\documentclass[reqno]{amsart} %\pdfoutput=1\relax\pdfpagewidth=8.26in\pdfpageheight=11.69in\pdfcompresslevel=9 \usepackage{hyperref} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2004(2004), No. 57, pp. 1--17.\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 2004 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2004/57\hfil Quasilinear evolution hemivariational inequalities] {Existence and comparison results for quasilinear evolution hemivariational inequalities} \author[Siegfried Carl, Vy K. Le, \& Dumitru Motreanu\hfil EJDE-2004/57\hfilneg] {Siegfried Carl, Vy K. Le, \& Dumitru Motreanu} % in alphabetical order \address{Siegfried Carl \hfill\break Fachbereich Mathematik und Informatik, Institut f\"ur Analysis \\ Martin-Luther - Universit\"at Halle-Wittenberg \\ 06099 Halle, Germany} \email{carl@mathematik.uni-halle.de} % Tel: +49 345 5524639 Fax: +49 345 5527003} \address{Vy K. Le \hfill\break Department of Mathematics and Statistics \\ University of Missouri - Rolla, Rolla, MO 65401, USA} \email{vy@umr.edu} \address{Dumitru Motreanu \hfill\break D\'epartement de Math\'ematiques, Universit\'e de Perpignan \\ 52 Avenue Paul Alduy, 66860 Perpignan, France} \email{motreanu@univ-perp.fr} \date{} \thanks{Submitted November 17, 2003. Published April 13, 2004.} \subjclass[2000]{35A15, 35K85, 49J40} \keywords{Evolution hemivariational inequality, quasilinear, subsolution, \hfill\break\indent supersolution, extremal solution, existence, comparison, compactness} \begin{abstract} We generalize the sub-supersolution method known for weak solutions of single and multivalued nonlinear parabolic problems to quasilinear evolution hemivariational inequalities. To this end we first introduce our basic notion of sub- and supersolutions on the basis of which we then prove existence, comparison, compactness and extremality results for the hemivariational inequalities under considerations. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{definition}[theorem]{Definition} \newtheorem{remark}[theorem]{Remark} \newtheorem{example}[theorem]{Example} \section{Introduction} \label{S1} Let $\Omega\subset \mathbb{R}^N$ be a bounded domain with Lipschitz boundary $\partial\Omega$, $Q=\Omega\times(0,\tau)$, and $\Gamma=\partial\Omega\times(0,\tau)$, with $\tau>0$. In this paper, we study the following quasilinear evolution hemivariational inequality: \begin{equation}\label{101} \begin{gathered} u\in W_0,\ u(\cdot,0)=0\quad \mbox{ in }\Omega\\ \langle \frac{\partial u }{\partial t}+Au-f, v-u\rangle +\int_{Q}j^o(u;v-u)\,dx\,dt\ge 0, \quad\forall\ v\in V_0, \end{gathered} \end{equation} where $ V_0=L^p(0,\tau;W^{1,p}_0(\Omega))$, $2\le p <\infty$, with the dual $V_0^*=L^q(0,\tau;W^{-1,q}(\Omega))$, $ W_0=\{w\in V_0 : \partial w/\partial t\in V_0^*\}$, and $\langle\cdot,\cdot\rangle$ denotes the duality pairing between $V_0^*$ and $V_0$. The real $q$ is the conjugate to $p$ satisfying $1/p+1/q=1$. By $j^o(s;r)$ we denote the generalized directional derivative of the locally Lipschitz function $j:\mathbb{R}\to \mathbb{R}$ at $s$ in the direction $r$ given by \begin{equation}\label{102} j^o(s;r)=\limsup_{y\to s,\; t\downarrow 0} \frac{j(y+t\,r)-j(y)}{t}, \end{equation} cf., e.g., \cite[Chap.\ 2]{Clarke}. The operator $A: V\to V_0^*$ is assumed to be a second order quasilinear differential operator in divergence form of Leray-Lions type \begin{equation}\label{103} Au(x,t)=-\sum_{i=1}^N \frac{\partial}{\partial x_i} a_i(x,t,u(x,t),\nabla u(x,t)), \end{equation} where $\nabla u=(\frac{\partial u}{\partial x_1},\dots,\frac{\partial u }{\partial x_N})$. Let $\partial j: {\mathbb R}\to 2^{\mathbb R}\setminus \{\emptyset\}$ denote Clarke's generalized gradient of $j$ defined by \begin{equation}\label{104} \partial j(s):=\{\zeta\in {\mathbb R} : j^o(s;r)\ge \zeta\,r,\; \forall r\in {\mathbb R}\}. \end{equation} A method of super-subsolutions has been established recently in \cite{CM1} for quasilinear parabolic differential inclusion problems in the form \begin{equation}\label{105} \frac{\partial u }{\partial t}+Au+\partial j(u)\ni f,\ \mbox{ in } \ Q,\quad u=0\ \mbox{ on }\ \Gamma,\ \ u(\cdot,0)=0\ \mbox{ in }\ \Omega. \end{equation} One can show that any solution of (\ref{105}) is a solution of the hemivariational inequality (\ref{101}). The reverse is true only if the function $j$ is regular in the sense of Clarke which means that the one-sided directional derivative and the generalized directional derivative coincide, cf. \cite[Chap.\ 2.3]{Clarke}. The main goal of this paper is to generalize the sub-supersolution method to the general case of evolution hemivariational inequalities (\ref{101}). This extension is by no means a straightforward generalization of the theory developed for the multivalued problems (\ref{105}) because of the intrinsic asymmetry of hemivariational inequalities compared with the symmetric structure of the multivalued equation (\ref{105}). In this paper we introduce our basic notion of sub- and supersolutions for inequalities in the form (\ref{101}) in a unified and coherent way which is inspired by recent papers on the sub-supersolution method for variational inequalities, see \cite{Le1, Le2}. The plan of the paper is as follows: In Section 2 we introduce the notion of sub-supersolution, and in Section 3 we provide some preliminary results used later. In Section 4 we prove an existence and comparison result in terms of sub- and supersolutions. Topological and extremality results of the solution set within the interval formed by sub- and supersolutions are given in Section 5. The theory developed in this paper can be extended to evolution hemivariational inequalities involving even more general quasilinear operators of Leray-Lions type and functions $j: Q\times\mathbb{R}\to\mathbb{R}$ depending, in addition, on the space-time variables $(x,t)$. Moreover, without loss of generality homogeneous initial and boundary data have been assumed. \section{Notation and hypotheses} \label{S2} Let $W^{1,p}(\Omega)$ denote the usual Sobolev space and $(W^{1,p}(\Omega))^*$ its dual space, and let us assume $2\le p<\infty$. Then $W^{1,p}(\Omega)\subset L^2(\Omega)\subset (W^{1,p}(\Omega))^*$ forms an evolution triple with all the embeddings being continuous, dense and compact, cf. \cite{ZII}. We set $V = L^p(0,\tau; W^{1,p}(\Omega))\,$, whose dual space is $V^* =L^q(0,\tau; (W^{1,p}(\Omega))^*)$, and define a function space $$ W =\{u\in V : u_t\in V^*\}\,, $$ where the derivative $u':=u_t=\partial u/\partial t$ is understood in the sense of vector-valued distributions, cf. \cite{ZII}, which is characterized by $$ \int_0^\tau u'(t)\phi(t)\,dt=-\int_0^\tau u(t)\phi'(t)\,dt,\quad\forall\ \phi\in C_0^\infty(0,\tau). $$ The space $W$ endowed with the graph norm $$ \|u\|_{W} =\|u\|_{V}+\|u_t\|_{ V^*}$$ is a Banach space which is separable and reflexive due to the separability and reflexivity of $ V$ and $ V^*$, respectively. Furthermore it is well known that the embedding $ W\subset C([0,\tau],\,L^2(\Omega))$ is continuous, cf. \cite{ZII}. Finally, because $W^{1,p}(\Omega)$ is compactly embedded in $ L^p(\Omega)$, we have by Aubin's lemma a compact embedding of $ W\subset L^p(Q)$\,, cf. \cite{ZII}. By $ W^{1,p}_0(\Omega)$ we denote the subspace of $W^{1,p}(\Omega)$ whose elements have generalized homogeneous boundary values. Let $W^{-1,q}(\Omega)$ denote the dual space of $ W^{1,p}_0(\Omega)$. Then obviously $ W^{1,p}_0(\Omega)\subset L^2(\Omega)\subset W^{-1,q}(\Omega)$ forms an evolution triple and all statements made above remain true also in this situation when setting $ V_0 = L^p(0,\tau; W^{1,p}_0(\Omega))\,, V^*_0 = L^{q}(0,\tau; W^{-1,q}(\Omega))$ and $W_0 =\{u\in V_0 : u_t\in V^*_0\}$. Let $\|\cdot\|_V$ and $\|\cdot\|_{V_0}$ be the usual norms defined on $V$ and $V_0$ (and similarly on $V^*$ and $V_0^*$): $$ \| u\|_V = \Big( \int_0^\tau \| u (t)\|_{W^{1,p}(\Omega)}^p \, dt \Big)^{1/p}, \quad \| u\|_{V_0} = \Big( \int_0^\tau \| u (t) \|_{W^{1,p}_0(\Omega)}^p \, dt \Big)^{1/p}. $$ We use the notation $\langle \cdot, \cdot \rangle$ for any of the dual pairings between $V$ and $V^*$, $V_0$ and $ V_0^*$, $W^{1,p}(\Omega)$ and $[W^{1,p}(\Omega)]^*$, and $W^{1,p}_0(\Omega)$ and $ W^{-1,q}(\Omega)$. For example, with $f\in V^*, u\in V$, $$ \langle f,u\rangle =\int_0^\tau \langle f(t), u(t) \rangle \, dt . $$ Let $L:=\partial/\partial t$ and its domain of definition $D(L)$ given by $$ D(L) = \left\{ u\in V_0 : u_t\in V_0^* \mbox{ and } u(0) =0 \right\}. $$ The linear operator $L: D(L)\subset V_0\to V_0^*$ can be shown to be closed, densely defined and maximal monotone, e.g., cf. \cite[Chap. 32]{ZII}. We assume $f\in V_0^*$ and impose the following hypotheses of Leray-Lions type on the coefficient functions $a_i$, $i=1,\dots ,N$, of the operator $A$: \begin{itemize} \item[(A1)] $a_i:Q\times \mathbb{R}\times\mathbb{R}^N\to \mathbb{R}$ are Carath\'eodory functions, i.e. $a_i(\cdot,\cdot,s,\xi):Q\to\mathbb{R}$ is measurable for all $(s,\xi)\in\mathbb{R}\times\mathbb{R}^N$ and $a_i(x,t,\cdot,\cdot):\mathbb{R}\times\mathbb{R}^N\to\mathbb{R}$ is continuous for a.e. $(x,t)\in Q$. In addition, one has $$ |a_i(x,t,s,\xi)|\leq k_0(x,t)+c_0\left(|s|^{p-1}+|\xi|^{p-1}\right) $$ for a.e. $(x,t)\in Q$ and for all $(s,\xi)\in\mathbb{R}\times\mathbb{R}^N$, for some constant $c_0>0$ and some function $k_0\in L^q(Q)$. \item[(A2)] $\displaystyle \sum_{i=1}^N (a_i(x,t,s,\xi)-a_i(x,t,s,\xi'))(\xi_i-\xi'_i)>0$ for a.e. $(x,t)\in Q$, for all $s\in \mathbb{R}$ and all $\xi, \xi' \in\mathbb{R}^N$ with $\xi\not=\xi'$. \item[(A3)] $\displaystyle \sum_{i=1}^N a_i(x,t,s,\xi)\xi_i\geq \nu|\xi|^p-k_1(x,t)$ for a.e. $(x,t)\in Q$ and for all $(s,\xi)\in \mathbb{R}\times\mathbb{R}^N$, for some constant $\nu>0$ and some function $k_1\in L^1(Q)$. \item[(A4)] $\displaystyle |a_i(x,t,s,\xi)-a_i(x,t,s',\xi)|\leq [k_2(x,t) +|s|^{p-1}+|s'|^{p-1}+|\xi|^{p-1}] \omega(|s-s'|)$ for a.e. $(x,t)\in Q$, for all $s, s'\in \mathbb{R}$ and all $\xi\in\mathbb{R}^N$, for some function $k_2\in L^q(Q)$ and a continuous function $\omega:[0,+\infty)\to[0,+\infty)$ satisfying $$ \int_{0^+}\frac{1}{\omega(r)}\,dr=+\infty. $$ \end{itemize} For example, we can take $\omega(r)=cr$, with $c>0$, in (A4). The operator $A: V\to V^*\subset V_0^*$ related with the quasilinear elliptic operator is defined as follows: \begin{equation}\label{200} \langle A(u),v\rangle = \sum_{i=1}^N \int_Q a_i(\cdot,\cdot,u, \nabla u) v_{x_i} \, dx dt , \end{equation} for all $v,\,u\in V$. Due to (A1) the operator $A: V\to V^*\subset V_0^*$ is continuous and bounded, and due to (A2) and (A3) the operator $A: D(L)\subset V_0\to V_0^*$ is pseudomonotone with respect to the graph norm topology of $D(L)$ (with respect to $D(L)$ for short), and coercive, see, e.g., \cite[Theorem E.3.2]{CARL-HEIKKILA}. Thus the evolution hemivariational inequality (\ref{101}) may be rewritten as: \begin{equation} \label{201} u\in D(L): \langle L u + A(u)-f , v-u \rangle +\int_{Q}j^o(u;v-u)\,dx dt\ge 0, \quad\forall v\in V_0. \end{equation} A partial ordering in $L^p(Q)$ is defined by $u\le w$ if and only if $w-u$ belongs to the positive cone $L_+^p(Q)$ of all nonnegative elements of $L^p(Q)$. This induces a corresponding partial ordering also in the subspace $ W$ of $L^p(Q)$, and if $ u,\,w\in W$ with $ u\le w$ then $$[u, w]=\{v\in W : u\le v\le w\}$$ denotes the order interval formed by $u$ and $w$. Further, for $u,v\in V$, and $U_1,U_2\subset V$, we use the notation $u\wedge v = \min\{u,v\}$, $u\vee v = \max\{u,v\}$, $U_1\ast U_2 = \{u \ast v : u\in U_1 , v\in U_2\}$, $u\ast U_1 = \{u\}\ast U_1$ with $\ast\in\{\wedge, \vee\}$. Our basic notion of sub-and supersolution of (\ref{101}) is defined as follows: \begin{definition}\label{D201} \rm A function $\underline u\in W$ is called a {\it subsolution} of (\ref{101}) if the following holds: \begin{itemize} \item[(i)] $\underline u(\cdot,0)\le 0$ in $\Omega$, $\underline u\le 0$ on $\Gamma$, \item[(ii)] $\langle \underline u_t+A\underline u-f, v-\underline u\rangle +\int_{Q}j^o(\underline u;v-\underline u)\,dx\,dt\ge 0, \quad\forall\ v\in \underline u\wedge V_0$. \end{itemize} \end{definition} \begin{definition}\label{D202} \rm $\bar u\in W$ is a {\it supersolution} of (\ref{101}) if the following holds: \begin{itemize} \item[(i)] $\bar u(\cdot,0)\ge 0$ in $\Omega$, $\bar u\ge 0$ on $\Gamma$, \item[(ii)] $\langle \bar u_t+ A\bar u-f, v-\bar u\rangle +\int_{Q}j^o(\bar u;v-\bar u)\,dx\,dt\ge 0, \quad\forall\ v\in \bar u\vee V_0$. \end{itemize} \end{definition} We assume the following hypothesis for $j$: \begin{itemize} \item[(H)] The function $j: \mathbb{R}\to \mathbb{R}$ is locally Lipschitz and its Clarke's generalized gradient $\partial j$ satisfies the following growth conditions: \begin{itemize} \item[(i)] there exists a constant $c_1\ge 0$ such that $$ \xi_1\le \xi_2+c_1(s_2-s_1)^{p-1} $$ for all $\xi_i\in\partial j(s_i),\ i=1,2$, and for all $s_1,\ s_2$ with $s_1\bar{u}(x,t),\\ 0 & \mbox{if } \underline u(x,t)\le s\leq\bar{u}(x,t),\\ -(\underline u(x,t) -s)^{p-1}& \mbox{if } s < \underline u(x,t). \end{cases} \end{equation} One readily verifies that $b$ is a Carath\'eodory function satisfying the growth condition \begin{equation}\label{403} |b(x,t,s)|\leq k_2(x,t)+c_3\,|s|^{p-1} \end{equation} for a.e. $(x,t)\in Q$, for all $s\in {\mathbb R}$, with some function $k_2\in L^q_+(Q)$ and a constant $c_3>0$. Moreover, one has the following estimate \begin{equation}\label{404} \int_{Q} b(x,t,u(x,t))\,u(x,t)\,dx dt \geq c_4\,\|u\|_{L^p(Q)}^p-c_5, \ \ \forall u\in L^p(Q), \end{equation} where $c_4$ and $c_5$ are some positive constants. In view of (\ref{403}) the Nemytskij operator $B:L^p(Q)\to L^q(Q)$ defined by $$ Bu(x,t)=b(x,t,u(x,t)) $$ is continuous and bounded, and thus due to the compact embedding $W_0\subset L^p(Q) $ it follows that $B: W_0\to L^q(Q)\subset V_0^*$ is completely continuous, which implies that $B: V_0\to V_0^*$ is compact with respect to $D(L)$. Let us consider the following auxiliary evolution hemivariational inequality: \begin{equation}\label{405} u\in D(L): \langle L u + A(u)+\lambda\,B(u)-f , v-u \rangle +\int_{Q}j^o(u;v-u)\,dx\,dt\ge 0, \quad\forall\ v\in V_0, \end{equation} where $\lambda$ is some positive constant to be specified later. The existence of solutions of (\ref{405}) will be proved by using Theorem \ref{T301}. To this end consider the multivalued operator $A+\lambda\, B+\partial (J|_{V_0}): V_0\to 2^{V_0^*}$, where $J$ is the locally Lipschitz functional defined in (\ref{300}) and $\partial (J|_{V_0})$ is the generalized Clarke's gradient of the restriction $J|_{V_0}$. By Corollary \ref{C301} and the property of $B$ we readily see that $A+\lambda\, B+\partial (J|_{V_0}): V_0\to 2^{V_0^*}$ is pseudomonotone with respect to $D(L)$ and bounded. In order to apply Theorem \ref{T301} we need to show the coercivity of $A+\lambda\, B+\partial (J|_{V_0}): V_0\to 2^{V_0^*}$. For any $v\in V_0\setminus\{0\}$ and any $w\in \partial (J|_{V_0})(v)$ we obtain by applying (A3), (H) (ii) and (\ref{404}) the estimate \begin{align*} &\frac{1}{\|v\|_{V_0}}\langle Av+\lambda B(v)+w,v\rangle\\ &=\frac{1}{\|v\|_{V_0}}\Big[\int_Q \sum_{i=1}^N a_i(\cdot,\cdot,v,\nabla v)\frac{\partial v}{\partial x_i}\,dx\,dt+\lambda \langle B(v),v\rangle+\int_Q wv\,dx\,dt\Big]\\ &\geq \frac{1}{\|v\|_{V_0}}\Big[\nu\int_Q |\nabla v|^p\,dx\,dt-\int_Q k_1\,dx\,dt+c_4\lambda \|v\|_{L^p(Q)}^p\\ &\quad -c_5\lambda-c_2\int_Q(1+|v|^{p-1})|v|\,dx\,dt\Big]\\ & \geq \frac{1}{\|v\|_{V_0}}\big[\nu\|v\|_{V_0}^p-C_0\big], \end{align*} for some constant $C_0>0$, by choosing the constant $\lambda$ sufficiently large such that $c_4\lambda>c_2$, which implies the coercivity. Thus we may apply Theorem \ref{T301} to ensure that $range\,(L+A+\lambda\, B+\partial (J|_{V_0}))=V_0^*$, which yields the existence of an $u\in D(L)$ such that $f\in Lu+A(u)+\lambda\, B(u)+\partial (J|_{V_0})(u)$, i.e., there exists an $\xi\in \partial (J|_{V_0})(u)$ such that \begin{equation}\label{406} u\in D(L):\quad Lu+A(u)-f+\lambda B(u)+\xi=0\quad\mbox{in } V_0^*. \end{equation} Since $V_0$ is dense in $L^p(Q)$ we get $\xi \in \partial J(u)$ and thus by the characterization (\ref{301}) of $\partial J(u)$ it follows that $\xi\in L^q(Q)$ and $\xi(x,t)\in \partial j(u(x,t))$, so that from (\ref{406}) we get \begin{equation}\label{407} \langle Lu+A(u)-f+\lambda B(u),\varphi\rangle +\int_Q\xi(x,t)\varphi(x,t)\,dx\,dt=0,\quad \forall\, \varphi\in V_0. \end{equation} By definition of Clarke's generalized gradient $\partial j$ it follows \begin{equation}\label{408} \int_{Q}\xi(x,t)\,\varphi(x,t)\,dx\,dt\le \int_{Q}j^o(u(x,t);\varphi(x,t))\,dx\,dt ,\quad\forall\, \varphi \in V_0. \end{equation} In view of (\ref{407}) and (\ref{408}), (\ref{405}) has a solution. Next we shall show that any solution $u$ of the auxiliary evolution hemivariational inequality (\ref{405}) satisfies $\underline u\le u\le \bar u$.\smallskip \noindent (b) Comparison: $u\in [\underline u,\bar u].$ \noindent Let $u$ be any solution of (\ref{405}). We are going to show that $\underline u_k\le u\le \bar u_j$ holds, where $k,\,j=1,2$, which implies the assertion. Let us first prove that $u\le \bar u_j$ is true. By Definition \ref{D202} $\bar u_j$ satisfies $\bar u_j(\cdot,0)\ge 0$ in $\Omega$, $\bar u_j\ge 0$ on $\Gamma$, and \begin{equation}\label{409} \langle \frac{\partial\bar u_j}{\partial t}+ A\bar u_j-f, v-\bar u_j\rangle +\int_{Q}j^o(\bar u_j;v-\bar u_j)\,dx\,dt\ge 0, \quad\forall v\in \bar u_j\vee V_0, \end{equation} which implies due to $v=\bar u_j\vee\varphi=\bar u_j+(\varphi-\bar u_j)^+$ with $\varphi\in V_0$ and $w^+=w\vee 0$ the following inequality \begin{equation}\label{410} \langle \frac{\partial\bar u_j}{\partial t}+ A\bar u_j-f, (\varphi-\bar u_j)^+\rangle +\int_{Q}j^o(\bar u_j;(\varphi-\bar u_j)^+)\,dx\,dt\ge 0, \quad\forall\, \varphi\in V_0. \end{equation} Let $M:=\{(\varphi-\bar u_j)^+ : \varphi\in V_0\}$, then one can show that the closure $\overline{M}^{V_0}=V_0\cap L^p_+(Q)$. Since $s\mapsto j^o(r;s)$ is continuous, we get from (\ref{410}) by using Fatou's lemma the inequality \begin{equation}\label{411} \langle \frac{\partial\bar u_j}{\partial t}+ A\bar u_j-f, \psi\rangle +\int_{Q}j^o(\bar u_j; \psi)\,dx\,dt\ge 0, \quad\forall\, \psi\in V_0\cap L^p_+(Q). \end{equation} Taking in (\ref{405}) the special test function $v=u-\psi$ and adding (\ref{405}) and (\ref{411}) we obtain: \begin{equation}\label{412} \langle\frac{\partial u}{\partial t}-\frac{\partial \bar u_j}{\partial t}+A(u)-A(\bar{u}_j)+\lambda\,B(u),\psi\rangle\le \int_{Q}\Bigl(j^o(\bar u_j; \psi)+j^o(u; -\psi)\Bigr)\,dx\,dt \end{equation} for all $\psi\in V_0\cap L^p_+(Q)$. Now we construct a special test function in (\ref{412}). By (A4), for any fixed $\varepsilon>0$ there exists $\delta(\varepsilon)\in (0,\varepsilon)$ such that $$ \int_{\delta(\varepsilon)}^\varepsilon \frac{1}{\omega(r)}\,dr=1. $$ We define the function $\theta_\varepsilon: \mathbb{R}\to\mathbb{R}_+$ by $$ \theta_\varepsilon(s)=\begin{cases} 0 & \mbox{if } s<\delta(\varepsilon)\\ \displaystyle\int_{\delta(\varepsilon)}^s \frac{1}{\omega(r)}\,dr & \mbox{if } \delta(\varepsilon)\leq s\leq \varepsilon\\ 1 & \mbox{if } s>\varepsilon. \end{cases} $$ We readily verify that, for each $\varepsilon > 0$, the function $\theta_\varepsilon$ is continuous, piecewise differentiable and the derivative is nonnegative and bounded. Therefore the function $\theta_\varepsilon$ is Lipschitz continuous and nondecreasing. In addition, it satisfies \begin{equation}\label{413} \theta_\varepsilon\to \chi_{\{s>0\}} \quad \mbox{as } \varepsilon\to 0, \end{equation} where $\chi_{\{s>0\}}$ is the characteristic function of the set $\{s>0\}$. Moreover, one has $$ \theta'_\varepsilon(s)=\begin{cases} 1/\omega(s) & \mbox{if } \delta(\varepsilon) < s < \varepsilon\\ 0 & \mbox{if } s\not\in [\delta(\varepsilon),\varepsilon]. \end{cases} $$ Taking in (\ref{412}) the test function $\theta_\varepsilon(u-\bar{u}_j)\in V_0\cap L_+^p(Q)$ we get \begin{equation}\label{414} \begin{aligned} &\langle\frac{\partial (u-\bar{u}_j)}{\partial t}, \theta_\varepsilon (u-\bar{u}_j)\rangle +\langle A(u)-A(\bar u_j),\theta_\varepsilon (u-\bar{u}_j)\rangle\\ &+\lambda\int_QB(u)\,\theta_\varepsilon (u-\bar{u}_j)\,dx\,dt \\ &\le \int_{Q}\Bigl(j^o(\bar u_j; \theta_\varepsilon (u-\bar{u}_j))+j^o(u; -\theta_\varepsilon (u-\bar{u}_j))\Bigr)\,dx\,dt. \end{aligned}\end{equation} Let $\Theta_\varepsilon$ be the primitive of the function $\theta_\varepsilon$ defined by $$ \Theta_\varepsilon(s)=\int_0^s \theta_\varepsilon(r)\,dr. $$ We obtain for the first term on the left-hand side of (\ref{414}) (cf., e.g., \cite{CR}) that \begin{equation}\label{415} \langle\frac{\partial (u-\bar{u}_j)}{\partial t},\theta_\varepsilon(u-\bar{u}_j)\rangle= \int_\Omega\Theta_\varepsilon(u-\bar{u}_j)(x,\tau)\,dx\geq 0. \end{equation} Using (A4) and (A2), the second term on the left-hand side of (\ref{414}) can be estimated as follows \begin{equation}\label{416} \begin{aligned} &\langle A(u)-A(\bar u_j),\theta_\varepsilon (u-\bar{u}_j)\rangle\\ &=\sum_{i=1}^N\int_Q(a_i(x,t,u,\nabla u)-a_i(x,t,\bar u_j,\nabla \bar u_j))\frac{\partial}{\partial x_i}\theta_\varepsilon (u-\bar{u}_j)\,dx\,dt\\ & \geq\sum_{i=1}^N\int_Q(a_i(x,t,u,\nabla u)-a_i(x,t,u,\nabla \bar u_j))\frac{\partial(u-\bar{u}_j)}{\partial x_i}\theta'_\varepsilon (u-\bar{u}_j)\,dx\,dt\\ &-N\int_Q(k_2+|u|^{p-1}+|\bar u_j|^{p-1}+|\nabla \bar u_j|^{p-1})\,\omega(|u-\bar{u}_j|)\theta'_\varepsilon (u-\bar{u}_j)|\nabla (u-\bar{u}_j)|\,dx\,dt\\ & \geq -N\int_{\{\delta(\varepsilon)0\}}\,dx\,dt. \end{equation} Again by applying Fatou's lemma and the continuity of $s\mapsto j^o(r;s)$ we obtain the following estimate for the right-hand side of (\ref{414}) \begin{equation}\label{418} \begin{aligned} &\limsup_{\varepsilon\to 0}\Big( \int_{Q}\Bigl(j^o(\bar u_j; \theta_\varepsilon (u-\bar{u}_j))+j^o(u; -\theta_\varepsilon (u-\bar{u}_j))\Bigr)\,dx\,dt\Big)\\ &\le \int_{Q}\Bigl(j^o(\bar u_j; \chi_{\{u-\bar{u}_j>0\}})+j^o(u; -\chi_{\{u-\bar{u}_j>0\}})\Bigr)\,dx\,dt. \end{aligned}\end{equation} Finally from (\ref{414})--(\ref{418}) one gets the inequality: \begin{equation}\label{419} \lambda\int_QB(u) \chi_{\{u-\bar{u}_j>0\}}\,dx\,dt \le \int_{Q}\Bigl(j^o(\bar u_j; \chi_{\{u-\bar{u}_j>0\}})+j^o(u; -\chi_{\{u-\bar{u}_j>0\}})\Bigr)\,dx\,dt. \end{equation} Note that $\bar u=\min\{\bar u_1,\bar u_2\}$, which by definition of the operator $B$ yields \begin{equation}\label{420} \lambda\int_QB(u) \chi_{\{u-\bar{u}_j>0\}}\,dx\,dt=\lambda \int_{\{u>\bar u_j\}}(u-\bar u)^{p-1} dx\,dt\ge \lambda \int_{\{u>\bar u_j\}}(u-\bar u_j)^{p-1}dx\,dt. \end{equation} The function $r\mapsto j^o(s;r)$ is finite and positively homogeneous, $\partial j(s)$ is a nonempty, convex and compact subset of $\mathbb{R}$, and one has \begin{equation}\label{421} j^o(s;r)=\max\{\xi\,r : \xi\in \partial j(s)\}. \end{equation} By using (H)(i), (\ref{421}) and the properties of $j^o$ and $\partial j$ we get for certain $\xi(x,t)\in\partial j(u(x,t))$ and $\bar \xi_j(x,t)\in\partial j(\bar u_j(x,t))$ with $\xi,\,\bar\xi_j \in L^q(Q)$ the following estimate: \begin{equation}\label{422} \begin{aligned} &\int_{Q}\Bigl(j^o(\bar u_j; \chi_{\{u-\bar{u}_j>0\}})+j^o(u; -\chi_{\{u-\bar{u}_j>0\}})\Bigr)\,dx\,dt\\ &= \int_{\{u>\bar u_j\}}\Bigl(j^o(\bar u_j; 1)+j^o(u; -1)\Bigr)\,dx\,dt\\ &=\int_{\{u>\bar u_j\}}(\bar \xi_j(x,t)-\xi(x,t))\,dx\,dt\le c_1 \int_{\{u>\bar u_j\}}(u(x,t)-\bar u_j(x,t))^{p-1}\,dx\,dt. \end{aligned}\end{equation} Thus (\ref{419}), (\ref{420}) and (\ref{422}) result in \begin{equation}\label{423} (\lambda-c_1)\int_{\{u>\bar u_j\}}(u-\bar u_j)^{p-1}\,dx\,dt\le 0. \end{equation} Selecting $\lambda$ large enough such that $\lambda > c_1$, then (\ref{423}) implies that meas$\,\{u>\bar u_j\}=0$, and thus $u\le \bar u_j$ in $Q$, where $j=1,2$, which shows that $u\le \bar u$. The proof of the inequality $\underline u\le u$ can be done analogously. \smallskip \noindent (c) Completion of the proof of the theorem. \noindent From steps (a) and (b) it follows that any solution $u$ of the auxiliary evolution hemivariational inequality (\ref{405}) with $\lambda > 0$ sufficiently large satisfies $u\in [\underline u,\bar u]$, which implies $B(u)=0$, and hence $u$ is a solution of the original evolution hemivariational inequality (\ref{101}) within the interval $[\underline u,\bar u]$. This completes the proof of Theorem \ref{T401}. \end{proof} The following corollaries are immediate consequences of Theorem \ref{T401}. \begin{corollary}\label{C401} Let $\underline w$ and $\bar w$ be any subsolution and supersolution, respectively of (\ref{101}) satisfying $\underline w\le \bar w$. Then there exist solutions of (\ref{101}) within the order interval $[\underline w,\bar w]$. \end{corollary} \begin{proof} Set $\underline w=\underline u_1=\underline u_2$ and $\bar w=\bar u_1=\bar u_2$ and apply Theorem \ref{T401}. \end{proof} Let $\mathcal{S}$ denote the set of all solutions of (\ref{101}) within the interval $[\underline w,\bar w]$ of an ordered pair of sub- and supersolutions. We introduce the following notion from set theory. \begin{definition}\label{D401} \rm Let $(\mathcal{P},\le )$ be a partially ordered set. A subset $\mathcal{C}$ of $\mathcal{P}$ is said to be {\it upward directed} if for each pair $x,y\in \mathcal{C}$ there is a $z\in \mathcal{C}$ such that $x\le z$ and $y\le z$, and $\mathcal{C}$ is {\it downward directed} if for each pair $x,y\in \mathcal{C}$ there is a $w\in \mathcal{C}$ such that $w\le x$ and $w\le y$. If $\mathcal{C}$ is both upward and downward directed it is called {\it directed}. \end{definition} \begin{corollary}\label{C402} The solution set $\mathcal{S}$ of (\ref{101}) is a directed set. \end{corollary} \begin{proof} Let $u_1,\,u_2\in \mathcal{S}$. Since any solution of (\ref{101}) is a subsolution and a supersolution as well, by Theorem \ref{T401} there exist solutions of (\ref{101}) within $[\max\{u_1,u_2\},\bar w]$ and also within $[\underline w,\min\{u_1,u_2\}]$, which proves the directedness. \end{proof} \section{Compactness and Extremality Results} \label{S5} In this section we show that the solution set $\mathcal{S}$ of (\ref{101}) within the interval of an ordered pair of sub-and supersolutions $[\underline w,\bar w]$ possesses the smallest and greatest elements with respect to the given partial ordering. The smallest and greatest element of $\mathcal{S}$ are called the {\it extremal solutions} of (\ref{101}) within $[\underline w,\bar w]$. We shall assume hypotheses (A1)--(A4) and (H) throughout this section. \begin{theorem}\label{T501} The solution set $\mathcal{S}$ is weakly sequentially compact in $W_0$ and compact in $V_0$. \end{theorem} \begin{proof} The solution set $\mathcal{S}\subset [\underline w,\bar w]$ is bounded in $L^p(Q)$. We next show that $\mathcal{S}$ is bounded in $W_0$. Let $u\in \mathcal{S}$ be given, and take as a special test function in (\ref{101}) $v=0$. This leads to \begin{equation}\label{the1} \langle u_t+Au, u\rangle\le \langle f,u\rangle +\int_{Q}j^o(u;-u)\,dx\,dt. \end{equation} Since $$\langle u_t, u\rangle=\frac12\|u(\cdot,\tau)\|^2_{L^2(\Omega)}\ge 0,$$ and $$\int_{Q}j^o(u;-u)\,dx\,dt\le c_2\int_Q(1+|u|^{p-1})\,|u|\, dx\,dt,$$ we get from (\ref{the1}) by means of (A3) and taking the $L^p(Q)$-boundedness of $\mathcal{S}$ into account the following uniform estimate \begin{equation}\label{the2} \|u\|_{V_0}\le C,\quad \forall\, u\in \mathcal{S}. \end{equation} Taking in (\ref{101}) the special test function $v=u-\varphi$, where $\varphi \in B=\{v\in V_0 : \|v\|_{V_0}\le 1\}$ we obtain \begin{equation}\label{the3} |\langle u_t,\varphi\rangle|\le |\langle f,\varphi\rangle|+|\langle Au,\varphi\rangle| +\big|\int_{Q}j^o(u;-\varphi)\,dx\,dt\big|. \end{equation} In view of (\ref{the2}), we obtain from (\ref{the3}) \begin{equation}\label{the4} |\langle u_t,\varphi\rangle|\le \mbox{\rm const},\quad \forall\,\varphi\in B, \end{equation} where the constant on the right-hand side of (\ref{the4}) does not depend on $u$, and thus from (\ref{the2}) and (\ref{the4}) we get \begin{equation}\label{the5} \|u\|_{W_0}\le C,\quad \forall\,u\in \mathcal{S}. \end{equation} Now let $(u_n)\subset\mathcal{S}$ be any sequence. Then by (\ref{the5}) there exists a weakly convergent subsequence $(u_k)$ with $$ u_k\rightharpoonup u\ \mbox{ in }\ W_0. $$ Since $u_k$ are solutions of (\ref{101}), we have \begin{equation}\label{the6} \langle \frac{\partial u_k }{\partial t}+Au_k-f, v-u_k\rangle +\int_{Q}j^o(u_k;v-u_k)\,dx\,dt\ge 0, \quad\forall\ v\in V_0. \end{equation} Taking as special test function the weak limit $u$ we get \begin{equation}\label{the7} \begin{aligned} \langle Au_k, u_k-u\rangle &\le \langle \frac{\partial u_k }{\partial t}-f, u-u_k\rangle +\int_{Q}j^o(u_k;u-u_k)\,dx\,dt \\ &\le \langle \frac{\partial u }{\partial t}-f, u-u_k\rangle +\int_{Q}j^o(u_k;u-u_k)\,dx\,dt. \end{aligned} \end{equation} The weak convergence of $(u_k)$ in $W_0$ implies $u_k\to u$ in $L^p(Q)$ due to the compact embedding $W_0\subset L^p(Q)$, and thus by applying (H) (ii) the right-hand side of (\ref{the7}) tends to zero as $k\to\infty$, which yields \begin{equation}\label{the8} \limsup_k\langle Au_k,u_k-u\rangle \le 0. \end{equation} Since $A$ is pseudomonotone with respect to $D(L)$, from (\ref{the8}) we get \begin{equation}\label{the9} Au_k\rightharpoonup Au\quad\mbox{and}\quad \langle Au_k,u_k\rangle \to \langle Au,u\rangle, \end{equation} and, moreover, because $A$ has the (S$_+)-$property with respect to $D(L)$ the strong convergence $u_k\to u$ in $V_0$ holds, see, e.g., \cite[Theorem E.3.2]{CARL-HEIKKILA}. The convergence properties of the subsequence $(u_k)$ obtained so far and the upper semicontinuity of $j^o: \mathbb{R}\times\mathbb{R}\to\mathbb{R}$ finally allow the passage to the limit in (\ref{the6}), which completes the proof. \end{proof} \begin{theorem}\label{T502} The solution set $\mathcal{S}$ possesses extremal elements. \end{theorem} \begin{proof} We prove the existence of the greatest solution of (\ref{101}) within $[\underline w,\bar w]$, i.e., the greatest element of $\mathcal{S}$. The proof of the smallest element can be done in a similar way. Since $W_0$ is separable, $\mathcal{S}\subset W_0$ is separable as well, and there exists a countable, dense subset $Z=\{z_n\,:\ n\in\mathbb{N}\}$ of $\mathcal{S}$. By Corollary \ref{C402} $\mathcal{S}$ is a directed set. This allows the construction of an increasing sequence $(u_n)\subset \mathcal{S}$ as follows. Let $u_1=z_1$. Select $u_{n+1}\in \mathcal{S}$ such that $$ \max\{z_n,u_n\}\leq u_{n+1}\leq \overline{w}. $$ The existence of $u_{n+1}$ is due to Corollary \ref{C402}. Since $(u_n)$ is increasing and both bounded and order-bounded, we deduce by applying Lebesgue's dominated convergence theorem that $u_n\to w:=\sup_nu_n$ strongly in $L^p(Q)$. By Theorem \ref{T501} we find a subsequence $(u_k)$ of $(u_n)$, and an element $u\in \mathcal{S}$ such that $u_k\rightharpoonup u$ in $W_0$, and $u_k\to u$ in $L^p(Q)$ and in $V_0$. Thus $u=w$ and each weakly convergent subsequence must have the same limit $w$, which implies that the entire increasing sequence $(u_n)$ satisfies: \begin{equation}\label{the10} u_n,\, w\in \mathcal{S}:\quad u_n\rightharpoonup w\ \mbox{ in } W_0,\ \ u_n\to w\ \mbox{ in }\ V_0. \end{equation} By construction, we see that $\max\{z_1,z_2,\dots,z_n\}\leq u_{n+1}\leq w$, for all $n$; thus $Z\subset [\underline{w},w]$. Since the interval $[\underline{w},w]$ is closed in $W_0$, we infer $$ \mathcal{S}\subset \overline{Z}\subset \overline{[\underline{w},w]}=[\underline{w},w], $$ which in conjunction with $w\in \mathcal{S}$ ensures that $w$ is the greatest element of $\mathcal{S}$. \end{proof} \begin{remark} \label{rmk5.1} \rm It should be noted that our main results of Section 4 and Section 5 remain valid also in case that the operator $A$ involves quasilinear first order terms, i.e., operators $A$ in the form \begin{equation} Au(x,t)=-\sum_{i=1}^N \frac{\partial}{\partial x_i} a_i(x,t,u(x,t),\nabla u(x,t))+ a_0(x,t,u(x,t),\nabla u(x,t)), \end{equation} where $a_0: Q\times\mathbb{R}\times\mathbb{R}^N\to \mathbb{R}$ satisfies the same regularity and growth condition as $a_i$, $i=1,\dots,N$. \end{remark} Next we provide examples to demonstrate the applicability of the theory developed in this paper. \begin{example} \label{ex5.1} \rm Let $c_P$ denote the best constant in Poincar\'e's inequality, i.e., $$ \int_Q|\nabla v|^p\,dx\,dt\ge c_P \int_Q|v|^p\,dx\,dt,\quad \forall v\in V_0. $$ Assume that (A1)--(A4) and (H) hold, and suppose in addition \begin{itemize} \item[(a)] $a_i(x,t,0,0)=0$ for a.e. $(x,t)\in Q$, $i=1,\dots ,N$. \item[(b)] $f\in L^q(Q)$ satisfying $f(x,t)\ge \max\{0,\min_{\zeta\in \partial j(0)}\zeta\}$ for a.e. $(x,t)\in Q$. \item[(c)] $k_1=0$ in assumption (A3). \item[(d)] $c_P\,\nu > c_2$, where $\nu$ and $c_2$ are the constants in (A3) and (H) (ii), respectively. \end{itemize} Under these assumptions, problem (\ref{101}) admits an extremal nonnegative solution. First, we check that $\underline u=0$ is a subsolution of problem (1.1). Indeed, using Definition \ref{D201} we have to check the inequality $$ \langle A0-f, v\rangle +\int_{Q}j^o(0;v)\,dx\,dt\ge 0, $$ for all $v\in 0\wedge V_0=\{\min\{0,w\}: w\in V_0\}=\{-w^-: w\in V_0\}$ (where $w^-=\max\{0,-w\}$). Taking into account assumption (a), this reduces to $$ \int_{Q}(j^o(0;-1)+f)w^-\,dx\,dt\ge 0, \quad\forall\ w\in V_0. $$ This is true due to assumption (b) because $$ f(x,t)\ge \min_{\zeta\in \partial j(0)}\zeta =-\max_{\zeta\in \partial j(0)}\zeta (-1)=-j^o(0;-1) \mbox{ for a.e. $(x,t)\in Q$}. $$ The claim that $\underline u=0$ is a subsolution of (\ref{101}) is verified. Consider now the initial boundary value problem \begin{equation}\label{e1} \begin{gathered} \frac{\partial u}{\partial t}- \sum_{i=1}^N \frac{\partial}{\partial x_i} a_i(x,t,u,\nabla u)-c_2(1+|u|^{p-1})=f \quad \mbox{in $Q$},\\ u(\cdot,0)=0 \quad \mbox{in $\Omega$}, \\ u=0 \quad \mbox{on $\Gamma$}, \end{gathered} \end{equation} which may be rewritten as the following abstract problem: \begin{equation}\label{e2} u\in D(L): Lu+A(u)+G(u)=f\quad\mbox{in }\ V_0^*, \end{equation} where $G: V_0\to V_0^*$ is defined by $$ \langle G(u),v\rangle=-c_2\int_Q(1+|u|^{p-1}) v\,dx\,dt. $$ One easily verifies that $A+G: V_0\to V_0^*$ is bounded, continuous and pseudomonotone with respect to $D(L)$, and due to condition (d) given above $A+G: V_0\to V_0^*$ is also coercive. Thus $L+A+G: D(L)\subset V_0\to V_0^*$ is surjective, which implies that (\ref{e2}) and hence (\ref{e1}) possesses solutions. We are going to show that any solution of (\ref{e1}) is nonnegative and a supersolution of (\ref{101}). Let $\bar u\in W_0$ be any solution of (\ref{e1}). Testing the equation by $-\bar u^-$ we find \begin{align*} &\int_{Q}\frac{\partial\bar u}{\partial t}(-\bar u^-)\,dx\,dt +\sum_{i=1}^N\int_{Q}a_i(x,t,\bar u,\nabla \bar u) \frac{\partial}{\partial x_i}(-\bar u^-)\,dx\,dt \\ &=\int_{Q}(c_2(1+|\bar u|^{p-1})+f)(-\bar u^-)\,dx\,dt. \end{align*} Since $$ \int_{Q}\frac{\partial\bar u}{\partial t}(-\bar u^-)\,dx\,dt =\frac{1}{2}\int_{\Omega}(\bar u^-)^2(x,\tau)\,dx \ge 0 $$ and using assumption (A3), it follows that \begin{align*} &\nu \int_{\{\bar u\le 0\}}|\nabla \bar u|^p\,dx\,dt +c_2\int_{\{\bar u\le 0\}}|\bar u|^p\,dx\,dt\\ &\le c_2 \int_{\{\bar u\le 0\}}\bar u\,dx\,dt +\int_{\{\bar u\le 0\}}f\bar u\,dx\,dt\leq 0. \end{align*} Here we used also the assumptions (b) and (c). Taking into account that $\nu >0$ we conclude that $\bar u\ge 0$. To obtain the desired conclusion concerning the existence of extremal nonnegative solutions of (\ref{101}), it is sufficient to show that $\bar u$ is a supersolution of problem (\ref{101}). Towards this, we see that every $v\in \bar u\vee V_0$ can be written as $v=\bar u+(w-\bar u)^+$ with $w\in V_0$. Then we have \begin{align*} &\langle \frac{\partial\bar u}{\partial t}+A\bar u -f,(w-\bar u)^+\rangle +\int_{Q} j^o(\bar u; (w-\bar u)^+) \,dx\,dt \\ &\ge \langle \frac{\partial\bar u}{\partial t}+A\bar u -f,(w-\bar u)^+\rangle -c_2\int_{Q} (1+|\bar u|^{p-1}) (w-\bar u)^+ \,dx\,dt=0, \quad \forall \, w\in V_0, \end{align*} where hypothesis (H) (ii) has been used as well as the fact that $\bar u$ solves the initial boundary value problem (\ref{e1}). Therefore, $\bar u\ge 0$ is a supersolution of problem (\ref{101}). Consequently, Theorem \ref{T502} yields extremal solutions in the order interval $[0,\bar u]$. \end{example} \begin{remark} \label{rmk5.2} \rm In case we have $p=2$ in Example 1 then condition (d) is not needed. \end{remark} \begin{example} \label{ex5.2} \rm Here we provide sufficient conditions for sub-supersolutions as constants. Let us assume that $a_i(x,t,u,0)=0$ for a.e.\ $(x,t)\in Q$, all $u\in \mathbb{R}$, $i=1,\dots ,N$. Then we have the following proposition. \begin{proposition}\label{P501} Let $D\in \mathbb{R}$. \begin{itemize} \item[(a)] If $D\le 0$ and $f(x,t) \ge - j^o (D; -1)$ for a.e.\ $(x,t)\in Q$, then $\underline{u}=D$ is a subsolution of (1.1). \item[(b)] If $D\ge 0$ and $f(x,t) \le j^o (D; 1)$ for a.e.\ $(x,t)\in Q$, then $\bar{u}=D$ is a supersolution of (1.1). \end{itemize} \end{proposition} \begin{proof} (a) We only need to check (ii) in Definition \ref{201}. Note that $\underline{u}_t = 0$ and $A\underline{u} =0$. Let $v\in D \wedge V_0$. Since $v - \underline{u} \le 0$ in $Q$, we have \begin{align*} & \langle \underline{u}_t + A\underline{u} -f , v-\underline{u} \rangle + \int_Q j^o (\underline{u}; v-\underline{u}) dx\,dt \\ &= \int_Q [ j^o (D; v- \underline{u}) - f (v-\underline{u})] dx\, dt \\ &= \int_Q [ j^o (D; -1 ) + f ] | v-\underline{u} | dx\, dt \ge 0 . \end{align*} (b) Similarly, in the second case, we have $v-D \ge 0$ for $v\in D\vee V_0$ and \begin{align*} & \langle \bar{u}_t + A\bar{u} -f , v-\bar{u} \rangle + \int_Q j^o (\bar{u}; v-\bar{u}) dx\,dt \\ &= \int_Q [ j^o (D; v- \bar{u}) - f (v-\bar{u})] dx\, dt \\ &= \int_Q [ j^o (D; 1 ) - f ] ( v-\bar{u} ) dx\, dt \ge 0 . \end{align*} \end{proof} As consequence, for example, if there exists $D >0$ such that \begin{equation}\label{e3} - j^o (0; -1 ) \le f(x,t) \le j^0 (D; 1) \;\mbox{ for a.e.\ $(x,t)\in Q$,} \end{equation} then (\ref{101}) has a nonnegative bounded solution (in the interval $[0,D]$). Similarly, if there is $D<0$ such that \begin{equation}\label{e4} - j^o (D; -1 ) \le f(x,t) \le j^0 (0; 1) \quad \mbox{for a.e.\ $(x,t)\in Q$,} \end{equation} then (\ref{101}) has a nonpositive bounded solution (in $[D,0]$). \end{example} It should be noted that, e.g., condition (\ref{e3}) may also formulated in terms of the generalized gradient as follows: \begin{equation}\label{e5} \min_{\zeta\in \partial j(0)}\zeta\leq f(x,t)\leq \max_{\zeta\in \partial j(D)}\zeta \ \ \mbox{ for a.e. $(x,t)\in Q$}. \end{equation} \begin{example} \label{5.3} \rm Finally, here we characterize a class of locally Lipschitz functions $j$ satisfying the hypothesis (H). Let $j_1:(-\infty,0)\to \mathbb{R}$ be a convex function and let $j_2:[0,+\infty)\to \mathbb{R}$ be a continuously differentiable function such that \begin{itemize} \item[(1)] $\lim_{s\to 0}j_1(s)=j_2(0)$; \item[(2)] For all $t<0$ and all $s\geq 0$, $$ -c_2(1+|t|^{p-1})\leq \min_{\xi\in \partial j_1(t)}\xi \leq \max_{\xi\in \partial j_1(t)}\xi \leq j_2'(s)\leq c_2(1+|s|^{p-1}) $$ \item[(3)] $$ \sup_{0\leq s_1