DÁNG ĐIỆU NGHIỆM CỦA CÁC BẤT ĐẲNG THỨC VI BIẾN PHÂN ttta

MINISTY OF EDUCATION AND TRAINING
HANOI NATIONAL UNIVERSITY OF EDUCATION
——————— * ———————

NGUYEN THI VAN ANH

BEHAVIOUR OF SOLUTIONS TO
DIFFERENTIAL VARIATIONAL INEQUALITIES
Speciality: Integral and Differential Equations
Code: 9 46 01 03

SUMMARY OF DOCTORAL THESIS IN MATHEMATICS

Hanoi - 2019


This thesis has been completed at the Hanoi National University of Education

Scientific Advisor: Assoc.Prof. PhD. Tran Dinh Ke

Referee 1: Prof. Dr. Sci. Nguyen Minh Tri, Vietnam Academy of Science and
Technology, Institute of Mathematics.
Referee 2: Assoc.Prof. Dr. Nguyen Xuan Thao, Hanoi University of Science and
Technology.
Referee 3: Assoc.Prof. Dr. Nguyen Sinh Bay, Vietnam University of Commerce.

The thesis shall be defended before the University level Thesis Assessment Council
at.............. on.......

The thesis can be found in the National Library and the Library of Hanoi National
University of Education.

1

INTRODUCTION

1. Motivation and outline

Quantitative theory of ordinary differential equation (ODE) is one of the
basic theories of mathematics which has existed for centuries and gives us many
models describing mechanics of motion in nature and engineering. In the several decades by the end of the 20th century, ODE has generalized as differential
algebraic equation (DAE) and then it has been extensively studied. Therefore,
various problems of engineering and science such as in flexible mechanics, electrical circuit design and chemical process control, etc. have been mathematical
modeling based on the DAE. However, as evidenced by the growing literature
that has surfaced in recent years on multi-rigid-body dynamics with frictional
contacts and on hybrid engineering systems, ODE and DAE are vastly inadequate to deal with many naturally occurring engineering problems that contain
inequalities (for modeling unilateral constraints). Hence, in order to research the
differential model with unilateral constraints, that satisfies the demands mentioned above, mathematicians need to investigate a larger problem:Differential
variational inequalities, which includes differential complementarity problem.
The notion of differential variational inequality was firstly used by Aubin and
Cellina in 1984. In their book the authors considered the problem:




∀t ≥ 0, x(t) ∈ K,
supy∈K x (t) − f (x(t)), x(t) − y = 0,

 x(0) = x ,
0

(1)

where K is a convex closed subset in Rn . By using concept about normal cone of
subset K, the problem was replaced with the differential inclusion of the form:
f (t) ∈ F (x(t)),
x(0) = x0 .
Then the solvability of (1) can be studied by the topological tools of multivalued
analysis. After this work, the theory of DVIs was considered and expanded in
the work of Avgerinous and Papageorgiou in 1997. Moreover, Avgerinous and
Papageorgiou studied the periodic solutions to the DVI of the form
−x (t) ∈ NK(t) (x(t)) + F (t, x(t)), a.e. t ∈ [0, b],
x(0) = x(b).
where NK(t) (x(t)) denotes the normal cone of the convex closed set K(t) at the
point x(t).
However, DVIs were first systematically studied by Pang and Stewart in 2008.
Differential variational inequality is the problem to find an absolutely continuous


2

function x : [0, T ] → Rn and an integrable function u : [0, T ] → Rm such that for
almost all t ∈ [0, T ], one has
x (t) = f (t, x(t), u(t)),
v − u(t), F (t, x(t), u(t) ≥ 0, a.e. t ∈ [0, T ]; ∀v ∈ K.
u(t) ∈ K.

(2)
(3)
(4)

The name DVI is based on the fact that an ordinary differential equation is linked
together with an algebraic constraint represented by the inequality. By the mean
of Pang and Stewart, the derivative of u(·) does not appear in (3), therefore u is
called an algebraic variable. On the other hand, x is called a differential variable.
Denote SOL(K, φ) is solution set of the variational inequality
v − u, φ(u) ≥ 0, ∀v ∈ K
Then u(t) solves variational inequality (3) with φ = F (t, x(t), ·). Therefore, the
properties of the solution mapping (t, x) ⇒ SOL(K, F (t, x(t), ·)) will play a key
role in our consideration. We convert the problem to a differential inclusion
x (t) = f (t, x(t), SOL(K, F (x(t), ·))),
The general boundary condition is considered by
Γ(x(0), x(T )) = 0,

(5)

In the stated form, the problem is a two-point boundary-value problem (BVP) in
the sense that, linked by the abstract function Γ, both the initial state x(0) and
the terminal state x(T ) are unknown variables to be computed; in particular, the
former variable x(0) is not completely given. The initial-value (IVP) version of
the problem corresponds to the special case where Γ(x, y) ≡ x − x0 , with x0 being
given.
One of the important problem generated by DVIs is differential complementarity problem (DCP), where K is a cone C.
x (t) = f (t, x(t), u(t)),
C u(t) ⊥ F (t, x(t), u(t)) ∈ C ∗ .
In turn, a proper specialization of the latter DCP yields the linear complementarity system, which is studied extensively in many previous papers. In the
paper of Pand and Stewart, DVIs comes from many applications including linear
complementarity systems, differential complementarity problems, and variational
inequalities of evolution.
After the work of Pang and Stewart, more and more scholars are attracted to

boost the development of theory and applications for (DVIs). For instance, Chen
- Wang (2014), Liu et al. in 2013 studied the existence and global bifurcation
problems for periodic solutions to a class of differential variational inequalities
in finite dimensional spaces by using the topological methods from the theory of
multivalued maps and some versions of the method of guiding functions, Gwinner in 2013 obtained a stability result of a new class of differential variational
inequalities by using the monotonicity method and the technique of the Mosco


3

convergence, and Chen-Wang in 2014 used the idea of (DVIs) to investigate a
dynamic Nash equilibrium problem of multiple players with shared constraints
and dynamic decision processes.
On the other hand, many applied problems of DVIs in engineering, operations
research, economical dynamics, and physical sciences, etc., are more precisely
described by partial differential equations. Based on this motivation, recently,
Liu–Zeng–Motreanu in 2016 and Liu et al. in 2017 proved the existence of solutions for a class of differential mixed variational inequalities in Banach spaces
through applying the theory of semigroups, the Filippov implicit function lemma
and fixed point theorems for condensing set-valued operators. However, until
now, only one reference, Liu et al. , considered a differential hemivariational inequality in Banach spaces which is constituted by a nonlinear evolution equation
and a hemivariational inequality of elliptic type rather than of parabolic type.
Also, in the paper, the authors required that the constraint set K is bounded,
the nonlinear function u → f (t, x, u) maps convex subsets of K to convex sets
and the C0 -semigroup eAt is compact.
The thesis concerns one of the important problem related to the dynamic system linked to variational inequalities. That is, we study the behaviour of solution
of differential variational inequalities when time tends to infinity. The recent results of DVI in finite spaces have investigated in some works, including Liu (2013),
Loi (2015)... There are many open questions concerning to study the DVIs, such
as the stability in the sense of Lyapunov, the exsitence of decay solution and
periodic solution, the existence of global attractors of m-semiflow generated by
differential variational inequalities. In addition, DVI in Banach space is also attractive and has some open problem. The main difficulty in research the infinite

system is the fact that we do not what exactly the solvability and properties of
solutions of linked variational inequalities. If the solution map of variational inequality is not suiable regularity, it is very hard to study the behaviour of solution
of DVI via diffential inclusion theory.
2. Purpose, objects and scope of the thesis

2.1. Purpose: The thesis focus on studying qualtitive behavior of solutions to
differential variational inequalities in finite dimensional space and infinite dimensional space.
2.2. Objects In this thesis, we consider three problems as follows:
1) Establish sufficient conditions ensuring the existence of mila solutions of multi
- valued dynamical systems generated by differential variational inequalities,
and then prove the existence of global attractors.
2) The existence of decay solutions for a class differential variational inequalities.
2.2. Objects: In the thesis, we consider two types of nonautonomous semilinear
differential inclusions in Banach spaces:
∗ The first type: Differential variational inequalities in finite dimensional
space;
∗ The second one: Differential variational inequalities of parabolic-elliptic
type in infinite spaces;


4

∗ The thirst one: Differential variational inequalities of parabolic-parabolic
type in infinite spaces;
2.3. Scope: The scope of the thesis is defined by the following contents
• Content 1: Study the solvability of differential variational inequalities;
• Content 2: Study the existence of decay solutions for these differential
variational inequalities;
• Content 3: Study the existence global attractors for differential variational
inequalities.

3. Research Methods

The thesis use the tools of multivalued analysis, fixed point theorem, oneparameter semigroup theory to study the contents. Moreover, we use some special
technique to get our purpose:
◦ To prove the existence of solutions to differential variational inequalities, we
employ the semigroup theory (see MNC’s estimates (see Bothe (1998) or
Kamenskii et al. (2001)).
◦ To prove the existence of solutions to differential variational inequalities: we
use fixed point theory for condensing multimaps (see Kamenskii et al (2001)).
◦ In order to do research on the existence of global attractor for multi-valued
dynamical systems, we use the frame work proposed by Melnik and Valero
(1998). In which, we estimated measure of noncompactness to get the asymptotically compactness of the process generated by differential inclusions.
4. Structure and Results

Together with the Introduction, Conclusion, Author’s works related to the thesis that have been published and References, the thesis includes four chapters:
Chapter 1 is devoted to present some preliminaries. In Chapter 2, we present the
solvability and the existence of global attractor for a class of differential variational inequality in finite dimensional space. Chapter 3 presents a sufficient condition ensuring the existence of attractor for differential variational of parabolicelliptic type in Banach space. Chapter 4 presents a sufficient condition ensuring
the existence of attractor for differential variational of parabolic-parabolic type
in infinite dimensional spaces.


5

Chapter 1
PRELIMINARIES

In this chapter, we present some preliminaries including: some functional
spaces; measure of noncompactness; multi-valued calculus and fixed point principles; global attractor for multi-valued autonomous dynamical systems, some
auxiliary results related to some inequalities and theorems.
1.1.

ONE PARAMETER SEMIGROUP

In this section, we present the basic knowledge about semigroup theory, including linear and nonlinear semigroup.
1.1.1. Linear semigroup
1.1.2. Nonlinear semigroup
1.2.

MEASURE OF COMPACTNESS (MNC) AND MNC ESTIMATES

In this section, we recall some notions and facts related to measure of noncompactness (MNC) and Hausdorff MNC, followed by some MNC estimate which is
necessary for the next chapters.
1.3.

MULTIVALUED CALCULUS AND SOME FIXED POINT THEOREMS

1.3.1. Multivalued calculus

In this subsection, we present some definitions and results in multivalued calculus, including concept of a selector and the existence of a selection function.
1.3.2. Condensing map and some fixed point theorems

In this section, we recall some notions of set-valued analysis and condensing
map, then introduce some fixed point theorem for multivalued maps.
1.4.

GLOBAL ATTRACTORS FOR MULTIVALUED SEMIFLOWS

In this section, we present some definitions and results on global attractors for
multivalued semiflows developed by Menik and Valero (1998) and the frame work
for the existence of a compact global attractor for m-semiflows generated by a

differential inclusion.
1.5.

SOME AUXILIARY RESULTS

In this section, we recall some notions and facts related to wellknown inequalities, consist of Gronwall inequality, Hanalay inequality and some theorems such as
Mazur Lemma, Arzela Ascoli theorem. In addition, we hightlight some essential
functional spaces which are used in this thesis.


6

1.5.1.

Some auxiliary inequalites

1.5.2.

Some auxiliary theorems

1.5.3.

Some functional spaces


7

Chapter 2
DIFFERENTIAL VARIATIONAL INEQUALITY IN FINITE SPACE

In this chapter, we study behaviour of solutions of differential variational
inequalities in finite space with delay. Our purpose was to give the sufficient
coditions to ensure the existence of solution and the stability of DVIs. Thus, the
existence of decay solution and a global attractor for m-semiflow generated by
DVIs were proved.
The content of this chapter is written based on the paper [1] in the author’s
works related to the thesis that has been published.
2.1.

Problem setting

We consider the following problem:
x (t) = Ax(t) + h(x(t)) + B(x(t), xt )u(t), t ∈ J = [0, T ],
v − u(t), F (x(t)) + G(u(t)) ≥ 0, ∀v ∈ K, for a.e. t ∈ J,
x(s) = ϕ(s), s ∈ [−τ, 0],

(2.1)
(2.2)
(2.3)

where x(t) ∈ Rn , u(t) ∈ K with K being a closed convex subset in Rm , xt stands
for the history of the state function up to time t, i.e. ut (s) = u(t + s) for
s ∈ [−h, 0]; A, B, F, G and h are given maps which will be specified in the next
section.
2.2.

Solvability

In this section, we will show the global existence of integral solution to problem
(2.1)-(2.2) on J = [τ, T ] under the following assumptions: Put

J = [0, T ], CT = C([0, T ]; Rn ), Cτ = C([−τ, 0]; Rn ), C = C([−τ, T ]; Rn ).
In what follows, we use the assumptions that:
(H1) A is a linear operator on Rn .
(H2) B : Rn ×Cτ → Rn×m is a continuous map such that there exist positive constants
ηB , ζB verifying that:

B(v, w) ≤ ηB ( v + w

Cτ )

+ ζB ,

for all v ∈ Rn , w ∈ Cτ .
(H3) The function F : Rn → Rm is continuous and there is a positive number ηF such
that F (v) ≤ ηF for all v ∈ Rn .
(H4) G : K → Rm is a continuous function such that
1) G is monotone on K , i.e.

u − v, G(u) − G(v) ≥ 0, ∀u, v ∈ K;


8

2) there exists v0 ∈ K such that

lim

v∈K, v →∞

v − v0 , G(v)

> 0.
2
v

(H5) h : Rn → Rn is continuous such that there are positive constants ηh , ζh verifying

h(u) ≤ ηh u + ζh , ∀u ∈ Rn .
We have the following definition of integral solution to (2.1)-(2.2).
Definition 2.1. A pair of functions (x, u), where x : [−τ, T ] → Rn is continuous
and u : [0, T → K] is integrable, called a solution of (2.1) − (2.3) iff the following
equalities hold
t

x(t) = etA ϕ(0) +

t

e(t−s)A B(x(s), xs )u(s)ds +
0

e(t−s)A h(x(s))ds, t ∈ J,
0

v − u(t), F (x(t)) + G(u(t)) ≥ 0, for a.e t ∈ J, ∀v ∈ K,
x(s) = ϕ(s), s ∈ [−τ, 0].
We denote
SOL(K, Q) = {v ∈ K : w − v, Q(v) ≥ 0, ∀w ∈ K},

(2.4)

where Q : Rm → Rm is a given mapping.
Lemma 2.1. Suppose that (H4) holds. Then, for every z ∈ Rm , the solution set
SOL(K, z + G(·)) of 2.4 is nonempty, convex and compact. Moreover, there exists
a number ηG > 0 such that
v ≤ ηG (1 + z ), ∀v ∈ SOL(K, z + G(·)).

(2.5)

Denote
U (z) = SOL(K, z + G(·)), z ∈ Rm .
By Lemma 2.1, the operator U : Rm → P(Rm ) has compact, convex values and
U is upper semicontinuous.
Now we define Φ : Rn × Cτ → P(Rn ) as follows
Φ(v, w) = {B(v, w)y + h(v) : y ∈ U (F (v))}.

(2.6)

Then the composition multimap Φ is upper semicontinuous.
By above setting, the differential variational inequality (2.1)-(2.3) is converted
to the differential inclusion as follows
x (t) ∈ Ax(t) + Φ(x(t), xt ), t ∈ J,
x(t) = ϕ(t), t ∈ [−τ, 0].

(2.7)
(2.8)

Denote
PΦ (x) = {f ∈ L1 (J; Rn ) : f (t) ∈ Φ(x(t), xt )}, with x ∈ C.
Thanks to Lemma 2.1, we have
Φ(v, w) ≤ ηG (1 + ηF )[ηB ( v + w

Cτ )

+ ζB ] + η h v + ζh .

(2.9)


9

Because Φ is upper semicontinuous with convex, compact values, the multimap
Λ(t) = Φ(x(t), xt ) is strongly measuable by using Proposition 1.3.1(Measuable
of Noncompactness- Kamenskii et al). Thus, Φ has a Castaing represent and it
implies PΦ (x) = ∅ for every x ∈ C.
Let y ∈ CT and ϕ ∈ Cτ , we define the function y[ϕ] ∈ C by
y[ϕ](t) =

y(t), if t ∈ [0, T ],
ϕ(t), if t ∈ [−τ, 0].

We consider the Cauchy operator
W : L1 (J; Rn ) → CT
t

e(t−s)A f (s)ds.

W(f )(t) =

(2.10)

0

For given ϕ ∈ Cτ , we call the sollution multimap F : CT → P(CT ) as the following
F(y)(t) = {etA ϕ(0) + W(f )(t) : f ∈ PΦ (y[ϕ])}, t ∈ J.
It is obvious that y ∈ CT is a fixed point of F if and only if y[ϕ] is a solution of
(2.1)-(2.3).
Lemma 2.2. Suppose that (H1)-(H5) hold. Then the multimap PΦ is welldefined and weakly upper semicontinuous.
Lemma 2.3. The operator W defined by (2.10) is compact.
Lemma 2.4. Let (H1)-(H5) hold. Then the solution map F is compact and has
a closed graph.
Theorem 2.1. Let (H1)-(H5) hold. Then problem (2.1)-(2.2) has at least one
solution. Moreover, the solution of the dynamic system generated by (2.1)-(2.2)
is a compact set.
2.3.

Decay solution

In this section, we consider the solution operator F on BC(0, ∞; Rn ). For a
positive number γ and ϕ ∈ Cτ , denote
Bϕγ (R) = {x ∈ C([0, ∞); Rn ) : x(0) = ϕ(0), eγt x(t) ≤ R for all t ≥ 0}.
Then Bϕγ (R) is a closed bounded convex subset of BC(0, ∞; Rn ). We need to
replace the assumptions (H1), (H2) and (H5) by stronger ones:
(H1*) A is a linear operator on Rn such that there exists a > 0 : −Az, z ≥ a z
all z ∈ Rn .

2

for

(H2*) B satisfies (H2) with ζB = 0.

(H5*) h fulfills (H5) with ζh = 0.

Lemma 2.5. Under hypotheses (H1*), (H2*), (H3)-(H4) and (H5*) F(Bϕγ (R)) ⊂
Bϕγ (R) for some R > 0, provided that
ηG (1 + ηF )ηB (1 + eγτ ) + ηh + γ < a.

(2.11)


10

Denote Π : BC([−τ, ∞]; Rn )×L1loc (R+ ; R) → BC([−τ, ∞]; Rn ) defined by Π(x, u) :=
x.
Theorem 2.2. Assume that (H1*), (H2*), (H3)-(H4) and (H5*) take place and
there exists γ > 0 such that
ηG (1 + ηF )ηB (1 + eγτ ) + ηh + γ < a.
Then the solution set S of (2.1)-(2.3) is a nonempty set. Moreover Π(S) is a
nonempty compact subset in BC([−τ, ∞]; Rn ) and

eγt x(t) = O(1) as t → ∞,
for all x ∈ Π(S).
2.4.

Global attractor

The m-semiflow governed by the dynamical system linked to our model DVI
(2.1) − (2.3) is defined as follows
G : R+ × Cτ → P(Cτ )
G(t, ϕ) = {xt : x[ϕ] is a solution of (2.1) − (2.3) on [−τ, T ] for any T > 0},
We get that

G(t1 + t2 , ϕ) = G(t1 , G(t2 , ϕ)), for all t1 , t2 ∈ R+ , ϕ ∈ Cτ .
For each ϕ ∈ Cτ , we denote
Σ(ϕ) = {x ∈ C([0, ∞); Rn ) : x[ϕ] is a solution of (2.1)-(2.3)
on [−τ, T ] for all T > 0}.
By Melnik and Valero (1998), we obtain that
πt ◦ Σ(ϕ) ⊂ S(·)ϕ(0) + W ◦ PΦ (πt ◦ Σ(ϕ)[ϕ]).
Moreover, we have
G(t, ϕ) = {x[ϕ]t : x ∈ Σ(ϕ)},
and by Theorem 2.1, πt ◦ Σ(ϕ) is a compact subset on C([0, t]; Rn ) for each t > 0.
Then, it implies G(t, ϕ) is a compact subset in Cτ , and G(t, ·) has compact values.
We obtain the results related to the regularity of G(t, ·) by the following lemma.
Lemma 2.6. Let the hypotheses (H1)-(H5) hold. Then G(t, ·) is a compact
multimap for each t > τ .
Lemma 2.7. Let the hypotheses (H1)-(H5) hold. Then G(t, ·) is upper semicontinuous for each t ≥ 0.
Lemma 2.8. Let (H1*) and (H2)-(H5) hold. Then the m-semiflow G admits
an absorbing set, provided that
2ηB ηG (1 + ηF ) + ηh < a
.
Theorem 2.3. Let (H1*) and (H2)-(H5) hold. Then the m-semiflow G generated by (2.1)-(2.3) admits a compact global attractor provided that
2ηB ηG (1 + ηF ) + ηh < a.


11

Chapter 3
DIFFERENTIAL VARIATIONAL INEQUALITY OF PARABOLIC-ELLIPTIC
TYPE

In this chapter we consider a model of infinite dimensional differential variational inequalities formulated by a parabolic differential inclusion and an elliptic
variational inequality. The existence of global solution and global attractor for the

semiflow governed by our system is proved by using measure of noncompactness.
The content of this chapter is written based on the paper [2] in the author’s
works related to the thesis that has been published.
3.1.

Setting problem

Let (X, · ) be a Banach space and (U, · U ) be another reflexive Banach
space with the dual U ∗ , we consider the following problem:
x (t) − Ax(t) ∈ F (x(t), u(t)), x(t) ∈ X, t ≥ 0,
B(u(t)) + ∂φ(u(t)) g(x(t), u(t)), u(t) ∈ U, t ≥ 0,
x(0) = ξ,

(3.1)
(3.2)
(3.3)

where x is the state function with values in X, u is a control function taking
values in U , φ : U → R is a proper, convex, lower semicontinuous function with
the subdifferential ∂φ ⊂ U × U ∗ .
By PF we will denote the set of Bochner integrable selections of F (·, ·), that
means
PF : C(J; X) × L1 (J; U ) → P(L1 (J; X)),
PF (x, u) = {f ∈ L1 (J; X) : f (t) ∈ F (x(t), u(t)) for a.e. t ∈ J}.

(3.4)

We mention here the definition of mild solution of the problem (3.1)− (3.3).
Definition 3.1. A pair of continuous functions (x, u), where x : [0, T ] → X,
u : [0, T ] → U , is a mild solution of (3.1) − (3.3) iff there exists a selection

f ∈ PF (x, u) such that
t

S(t − s)f (s)ds, t ∈ J,

x(t) = S(t)ξ +
0

Bu(t) + ∂φ(u(t))
3.2.

g(x(t), u(t)), t ∈ J.

Solvability

We consider the problem (3.1)-(3.3) with the following assumptions
(A) A is a closed linear operator generating a C0 −semigroup (S(t))t≥0 .
(F) F : X × U → Pc (X) is u.s.c with weakly compact and convex values and


12

(1) χ(F (C, D)) ≤ pχ(C)+qU(D) for all bounded set C ⊂ X and D ⊂ U , where
p, q are positive constants; here χ and U stand for the Hausdorff MNC in
the spaces X and H, respectively.
(2) F (x, u) := sup{ ξ X : ξ ∈ F (x, u)} ≤ a x X + b u U + c, for all x ∈ X ,
y ∈ U , where a, b, c are nonnegative constants.
(B) B is a linear continuous operator from U to U ∗ −the dual of U such that B is
defined by the equation

u, Bv = b(u, v), ∀u, v ∈ U,
where b : U × U → R is a bilinear continuous function on U × U and there exists
a positive number ηB satisfying

b(u, u) ≥ ηB u

2
U , ∀u

∈ U.

(G) g : X × U → U ∗ is Lipschitzian, i.e there exist two positive constants η1 and η2
such that

g(x, u) − g(x , u )

U∗

≤ η1 x − x

X

+ η2 u − u

U.

We consider the solution set of the elliptic variational inequality
S(z) = {u ∈ U : Bu + ∂φ(u)

z}.

By Corrolary 2.9 (V.Barbu), we have the following lemma.
Lemma 3.1. Suppose that the hypothesis (B) holds. Then for each z ∈ U ∗ ,
the solution set S(z) is single-valued. Moreover, the corresponding z → S(z) is
Lipchizian from U ∗ to U .
In fact, the variational inequality (3.2)
Bu + ∂φ(u)

g(y, u), for each y.

(3.5)

We consider the elliptic variational inequality
Bu + ∂φ(u)

g(y, u), for given y.

(3.6)

We have the result related to the solution of elliptic variational inequality (3.3)
as follows.
Lemma 3.2. Suppose that (B) and (G) hold. In addition, we assume that η2 <
ηB . Then for each y ∈ X, there exists u ∈ U of (3.6). Moreover, the solution
map
V:X→U

y → u,
is Lipschitzian, i.e.
V(y1 ) − V(y2 )

U

≤

η1
y 1 − y2
ηB − η2

X , ∀y1 , y2

∈ X.

(3.7)


13

Consider the multivalued map
G(y) := F (y, V(y)), y ∈ X.
Then we have
χ(G(B)) = χ(F (B, V(B)))
≤ pχ(B) + qU(V(B))
qη1
≤ (p +
)χ(B).
ηB − η2

(3.8)

and

G(y) := sup{ z X , z ∈ G(y)}
≤ a y X + b V(y) U + c
bη1
≤a y X+
y X + V(0)
ηB − η2
bη1
) y X + d.
≤ (a +
ηB − η2

U

+c
(3.9)

We convert the DVI (3.1)-(3.3) to differential inclusion
x (t) − Ax(t) ∈ G(x(t)), t ∈ [0, T ],
x(0) = ξ.

(3.10)
(3.11)

We define
RG : C([0, T ]; X) → P(L1 (0, T ; X)),
RG (x) = {f ∈ L1 (0, T ; X) : f (t) ∈ G(x(t)), a.e. t ∈ [0, T ]}.
Proposition 3.1. Under assumptions (B), (F) and (G), the map RG is weakly
upper semicontinuous with convex and weakly compact values.
Consider Cauchy operator
W : L1 (0, T ; X) → C([0, T ]; X)

t

W(f )(t) =

S(t − s)f (s)ds.
0

Proposition 3.2. Suppose that (A) satisfies. If the set D ⊂ L1 (0, T ; X) is semicompact then W(D) is relatively compact in C([0, T ]; X). In particular, if the
sequence {fn } is semicompact and fn
f ∗ in L1 (0, T ; X) then W(fn ) → W(f ∗ )
in C([0, T ]; X).
Theorem 3.1. Let the hypotheses (A), (B), (F) and (G) hold. Then the problem
(3.1)− (3.3) has at least one mild solution for each initial datum ξ ∈ X.
Let πT , T > 0, be the truncation operator to [0, T ] imposed on C([0, +∞); X),
that is, for z ∈ C([0, +∞); X), πT (z) is the restriction of z on the interval [0, T ].
Denote
Σ(ξ) = {x ∈ C([0, +∞); X) : x(0) = ξ, x is a mild solution
of (3.1)-(3.2) on [0, T ] for any T > 0}.


14

Obviously,
πT ◦ Σ(ξ) ⊂ S(·)ξ + W ◦ PG (πT ◦ Σ(ξ)),
(3.12)
for all T > 0 and πT ◦ Σ(ξ) =Fix(F), the fixed points set of the solution operator
F of (3.1) - (3.3) in Cξ .
Lemma 3.3. Under the assumptions (A), (B), (F) and (G), πT ◦ Σ({ξn }) is relatively compact in C([0, T ]; X), where {ξn } ⊂ X is a convergent sequence. In
particular, πT ◦ Σ(ξ) is a compact set for each ξ ∈ X.
3.3.

Existence of global attractor

The m-semiflow governed by (3.1)-(3.3) is defined as follows
G : R+ × X → P(X),
G(t, ξ) = {x(t) : x is a mild solution of (3.1) − (3.3), x(0) = ξ}.
In this section, we need an additional assumption as following.
(A∗ ) {S(t)}t≥0 is exponentially stable with exponent α, and is χ-decreasing with exponent β , that is
S(t)

L(X)

where α, β > 0, N, P ≥ 1,

≤ N e−αt , S(t)
·

χ

χ

≤ P e−βt , ∀t > 0,

is the χ-norm.

Lemma 3.4. Under the assumptions (A), (B), (F) and (G), G(t, ·) is u.s.c with
compact values for each t > 0.
Lemma 3.5. Let the hypotheses (A∗ ), (B), (F) and (G) hold. If β − 4N (p +
qη1
) > 0, then there exist a number T0 > 0 and a number ζ ∈ [0, 1) such that

ηB −η2
for all T ≥ T0 we have
χ(GT (B)) ≤ ζ · χ(B), for every bounded set B ⊂ X.
Lemma 3.6. Assume that (A∗ ), (B), (F) and (G) hold. Then G has an absorbing
1
set, provided that α > N (a + ηBbη−η
).
2
Lemma 3.7. Let (A∗ ), (B), (F) and (G) hold. If β − 4P (p +
is asymptotically upper semicompact.

qη1
)
ηB −η2

> 0, then G

Theorem 3.2. Let the hypotheses (A∗ ), (B), (F) and (G) hold. Then the msemiflow G generated by the system (3.1)-(3.3) admits a compact global attractor
provided that
min{α − N (a +

bη1
qη1
), β − 4P (p +
)} > 0.
ηB − η2
ηB − η2


15

3.4.

Application

Let Ω be a bounded domain in Rn with C 2 boundary. We consider the following
system:
∂Z
(t, x) − ∆x Z(t, x) = f (t, x), t ≥ 0, x ∈ Ω,
∂t
f (t, x) ∈ [f1 (x, Z(t, x), u(t, x)), f2 (x, Z(t, x), u(t, x))], t > 0, x ∈ Ω,
∆x u(t, y) + β(u(t, x) − ψ(x)) g¯(x, Z(t, x), u(t, x)), t ≥ 0, x ∈ Ω,
Z(t, x) = u(t, x) = 0, t ≥ 0, x ∈ ∂Ω,
Z(0, x) = ϕ(x), x ∈ Ω,

(3.13)
(3.14)
(3.15)
(3.16)
(3.17)

where f1 , f2 , g : Ω × R × R → R are continuous functions, ψ ∈ H 2 (Ω) and
β : R → 2R is a maximal monotone graph


0
β(r) = R−

∅

if r > 0,
if r = 0,
if r < 0.

This system describes a diffusion process subject to an elliptic VI that models a
free boundary problem [20]. Let X = L2 (Ω), U = V = H01 (Ω), H = L2 (Ω), V =
H −1 (Ω). The norm in X and U are defined by
|u(y)|2 dy, ∀u ∈ L2 (Ω),

|u|2 =
Ω

|∇u(y)|2 dy, ∀u ∈ H01 (Ω),

u =
Ω

We define the multivalued map
F : X × U → P(X),
¯ u¯)(x) = {λf1 (x, Z(x),
¯
¯
F (Z,
u¯(x)) + (1 − λ)f2 (x, Z(x),
u¯(x)) : λ ∈ [0, 1]}.
The problem (3.13)-(3.14) is rewritten by
Z (t) − AZ(t) ∈ F (Z(t), u(t)), t ≥ 0,
where A = ∆, D(A) = H 2 (Ω) ∩ H01 (Ω), Z(t) ∈ X, u(t) ∈ U such that Z(t)(x) =
Z(t, x), u(t)(x) = u(t, x). By Theorem 7.2.5 and Theorem 7.2.8 in Vrabie, the
semigroup S(t) = etA generated by the operator A is compact and exponentially

stable, that is
S(t) L(X) ≤ e−λ1 t ,
where λ1 := inf{ ∇u 2X : u ∈ H01 (Ω), u X = 1}. The assumption (A∗ ) is
satisfied.
Suppose that there exist nonnegative functions a1 , a2 , b1 , b2 ∈ L∞ (Ω), c1 , c2 ∈
L2 (Ω) such that
|f1 (x, p, q)| ≤ a1 (x)|p| + b1 (x)|q| + c1 (x),
|f2 (x, p, q)| ≤ a2 (x)|p| + b2 (x)|q| + c2 (x), ∀x ∈ Ω, p, q ∈ R,


16

Because f1 and f2 are continuous, F has closed graph. In addition, if {Z¯n } ⊂ X
and {¯
un } ⊂ U then we can find a sequence fn ∈ F (Z¯n , u¯n ) converging to X.
Therefore, F is upper semicontinuous. Thus, the condition (F) is testified.
We consider the elliptic variational inequality (3.15). Let B := −∆ : V → V ,
where −∆ is Laplace operator
∇u∇vdy, for each u, v ∈ H01 (Ω).

u, −∆v :=
Ω

It is easy to see that u, Bu = u 2H 1 (Ω) ≥ λ1 u 2X . Thus, assumption (B) hold
0
with ηB = λ1 .
In terms of g, suppose that there exist nonnegative functions η1 , η2 ∈ L∞ (Ω)
such that
|g(x, p, q) − g(x, p , q )| ≤ η1 (x) p − p + η2 (x)|q − q |, ∀x ∈ Ω, p, q, p , q ∈ R.
Then we arrive at the main theorem in this problem.

Theorem 3.3. If η2

2
∞

λ1 > max{ a1

< λ1 and
∞,

a2

∞}

+ max{ b1

∞,

b2

∞

}√

η1 ∞
λ1 − η2

∞

then there exists a global attractor for m-semiflow G governed by (3.13)− (3.17).

17

Chapter 4
DIFFERENTIAL VARIATIONAL INEQUALITY OF PARABOLIC-PARABOLIC
TYPE

In this chapter, we consider a coupled parabolic-parabolic model formulated
by a parabolic differential inclusion and a parabolic variational inequality. We
study the existence of solution for this problem in infinite spaces. In addition,
the existence of global attractor for the m-semiflow generated by our system is
given via the techniques of measure of noncompactness.
The content of this chapter is written based on the paper [3] in the author’s
works related to the thesis that has been submitted.
4.1.

Setting problem

Let (X, · ) be a Banach space, U and H are real Hilbert spaces such that U
is dense in H and U ⊂ H ⊂ U algebraically and topologically. We denote by | · |
and · U the norms of H and U , respectively, and by ·, · the scalar product in
H and the pairing between U and its dual U . The space H is identified with its
own dual. We consider the following problem.
x (t) ∈ Ax(t) + F (x(t), u(t)),
u (t) + Bu(t) + ∂φ(u(t)) h(x(t)),
x(0) = x0 v`a u(0) = u0 ,

(4.1)
(4.2)

(4.3)

where φ : H → R is a proper, convex, lower semicontinuous function.
By the definition of subdifferential ∂φ, evolution inclusion (4.2) can be rewritten in a form of inequality as follow
u (t) + Bu(t) − h(x(t)), u(t) − v + φ(u(t)) − φ(v), ∀v ∈ H.
Denote BH is a truncated operator of B as follows
BH : D(BH ) ⊂ U → H,

BH u = Bu, Bu ∈ H,

D(BH ) = {u ∈ U : Bu ∈ H}.

We need the assumptions for the problem (4.1)-(4.3) as following:
(A) A generates a C0 - semigroup {S(t)}t≥0 .
(B) B : U → U is a linear, symmetric, continuous operator satisfying
(B1) corecive condition

Bu, u ≥ ω u

2
U

for some ω > 0 ;
(B2) U ∩ D(φ) = ∅ and there exists h ∈ H such that
φ((I + λBH )−1 (x + λh)) ≤ φ(x) + Cλ(1 + φ(x)), ∀x ∈ D(φ), λ > 0,


18

(F ) F : X × H → P(X) is u.s.c multimap with compact, convex values and

sastisfying
(F 1) There exist η1F > 0, η2F > 0, a ≥ 0, such that

F (x) := sup{ ξ

X

: ξ ∈ F (x)} ≤ η1F x

X

+ η2F |u| + a,

(F 2) there exist p > 0, q > 0 such that

χ(F (B, D)) ≤ pχ(B) + qϑ(D), ∀B ∈ Pb (X), D ∈ Pb (H),
where χ and ϑ is MNC in X and H, responsibility.
(H) h : X → H is a continuous satisfying h(0) ∈ ∂φ(0) and there exist ηh > 0, b ≥

0 such that
h(x)

H

≤ ηh x

X

+ b.

Consider the corresponding PF as follows
PF : C([0, T ]; X) × C([0, T ]; H) → P(L1 (0, T ; X)),
PF (x, u) = {f ∈ L1 (0, T ; X) : f (t) ∈ F (x(t), u(t)), a.e. t ∈ [0, T ]}.
We mention here the definition of solution of the problem (4.1)-(4.3).
Definition 4.1. A pair of functions (x, u) where x ∈ C([0, T ]; X) and u ∈
L2 (0, T ; U ) ∩ W 1,2 (0, T, H) ∩ C([0, T ]; H), Bu(t) ∈ H almost everywhere t ∈ [0, T ]
is a solution of the problem if there exist a selection f ∈ PF (x, u) such that
t

S(t − s)f (s)ds, t ∈ [0, T ],

x(t) = S(t)x0 +
0

u (t) + Bu(t) + ∂φ(u(t))
u(0) = u0 .

h(x(t)), ∀z ∈ U, a.e. t,

Denote V I(u0 , f ) is the following problem: For each u0 , f given, find u such
that
u (t) + Bu(t) + ∂φ(u(t)) f (t), u(0) = u0 .
Take φ0 : H → (−∞, ∞] such that
φ0 (u) =

1
2

Bu, u , u ∈ U,

+∞, otherwise.

We call that ν is a positive number suct that |u|2 ≤ ν u

2
U

for all u ∈ U .

Proposition 4.1. We have the statements
(a) BH is a m-accretive set in H × H (maximal monotone in H × H). Moreover,

BH is

ω
ν

− m-accretive in H × H and BH = ∂φ0 ;

(b) BH + ∂φ is maximal monotone in H × H, then it implies BH + ∂φ is

accretive set in H × H with domain
D(BH + ∂φ) = D(BH ) ∩ D(φ);

ω
ν

− m-

19

(c) BH + ∂φ = ∂ψ, where ψ = φ0 + φ1 , ψ : H → (−∞, ∞] and

ψ(u) =

1
2

Bu, u + φ(u), u ∈ U,
+∞,
otherwise.

(d) −(BH +∂φ) generates a semigroup S1 (t) which is equicontinuous in D(BH + ∂φ).
(e) −(BH + ∂φ) generates a semigroup S1 (t) which is compact in D(BH + ∂φ).
(f) S1 (t) is the semigroup

ω
ν

exponential stability in D(BH + ∂φ).

Proposition 4.2. Let (B) be hold. Take u0 ∈ D(φ) ∩ U and f ∈ L2 (0, T, H).
Then the problem
u (t) + Bu(t) + ∂φ(u(t))

f (t),

has a unique solution u such that u(t) ∈ D(BH ) a.e. t ∈ [0, T ] and u ∈
L2 (0, T ; U )∩W 1,2 (0, T ; H)∩C([0, T ]; H). Moreover, the corresponding (u0 , f ) → u

is Lipschitzian from H × L2 (0, T, H) to C([0, T ]; H) ∩ L2 (0, T ; H).
Lemma 4.1. Suppose that (B) and (H) satisfy. Then, for each x(·) ∈ C([0, T ]; X)
and u0 ∈ D(φ) ∩ U , the problem
u (t) + Bu(t) + ∂φ(u(t))

h(x(t)),

u(0) = u0 ,
has a solution u(·) such that u(t) ∈ D(BH ) a.e. t ∈ [0, T ] and u ∈ L2 (0, T ; U ) ∩
W 1,2 (0, T ; H) ∩ C([0, T ]; H).
Lemma 4.2. For each x ∈ C([0, T ]; H) and B, h satisfy (B) and (H), parabolic
variational inequality
u (t) + Bu(t) + ∂φ(u(t))

h(x(t)), u(0) = u0 ,

(4.4)

has a solution u(·) and we have the following estimate:
ω

|u(t)| ≤ e
4.2.

−ω
νt

1 − e− ν t
bν + ηh
|u0 | +

ω

t

−ω
ν (t−s)

e

x(s)

X ds.

(4.5)

0

THE SOLVABILITY

Let
CxX0 = {x ∈ C([0, T ]; X) : x(0) = x0 };
CuH0 = {u ∈ C([0, T ]; H) : u(0) = u0 }.
Proposition 4.3. Suppose that the hypothesis (F ) satisfies. Then SF (x, u) = ∅
for each x ∈ CxX0 , u ∈ CuH0 . Moreover, the multimap SF is weakly u.s.c and has
convex, weakly compact values.
We introduce the solution operator for given (x0 , u0 ).
Φ : C([0, T ]; X) × C([0, T ]; H) → P(C([0, T ]; X) × C([0, T ]; H)),
Φ(x, u) := S(·)x0 +

t

0

S(t − s)f (s)ds, f ∈ SF (x, u)
W ◦ h(x(·))


20

where
W : L1 (0, T, X) → C([0, T ]; X),
W (g)(t) = u(t, u0 , g), u is a unique strong solution of V I(u0 , g).
Proposition 4.4. (1) If Ω ⊂ L1 (0, T ; H) is integral bounded then W (Ω) is equacontinuous in C([0, T ]; H)
t

ϑ({W (fn (t))}) ≤

ϑ{fn (s)}ds.
0

(2) W is a compact operator.
Consider Cauchy operator
Q : L1 (0, T, X) → C([0, T ]; X),
T

Q(f )(t) =

S(t − s)f (s)ds.
0

The solution map Φ is rewritten by

Φ(x, u) :=

S(·)x0 + Q ◦ SF (x, u)
W ◦ h(x(·))

We have the following result related to the operator Q.
Proposition 4.5. Let (A) hold. If D ⊂ L1 (0, T ; X) is semicompact, then Q(D)
is relatively compact in C(J; X). In particular, if sequence {fn } is semicompact
and fn
f ∗ in L1 (0, T ; X) then Q(fn ) → Q(f ∗ ) in C([0, T ]; X).
Theorem 4.1. Suppose that (A), (B), (F ) and (H) hold. Then the DVIs (4.1)(4.3) has a solution (x(·), u(·)) for each x0 ∈ X, u0 ∈ D(φ) ∩ U .
x
−A 0
x
g(x, u) + F (x)
Let Y = u , Y0 = u0 , A = 0 B , F(Y ) = h(x) + ∂φ(u) , we convert
0
the DVIs to the following system
dY
+ AY ∈ F(Y )
dt
Y (0) = Y0 ,

(4.6)
(4.7)

We give here the universal space to study behaviour of (4.6)-(4.7) as follows
X := X × D(φ) ∩ U .
The m-semiflow generated by (4.6)-(4.7) is given
G : R+ × X → X ,

G(t, x0 , u0 ) = {(x(t), u(t)) : Y is a solution of (4.6) − (4.7),
x(0) = x0 , u(0) = u0 }.


21

Denote that Σ(x0 , u0 , T ) is a set which contain all solution in [0, T ] of the
problem with the initial condition (x0 , u0 ) and let
Σ(x0 , u0 ) = ∪T >0 Σ(x0 , u0 , T ).
We have
G(t, x0 , u0 ) = {(x(t), u(t)) : (x(·), u(·)) ∈ Σ(x0 , u0 ), t ≥ 0, x0 ∈ X,
u0 ∈ D(φ) ∩ U }.
Proposition 4.6. For each {(ξn , ηn )} ⊂ X such that ξn → ξ, ηn → η w.r.t.
in X and H. Then Σ({(ξn , ηn , T )}) is a relatively compact in C([0, T ]; X) ×
C([0, T ]; H). In particular, Σ(ξ, η, T ) is a compact subset for each (ξ, η) ∈ X .
Corollary 4.1. The multimap G has compact values in X × H.
Lemma 4.3. G is a strict m-semiflow.
4.3.

Global attractor

We need extra assumption:
(A∗ ) The semigroup {S(t)}t≥0 is exponential stability with rate α, and χ-decreasing
with exponential coefficient β, i.e.
S(t)

L(X)

≤ N e−αt , S(t)

where α, β > 0, N, P ≥ 1,
norm.

·

χ

χ

≤ P e−βt , ∀t > 0,

is norm of operator in the sense of MNC

Lemma 4.4. Suppose that (A∗ ), (B), (F ) and (H) hold. If we have
β > 4P p,
then there exist positive numbers T0 > 0 and ζ ∈ [0, 1) such that for all T ≥ T0 ,
we have
χ∗ (G(T, C, D)) ≤ ζχ(C),
for all bounded subset (C, D) ∈ X . Thus, G is asymptotically upper semicompact.
Lemma 4.5. For each t > 0, the m-semiflow G(t, ·, ·) is u.s.c.
Lemma 4.6. Ther exist an absorbing set of Gif the coefficients α, η1F , η2F , ηh , ω
satisfying
ω
min{ , α} > max{η1F + ηh , η2F }.
ν
Theorem 4.2. There exists a global attractor of m-semiflow G if the conditions
of Lemma 4.4, 4.5 and 4.6 hold.


22

4.4.

Application

Let Ω ⊂ Rn be bounded domain with C 2 boundary. Consider the following
problem
∂Z
(t, x) − ∆x Z(t, x) = f (t, x),
∂t
f (t, x) ∈ [f1 (x, Z(t, x), u(t, x)), f2 (x, Z(t, x), u(t, x))],
∂u
(t, x) − ∆x u(t, x) + β(u(t, x) − ψ(x)) h(x, Z(t, x)),
∂t
Z(t, x) = u(t, x) = 0, x ∈ ∂Ω, t ≥ 0,
Z(0, x) = Z0 (x), u(0, x) = u0 (x).

(4.8)
(4.9)
(4.10)
(4.11)
(4.12)

where f1 , f2 , h : Ω × R → R are continuous functions, the functions ψ is in H 2 (Ω),
ψ ≤ 0 in ∂Ω and β : R → 2R is a maximal monotone graph


0
β(r) = R−


∅

if r > 0,
if r = 0,
if r < 0.

Note that, parabolic variational inequality (4.10) can be read as follows
∂u
(t, x) − ∆x u(t, x) = h(x, Z(t, x)) in {(t, x) ∈ Q := (0, T ) × Ω : u(t, x) ≥ ψ(x)},
∂t
∂u
(t, x) − ∆x u(t, x) ≥ h(x, Z(t, x)), in Q,
∂t
u(t, x) ≥ ψ(x), ∀(t, x) ∈ Q.
which represents a rigorous and efficient way to treat dynamic diffusion problems
with a free or moving boundary. This model is called the obstacle parabolic problem
(see V.Barbu(2010)).
Let X = H = L2 (Ω), U = H01 (Ω) and U = H −1 (Ω), the norm in X and H is
given by
|u|2 =

u2 (x)dx, u ∈ L2 (Ω).
Ω

The norm in H01 (Ω) is given by
u

2

|∇u(x)|2 dx.

=
Ω

By Poincar´e inequality, we get that u
Define the multi-valued function

−1/2

U

≤ λ1

|u| with inf{ ∇u

2
X

: u

X

F : X × H → P(X)
F (¯
u, y¯) = {λf1 (x, u¯(x), y¯(x)) + (1 − λ)f2 (x, u¯(x), y¯(x)) : λ ∈ [0, 1]}.
and the operator
A = ∆ : U → U ; D(A) = {H 2 (Ω) ∩ H01 (Ω)}.

= 1}.

23

Then (4.8)-(4.9) can be reformulated as
Z (t) − AZ(t) ∈ F (Z(t), u(t))
where Z(t) ∈ X, u(t) ∈ H such that Z(t)(x) = Z(t, x) and u(t)(x) = u(t, x). It is
known that in Vrabie (1987), S(t) generated by A is compact and exponentially
stable, that is,
S(t) L(X) ≤ e−λ1 t ,
The assumption (A∗ ) is satisfied.
We assume, in addition, that there exist nonnegative functions a1 , a2 ∈ L∞ (Ω)
such that
|f1 (x, p, q)| ≤ a1 (x)|p| + b1 (x)|q| + c1 (x),
|f2 (x, p, q)| ≤ a2 (x)|q| + b2 (x)|q| + c2 (x), ∀x ∈ Ω, p ∈ R.
It is easy to see that F is a multimap with closed convex and compact values.
Moreover,
¯ u¯) ≤ max{ a1 ∞ , a2 ∞ } Z¯ X + max{ b1 ∞ , b2 ∞ } u¯ H
F (Z,
+ max{|c1 |, |c2 |},
which implies (F ).
Consider the parabolic variational inequality (4.10), putting B = −∆, where
−∆ is Laplace operator
u, −∆v :=

∇u(x)∇v(x)dx,
Ω

then Bu, u ≥ λ1 u 2X . So, the assumption (B) is testified with ω = λ1 .
The map h : Ω × R → R satisfies h(x, 0) = 0, ∀x ∈ Ω and
|h(x, p1 )| ≤ b(x)|p1 | + c(x), ∀x ∈ Ω, p ∈ R,

where b, c is nonnegative functions in L∞ (Ω).
¯
¯
Let h : X → H, h(Z)(x)
= h(x, Z(x)),
we obtain
¯ H≤ b
|h(Z)|

∞

Z¯

X

+ |c|.

Then the PVI (4.10) can be read as
u (t) + Bu(t) + ∂IK (u(t)) h(Z(t)),
where
K = {u ∈ L2 (Ω) : u(y) ≥ ψ(x), for a.e. x ∈ Ω},
∂IK (u) = u ∈ L2 (Ω) :

u(x)(v(x) − z(x))dx ≥ 0, ∀z ∈ K ,
Ω

= {u ∈ L2 (Ω) : u(x) ∈ β(v(x) − ψ(x)), for a.e. x ∈ Ω}.
It is easy to see that h(0) = 0 ∈ ∂IK (0) and we have (H).
We have the following result due to Theorem 4.2.
Theorem 4.3. The m-semiflow generated by (4.8)-(4.12) admits a compact global

attractor in L2 (Ω) × K provided that
λ1 > max

b

∞

+ max{ a1

∞,

a2

∞ }; max{

b1

∞,

b2

∞}

.