Convergence of a Time Discretization for Nonlinear Second Order Inclusion

TL;DR

This paper proves the convergence of a time discretization scheme for a class of nonlinear second order inclusions, including applications to nonsmooth differential inclusions with Clarke subdifferentials.

Contribution

It establishes the existence of solutions for nonlinear second order inclusions using a novel time discretization approach and applies it to nonsmooth, nonconvex problems involving Clarke subdifferentials.

Findings

Weak convergence of approximate solutions to the exact solution

Applicability to nonsmooth, nonconvex differential inclusions

Demonstrated through two specific examples

Abstract

We study an abstract second order inclusion involving two nonlinear single-valued operators and a nonlinear multivalued term. Our goal is to establish the existence of solutions to the problem by applying numerical scheme based on time discretization. We show that the sequence of approximate solution converges weakly to a solution of the exact problem. We apply our abstract result to a dynamic, second order in time differential inclusion involving Clarke subdifferential of a locally Lipschitz, possibly nonconvex and nonsmooth potential. In two presented examples the Clarke subdifferential appears either in a source term or in a boundary term.

Equations

$$\{\begin{array}{l}{u}^{\prime \prime }(t)+A(t,u^{\prime }(t))+B(t,u(t))+{\gamma }^{\ast }M(\gamma u^{\prime }(t))\ni f(t)\\ u(0)=u_0,u^{\prime }(0)=v_0\end{array}$$

$$\displaystyle\frac{2}{\tau_{n+1}+\tau_{n}}\bigg{(}\frac{u^{n+1}-u^{n}}{\tau_{n+1}}-\frac{u^{n}-u^{n-1}}{\tau_{n}}\bigg{)}+A\bigg{(}t_{n},\frac{u^{n+1}-u^{n}}{\tau_{n+1}}\bigg{)}$$

$$\displaystyle+B(t_{n},u^{n})+\gamma^{*}M\bigg{(}\gamma\frac{u^{n+1}-u^{n}}{\tau_{n+1}}\bigg{)}\ni f^{n},\quad n=1,...,N-1,$$

$$\|x\|_{BV(I;X)}=\sup_{\pi\in\mathcal{F}}\bigg{\{}\sum_{\sigma_{i}\in\pi}\|x(b_{i})-x(a_{i})\|_{X}\bigg{\}}.$$

$$\|x\|^{q}_{BV^{q}(I;X)}=\sup_{\pi\in\mathcal{F}}\bigg{\{}\sum_{\sigma_{i}\in\pi}\|x(b_{i})-x(a_{i})\|_{X}^{q}\bigg{\}}$$

$$B{V}^q(I;X)=\{x:I\to X;\Vert x{\Vert }_{B{V}^q(I;X)}<\infty \}.$$

$$M^{p,q}(I;X,Z)=L^p(I;X)\cap B{V}^q(I;Z).$$

$$W\subseteq V\subseteq H\subseteq V^{\ast }\subseteq W^{\ast },$$

$$u^{\prime \prime }(t)+A(t,u^{\prime }(t))+B(t,u(t))+{\gamma }^{\ast }M(\gamma u^{\prime }(t))\ni f(t)\text{mbox}a.e.t\in (0,T),$$

$$u(0)=u_0,u^{\prime }(0)=v_0.$$

$$u^{\prime \prime }(t)+A(t,u^{\prime }(t))+B(t,u(t))+{\gamma }^{\ast }\eta (t)=f(t)\text{mbox}a.e.t\in (0,T),$$

$$\eta (t)\in M(\gamma u^{\prime }(t))\text{mbox}a.e.t\in (0,T).$$

$$⟨B_{0}v,v⟩⩾{\mu }_B\Vert v{\Vert }_V^2,\Vert B_{0}v\Vert ⩽{\beta }_B\Vert v{\Vert }_V.$$

$$\underset{n\to \infty }{\mathrm{limsup}}⟨\mathcal{A}{v}_n,v_n-v{⟩}_{{\mathcal{W}}^{\ast }\times \mathcal{W}}⩽0.$$

$$0=t_0<t_1\dots t_{n-1}<t_n^N=T,{\tau }_n=t_n-t_{n-1}\text{for}n=1,\dots ,N.$$

$${\tau }_{n+\frac{1}{2}}=\frac{{\tau }_n+{\tau }_{n+1}}{2},t_{n+\frac{1}{2}}=t_n+\frac{1}{2}{\tau }_{n+1}\text{for}n=1,\dots ,N-1,$$

$$r_{n+1}=\frac{{\tau }_{n+1}}{{\tau }_n}\text{for}n=1,\dots ,N-1,$$

$${\gamma }_n:=\max \left(0,\frac{1}{r_n}-\frac{1}{r_{n-1}}\right)\text{for}n=2,\dots ,N,$$

$${\tau }_{max}=\underset{n=1,\dots ,N}{\max }{\tau }_n,r_{max}:=\underset{n=2,\dots ,N}{\max }{r}_n,r_{min}:=\underset{n=2,\dots ,N}{\max }{r}_n,$$

$$c_{\gamma }:=\underset{n=3,\dots ,N}{\max }\frac{{\gamma }_n}{{\tau }_n},\sigma (\tau )=\frac{1}{2}\sum _{j=1}^{N-1}\frac{({\tau }_{j+1}-{\tau }_j{)}^2}{{\tau }_{j+1}+{\tau }_j}.$$

$$v^n=\frac{1}{{\tau }_{n+1}}(u^{n+1}-u^n),n=0,1,\dots ,N-1,$$

$$\frac{1}{{\tau }_{n+\frac{1}{2}}}(v^n-v^{n-1})+A(t_n,v^n)+B(t_n,u^n)+{\gamma }^{\ast }{\eta }^n=f^n,$$

$$n=1,2,\dots ,N-1,$$

$${\eta }^n\in M(\gamma v^n),$$

$$u^0=u_{\tau }^0,v^0=v_{\tau }^0.$$

$$Tv=\frac{1}{{\tau }_{n+\frac{1}{2}}}v+A(t_n,v)+{\gamma }^{\ast }M(\gamma v),\text{for}v\in W.$$

$$⟨z,v{⟩}_{W^{\ast }\times W}=\frac{1}{{\tau }_{n+\frac{1}{2}}}(v,v{)}_H+⟨A(t_n,v),v{⟩}_{W^{\ast }\times W}+⟨\eta ,\gamma v{⟩}_{U^{\ast }\times U}$$

$$u^n$$

$$\tau_{max}<\min\Bigg{\{}\frac{2\left(\mu_{A}-c_{M}\|\gamma\|_{\mathcal{L}(W,U)}^{p}\right)}{\beta_{B}\|i_{WV}\|_{\mathcal{L}(W,V)}},\frac{1}{2\beta}\Bigg{\}}.$$

$$\Vert u^{n+1}{\Vert }_V^2+|v^n{|}^2+\sum _{j=1}^n|v^j-v^{j-1}{|}^2+\sum _{j=1}^n{\tau }_{j+\frac{1}{2}}\Vert v^j{\Vert }_W^p+\sum _{j=1}^n{\tau }_{j+\frac{1}{2}}\Vert {\eta }^j{\Vert }_{U^{\ast }}^q$$

$$⩽$$

$$\sum_{j=1}^{n}\tau_{j+\frac{1}{2}}\bigg{\|}\frac{1}{\tau_{j+\frac{1}{2}}}(v^{j}-v^{j-1})\bigg{\|}_{W^{*}}^{q}\leqslant c.$$

$$\frac{1}{{\tau }_{j+\frac{1}{2}}}(v^n-v^{n-1},v^n)$$

$$⟨A(t_n,v^n),v^n{⟩}_{W^{\ast }\times W}$$

$$⟨B(t_n,u^n),v^n{⟩}_{W^{\ast }\times W}$$

Full text

Convergence of a Time Discretization for Nonlinear Second Order Inclusion††thanks: The research was

supported by the Marie Curie International Research Staff Exchange Scheme Fellowship within the 7th European Community Framework Programme under Grant Agreement No. 295118, the International Project co-financed by the Ministry of Science and Higher Education of Republic of Poland under grant no. W111/7.PR/2012 and the National Science Center of Poland under Maestro Advanced Project no. DEC-2012/06/A/ST1/00262.

Krzysztof Bartosz* 1*,   Leszek Gasiński* 1111Corresponding author email: Leszek.Gasinski@ii.uj.edu.pl,   Zhenhai Liu 2*   

and Paweł Szafraniec* 1*,

 

1 Jagiellonian University, Faculty of Mathematics and Computer Science

ul. Lojasiewicza 6, 30348 Krakow, Poland

 

2 College of Sciences, Guangxi University for Nationalities

Nanning, Guangxi 530006, People’s Republic of China

Abstract. We study an abstract second order inclusion involving two nonlinear single-valued operators and a nonlinear multivalued term. Our goal is to establish the existence of solutions to the problem by applying numerical scheme based on time discretization. We show that the sequence of approximate solution converges weakly to a solution of the exact problem. We apply our abstract result to a dynamic, second order in time differential inclusion involving Clarke subdifferential of a locally Lipschitz, possibly nonconvex and nonsmooth potential. In two presented examples the Clarke subdifferential appears either in a source term or in a boundary term.

Keywords: time discretization; differential inclusions; nonconvex potential; weak solution; Rothe method

2010 Mathematics Subject Classification: 49J40, 47J20, 47J22

1 Introduction

In this paper we study the following inclusion problem

[TABLE]

and we deal with the existence of its solution in an appropriate function space. In the above problem, $A$ and $B$ are single-valued, nonlinear operators, $M$ is a multivalued term, $\gamma$ is a linear, continuous and compact operator and $\gamma^{*}$ denotes its adjoint operator. Our goal is to generalize the result obtained in [6], where the second order equation has been studied. In our case, because of the presence of multivalued term $M$, we need to apply a technique taken from set valued analysis. Moreover, in [6], the operator $A$ is assumed to be hemicontinuous and monotone with respect to the second variable. In comparison to [6], we assume that it is only pseudomonotone which allows to deal with a larger class of operators. On the other hand, it forces to use more advanced approach. Similarly as in [6], the operator $B$ is assumed to be a nonlinear perturbation of a linear principal part $B_{0}$. Using the idea presented in [6], we start with the following numerical scheme

[TABLE]

with initial condition. We obtain a solution $\{u^{n}\}$ applying an existence result for a corresponding elliptic inclusion in each fixed time step. To this end, we use a surjectivity result for a pseudomonotone, coercive multivalued operator. Having the solution of the time-semidiscrete problem (1.2), we construct a sequence $u_{\tau}$ of piecewise constant functions in order to approximate a solution of (1.1) and sequences $v_{\tau}$ and $\hat{v}_{\tau}$ of piecewise constant and piecewise linear functions in order to approximate its time derivative. First, using a priori bounds in reflexive functional spaces, we obtain a weak limit for the approximate sequences. Then we pass to convergence analysis in order to prove that the limit function satisfies (1.1).

This kind of approach, known also as the Rothe method, has been used for solving many types of evolution partial differential equations or variational inequalities. We refer to [14] as for a basic handbook concerning this subject. The Rothe method for evolution inclusion has been applied first in [7] and then developed in [3, 4, 5, 8, 9, 11, 12, 13].

There are two main difficulties arising in our problem, both concern the analysis of convergence. The first issue comes from the fact that the operator $A$ is assumed to be pseudomonotone with respect to the second variable, which is a relatively weak assumption in comparison to [6]. Moreover, we have to provide an analogous property of its Nemytskii operator $\mathcal{A}$. To this end, we use Lemma 3.3, which requires to know that the considered sequence of piecewise constant functions is bounded in space $M^{p,q}(0,T;W,W^{*})$, thus, in particular that they have a bounded total variation.

The second main difficulty appears when passing to the limit with multivalued term. In this part, we use the Aubin-Celina convergence theorem. However, to do this, we need to have a strong convergence of the sequence $\gamma v_{\tau}$ in an appropriate space, where the functions are piecewise constant, and in a consequence, their time derivatives are not regular enough to apply classical Lions-Aubin compactness results in our case. Thus, we apply more general result of Lemma 2.5, which requires only that functions $v_{\tau}$ have bounded total variations, instead of bounded time derivative in a space of type $L^{q}$ with respect to time. We remark that the compactness of the operator $\gamma$ is a crucial assumption, which allows to use Lemma 2.5. In examples $\gamma$ is either the compact embedding $W_{0}^{1,p}(\Omega)\subset L^{p}(\Omega)$ or the trace operator $\gamma:W^{1,p}(\Omega)\to L^{p}(\partial\Omega)$.

The rest of the paper is organized as follows. In Section 2, we introduce basic definitions and recall some useful results. In Section 3, we formulate an abstract problem and establish assumptions on its data. In Section 4, we state a discrete problem and obtain a priori estimates on its solution. In Section 5, we study convergence of solutions of the discrete problem to a solution of exact one. Finally, in Section 6, we show two examples for which our main result is applicable.

2 Preliminaries

In this section we introduce the basic definitions and recall results used in the sequel. We start with the definition of a pseudomonotone operator in both single valued and multivalued case.

Definition 2.1**.**

Let $X$ be a real Banach space. A single valued operator $A\colon X\to X^{*}$ is called pseudomonotone, if for any sequence $\{v_{n}\}_{n=1}^{\infty}\subset X$ such that $v_{n}\to v$ weakly in $X$ and $\limsup\limits_{n\rightarrow\infty}\langle Av_{n},v_{n}-v\rangle\leqslant 0$ we have $\langle Av,v-y\rangle\leqslant\liminf\limits_{n\rightarrow\infty}\langle Av_{n},v_{n}-y\rangle$ for every $y\in X$.

Definition 2.2**.**

Let $X$ be a real Banach space. The multivalued operator $A\colon X\to 2^{X^{*}}$ is called pseudomonotone if the following conditions hold:

$A$* has values which are nonempty, weakly compact and convex,*

• 2)

$A$* is upper semicontinuous from every finite dimensional subspace of $X$ into $X^{\ast}$ furnished with weak topology,*

• 3)

if $\{v_{n}\}_{n=1}^{\infty}\subset X$ and $\{v_{n}^{*}\}_{n=1}^{\infty}\subset X^{*}$ are two sequences such that $v_{n}\rightarrow v$ weakly in $X$, $v_{n}^{\ast}\in A(v_{n})$ for all $n\geqslant 1$ and $\limsup\limits_{n\rightarrow\infty}\langle v_{n}^{\ast},v_{n}-v\rangle\leqslant 0$, then for every $y\in X$ there exists $u(y)\in A(v)$ such that $\langle u(y),v-y\rangle\leqslant\liminf\limits_{n\rightarrow\infty}\langle v_{n}^{\ast},v_{n}-y\rangle$.

Now we recall two important results concerning properties of pseudomonotone operators.

Proposition 2.3**.**

Assume that $X$ is a reflexive Banach space and $A_{1},A_{2}\colon X\to 2^{X^{*}}$ are pseudomonotone operators. Then $A_{1}+A_{2}\colon X\to 2^{X^{*}}$ is a pseudomonotone operator.

Theorem 2.4**.**

Let $X$ be a reflexive Banach space and let $A\colon X\to 2^{X^{*}}$ be a pseudomonotone and coercive operator. Then $A$ is surjective, i.e. $R(A)=X^{*}$.

Let $X$ be a Banach space and let $I=(0,T)$ be a time interval. We introduce the space $BV(I;X)$ of functions of bounded total variation on $I$. Let $\pi$ denote any finite partition of $I$ by a family of disjoint subintervals $\{\sigma_{i}=(a_{i},b_{i})\}$ such that $\overline{I}=\cup_{i=1}^{n}\overline{\sigma}_{i}$. Let $\mathcal{F}$ be the family of all such partitions. Then for a function $x\colon I\to X$ we define its total variation by

[TABLE]

As a generalization of the above definition, for $1\leqslant q<\infty$, we define a seminorm

[TABLE]

and the space

[TABLE]

For $1\leqslant p\leqslant\infty$, $1\leqslant q<\infty$ and Banach spaces $X$ and $Z$ such that $X\subset Z$, we introduce the following space

[TABLE]

Then $M^{p,q}(I;X,Z)$ is also a Banach space with the norm given by $\|\cdot\|_{L^{p}(I;X)}+\|\cdot\|_{BV^{q}(I;Z)}$.

Finally, we recall a compactness result, which will be used in the sequel. For its proof, we refer to [7].

Proposition 2.5**.**

Let $1\leqslant p,q<\infty$. Let $X_{1}\subset X_{2}\subset X_{3}$ be real Banach spaces such that $X_{1}$ is reflexive, the embedding $X_{1}\subset X_{2}$ is compact and the embedding $X_{2}\subset X_{3}$ is continuous. Then the embedding $M^{p,q}(0,T;X_{1};X_{3})\subset L^{p}(0,T;X_{2})$ is compact.

3 Problem statement

In this section we formulate an abstract problem and give a list of assumptions concerning the data of the problem. For a Banach space $X$ by $X^{*}$ we denote its topological dual, by $\left\langle\cdot,\cdot\right\rangle_{{X^{*}\times X}^{*}\times X^{*}\times X}$ the duality pairings for the pair $(X,X^{*})$ and by $i_{XY}\colon X\to Y$ we will denote the embedding operators of $X$ into $Y$ provided that $X\subseteq Y$.

First we introduce appropriate spaces. Let $(W,\|\cdot\|_{W})$ be a reflexive Banach space densely and continuously embedded in a reflexive Banach space $(V,\|\cdot\|_{V})$, and let $(V,\|\cdot\|_{V})$ be densely and continuously embedded in a Hilbert space $(H,(\cdot,\cdot),|\cdot|)$. We also assume that the embedding $W\subseteq H$ is compact. We have

[TABLE]

where $V^{*}$ and $W^{*}$ denote the dual spaces to $V$ and $W$, respectively. Let $(U,\|\cdot\|_{U})$ be a Banach space such that there exists a compact mapping $\gamma\colon W\to U$. For $T>0$, $p\geqslant 2$ we define the spaces $\mathcal{W}=L^{p}(0,T;W)$, $\mathcal{V}=L^{p}(0,T;V)$, ${\mathcal{H}}=L^{2}(0,T;H)$, $\mathcal{U}=L^{p}(0,T;U)$. We knot that their dual spaces are $\mathcal{W}^{*}=L^{q}(0,T;W^{*})$, $\mathcal{V}^{*}=L^{q}(0,T;V^{*})$, ${\mathcal{U}}^{*}=L^{q}(0,T;U^{*})$, respectively (where $\frac{1}{p}+\frac{1}{q}=1$). We identify the space $\mathcal{H}$ with its dual and denote by $(\cdot,\cdot)_{\mathcal{H}}$ the scalar product in $\mathcal{H}$.

We are concerned with the following problem.

Problem ${\mathcal{P}}$. Find $u\in\mathcal{W}$ with $u^{\prime}\in\mathcal{W}$ and $u^{\prime\prime}\in\mathcal{W}^{*}$ such that

[TABLE]

A solution of Problem ${\mathcal{P}}$ will be understood in the following sense.

Definition 3.1**.**

The function $u\in{\mathcal{W}}$ is said to be a solution of Problem ${\mathcal{P}}$ if $u^{\prime}\in\mathcal{W}$, $u^{\prime\prime}\in\mathcal{W}^{*}$, $u$ satisfies (3.2), and there exists a function $\eta\in{\mathcal{U}^{*}}$ such that

[TABLE]

We impose the following assumptions on the data of Problem $\mathcal{P}$.

${\underline{H(A)}:}A\colon[0,T]\times W\to W^{*}$ is such that

• (i)

for all $v\in W$, the mapping $t\to A(t,v)$ is continuous,

• (ii)

\|A(t,v)\|_{W^{*}}\leqslant\beta_{A}\big{(}1+\|v\|_{W}^{p-1}\big{)} for a.e. $t\in(0,T)$, all $v\in W$ with $\beta_{A}>0$,

• (iii)

$\langle A(t,v),v\rangle_{W^{*}\times W}\geqslant\mu_{A}\|v\|_{W}^{p}-\beta|u|^{2}-\lambda$ for all $v\in W$ with $\mu_{A}>0$, ${\beta,\lambda\in\mathbb{R}}$,

• (iv)

$v\to A(t,v)$ is pseudomonotone for all $t\in[0,T]$.

We assume that $B\colon[0,T]\times V\to W^{*}$ has a decomposition $B(t,v)=B_{0}(v)+C(t,v)$, where

${\underline{H(B_{0})}:}B_{0}\colon\in\mathcal{L}(V,V^{*})$ is symmetric and strongly positive, with constants $\mu_{B},\beta_{B}>0$ such that

[TABLE]

${\underline{H(C)}:}C\colon[0,T]\times V\to W^{*}$ is such that

• (i)

for all $v\in V$, the function $t\to C(t,v)$ is continuous,

• (ii)

$\|C(t,v)\|_{W^{*}}\leqslant\beta_{C}(1+\|v\|_{V}^{\frac{2}{q}})$ for a.e. $t\in(0,T)$, all $v\in V$ with $\beta_{C}>0$,

• (iii)

$\|C(t,v)-C(t,w)\|_{W^{*}}\leqslant\alpha(\max(\|v\|_{V},\|w\|_{V}))|v-w|^{\frac{1}{q}}$ for all $t\in[0,T]$, all $v,w\in V$, where $\alpha\colon\mathbb{R}_{+}\to\mathbb{R}_{+}$ is a monotonically increasing function.

${\underline{H(M)}:}M\colon U\to 2^{U^{*}}$ is such that

• (i)

for all $u\in U$, $M(u)$ is a nonempty, closed and convex set,

• (ii)

$M$ is upper semicontinuous in $(s\textrm{-}U\times w\textrm{-}U^{*})$-topology,

• (iii)

$\|\eta\|_{U^{*}}\leqslant c_{M}(1+\|w\|_{U}^{p-1})$ for all $w\in U$, all $\eta\in M(w)$.

${\underline{H(f)}}$ $f\in{\mathcal{W}}^{*}$.

${\underline{H(\gamma)}}$: $\gamma\colon W\to U$ is linear, continuous and compact and its Nemytskii operator $\overline{\gamma}\colon M^{p,q}(0,T;W,W^{*})\to L^{q}(0,T;U^{*})$ is compact.

${\underline{H_{0}}:}\mu_{A}>c_{M}\|\gamma\|_{\mathcal{L}(W,U)}^{p}$.

Now, we provide a result concerning pseudomonotonicity of the superposition $\gamma^{*}M(\gamma)$.

Lemma 3.2**.**

Let the multivalued operator $M\colon U\to 2^{U^{*}}$ satisfy assumption $H(M)$ and the operator $\gamma\colon W\to U$ be linear, continuous and compact. Then the operator $W\ni v\to\gamma^{*}M(\gamma u)\in W^{*}$ is pseudomonotone.

The proof of Lemma 3.2 can be obtained similarly as the proof of Proposition 5.6 in [2].

We complete this section with a lemma, which will play a crucial role in the convergence of numerical scheme presented below.

Lemma 3.3**.**

Let $A\colon[0,T]\times W\to W^{*}$ be an operator satisfying hypotheses $H(A)$ and $\mathcal{A}\colon\mathcal{W}\to\mathcal{W}^{*}$ be a Nemytskii operator corresponding to $A$ defined by $(\mathcal{A}v)(t)=A(t,v(t))$ for all $t\in[0,T]$, all $v\in\mathcal{W}$. Assume that $\{v_{n}\}\subset\mathcal{W}$ is a sequence bounded in $M^{p,q}(0,T;W,W^{*})$ and such that $v_{n}\to v$ weakly in $\mathcal{W}$ and

[TABLE]

Then $\mathcal{A}v_{n}\to\mathcal{A}v$ weakly in $\mathcal{W}^{*}$.

The proof of Lemma 3.3 can be obtained by standard techniques, cf. Lemma 2 in [9].

4 Discrete problem

In this section we consider a discrete problem corresponding to Problem $\mathcal{P}$.

For $N\in\mathbb{N}$ we consider an arbitrary fixed time grid

[TABLE]

We define the following discretization parameters

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

We also define $f^{n}=\frac{1}{\tau_{n+\frac{1}{2}}}\int_{t_{n-\frac{1}{2}}}^{t_{n+\frac{1}{2}}}f(t)\,dt$ for $n=1,...,N-1$.

Finally, in order to approximate the initial conditions, we introduce elements $u^{0}_{\tau}$, $v^{0}_{\tau}$, whose convergence to $u_{0}$ and $v_{0}$ will be specified later.

The discrete problem reads as follows.

Problem ${\mathcal{P}}_{\tau}$. Find sequences $\{u^{n}\}_{n=0}^{N}\subset W$ and $\{v^{n}\}_{n=0}^{N}\subset W$ such that

[TABLE]

In what follows, we formulate a theorem concerning existence of solution to Problem ${\mathcal{P}}_{\tau}$.

Theorem 4.1**.**

Under hypotheses $H(A)$, $H(B_{0})$, $H(C)$, $H(M)$, $H_{0}$ and $\tau_{max}<\frac{1}{\beta}$ there exist sequences $\{u^{n}\}_{n=0}^{N}$ and $\{v^{n}\}_{n=0}^{N}$ being a solution to Problem ${\mathcal{P}}_{\tau}$.

Proof.

We define the multivalued operator $T\colon W\to 2^{W^{*}}$ by

[TABLE]

First we show that $T$ is coercive. Let $v\in W$ and $z\in Tv$. Thus we have $z=\frac{1}{\tau_{n+\frac{1}{2}}}v+A(t_{n},v)+\gamma^{*}\eta$ with $\eta\in M(\gamma v)$. Using hypotheses $H(A)$, $H(B_{0})$, $H(C)$ and $H(M)$, we estimate

[TABLE]

Using $H_{0}$ and inequality $\tau_{max}<\frac{1}{\beta}$, we see that $T$ is coercive. From $H(A)(iv)$ and Lemma 3.2, we conclude that operator $T$ is pseudomonotone as a sum of three pseudomonotone operators. This allows to use Theorem 2.4 to conclude that $T$ is surjective, and as a result, we can establish the existence of $v^{n}$ for a given $v^{0},\ldots,v^{n-1}$ in Problem ${\mathcal{P}}_{\tau}$. Moreover, using

[TABLE]

we can recover the sequence $u^{1},u^{2},...,u^{n}$. This completes the proof. ∎

The next lemma concerns a priori estimate for solution of Problem ${\mathcal{P}}_{\tau}$. In what follows, we denote by $c$ a constant independent on $\tau$, which can vary from line to line. The dependence of $c$ on the other data or parameter will be specified if needed.

Lemma 4.2** (A priori estimate).**

Let hypotheses $H(A)$, $H(B_{0})$, $H(C)$, $H(M)$, $H_{0}$ hold and the time grid satisfy the following constraint

[TABLE]

Then, for $n=1,2,\ldots,N-1$, we have

[TABLE]

where $c=c(r_{min},r_{max},c_{\gamma},T)>0$. Moreover

[TABLE]

Proof.

We test (4.3) by $v^{n}$ and calculate

[TABLE]

We introduce the inner product $\langle\cdot,\cdot\rangle_{B}\colon V\times V\to\mathbb{R}$ by

[TABLE]

and the corresponding norm

[TABLE]

Note that the norms $\|u\|_{B}$ and $\|u\|_{V}$ are equivalent since $\mu_{B}\|u\|_{V}^{2}\leqslant\|u\|_{B}^{2}\leqslant\beta_{B}\|v\|_{V}^{2}$ for all $u\in V$. We have

[TABLE]

From hypotheses $H(C)$, using Young inequality, for any fixed $\varepsilon>0$, we find that

[TABLE]

We come to the multivalued term

[TABLE]

and

[TABLE]

We test (4.3) with $v^{n}$, apply (4.10)-(4.17), replace $n$ with $j$ and multiply by $2\tau_{j+\frac{1}{2}}$ to obtain

[TABLE]

We sum up (4.20) for $j=1,\ldots,n$, to obtain

[TABLE]

Note that

[TABLE]

and

[TABLE]

From $\frac{\tau_{j+1}}{\tau_{j}}=r_{j+1}$ is follows that $\tau_{j}\leqslant\frac{\tau_{j+1}}{r_{min}}$. Thus

[TABLE]

[TABLE]

and

[TABLE]

Using (4.27)-(4.36) in (4.24), we get

[TABLE]

Using $H_{0}$ and (4.6), we see that for $\varepsilon>0$ small enough, we can use the Gronwall lemma for the last inequality. This, together with hypothesis $H(M)(iii)$, gives (4.6).

As for (4.7), we use (4.3) and get

[TABLE]

with a positive constant $c$. From growth conditions on $A$, $B_{0}$ and $C$, we estimate

[TABLE]

Using (4.39)-(4.41) in (4.38), multiplying (4.38) by $\tau_{j+{\frac{1}{2}}}$ and summing up with $j=1,\ldots,n$, we have

[TABLE]

Finally, using (4.7), we get (4.9) from (4.43). This completes the proof of the lemma. ∎

Now, we use the solution $\{u^{n}\}_{n=0}^{N},\{v^{n}\}_{n=0}^{N}$ of (4.1)-(4.5) to define piecewise constant and piecewise linear functions whose convergence will be studied in next section.

[TABLE]

Note that the above functions depend on the parameter $N$. However, for the sake of simplicity, we omit the symbol $N$ in their notation.

It is well known (see Remark 8.15 in [14]) that

[TABLE]

5 Convergence of the scheme

In this section we study the behaviour of sequences $u_{\tau}$, $v_{\tau}$, $\hat{u}_{\tau}$, $\eta_{\tau}$ and $f_{\tau}$ with respect to the increasing number of time grids $N$. In what follows, all convergences, unless it is specified differently, will be understood with respect to $N\to\infty$. In particular, we impose the following, additional assumptions.

${\underline{H(\tau)}:}$

• (1)

$\tau_{max}\to 0$,

• (2)

$\tau_{max}\leqslant D\tau_{min}$ with a constant $D>0$ independent on $N$,

• (3)

$\sigma_{\tau}\to 0.$

${\underline{H_{1}}:}$

• (1)

$u_{\tau}^{0}\to u_{0}$ in $V$,

• (2)

$v_{\tau}^{0}\to v_{0}$ in $H$,

• (3)

$\sup\limits_{N\in\mathbb{N}}\tau_{max}\|v_{\tau}^{0}\|_{V}^{2}<\infty$.

We introduce the integral operator $K\colon\mathcal{V}\to\mathcal{V}$ defined by

[TABLE]

Lemma 5.1** (Convergences).**

Under hypotheses $H(A)$, $H(B)$, $H(M)$, $H(\gamma)$, $H(\tau)$, $H_{0}$, and $H_{1}$, there exists $u\in C(0,T;V)$ and $v\in\mathcal{W}$ with $v^{\prime}\in\mathcal{W}^{*}$ such that $u=u_{0}+Kv$ and for a subsequence, we have

1. (a)

$u_{\tau}\to u$* weakly*∗* in $L^{\infty}(0,T;V)$,* 2. (b)

$v_{\tau}\to v$* weakly in $\mathcal{W}$ and weakly*∗* in $L^{\infty}(0,T;H)$,* 3. (c)

$\hat{v}_{\tau}\to v$* weakly in $\mathcal{W}$ and weakly*∗* in $L^{\infty}(0,T;H)$,* 4. (d)

$\hat{v}_{\tau}^{\prime}\to v^{\prime}$* weakly in $\mathcal{W}^{*}$,* 5. (e)

$Kv_{\tau}\to Kv$* weakly*∗* in $L^{\infty}(0,T;W)$,* 6. (f)

$u_{0}+Kv_{\tau}-u_{\tau}\to 0$* in $L^{r}(0,T;V)$ for $r\in[1,\infty)$,* 7. (g)

$\eta_{\tau}\to\eta$* weakly in $\mathcal{U}^{*}$,* 8. (h)

$\hat{v}_{\tau}\to v$* in $L^{r}(0,T;H)$ for $r\in[1,\infty)$,* 9. (i)

$v_{\tau}\to v$* in $L^{r}(0,T;H)$ for $r\in[1,\infty)$,* 10. (j)

$u_{\tau}\to u$* in $L^{r}(0,T;H)$ for $r\in[1,\infty)$,* 11. (k)

$u_{0}+Kv_{\tau}\to u$* w $C(0,T;H)$,* 12. (l)

$v_{\tau}$* is bounded in $M^{p,q}(0,T;W;W^{*})$.*

Proof.

By estimates (4.7) and (4.9), we easily get

[TABLE]

The convergences $(a)$-$(d)$ and $(g)$ follow from (5.1)-(5.4) and (5.5), respectively. However, we need to show that limits obtained in $(b)$ and $(c)$ coincide. Note that

[TABLE]

So $\hat{v}_{\tau}-v_{\tau}\to 0$ in $\mathcal{H}$. Now since $v_{\tau}-\hat{v}_{\tau}\to v-\hat{v}$ weakly in $\mathcal{W}$, it follows that $v_{\tau}-\hat{v}_{\tau}\to v-\hat{v}$ weakly in $\mathcal{H}$. From the uniqueness of the limit we obtain $v=\hat{v}$.

Now we prove $(e)$. Let $g\in L^{1}(0,T;W^{*})$. Then

[TABLE]

where $G(s)=\int_{s}^{T}g(t)dt$ for $s\in[0,T]$. Since $G\in L^{\infty}(0,T;W^{*})$, we have $G\in\mathcal{W}^{*}$. This proves $(e)$.

To prove $(f)$ we first estimate integrals

[TABLE]

For $n=1,...,N-1$, we have

[TABLE]

Hence

[TABLE]

Finally, we estimate the integral

[TABLE]

Now, using Cauchy-Schwartz inequality, we estimate

[TABLE]

Since $p\geqslant 2$, we have $s^{2}\leqslant 1+s^{p}$ for all $s\in\mathbb{R}$. Thus, we have

[TABLE]

Therefore, by (5.9)-(5.20) and hypothesis $H(\tau)$ we obtain $u_{0}+Kv_{\tau}-u_{\tau}\to 0$ in $L^{2}(0,T;V)$. Since $v_{\tau}$ is bounded in $\mathcal{W}$ it is also bounded in $\mathcal{V}$, so $Kv_{\tau}$ in bounded in $L^{\infty}(0,T;V)$. Moreover $u_{\tau}$ is bounded in $L^{\infty}(0,T;V)$. So $u_{0}+Kv_{\tau}-u_{\tau}$ is bounded in $L^{\infty}(0,T;V)$. For $r>2$, we have

[TABLE]

Therefore $u_{0}+Kv_{\tau}-u_{\tau}\to 0$ in $L^{r}(0,T;V)$ for all $r\in[1,\infty)$, which completes the proof of $(f)$.

From $(a)$, $(e)$, $(f)$ and uniqueness of the weak limit in $L^{2}(0,T;V)$ we obtain

[TABLE]

and, in particular, $u\in C(0,T;V)$. From $(c)$, $(d)$, compactness of embedding $W\subset H$ and the Lions-Aubin lemma, we have

[TABLE]

Again, since $\hat{v}_{\tau}$ is bounded in $L^{\infty}(0,T;H)$ it follows that

[TABLE]

which proves $(h)$.

From (5.8) and $(h)$ we have $v_{\tau}\to v$ in $L^{2}(0,T;H)$, and also $v_{\tau}\to v$ in $L^{r}(0,T;H)$ with $r\in[1,\infty]$, since $v_{\tau}$ is bounded in $L^{\infty}(0,T;H)$.

Thus $(i)$ holds. Now, using $(i)$, we calculate,

[TABLE]

From (5.21) we have

[TABLE]

Combining $(f)$, (5.22) and (5.23), we obtain $(j)$.

Moreover, by (5.21) we have $\|u_{0}+Kv_{\tau}-u\|_{C(0,T;H)}=\|Kv_{\tau}-Kv\|_{C(0,T;H)}$. This together with (5.22) gives $(k)$.

It remains to show $(l)$. Taking into account (5.2), it is enough to estimate the seminorm $\|v_{\tau}\|_{BV^{q}(0,T;W^{*})}$. Since the function $v_{\tau}$ is piecewise constant, the seminorm will be measured by means of jumps between elements of sequence $\{v_{\tau}^{k}\}_{k=1}^{N}$. Namely, let $\{m_{i}\}_{i=0}^{n}\subset\{1,...,N\}$ be an increasing sequence of numbers such that $m_{0}=0$, $m_{n}=N$ and

[TABLE]

In what follows, we estimate

[TABLE]

We now combine (4.9) with (5.24) and (5.25) to see that $\|v_{\tau}\|^{q}_{BV^{q}(0,T;W^{*})}$ is bounded. This completes the proof of the lemma. ∎

Now we formulate the existence theorem which is the main result of the paper.

Theorem 5.2**.**

Let hypotheses $H(A)$, $H(B_{0})$, $H(C)$, $H(\gamma)$, $H_{0}$ hold and $u_{0}\in V,v_{0}\in H,f\in L^{q}(0,T;W^{*})$. Then Problem $\mathcal{P}$ has a solution such that $u\in C([0,T];V)$.

Proof.

We define Nemytskii operators $\mathcal{A}\colon\mathcal{W}\to\mathcal{W}^{*}$, $\mathcal{B}_{0}\colon\mathcal{V}\to\mathcal{V}^{*}$, $\mathcal{C}\colon\mathcal{V}\to\mathcal{W}^{*}$ and $\overline{\gamma}\colon\mathcal{W}\to\mathcal{U}$, by

[TABLE]

Moreover, we approximate the operators $\mathcal{A}$ and $\mathcal{C}$ by their piecewise constant interpolates given by

[TABLE]

[TABLE]

Let $u_{\tau}$, $v_{\tau}$, $\hat{v}_{\tau}$, $\eta_{\tau}$ and $f_{\tau}$ be the functions defined in Section 4. Now, Problem ${\mathcal{P}}_{\tau}$ is equivalent to

[TABLE]

We will pass to the weak limit in $\mathcal{W}^{*}$ with (5.32). From Lemma 5.1$\,\,(d)$, we have

[TABLE]

Next, from Lemma 5.1$(a)$, we obtain

[TABLE]

Thus, by continuity of $B_{0}$, we also have

[TABLE]

Next we will show that $\mathcal{C}_{\tau}u_{\tau}\to\mathcal{C}u$ in $\mathcal{W}^{*}$. First we will show that $\mathcal{C}_{\tau}u\to\mathcal{C}u$ in $\mathcal{W}^{*}$. We will use Lebesgue dominated convergence theorem. We show the pointwise convergence, which follows from $H(C)(i)$, namely

[TABLE]

Next, we show boundedness, as follows

[TABLE]

The function $t\to c(1+\|u(t)\|_{V}^{2})$ is integrable, because $u\in L^{2}(0,T;V)$. By the Lebesgue dominated convergence theorem, we have $\mathcal{C}_{\tau}u\to\mathcal{C}u$ in $\mathcal{W}^{*}$. From hypotheses $H(C)(iii)$ and Lemma 5.1, we obtain

[TABLE]

Since $\|\mathcal{C}_{\tau}u_{\tau}-\mathcal{C}u\|_{\mathcal{W}^{*}}\leqslant\|\mathcal{C}_{\tau}u_{\tau}-\mathcal{C}_{\tau}u\|_{\mathcal{W}^{*}}+\|\mathcal{C}_{\tau}u-\mathcal{C}u\|_{\mathcal{W}^{*}}\to 0$ we get

[TABLE]

By Lemma 5.1$(g)$ and the continuity of $\overline{\gamma}^{*}$, we infer that

[TABLE]

It remains to show that

[TABLE]

In order to prove (5.39), we proceed in two steps. First, we show that

[TABLE]

To this end, let $w\in\mathcal{W}$. We define the function

[TABLE]

and note that $\langle\mathcal{A}_{\tau}v_{\tau}-\mathcal{A}v_{\tau},w\rangle_{\mathcal{W}^{*}\times\mathcal{W}}=\int_{0}^{T}h_{\tau}(t)\,dt$. Let $S\subset[0,T]$ be a set of measure zero, such that the function $w$ is well defined on the set $[0,T]\setminus S$. Let $t\in[0,T]\setminus S$ and $n\in\mathbb{N}$ be such that $t\in[t_{n-\frac{1}{2}},t_{n+\frac{1}{2}}]$. We estimate

[TABLE]

By hypothesis $H(\tau)$, it is clear that $t_{n}\to t$. Thus, by hypothesis $H(A)(i)$, we have $\|A(t_{n},v_{\tau}(t))-A(t,v_{\tau}(t))\|_{W^{*}}\to 0$, so $h_{\tau}(t)\to 0$ for a.e. $t\in[0,T]$. Moreover, we have

[TABLE]

By the Hölder inequality, the right hand side is integrable on $[0,T]$, so we can use Lebesgue dominated convergence theorem and we $\langle\mathcal{A}_{\tau}v_{\tau}-\mathcal{A}v_{\tau},w\rangle_{\mathcal{W}^{*}\times\mathcal{W}}\to 0.$ Since the function $w$ is arbitrary, we obtain (5.40).

In the second step, we calculate

[TABLE]

Using (5.40), we have

[TABLE]

Analogously as in the proof of (5.42), we show that

[TABLE]

From (5.32) we get

[TABLE]

From (4.44) and Lemma 5.1(d), we have

[TABLE]

Moreover, we have

[TABLE]

From Lemma 5.1(c) and (d) and from continuity of the embedding

[TABLE]

we have $v_{\tau}\to v$ that $v_{\tau}(t)\to v(t)$ weakly in $H$. From hypothesis $H_{1}(2)$ and the uniqueness of the weak limit, we have

[TABLE]

The continuity of the norm implies $|v_{\tau}(0)|\to|v(0)|$. Moreover, $v_{\tau}(T)\to v(T)$ weakly in $H$ and by the weak lower semicontinuity of norm, we have $|v(T)|\leqslant\liminf|v_{\tau}(T)|$. Summarizing, we conclude that

[TABLE]

Since the operator $B_{0}\colon V\to V^{*}$ defines the inner product on $V$ and since $(Kw)^{\prime}=w$ for all $w\in L^{2}(0,T;V)$, integrating by parts, we get

[TABLE]

By (5.21) and (5.48) we have

[TABLE]

From Lemma 5.1$(b)$, it follows that $v_{\tau}\to v$ weakly in $\mathcal{V}$. Thus

[TABLE]

Moreover, by Lemma 5.1$(f)$, we also have

[TABLE]

Thus, it follows from (5.53), that

[TABLE]

From (5.37) and Lemma 5.1$(b)$, we obtain

[TABLE]

From Lemma 5.1 $(g)$, $(l)$ and hypothesis $H(\gamma)$, passing to a subsequence if necessary, we have

[TABLE]

Applying (5.47)-(5.56) in (5.44) we have

[TABLE]

From (5.42), (5.43), (5.57) in (5.41), we get

[TABLE]

Now, from Lemma 5.1$(b)$, $(l)$, (5.58) and Lemma 3.3 we obtain

[TABLE]

By (5.59) and (5.40) we obtain (5.39). Using (5.34)-(5.39) we pass to the limit in (5.32) and obtain

[TABLE]

Next, we pass to the limit with inclusion (5.33). From Lemma 5.1$(l)$ and hypothesis $H(\overline{\gamma})$, we have that $\overline{\gamma}v_{\tau}\to\overline{\gamma}v$ in $\mathcal{U}$ and in consequence

[TABLE]

From Lemma 5.1 $(g)$, (5.61) and Aubin-Celina convergence theorem (cf. [1]), we get

[TABLE]

Moreover, by Lemma 5.1 we have $u=u_{0}+Kv$, thus $u(0)=u_{0}$ and $u^{\prime}=v$. Combining it with (5.46), we see that $u^{\prime}(0)=v_{0}$. This, together with (5.60) and (5.62) shows that $u$ is a solution of Problem $\mathcal{P}$. This completes the proof. ∎

6 Examples

In this section we consider two problems, for which the existence result obtained in Theorem 5.2 is applicable.

Let $\Omega$ be an open bounded subset of $\mathbb{R}^{N}$ with a Lipschitz boundary $\partial\Omega$. The boundary is divided in two parts $\Gamma_{1}$, $\Gamma_{2}$ such that $\overline{\Gamma_{1}}\cup\overline{\Gamma_{2}}={\partial\Omega}$ and the $N-1$ dimensional measure of $\Gamma_{1}$ is positive. We denote by $\nu$ the outward unit vector normal to $\partial\Omega$. Let $p\geq 2$, $T>0$ and let the functions $g\colon\mathbb{R}\to\mathbb{R}$, $j_{1}\colon\Gamma_{2}\to\mathbb{R}$, $j_{2}\colon\Omega\to\mathbb{R}$, $f_{1}\colon[0,T]\times\Omega\to\mathbb{R}$ and $f_{2}\colon[0,T]\times\Omega\to\mathbb{R}$ be given. We formulate two problems.

Problem $P_{1}$. Find $u\colon[0,T]\times\Omega\to\mathbb{R}$ such that

[TABLE]

Problem $P_{2}$. Find $u\colon[0,T]\times\Omega\to\mathbb{R}$ such that

[TABLE]

In the above problems $\partial j_{i}$ denotes the Clarke subdifferential of the function $j_{i}$, $i=1,2$, $\alpha_{i}>0$ are constants and $\delta\leqslant 1-\frac{2}{p}$.

We impose the following assumptions on the functions $g$, $j_{1}$ and $j_{2}$.

${\underline{H(g)}}$ $g\colon\mathbb{R}\to\mathbb{R}$ is such that

• (i)

$g$ is continuous,

• (ii)

$\inf\limits_{s\in\mathbb{R}}g(s)s>-\infty$,

• (iii)

$|g(s)|\leqslant c_{g}(1+|s|^{p-1})$ for all $s\in\mathbb{R}$ with $c_{g}>0$.

$\underline{H(j_{1})}$ $j_{1}\colon\Gamma_{2}\times\mathbb{R}\to\mathbb{R}$ is such that

• (i)

$j_{1}(\cdot,\xi)$ is measurable for all $\xi\in\mathbb{R}$ and $j_{1}(\cdot,0)\in L^{1}(\Gamma_{2})$,

• (ii)

$j_{1}(x,\cdot)$ is locally Lipschitz for a.e. $x\in\Gamma_{2}$,

• (iii)

$|\eta|\leqslant c_{j_{1}}(1+|\xi|^{p-1})$ for all $\eta\in\partial j_{1}(x,\xi)$, $x\in\Gamma_{2}$ with $c_{j1}>0.$

$\underline{H(j_{2})}$ $j_{2}\colon\Omega\times\mathbb{R}\to\mathbb{R}$ is such that

• (i)

$j_{2}(\cdot,\xi)$ is measurable for all $\xi\in\mathbb{R}$ and $j_{2}(\cdot,0)\in L^{1}(\Omega)$,

• (ii)

$j_{2}(x,\cdot)$ is locally Lipschitz for a.e. $x\in\Omega$,

• (iii)

$|\eta|\leqslant c_{j_{2}}(1+|\xi|^{p-1})$ for all $\eta\in\partial j_{2}(x,\xi)$, $x\in\Omega$ with $c_{j2}>0.$

We introduce the spaces $W_{1}=\{v\in W^{1,p}(\Omega),\,\,v=0\,\,\text{on}\,\,\Gamma_{1}\}$ and $W_{2}=W_{0}^{1,p}(\Omega)$ equipped with the norm

[TABLE]

Moreover, we define spaces $V_{1}=\{v\in H^{1}(\Omega):\ v=0\,\,\text{on}\,\,\Gamma_{1}\}$, $V_{2}=H_{0}^{1}(\Omega)$, $H=L^{2}(\Omega)$, equipped with the norm

[TABLE]

Finally, we take $U_{1}=L^{p}(\Gamma_{1})$ and $U_{2}=L^{p}(\Omega)$. Next, we consider operators $A_{1}\colon W_{1}\to W_{1}^{*}$, $A_{2}\colon W_{2}\to W_{2}^{*}$, $B_{1}\colon V_{1}\to W_{1}^{*}$ and $B_{2}\colon V_{2}\to W_{2}^{*}$ defined by

[TABLE]

We define the spaces $\mathcal{W}_{1}=L^{p}(0,T;W_{1})$, $\mathcal{W}_{2}=L^{p}(0,T;W_{2})$, $\mathcal{V}=L^{p}(0,T;V)$, ${\mathcal{H}}=L^{2}(0,T;H)$, $\mathcal{U}_{1}=L^{p}(0,T;U_{1})$ and $\mathcal{U}_{2}=L^{p}(0,T;U_{2})$.

We need the following assumptions on the right hand side of Problems $P_{1}$ and $P_{2}$.

$\underline{H(f_{1})}$: $f_{1}\in{\mathcal{W}}_{1}$.

$\underline{H(f_{2})}$: $f_{1}\in{\mathcal{W}}_{2}$.

We define the functionals $F_{1}\in W_{1}^{*}$ and $F_{2}\in W_{2}^{*}$ by

[TABLE]

Now we introduce the notion of a weak solution of Problems ${P}_{1}$ and ${P}_{2}$.

Definition 6.1**.**

A function $u\in\mathcal{W}_{1}$ is said to be a weak solution of Problem ${P_{1}}$ if $u^{\prime}\in\mathcal{W}_{1}$, $u^{\prime\prime}\in\mathcal{W}_{1}^{*}$ and satisfies

[TABLE]

Definition 6.2**.**

A function $u\in\mathcal{W}_{2}$ is said to be a weak solution of Problem ${P_{2}}$ if $u^{\prime}\in\mathcal{W}_{2}$, $u^{\prime\prime}\in\mathcal{W}_{2}^{*}$ and satisfies

[TABLE]

We remark that the weak formulations in Definitions 6.1 and 6.2 are obtained from equations in Problems $P_{1}$ and $P_{2}$, respectively, by multiplying them by a test function $v\in W_{1}$ ($v\in W_{2}$, respectively) and using the Green formula.

In what follows we will deal with the existence of weak solutions of Problems $P_{1}$ and $P_{2}$. First we define two auxiliary functionals $J_{1}\colon U_{1}\to\mathbb{R}$ and $J_{2}\colon U_{2}\to\mathbb{R}$ given by

[TABLE]

[TABLE]

Next, we define the multifunctions $M_{1}\colon U_{1}\to 2^{U_{1}^{*}}$ and $M_{2}\colon U_{2}\to 2^{U_{2}^{*}}$ given by $M_{i}(v)=\partial J_{i}(v)$ for all $v\in U_{i}$, $i=1,2$. Finally, let $\gamma_{1}\colon W_{1}\to U_{1}$ denote the trace operator and $\gamma_{2}\colon W_{2}\to U_{2}$ the embedding operator. Now, we formulate two auxiliary problems.

Problem ${\mathcal{P}}_{1}$. Find $u\in\mathcal{W}_{1}$ with $u^{\prime}\in\mathcal{W}_{1}$ and $u^{\prime\prime}\in\mathcal{W}_{1}^{*}$ such that

[TABLE]

Problem ${\mathcal{P}}_{2}$ Find $u\in\mathcal{W}_{2}$ with $u^{\prime}\in\mathcal{W}_{2}$ and $u^{\prime\prime}\in\mathcal{W}_{2}^{*}$ such that

[TABLE]

Remark 6.3**.**

By the properties of Clarke subdifferential of functionals $J_{i}$, it follows that each solution of Problem ${\mathcal{P}}_{i}$ is also a solution of Problem $P_{i}$, $i=1,2$.

We recall that the following Poincare inequalities hold

[TABLE]

with $\tilde{c}_{1},\tilde{c}_{2}>0$. Let us define the following constants

[TABLE]

where $|\Gamma_{2}|$ and $|\Omega|$ denote the surface measure of $\Gamma_{2}$ and the Lebesgue measure of $\Omega$, respectively. Now we formulate lemmata containing the properties of the operators $A_{1}$, $A_{2}$, $M_{1}$ and $M_{2}$.

Lemma 6.4**.**

If assumption $H(g)$ holds, then operator $A_{1}$ satisfies

• (i)

$\|A_{1}u\|_{W_{1}^{*}}\leq c_{A_{1}}(1+\|u\|_{W_{1}}^{p-1})$*    for all $u\in W_{1}$,*

• (ii)

$\left\langle A_{1}u,u\right\rangle_{{W_{1}}^{*}\times W_{1}}\geq\alpha_{1}\|u\|_{W_{1}}^{p}+\inf_{s\in\mathbb{R}}g(s)s|\Omega|$* for all $u\in W_{1}$,*

• (iii)

$A_{1}$* is pseudomonotone.*

Proof.

Condition $(i)$ follows from $H(g)(iii)$ and (6.3). Condition $(ii)$ follows directly from the definition of $A_{1}$ and $H(g)(ii)$. Finally, for the pseudomonotonicity of $A_{1}$, we refer to Chapter 2 of [14]. ∎

Lemma 6.5**.**

If assumption $H(g)$ holds, then operator $A_{2}$ satisfies

• (i)

$\|A_{2}u\|_{W_{2}^{*}}\leq c_{A_{2}}(1+\|u\|_{W_{2}}^{p-1})$*    for all $u\in W_{2}$,*

• (ii)

$\left\langle A_{2}u,u\right\rangle_{{W_{2}}^{*}\times W_{2}}\geq\alpha_{2}\|u\|_{W_{2}}^{p}+\inf_{s\in\mathbb{R}}g(s)s|\Omega|$*    for all $u\in W_{2}$,*

• (iii)

$A_{2}$* is pseudomonotone.*

The proof of Lemma 6.5 is analogous to the proof of Lemma 6.4.

Lemma 6.6**.**

If assumption $H(j_{1})$ holds, then operator $M_{1}$ satisfies

• (i)

for all $u\in U_{1}$, $M_{1}(u)$ is a nonempty, closed and convex set,

• (ii)

$M_{1}$* is upper semicontinuous in $(s\textrm{-}U_{1}\times w\textrm{-}U_{1}^{*})$-topology,*

• (iii)

$\|\eta\|_{U_{1}^{*}}\leqslant c_{M_{1}}(1+\|w\|_{U_{1}}^{p-1})$* for all $w\in U_{1}$, all $\eta\in M_{1}(w)$.*

Lemma 6.7**.**

If assumption $H(j_{2})$ holds, then operator $M_{2}$ satisfies

• (i)

for all $u\in U_{2}$, $M_{2}(u)$ is a nonempty, closed and convex set,

• (ii)

$M_{2}$* is upper semicontinuous in $(s\textrm{-}U_{2}\times w\textrm{-}U_{2}^{*})$-topology,*

• (iii)

$\|\eta\|_{U_{2}^{*}}\leqslant c_{M_{2}}(1+\|w\|_{U_{2}}^{p-1})$* for all $w\in U_{2}$, all $\eta\in M_{2}(w)$.*

Let $\overline{\gamma}_{i}\colon{\mathcal{W}}_{i}\to{\mathcal{U}}_{i}$ be Nemytskii operator corresponding to $\gamma_{i}$ defined by $(\overline{\gamma}_{i}v)(t)=\gamma_{i}v(t)$ for all $v\in{\mathcal{W}}_{i}$, $i=1,2$. The following lemmata deal with the properties of $\overline{\gamma}_{1}$ and $\overline{\gamma}_{2}$.

Lemma 6.8**.**

The Nemytskii operator $\overline{\gamma}_{1}\colon M^{p,q}(0,T;W_{1},W_{1}^{*})\to L^{q}(0,T;U_{1}^{*})$ is compact.

Proof.

Let $\varepsilon\in(0,\frac{1}{2})$. Then the embedding $i\colon W_{1}\to W^{1-\varepsilon,p}(\Omega)$ is compact. The trace operator $\tilde{\gamma}_{1}\colon W^{1-\varepsilon,p}(\Omega)\to W^{\frac{1}{2}-\varepsilon,p}(\partial\Omega)$ is linear and continuous and, finally, the embedding $j\colon W^{\frac{1}{2}-\varepsilon,p}(\partial\Omega)\to L^{p}(\partial\Omega)=U_{1}$ is also linear and continuous. Thus $\gamma_{1}=j\circ\tilde{\gamma}_{1}\circ i$ is linear, continuous and compact. Moreover, the spaces $V_{1}\subset W^{1-\varepsilon,p}(\Omega)\subset V_{1}^{*}$ satisfy assumptions of Proposition 2.5 so the embedding $M^{p,q}(0,T;V_{1},V_{1}^{*})\subset L^{p}(0,T;W^{1-\varepsilon,p}(\Omega))$ is compact. Since the embedding $L^{p}(0,T;W^{1-\varepsilon,p}(\Omega))\subset\mathcal{U}_{1}$ is continuous the Nemytskii operator corresponding to $\gamma_{1}$ is compact. ∎

Lemma 6.9**.**

The Nemytskii operator $\overline{\gamma}_{2}\colon M^{p,q}(0,T;W_{2},W_{2}^{*})\to L^{q}(0,T;U_{2}^{*})$ is compact.

Proof.

Use directly Proposition 2.5 to the triple of spaces $W_{2}$, $U_{2}^{*}$ and $W_{2}^{*}$. ∎

Now we impose additional assumptions on the constants of the problems.

$\underline{H_{0}^{1}}$: $\alpha_{1}>c_{M_{1}}\|\gamma_{1}\|^{p}_{\mathcal{L}(W_{1},U_{1})}$,

$\underline{H_{0}^{2}}$: $\alpha_{2}>c_{M_{2}}\|\gamma_{2}\|^{p}_{\mathcal{L}(W_{2},U_{2})}$.

We are in a position to formulate the existence results for Problems $P_{1}$ and $P_{2}$.

Theorem 6.10**.**

Let assumptions $H(g)$, $H(j_{1})$, $H(f_{1})$, $H_{0}^{1}$ ($H(j_{2})$, $H(f_{2})$, $H_{0}^{2}$, respectively) hold and $u_{0}\in V,v_{0}\in H$. Then Problem $P_{1}$ (Problem $P_{2}$, respectively) admits a weak solution.

Proof.

We apply Theorem 5.2 to Problems ${{\mathcal{P}}}_{1}$ and ${{\mathcal{P}}}_{2}$. To this end we observe that Lemmata 6.4 and 6.5 imply that operators $A_{1}$ and $A_{2}$ satisfy assumptions corresponding to $H(A)$. It is also clear that both operators $B_{1}$ and $B_{2}$ can be represented as a sum of linear term $B_{0}$ and nonlinear one $C$, which satisfy assumptions corresponding to $H(B_{0})$ and $H(C)$ (see example in [6]). Moreover, Lemmata 6.6 and 6.7 provide that the multivalued operators $M_{1}$ and $M_{2}$ satisfy assumptions analogous to $H(M)$. Similarly, Lemmata 6.8 and 6.9 guaranty that assumption $H(\gamma)$ is fulfilled in case of operators $\gamma_{1}$ and $\gamma_{2}$. Finally, assumptions $H_{0}^{1}$ and $H_{0}^{2}$ are analogous to assumption $H_{0}$ of Theorem 5.2. Including assumptions $H(f_{1})$ and $H(f_{2})$, we are in a position to use Theorem 5.2 and obtain the existence of solution to Problems ${{\mathcal{P}}}_{1}$ and ${{\mathcal{P}}}_{2}$. From Remark 6.3 and Definitions 6.1 and 6.2, we get that Problems $P_{1}$ and $P_{2}$ admit weak solutions. ∎