Nonlinear second order evolution inclusions with noncoercive viscosity term

TL;DR

This paper proves the existence of solutions for a second order nonlinear evolution inclusion with a noncoercive viscosity term using regularization and a priori bounds.

Contribution

It introduces a novel approach to handle noncoercive viscosity terms in second order evolution inclusions through regularization and limit processes.

Findings

Existence of solutions established for the nonlinear evolution inclusion.

Method handles nonmonotone, noncoercive viscosity terms effectively.

Provides a framework for similar nonlinear evolution problems.

Abstract

In this paper we deal with a second order nonlinear evolution inclusion, with a nonmonotone, noncoercive viscosity term. Using a parabolic regularization (approximation) of the problem and {\it a priori} bounds that permit passing to the limit, we prove that the problem has a solution.

Equations

$$\left\{\begin{array}[]{ll}u^{\prime\prime}(t)+A(t,u^{\prime}(t))+Bu(t)\in F(t,u(t),u^{\prime}(t))\ \mbox{for almost all}\ t\in T,\\ u(0)=u_{0},\ u^{\prime}(0)=u_{1}.\end{array}\right\}$$

$$X\hookrightarrow H=H^{\ast }\hookrightarrow X^{\ast }.$$

$$|\cdot |⩽{\hat{c}}_1||\cdot ||\text{mbox}and||\cdot |{|}_{\ast }⩽{\hat{c}}_2|\cdot |.$$

$$⟨\cdot ,\cdot ⟩{|}_{H\times X}=(\cdot ,\cdot ).$$

$$W_p(0,b)=\left\{u\in L^p(T,X):u^{\prime }\in L^{p^{\prime }}(T,X^{\ast })\right\}\left(\frac{1}{p}+\frac{1}{p^{\prime }}=1\right).$$

$$W_p(0,b)\subseteq A{C}^{1,p^{\prime }}(T,X^{\ast })=W^{1,p^{\prime }}((0,b),X^{\ast }).$$

$$||u|{|}_{W_p}={\left[||u|{|}_{L^p(T,X)}^p+||u^{\prime }|{|}_{L^{p^{\prime }}(T,X^{\ast })}^p\right]}^{\frac{1}{p}}\text{mbox}forallu\in W_p(0,b).$$

$$|u{|}_{W_p}=||u|{|}_{L^p(T,X)}+||u^{\prime }|{|}_{L^p(T,X^{\ast })}\text{mbox}forallu\in W_p(0,b).$$

$$W_p(0,b)\hookrightarrow C(T,H);$$

$$W_p(0,b)\hookrightarrow L^p(T,H)\text{mbox}andtheembeddingiscompact.$$

$$\mathrm{Gr}F=\{(\omega ,y)\in \Omega \times Y:y\in F(\omega )\}\in \Sigma \times B(Y),$$

$$\omega \mapsto d(y,F(\omega ))=\inf \{||y-v|{|}_Y:v\in F(\omega )\}$$

$$S_F^p=\{f\in L^p(\Omega ,Y):f(\omega )\in F(\omega )\mu \text{mbox}-almosteverywhere\}.$$

$${\chi }_A{f}_1+{\chi }_{A^c}{f}_2\in S_F^p.$$

$$w-\underset{n\to \infty }{\mathrm{limsup}}{C}_n=\{y\in Y:y=w-\underset{k\to \infty }{\lim }{y}_{n_k},y_{n_k}\in C_{n_k},n_1<n_2<\cdots <n_k<\cdots \}.$$

$$v_n\stackrel{w}{\to }v\text{mbox}inV,\underset{n\to \infty }{\mathrm{limsup}}⟨A(v_n),v_n-v⟩⩽0$$

$$⟨A(v),v-y⟩⩽\underset{n\to \infty }{\mathrm{liminf}}⟨A(v_n),v_n-y⟩\text{mbox}forally\in V.$$

$$P_{f(c)}(Z)=\{C\subseteq Z:C\text{mbox}isnonempty,closed(andconvex)\},$$

$$P_{(w)k(c)}(Z)=\{C\subseteq Z:C\text{mbox}isnonempty,(weakly-)compact(andconvex)\}.$$

$$\left\{\begin{array}[]{ll}u^{\prime\prime}(t)+A(t,u^{\prime}(t))+Bu(t)=f(t)\ \mbox{for almost all}\ t\in T,\\ u(0)=u_{0},u^{\prime}(0)=u_{1}.\end{array}\right\}$$

$$S(u_0,u_1)\subseteq C^1(T,H).$$

$$\mathcal{F}(x)=\{x^{\ast }\in X^{\ast }:⟨x^{\ast },x⟩=||x|{|}^2=||x^{\ast }|{|}_{\ast }^2\}.$$

$$K_r(y)=||y|{|}^{r-2}\mathcal{F}(y)\text{mbox}forally\in X.$$

$$\left\{\begin{array}[]{ll}u^{\prime\prime}(t)+A(t,u^{\prime}(t))+\epsilon K_{r}(u^{\prime}(t))+Bu(t)\in F(t,u(t),u^{\prime}(t))\ \mbox{for a.a.}\ t\in T,\\ u(0)=u_{0},\ u^{\prime}(0)=u_{1}.\end{array}\right\}$$

$$A_{ϵ}(t,y)=A(t,y)+ϵK_r(y)\text{mbox}forallt\in T,\text{mbox}andally\in X.$$

$$u_{ϵ}^{\prime }\in W_r(0,b).$$

$$||u|{|}_{C(T,X)},||u^{\prime }|{|}_{C(T,H)},{ϵ}^{\frac{1}{r}}||u^{\prime }|{|}_{L^r(T,X)},||u^{\prime \prime }|{|}_{L^2(T,X^{\ast })}⩽M_0.$$

$$u^{\prime \prime }(t)+A(t,u^{\prime }(t))+ϵK_r(u^{\prime }(t))+Bu(t)=f(t)\text{mbox}foralmostallt\in T.$$

$$⟨u^{\prime \prime }(t),u^{\prime }(t)⟩+⟨A(t,u^{\prime }(t)),u^{\prime }(t)⟩+ϵ⟨K_r(u^{\prime }(t)),u^{\prime }(t)⟩=(f(t),u^{\prime }(t))$$

$$\text{mbox}foralmostallt\in T\text{mbox}(see(\text{ref}eq2)).$$

$$⟨u^{\prime \prime }(t),u^{\prime }(t)⟩=\frac{1}{2}\frac{d}{dt}|u^{\prime }(t){|}^2\text{mbox}foralmostallt\in T.$$

$$⟨A(t,u^{\prime }(t)),u^{\prime }(t)⟩+ϵ⟨K_r(u^{\prime }(t)),u^{\prime }(t)⟩⩾ϵ||u^{\prime }(t)|{|}^r\text{mbox}foralmostallt\in T.$$

Full text

aa 2010 AMS Subject Classification: 35L90 (Primary); 35R70, 47H04, 47H05 (Secondary)

1. Introduction

Let $\displaystyle T=[0,b]$ and let $\displaystyle(X,H,X^{*})$ be an evolution triple of spaces, with the embedding of $\displaystyle X$ into $\displaystyle H$ being compact (see Section 2 for definitions).

In this paper, we study the following nonlinear evolution inclusion:

[TABLE]

In the past, such multi-valued problems were studied by Gasinski [3], Gasinski and Smolka [6, 7], Migórski et al. [11, 12, 13, 14], Ochal [15], Papageorgiou, Rădulescu and Repovš [16, 17], Papageorgiou and Yannakakis [18, 19]. The works of Gasinski [3], Gasinski and Smolka [6, 7] and Ochal [15], all deal with hemivariational inequalities, that is, $\displaystyle F(t,x,y)=\partial J(x)$ with $\displaystyle J(\cdot)$ being a locally Lipschitz functional and $\displaystyle\partial J(\cdot)$ denoting the Clarke subdifferential of $\displaystyle J(\cdot)$. In Papageorgiou and Yannakakis [18, 19], the multivalued term $\displaystyle F(t,x,y)$ is general (not necessarily of the subdifferential type) and depends also on the time derivative of the unknown function $\displaystyle u(\cdot)$. With the exception of Gasinski and Smolka [7], in all the other works the viscosity term $\displaystyle A(t,\cdot)$ is assumed to be coercive or zero. In the work of Gasinski and Smolka [7], the viscosity term is autonomous (that is, time independent) and $\displaystyle A:X\rightarrow X^{*}$ is linear and bounded.

In this work, the viscosity term $\displaystyle A:T\times X\rightarrow X^{*}$ is time dependent, noncoercive, nonlinear and nonmonotone in $\displaystyle x\in X$. In this way, we extend and improve the result of Gasinski and Smolka [7]. Our approach uses a kind of parabolic regularization of the inclusion, analogous to the one used by Lions [10, p. 346] in the context of semilinear hyperbolic equations.

2. Mathematical Background and Hypotheses

Let $\displaystyle V,Y$ be Banach spaces and assume that $\displaystyle V$ is embedded continuously and densely into $\displaystyle Y$ (denoted by $\displaystyle V\hookrightarrow Y$). Then we have the following properties:

(i) $\displaystyle Y^{*}$ is embedded continuously into $\displaystyle V^{*}$;

(ii) if $\displaystyle V$ is reflexive, then $\displaystyle Y^{*}\hookrightarrow V^{*}$.

The following notion is a useful tool in the theory of evolution equations.

Definition 1**.**

By an “evolution triple” (or “Gelfand triple”) we understand a triple of spaces $\displaystyle(X,H,X^{*})$ such that

• (a)

$\displaystyle X$* is a separable reflexive Banach space and $\displaystyle X^{*}$ is its topological dual;*

• (b)

$\displaystyle H$* is a separable Hilbert space identified with its dual $\displaystyle H^{*}$, that is, $\displaystyle H=H^{*}$ (pivot space);*

• (c)

$\displaystyle X\hookrightarrow H$.

Then from the initial remarks we have

[TABLE]

In what follows, we denote by $\displaystyle||\cdot||$ the norm of $\displaystyle X$, by $\displaystyle|\cdot|$ the norm of $\displaystyle H$ and by $\displaystyle||\cdot||_{*}$ the norm of $\displaystyle X^{*}$. Evidently we can find $\displaystyle\hat{c}_{1},\hat{c}_{2}>0$ such that

[TABLE]

By $\displaystyle(\cdot,\cdot)$ we denote the inner product of $\displaystyle H$ and by $\displaystyle\langle\cdot,\cdot\rangle$ the duality brackets for the pair $\displaystyle(X^{*},X)$. We have

[TABLE]

Let $\displaystyle 1<p<\infty$. The following space is important in the study of problem (1):

[TABLE]

Here $\displaystyle u^{\prime}$ is understood in the distributional sense (weak derivative). We know that $\displaystyle L^{p}(T,X)^{*}$ $\displaystyle=L^{p^{\prime}}(T,X^{*})$ (see, for example, Gasinski and Papageorgiou [4, p. 129]). Suppose that $\displaystyle u\in W_{p}(0,b)$. If we view $\displaystyle u(\cdot)$ as an $\displaystyle X^{*}$-valued function, then $\displaystyle u(\cdot)$ is absolutely continuous, hence differentiable almost everywhere and this derivative coincides with the distributional one. So, $\displaystyle u^{\prime}\in L^{p^{\prime}}(T,X^{*})$ and we can say

[TABLE]

The space $\displaystyle W_{p}(0,b)$ is equipped with the norm

[TABLE]

Evidently, another equivalent norm on $\displaystyle W_{p}(0,b)$ is

[TABLE]

With any of the above norms, $\displaystyle W_{p}(0,b)$ becomes a separable reflexive Banach space. We have that

[TABLE]

The elements of $\displaystyle W_{p}(0,b)$ satisfy an integration by parts formula which will be useful in our analysis.

Proposition 2**.**

If $\displaystyle u,v\in W_{p}(0,b)$ and $\displaystyle\xi(t)=(u(t),v(t))$ for all $\displaystyle t\in T$, then $\displaystyle\xi(\cdot)$ is absolutely continuous and $\displaystyle\frac{d\xi}{dt}(t)=\langle u^{\prime}(t),v(t)\rangle+\langle u(t),v^{\prime}(t)\rangle$ for almost all $\displaystyle t\in T$.

Now suppose that $\displaystyle(\Omega,\Sigma,\mu)$ is a finite measure space, $\displaystyle\Sigma$ is $\displaystyle\mu-complete$ and $\displaystyle Y$ is a separable Banach space. A multifunction (set-valued function) $\displaystyle F:\Omega\rightarrow 2^{Y}\backslash\{\emptyset\}$ is said to be “graph measurable”, if

[TABLE]

with $\displaystyle B(Y)$ being the Borel $\displaystyle\sigma$-field of $\displaystyle Y$.

If $\displaystyle F(\cdot)$ has closed values, then graph measurability is equivalent to saying that for every $\displaystyle y\in Y$ the $\displaystyle\mathbb{R}_{+}$-valued function

[TABLE]

is $\displaystyle\Sigma$-measurable.

Given a graph measurable multifunction $\displaystyle F:\Omega\rightarrow 2^{Y}\backslash\{\emptyset\}$, the Yankov-von Neumann-Aumann selection theorem (see Hu and Papageorgiou [8, p. 158]) implies that $\displaystyle F(\cdot)$ admits a measurable selection, i.e. that there exists $\displaystyle f:\Omega\rightarrow Y$ a $\displaystyle\Sigma$-measurable function such that $\displaystyle f(\omega)\in F(\omega)$ $\displaystyle\mu$-almost everywhere. In fact, we can find an entire sequence $\displaystyle\{f_{n}\}_{n\geqslant 1}$ of measurable selections such that $\displaystyle F(\omega)\subseteq\overline{\{f_{n}(\omega)\}}_{n\geqslant 1}$ $\displaystyle\mu$-almost everywhere.

For $\displaystyle 1\leqslant p\leqslant\infty$, we define

[TABLE]

It is easy to see that $\displaystyle S^{p}_{F}\neq\emptyset$ if and only if $\displaystyle\omega\mapsto\inf\{||v||_{Y}:v\in F(\omega)\}$ belongs to $\displaystyle L^{p}(\Omega)$. This set is “decomposable” in the sense that if $\displaystyle(A,f_{1},f_{2})\in\Sigma\times S^{p}_{F}\times S^{p}_{F}$, then

[TABLE]

Finally, for a sequence $\displaystyle\{C_{n}\}_{n\geqslant 1}$ of nonempty subsets of $\displaystyle Y$, we define

[TABLE]

For more details on the notions discussed in this section, we refer to Gasinski and Papageorgiou [4], Roubiček [20], Zeidler [21] (for evolution triples and related notations) and Hu and Papageorgiou [8] (for measurable multifunctions).

Let $\displaystyle V$ be a reflexive Banach space and $\displaystyle A:V\rightarrow V^{*}$ a map. We say that $\displaystyle A$ is “pseudomonotone”, if $\displaystyle A$ is continuous from every finite dimensional subspace of $\displaystyle V$ into $\displaystyle V^{*}_{w}$ (= the dual $\displaystyle V^{*}$ equipped with the weak topology) and if

[TABLE]

then

[TABLE]

An everywhere defined maximal monotone operator is pseudomonotone. If $\displaystyle V$ is finite dimensional, then every continuous map $\displaystyle A:V\rightarrow V^{*}$ is pseudomonotone.

In what follows, for any Banach space $\displaystyle Z$, we will use the following notations:

[TABLE]

The hypotheses on the data of problem (1) are the following:

$\displaystyle H(A):$ $\displaystyle A:T\times T\rightarrow X^{*}$ is a map such that

• (i)

for all $\displaystyle y\in X,t\mapsto A(t,y)$ is measurable;

• (ii)

for almost all $\displaystyle t\in T$, the map $\displaystyle y\mapsto A(t,y)$ is pseudomonotone;

• (iii)

$\displaystyle||A(t,y)||_{*}\leqslant a_{1}(t)+c_{1}||y||^{p-1}$ for almost all $\displaystyle t\in T$ and all $\displaystyle y\in X$, with $\displaystyle a_{1}\in L^{p^{\prime}}(T)$, $\displaystyle c_{1}>0$, $\displaystyle 2\leqslant p<\infty$;

• (iv)

$\displaystyle\langle A(t,y),y\rangle\geqslant 0$ for almost all $\displaystyle t\in T$ and all $\displaystyle y\in X$.

$\displaystyle H(B):$ $\displaystyle B\in\mathscr{L}(X,X^{*})$, $\displaystyle\langle Bx,y\rangle=\langle x,By\rangle$ for all $\displaystyle x,y\in X$ and $\displaystyle\langle Bx,x\rangle\geqslant c_{0}||x||^{2}$ for all $\displaystyle x\in X$ and some $\displaystyle c_{0}>0$.

$\displaystyle H(F):$ $\displaystyle F:T\times H\times H\rightarrow P_{f_{c}}(H)$ is a multifunction such that

• (i)

for all $\displaystyle x,y\in H$, $\displaystyle t\mapsto F(t,x,y)$ is graph measurable;

• (ii)

for almost all $\displaystyle t\in T$, the graph $\displaystyle{\rm Gr}\,F(t,\cdot,\cdot)$ is sequentially closed in $\displaystyle H\times H_{w}\times H_{w}$ (here $\displaystyle H_{w}$ denotes the Hilbert space $\displaystyle H$ furnished with the weak topology);

• (iii)

$\displaystyle|F(t,x,y)|=\sup\{|h|:h\in F(t,x,y)\}\leqslant a_{2}(t)(1+|x|+|y|)$ for almost all $\displaystyle t\in T$ and all $\displaystyle x,y\in H$ with $\displaystyle a_{2}\in L^{2}(T)_{+}$.

Definition 3**.**

We say that $\displaystyle u\in C(T,X)$ is a “solution” of problem (1) with $\displaystyle u_{0}\in X,\ u_{1}\in H$, if

• •

$\displaystyle u^{\prime}\in W_{p}(0,b)$* and*

• •

there exists $\displaystyle f\in S^{2}_{F(\cdot,u(\cdot),u^{\prime}(\cdot))}$ such that

[TABLE]

In what follows, we denote by $\displaystyle S(u_{0},u_{1})$ the set of solutions of problem (1). Recalling that $\displaystyle W_{p}(0,b)\hookrightarrow C(T,H)$ (see (3)), we have that

[TABLE]

By Troyanski’s renorming theorem (see Gasinski and Papageorgiou [4, p. 911]) we may assume without loss of generality that both $\displaystyle X$ and $\displaystyle X^{*}$ are locally uniformly convex. Let $\displaystyle\mathcal{F}:X\rightarrow X^{*}$ be the duality map of $\displaystyle X$ defined by

[TABLE]

We know that $\displaystyle\mathcal{F}(\cdot)$ is single-valued and a homeomorphism (see Gasinski and Papageorgiou [4, p. 316] and Zeidler [21, p. 861]).

For every $\displaystyle r\geqslant p$, let $\displaystyle K_{r}:X\rightarrow X^{*}$ be the map defined by

[TABLE]

3. Existence Theorem

Given $\displaystyle\epsilon>0$, we consider the following perturbation (parabolic regularization) of problem (1):

[TABLE]

Consider the map $\displaystyle A_{\epsilon}:T\times X\rightarrow X^{*}$ defined by

[TABLE]

This map has the following properties:

• (i)

for all $\displaystyle y\in X$, the map $\displaystyle t\mapsto A_{\epsilon}(t,y)$ is measurable;

• (ii)

for almost all $\displaystyle t\in T$, the map $\displaystyle y\mapsto A_{\epsilon}(t,y)$ is pseudomonotone;

• (iii)

$\displaystyle||A_{\epsilon}(t,y)||_{*}\leqslant\hat{a}_{1}(t)+\hat{c}_{1}||y||^{r-1}$ for almost all $\displaystyle t\in T$, all $\displaystyle y\in X$ and with $\displaystyle\hat{a}_{1}\in L^{p^{\prime}}(T)$, $\displaystyle\hat{c}_{1}>0$ (recall that $\displaystyle r\geqslant p$ and $\displaystyle\frac{1}{r}+\frac{1}{r^{\prime}}=1$);

• (iv)

$\displaystyle\langle A_{\epsilon}(t,y),y\rangle\geqslant\epsilon||y||^{r}$ for all $\displaystyle t\in T$, all $\displaystyle y\in X$.

So, in problem (1) the viscosity term $\displaystyle A_{\epsilon}(t,\cdot)$ is coercive. Therefore we can apply Theorem 1 of Papageorgiou and Yannakakis [18] and we obtain the following existence result for the approximate (regularized) problem (5).

Proposition 4**.**

If hypotheses $\displaystyle H(A),\ H(B),\ H(F)$ hold and $\displaystyle u_{0}\in X,u_{1}\in H$, then problem (5) admits a solution $\displaystyle u_{\epsilon}\in W^{1,r}((0,b),X)\cap C^{1}(T,H)$ with

[TABLE]

To produce a solution for the original problem (1), we have to pass to the limit as $\displaystyle\epsilon\rightarrow 0^{+}$. To do this, we need to have a priori bounds for the solutions $\displaystyle u_{\epsilon}(\cdot)$ which are independent of $\displaystyle\epsilon\in(0,1]$ and $\displaystyle r\geqslant p$.

Proposition 5**.**

If hypotheses $\displaystyle H(A),H(B),H(F)$ hold, $\displaystyle u_{0}\in X,u_{1}\in H$ and $\displaystyle u(\cdot)$ is a solution of (5), then there exists $\displaystyle M_{0}>0$ which is independent of $\displaystyle\epsilon\in(0,1]$ and $\displaystyle r\geqslant p$ for which we have

[TABLE]

Proof.

It follows from Proposition 4 that $\displaystyle u^{\prime}\in W_{r}(0,b)$ and that there exists $\displaystyle f\in S^{2}_{F(\cdot,u(\cdot),u^{\prime}(\cdot))}$ such that

[TABLE]

We act with $\displaystyle u^{\prime}(t)\in X$. Then

[TABLE]

We examine separately each summand on the left-hand side of (6). Recall that $\displaystyle u^{\prime}_{r}\in W_{r}(0,b)$. So from Proposition 2 (the integration by parts formula), we have

[TABLE]

Hypothesis $\displaystyle H(A)(iv)$ and the definition of the duality map, imply that

[TABLE]

By hypothesis $\displaystyle H(B)$, we have

[TABLE]

We return to (6) and use (7), (8), (9). We obtain

[TABLE]

Using hypothesis $\displaystyle H(F)(iii)$, we get

[TABLE]

Recall that $\displaystyle u\in W^{1,r}((0,b),X)$ (see Proposition 4). So, $\displaystyle u\in AC^{1,r}(T,H)$ and we can write

[TABLE]

We use (12) in (11) and obtain

[TABLE]

We use (3) in (10) and have

[TABLE]

Invoking Proposition 1.7.87 of Denkowski, Migórski and Papageorgiou [2, p. 128] we can find $\displaystyle M>0$ (independent of $\displaystyle\epsilon\in(0,1]$ and $\displaystyle r\geqslant p$) such that

[TABLE]

Using this bound in (14), we can find $\displaystyle M_{2}>0$ (independent of $\displaystyle\epsilon\in(0,1]$ and $\displaystyle r\geqslant p$) such that

[TABLE]

Finally, directly from (5), we see that there exists $\displaystyle M_{3}>0$ (independent of $\displaystyle\epsilon\in(0,1]$ and $\displaystyle r\geqslant p$) such that

[TABLE]

We set $\displaystyle M_{0}=\max\{M_{1},M_{2},M_{3}\}>0$ and get the desired bound. ∎

The bounds produced in Proposition 5 permit passing to the limit as $\displaystyle\epsilon\rightarrow 0^{+}$ to produce a solution for problem (1).

Theorem 6**.**

If hypotheses $\displaystyle H(A),H(B),H(F)$ hold and $\displaystyle u_{0}\in X,u_{1}\in H$, then $\displaystyle S(u_{0},u_{1})\neq\emptyset$.

Proof.

Let $\displaystyle\epsilon_{n}\rightarrow 0^{+}$ and let $\displaystyle u_{n}=u_{\epsilon_{n}}$ be solutions of the “regularized” problem (5) (see Proposition 4). Because of the bounds established in Proposition 5 and by passing to a suitable subsequence if necessary, we can say that

[TABLE]

Recall that $\displaystyle u_{n}\in AC^{1,r}(T,H)$ for all $\displaystyle n\in\mathbb{N}$ and so

[TABLE]

Since $\displaystyle u_{n}\in W_{r}(0,b)$ for all $\displaystyle n\in\mathbb{N}$, we have

[TABLE]

Let $\displaystyle a:L^{r}(T,X)\rightarrow L^{r^{\prime}}(T,X^{*})$ be the nonlinear map defined by

[TABLE]

Also, let $\displaystyle\hat{K}_{r}:L^{r}(T,X)\rightarrow L^{r^{\prime}}(T,X^{*})$ be defined by

[TABLE]

Both maps are continuous and monotone, hence maximal monotone (see Gasinski and Papageorgiou [4, Corollary 3.2.32, p. 320]).

Finally, let $\displaystyle\hat{B}\in\mathscr{L}(L^{r}(T,X),L^{r^{\prime}}(T,X^{*}))$ be defined by

[TABLE]

We have

[TABLE]

From (15) we have

[TABLE]

Also, we have

[TABLE]

From (15) and since $\displaystyle v=u^{\prime\prime}$, we have

[TABLE]

Finally, hypothesis $\displaystyle H(F)(iii)$ and Proposition 5 imply that

[TABLE]

By passing to a subsequence if necessary, we may assume that

[TABLE]

Invoking Proposition 3.9 of Hu and Papageorgiou [8, p. 694], we have

[TABLE]

From (15) we see that

[TABLE]

Recall that $\displaystyle W^{1,r^{\prime}}((0,b),X^{*})\hookrightarrow C(T,X^{*})$. So, it follows that

[TABLE]

On the other hand, by Proposition 5 we have

[TABLE]

So, by passing to a subsequence (a priori the subsequence depends on $\displaystyle t\in T$), we have

[TABLE]

Hence for the original sequence we have

[TABLE]

We know that $\displaystyle\{u_{n}\}_{n\geqslant 1}\subseteq W_{r}(0,b)$ is bounded (see Proposition 5) and recall that $\displaystyle W_{r}(0,b)\hookrightarrow L^{r}(T,H)$ compactly (see (4)). From this compact embedding and from (22), we obtain

[TABLE]

From (20), (22), (23) and hypothesis $\displaystyle H(F)(iii)$ we infer that

[TABLE]

In what follows, we denote by $\displaystyle((\cdot,\cdot))$ the duality brackets for the pair

[TABLE]

Acting with $\displaystyle u^{\prime}_{n}-u^{\prime}\in L^{r}(T,X)$ on (16), we have

[TABLE]

Note that

[TABLE]

Also we have

[TABLE]

Recall that

[TABLE]

Suppose that $\displaystyle r_{m}\rightarrow+\infty$, $\displaystyle r_{m}\geqslant p$ for all $\displaystyle m\in\mathbb{N}$. Then for every $\displaystyle n\in\mathbb{N}$, $\displaystyle\epsilon_{n}^{\frac{1}{r_{m}}}\rightarrow 1$ as $\displaystyle m\rightarrow\infty$. Invoking Problem 1.175 of Gasinski and Papageorgiou [5], we can find $\displaystyle\{m_{n}\}_{n\geqslant 1}$ with $\displaystyle m_{n}\rightarrow+\infty$ such that

[TABLE]

Therefore there exists $\displaystyle n_{0}\in\mathbb{N}$ such that

[TABLE]

On account of (15) and since $\displaystyle y=u^{\prime}$, we have

[TABLE]

Then from (26) and (27) it follows that

[TABLE]

In addition, we have

[TABLE]

By Proposition 5 and (27) it follows that

[TABLE]

Therefore we have

[TABLE]

If in (24) we pass to the limit as $\displaystyle n\rightarrow\infty$ and use (25), (28), (29), (30), then

[TABLE]

Invoking Theorem 2.35 of Hu and Papageorgiou [9, p. 41], we have

[TABLE]

In (24) we pass to the limit as $\displaystyle n\rightarrow\infty$ and use (15) (with $\displaystyle v=u^{\prime\prime}$) (27), (29), (31). We obtain

[TABLE]

The proof is now complete. ∎

3.1. An example

We illustrate the main abstract result of this paper with a hyperbolic boundary value problem. Let $\displaystyle\Omega\subseteq\mathbb{R}^{N}$ be a bounded domain. We consider the following boundary value problem

[TABLE]

with $\displaystyle u_{t}=\frac{\partial u}{\partial t}$, $\displaystyle 2\leqslant p\leqslant\infty$, $\displaystyle\gamma>0$.

The forcing term $\displaystyle f(t,z,\cdot)$ need not to be continuous. So, following Chang [1], to deal with (32), we replace it by a multivalued problem (partial differential inclusion), by filling in the gaps at the discontinuity points of $\displaystyle f(t,z,\cdot)$. So we define

[TABLE]

Then we replace (32) by the following partial differential inclusion

[TABLE]

Our hypotheses on the data of (33) are the following:

$\displaystyle H(a):$ $\displaystyle a\in L^{\infty}(T\times\Omega),a(t,z)\geqslant 0$ for almost all $\displaystyle(t,z)\in T\times\Omega$.

$\displaystyle H(\beta):$ $\displaystyle\beta\in L^{\infty}(\Omega)$, $\displaystyle\beta(z)\geqslant 0$ for almost all $\displaystyle z\in\Omega$.

$\displaystyle H(f):$ $\displaystyle f:T\times\Omega\times\mathbb{R}\rightarrow\mathbb{R}$ is a function such that

• (i)

$\displaystyle f_{l},\ f_{u}$ are superpositionally measurable (that is, for all $\displaystyle u:T\times\Omega\rightarrow\mathbb{R}$ measurable, the functions $\displaystyle(t,z)\mapsto f_{l}(t,z,u(t,z)),\ f_{u}(t,z,u(t,z))$ are both measurable);

• (ii)

there exists $\displaystyle a\in L^{2}(T\times\Omega)$ such that

[TABLE]

Let $\displaystyle X=W^{1,p}_{0}(\Omega)$, $\displaystyle H=L^{2}(\Omega)$ and $\displaystyle X^{*}=W^{-1,p^{\prime}}(\Omega)$. Then $\displaystyle(X,H,X^{*})$ is an evolution triple with $\displaystyle X\hookrightarrow H$ compactly (by the Sobolev embedding theorem).

Let $\displaystyle A:T\times X\rightarrow X^{*}$ be defined by

[TABLE]

Then $\displaystyle A(t,u)$ is measurable in $\displaystyle t\in T$, continuous and monotone in $\displaystyle u\in W^{1,p}_{0}(\Omega)$ (hence, maximal monotone) and $\displaystyle\langle A(t,u),u\rangle\geqslant 0$ for almost all $\displaystyle t\in T$, all $\displaystyle u\in W^{1,p}_{0}(\Omega)$.

Let $\displaystyle B\in\mathscr{L}(X,X^{*})$ be defined by

[TABLE]

Clearly, $\displaystyle B$ satisfies hypothesis $\displaystyle H(B)$.

Finally, let $\displaystyle G(t,z,x)=\left[f_{l}(t,z,x),f_{u}(t,z,x)\right]$ and set

[TABLE]

Hypothesis $\displaystyle H(f)$ implies that $\displaystyle F$ satisfies $\displaystyle H(F)$.

Using $\displaystyle A(t,u),Bu$ and $\displaystyle F(t,u,v)$ as defined above, we can rewrite problem (33) as the equivalent second order nonlinear evolution inclusion (1). Assuming that $\displaystyle u_{0}\in W^{1,p}_{0}(\Omega)$ and that $\displaystyle u_{1}\in L^{2}(\Omega)$, we can use Theorem 6 and infer that problem (30) has a solution $\displaystyle u\in C^{1}(T,L^{2}(\Omega))\cap C(T,W^{1,p}(\Omega))$ with $\displaystyle\frac{\partial u}{\partial t}\in L^{p}(\Omega,W^{1,p}_{0}(\Omega))$ and $\displaystyle\frac{\partial^{2}u}{\partial t}\in L^{p^{\prime}}(\Omega,W^{-1,p^{\prime}}(\Omega))$.

Note that if $\displaystyle a=0,f(t,z,x)=x$ and $\displaystyle\gamma=0$, then we have the Klein-Gordon equation. If $\displaystyle f(t,z,x)=f(x)=\eta\sin x$ with $\displaystyle\eta>0$, then we have the sine Gordon equation.

Acknowledgments. This research was supported in part by the Slovenian Research Agency grants P1-0292, J1-7025, J1-8131, and N1-0064. V.D. Rădulescu acknowledges the support through a grant of the Romanian Ministry of Research and Innovation, CNCS–UEFISCDI, project number PN-III-P4-ID-PCE-2016-0130, within PNCDI III.