlin-Heidelberg-NewYork, 1969. \bibitem{15} \textsc{Robert J.}, \textit{In\'{e}quations variationnelles paraboliques fortement non lin\'{e}aires}, J. Math. Pures Appl., 53 (1974), 299-321. \bibitem{16} \textsc{Simon J.}, \textit{Compact sets in the space }$% L^{p}(0,T;B)$, Ann. Mat. pura. 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}