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Manuscript Number: JMAA-09-699R1

Title: Global attractor for the m-semiflow generated by a quasilinear degenerate parabolic equation

Article Type: Regular Article

Keywords: quasilinear degenerate parabolic equation; m-semiflow;

global solution; global attractor; compact embedding

Corresponding Author: Mr Ke Tran Dinh, Ph.D

Corresponding Author's Institution: Hanoi National University of Education

First Author: Ke Tran Dinh, Ph.D

Order of Authors: Ke Tran Dinh, Ph.D; Anh Cung The, Ph.D; Chuong Nguyen Minh, Dr.Sci.

Abstract: Using theory of global attractors for multi-valued semiflows, we prove the existence of a

global attractor for the m-semiflow generated by a parabolic equation involving the nonlinear

degenerate operator in a bounded domain.

Dear Professors,

We have some modifications for our manuscript. Let us give the detail

descriptions.

1. In the definition 1.1, we have excluded the second inclusion (in old version)

and proved that, it’s a consequence. Other inclusions are kept to ensure that

the equation makes sense in the dual space.

2. The notation of the space of continuous functions has changed to C instead

of C^0.

3. In the proof of Proposition 2.4, we have corrected the argument. We use

similar argument as in the proof of Arzela-Ascoli theorem.

4. The proof of Proposition 2.5 has been omitted, since it can be refered to [4].

5. The proof of Lemma 3.4 has been made to be more explicit.

The authors would like to thank the reviewer for the helpful comments, which

help us to improve the manuscript.

Tran Dinh Ke,

Department of Mathematics and Informatics,

Hanoi National University of Education,

Vietnam.

Cover Letter/Response to Reviews

GLOBAL ATTRACTOR FOR THE M-SEMIFLOW GENERATED

BY A QUASILINEAR DEGENERATE PARABOLIC EQUATION

CUNG THE ANH, NGUYEN MINH CHUONG AND TRAN DINH KE

Abstract. Using theory of global attractors for multi-valued semiflows, weprove the existence of a global attractor for the m−semiflow generated by aparabolic equation involving the nonlinear degenerate operator in a boundeddomain.

1. Introduction

Let Ω be a bounded domain in RN (N > 2) with smooth boundary ∂Ω. We

consider the following problem

∂u

∂t− div(ρ|u|p−2∇u) + f(u) = g(x), x ∈ Ω, t > 0,

u|t=0 = u0(x), x ∈ Ω,

u|∂Ω = 0,

(1.1) e1

where p > 2 and the functions ρ, f, g satisfy some conditions specified later.The degeneracy of problem (1.1) is considered in the sense that the measurable,

nonnegative diffusion coefficient ρ(x), is allowed to have at most a finite number of(essential) zeroes at some points. Motivated by [2], where a semilinear degenerateelliptic problem was studied, we assume that the function ρ : Ω → R satisfies thefollowing assumptions

(Hα) ρ ∈ L1loc(Ω) and for some α ∈ (0, 2), lim infx→z |x − z|−αρ(x) > 0 for

every z ∈ Ω.

For the physical motivation of (Hα), we refer the reader to [1, 2]. Note that invarious diffusion processes, the equation involves diffusion ρ(x) ∼ |x|α, α ∈ (0, 2).

The existence of a global attractor for problem (1.1) in the case p = 2 (thesemilinear case), under some certain conditions of the nonlinear term, has beenstudied in [5, 6] and recently in [1]. In these works, the growth of the nonlinearitysatisfies

|f(u)| 6 c(1 + |u|γ), γ 6N + 2 − α

N − 2 + αand the conditions imposed on f ensure the uniqueness of solutions.

In this paper we study the long-time behavior of solutions to problem (1.1) inthe case p > 2, i.e. in the quasilinear case. Here the restrictions on the growthof the nonlinearity f are removed. Moreover, the conditions imposed on f provide

2000 Mathematics Subject Classification. 35B41, 35K65, 35D05.Key words and phrases. quasilinear degenerate parabolic equation; m-semiflow; global solution;

global attractor; compact embedding. Corresponding author: ketd@hn.vnn.vn.

1

* Manuscript

2 C.T. ANH, N.M.CHUONG AND T.D. KE

the global existence of a weak solution to the problem (1.1), but not uniqueness.More precisely, we assume that u0 ∈ L2(Ω) is given, g ∈ L2(Ω), and f : R → R

is a continuous function and there exist some positive constants C1, C2, C3 and C4

such that

|f(u)| 6 C1|u|q−1 + C2, (1.2) h1

uf(u) > C3|u|q − C4, (1.3) h2

where q > p > 2. Under the conditions (1.2)-(1.3), we prove the existence of aglobal solution and of a global attractor for the problem (1.1). It is clear that thesemigroup method for proving the existence of solutions is not valid here. Thanks tothe structure of the nonlinearity f , however, we may use the compactness method[8, Chapter 1] to prove the global existence of a weak solution. Since we onlyassume the conditions (1.2)-(1.3), the weak solution may not be unique. Thus, whenstudying the long-time behavior of solutions, in order to handle nonuniqueness ofsolutions, we need to use the theory of attractors for multi-valued semiflows.

Let D(u0) be the set of all global weak solutions of the problem (1.1) with the

initial data u0. We define the multi-valued map G : R+×L2(Ω) 7−→ 2L2(Ω) as follows

G(t, u0) = u(t) : u(·) ∈ D(u0).

We first prove that G is a strict multi-valued semiflow and then use the theory ofMelnik and Valero [7] to prove the existence of a compact global attractor in L2(Ω)for the multi-valued semiflow G.

The existence and uniqueness of weak solutions to problem (1.1) in a very specialcase, namely ρ = 1 and f = 0, was given in [8, §12 Chapter 1]. Noting that underthe above conditions of f , the existence of a global attractor for the problem (1.1)in the semilinear nondegenerate case, namely when ρ = 1 and p = 2, was studied in[4] (see also [3, 10] for related works). Thus, our results extend some known resultson the existence and long-time behavior of solutions to the problem type (1.1).

In order to study the problem (1.1) we introduce some weighted Sobolev spaces.

By D1,20 (Ω, ρ) we denote the closure of C∞

0 (Ω) with respect to the norm

‖v‖ =( ∫

Ω

ρ(x)|∇v|2) 1

2

and let V be the space of all functions w ∈ Lp(Ω) satisfying

w∣∣∂Ω

= 0 and∂

∂xi

(ρ∂w

∂xi

) ∈ Lp(Ω); i = 1, ..., N, (1.4) e3

equipped with the norm

‖w‖V =(∫

Ω

∣∣div(ρ∇w)∣∣p

) 1p

.

Remark. By results of Proposition 2.1 below, ‖.‖V is the well-defined norm. Indeed,it suffices to check that if ‖w‖V = 0 then w = 0. For w ∈ V, we have

−

∫

Ω

wdiv(ρ∇w) =

∫

Ω

ρ|∇w|2 = ‖w‖2D1,2

0 (Ω,ρ)> C1‖w‖2

L2(Ω).

GLOBAL ATTRACTOR FOR THE M-SEMIFLOW 3

In addition

−

∫

Ω

wdiv(ρ∇w) 6 ‖w‖L2(Ω)‖div(ρ∇w)‖L2(Ω)

6 C2‖w‖L2(Ω)‖div(ρ∇w)‖Lp(Ω)

for some C1, C2 > 0. Thus

‖w‖V >C2

C1

‖w‖L2(Ω).

Putting

a(u, ξ) =

∫

Ω

ρ(x)|u|p−2∇u∇ξ, for ξ ∈ V (1.5) e5

one can write

a(u, ξ) = −1

p − 1

∫

Ω

|u|p−2udiv(ρ∇ξ), (1.6) ea1

a(u, u) =4

p2

∫

Ω

ρ(x)

N∑

i=1

[ ∂

∂xi

(|u|

p−22 u

)]2

. (1.7) ea2

It’s easy to see that, if u ∈ Lp(Ω) and ξ ∈ V then a(u, ξ) is well-defined.

def0 Definition 1.1. We say that a function u(x, t) is a weak solution of (1.1)on (0, T )iff

u ∈ Lq(QT ) ∩ C([0, T ]; L2(Ω))

∂u

∂t∈ Lp′

(0, T ; V ∗) + Lq′

(QT ),

u|t=0 = u0 a.e. in Ω

and ∫ T

0

a(u, ξ)dt =

∫

QT

(g −

∂u

∂t− f(u)

)ξdxdt (1.8) e4

for all test functions ξ ∈ Lp(0, T ; V ) ∩ Lq(QT ).

Remark. 1. The equation in (1.1) can be seen as an equation in Lp′

(0, T ; V ∗) +

Lq′

(QT ). Here V ∗ is the dual space of V ; p′, q′ are the conjugate exponents of pand q, respectively.

2. If u is a weak solution of (1.1) then |u|p−22 u ∈ L2(0, T ;D1,2

0 (Ω, ρ)). Indeed,multiplying the first equation in (1.1) by u, we have

∫ T

0

a(u, u)dt =

∫

QT

(gu −

∂u

∂tu − f(u)u

)dxdt,

=

∫

QT

(gu − f(u)u

)dxdt +

1

2

(‖u(0)‖2

L2(Ω) − ‖u(T )‖2L2(Ω)

).

Since the right side makes sense, we obtain that the left side does. Taking equation(1.7) into account, we have the conclusion.

The rest of the paper is organized as follows. In the next section, we show somecompactness results for our purposes. In Section 3, we state and prove the mainresults. The last section gives some further remarks.

4 C.T. ANH, N.M.CHUONG AND T.D. KE

2. Preliminaries

We recall the results of Caldirolli and Musina [2] related to the space D1,20 (Ω, ρ).

p1 Proposition 2.1. Assume that Ω is a bounded domain in RN , N > 2, and ρ

satisfy (Hα). Then the following embeddings hold

: i) D1,20 (Ω, ρ) → L2∗

α(Ω) continuously,

: ii) D1,20 (Ω, ρ) → Lr(Ω) compactly if r ∈ [1, 2∗α),

where 2∗α = 2NN−2+α

.

The following is the important point for our arguments.

p2 Proposition 2.2. Let vn be a sequence such that a(vn, vn) 6 C, C > 0, for alln ∈ N. Then vn is precompact in Lp(Ω).

Proof. Denoting η(s) = |s|p−22 s for s ∈ R, we observe that the sequence η(vn)

is bounded in D1,20 (Ω, ρ). By Proposition 2.1, η(vn) is precompact in Lβ(Ω) for

all 1 < β < 2∗α and then η(vn) → χ strongly in Lβ(Ω) up to a subsequence. Thisensures that η(vn) → χ a.e. in Ω. In addition, since η(vn) is bounded in Lβ(Ω),

vn is bounded in Lpβ

2 (Ω). It follows that vn v in Lpβ

2 (Ω). By the monotonicityof η, we deduce that vn → η−1(χ) = v a.e. in Ω.

For any ǫ > 0, by Egorov’s theorem, there exists a subset E ⊂ Ω, |E| < ǫ, such

that vn → v uniformly in Ω\E. Taking r < pβ2 , we have

∫

Ω

|vn − v|r =

∫

Ω\E

|vn − v|r +

∫

E

|vn − v|r

6

∫

Ω\E

|vn − v|r +(∫

E

|vn − v|pβ

2

)θ

ǫ1−θ, θ =2r

pβ.

The last inequality shows that vn → v strongly in Lr(Ω). Choosing β such that2 < β < 2∗α and then taking r = p, we get the conclusion.

The next two propositions can be proved by using the arguments as in [8, §12Chapter 1] with some slight modifications.

p3 Proposition 2.3. For any ǫ > 0, there exists Cǫ > 0 such that

‖u − v‖p

Lp(Ω) 6 ǫ[a(u, u) + a(v, v)] + Cǫ‖u − v‖pV ∗ .

for all u, v ∈ S = w∣∣ |w|

p−22 w ∈ D1,2

0 (Ω, ρ).

Proof. Assume contrary that there exist ǫ0 > 0 and two sequences un, vn ∈ S suchthat

‖un − vn‖p

Lp(Ω) > ǫ0[a(un, un) + a(vn, vn)] + n‖un − vn‖pV ∗ .

Putting

un =un

[a(un, un) + a(vn, vn)]1p

, vn =vn

[a(un, un) + a(vn, vn)]1p

,

we have

‖un − vn‖p

Lp(Ω) > ǫ0 + n‖un − vn‖pV ∗ . (2.1) e6

GLOBAL ATTRACTOR FOR THE M-SEMIFLOW 5

Noting that a(tu, tv) = |t|pa(u, v) for all t ∈ R, we get

a(un, un) 6 1, a(vn, vn) 6 1.

Thus, by Proposition 2.2, there exist two subsequences unm, vnm

such that

unm→ u, vnm

→ v strongly in Lp(Ω).

Since p > 2 and V ⊂ Lp(Ω), it follows that Lp(Ω) ⊂ V ∗. Thanks to (2.1) we have

‖u − v‖V ∗ 6c

nm

,

and therefore u = v. Then ‖u − v‖Lp(Ω)

= 0, which contradicts (2.1).

p4 Proposition 2.4. Assume thatT∫0

a(un(t), un(t))dt 6 C and dun

dt is bounded in

Lp′

(0, T ; V ∗). Then un is precompact in Lp(QT ).

Proof. From Proposition 2.3, we have∫ T

0

‖un+ℓ − un‖p

Lp(Ω) 6 ǫ

∫ T

0

[a(un+ℓ, un+ℓ) + a(un, un)] + Cǫ

∫ T

0

‖un+ℓ − un‖pV ∗ .

Hence it suffices to show that the sequence un contains a subsequence, which isa Cauchy sequence in Lp(0, T ; V ∗).

We will prove a stronger claim: There exist a subsequence uµ such that

uµ → u in C([0, T ]; V ∗). (2.2) e7

By the hypotheses, there exists a set Z ⊂ [0, T ], |Z| = 0 such that, for t ∈ [0, T ]\Z,

a(un(t), un(t)) 6 Kt < ∞.

Hence for any t /∈ Z, there exists a subsequence (dependent of t) such that

uk(t) → u(t) strongly in V ∗ (2.3) econv1

Now let t1, t2, ... is a sequence which is dense in [0, T ], ti /∈ Z. Using (2.3) and adiagonal proceduce, we can extract a subsequence uµ such that

uµ(ti) → u(ti) strongly in V ∗, ∀i. (2.4) econv2

For all t ∈ [0, T ], we have

‖uµ(ti) − uµ(t)‖V ∗ =∥∥∥

∫ ti

t

u′µ(s)ds

∥∥∥

6

(∫ ti

t

‖u′µ(s)‖p′

V ∗ds) 1

p′

|ti − t|1p 6 C|ti − t|

1p .

Then

‖uµ+r(t) − uµ(t)‖V ∗ 6 ‖uµ+r(t) − uµ+r(ti)‖V ∗

+ ‖uµ+r(ti) − uµ(ti)‖V ∗ + ‖uµ(ti) − uµ(t)‖V ∗ ,

6 ‖uµ+r(ti) − uµ(ti)‖V ∗ + 2C|ti − t|1p

for all t ∈ [0, T ]. By density of ti in [0, T ], it follows that uµ is a Cauchysequence in V ∗ uniformly in t ∈ [0, T ]. The proof is complete.

6 C.T. ANH, N.M.CHUONG AND T.D. KE

We use the next proposition, which makes the initial condition in problem (1.1)meaningful.

p5 Proposition 2.5. If u ∈ Lp(0, T ; V ) ∩ Lq(QT ) anddu

dt∈ Lp′

(0, T ; V ∗) + Lq′

(QT )

then u ∈ C([0, T ]; L2(Ω)).

For the proof, we refer the readers to [4].

3. Main Results

3.1. Existence of a global solution.

th1 Theorem 3.1. Under the assumptions (1.2)-(1.3), the problem (1.1) has at leastone weak solution for each u0 ∈ L2(Ω).

Proof. Consider the approximating solution un(t) in the form

un(t) =

n∑

k=1

unk(t)ek,

where ej∞j=1 is a basis of D1,2

0 (Ω, ρ) ∩ Lp(Ω), which is orthogonal in L2(Ω). Weget un from solving the problem

〈dun

dt, ek〉 = −〈Aun, ek〉 − 〈f(un), ek〉 + 〈g, ek〉,

(un(0), ek) = (u0, ek),

where Au = div (ρ|u|p−2∇u).Since f ∈ C(R), theory of ODEs ensures the existence of un. We now establish

some a priori estimates for un. We have

1

2

d

dt‖un‖

2L2(Ω) + a(un, un) +

∫

Ω

f(un)un =

∫

Ω

gun.

Using hypothesis (1.2) and the Cauchy inequality, we get

1

2

d

dt‖un‖

2L2(Ω) + a(un, un) +C3

∫

Ω

|un|q 6 C4|Ω|+

1

2‖g‖2

L2(Ω) +1

2‖un‖

2L2(Ω). (3.1) e7a

It follows that

d

dt‖un‖

2L2(Ω) 6 ‖un‖

2L2(Ω) + ‖g‖2

L2(Ω) + 2C4|Ω|.

The Gronwall inequality implies that un is bounded in L∞(0, T ; L2(Ω)).Integrating (3.1) on [0, T ], we have

‖un‖2L2(Ω) + 2

∫ T

0

a(un, un)dt + 2C3‖un‖q

Lq(QT ) 6 ‖un(0)‖L2(Ω) + T ‖g‖2L2(Ω)

+ ‖un‖2L2(QT ) + 2C4|Ω|T.

The last inequality implies that

un is bounded in Lq(QT ), (3.2) e8

∫ T

0

a(un(t), un(t))dt is bounded. (3.3) e9

GLOBAL ATTRACTOR FOR THE M-SEMIFLOW 7

Taking (1.2) into account, we get the estimate∫

QT

|f(un)|q′

6

∫

QT

C(1 + |un|q−1)q′

6

∫

QT

C(1 + |un|q).

Then f(un) is bounded in Lq′

(QT ) and

f(un) χ in Lq′

(QT ). (3.4) e10

On the other hand, we rewrite the equation as

dun

dt= a(un, .) − f(un) + g

and implement the following estimates

|

∫ T

0

a(un, v)dt| =1

p − 1|

∫ T

0

dt

∫

Ω

|u|p−2udiv (ρ∇v)|

6 C

∫ T

0

‖un‖p

p′

Lp(Ω)‖v‖V dt 6 C‖un‖p

p′

Lp(QT )‖v‖Lp(0,T ;V )

|〈f(un), v〉| 6 ‖f(un)‖Lq′(QT )‖v‖Lq(QT ),

|〈g, v〉| 6 ‖g‖Lq′(QT )‖v‖Lq(QT ),

for all v ∈ Lp(0, T ; V ) ∩ Lq(QT ). Then it follows that dun

dt is bounded in

Lp′

(0, T ; V ∗) + Lq′

(QT ). Combining this with (3.3) and using Proposition 2.4 weconclude that un is precompact in Lp(QT ). Hence we can assume that

• un → u strongly in Lp(QT ),• un → u a.e. in QT .

Since f ∈ C(R), it follows that f(un) → f(u) a.e. in QT . Thanks to (3.4), one has

f(un) f(u) in Lq′

(QT ). (3.5) e11

Finally, passing to the limit as n → ∞ in the relation∫ T

0

〈u′n, v〉 +

∫ T

0

a(un, v) +

∫ T

0

dt

∫

Ω

f(un)v =

∫ T

0

dt

∫

Ω

gv

for v ∈ Lp(0, T ; V ) ∩ Lq(QT ), and using Proposition 2.5 we conclude that u is aweak solution of (1.1).

3.2. Existence of the global attractor. Let us recall some concepts and resultsrelated to the theory of multi-valued semiflows [7].

def1 Definition 3.1. Let E be a Banach space. The mapping

G : [0, +∞) × E → 2E

is called an m−semiflow if the following conditions are satisfied

(1) G(0, w) = w for arbitrary w ∈ E;(2) G(t1 + t2, w) ⊂ G(t1,G(t2, w)) for all w ∈ E, t1, t2 > 0.

It is called a strict m−semiflow if G(t1 + t2, w) = G(t1,G(t2, w)), for all w ∈E, t1, t2 ∈ R

+.

def2 Definition 3.2. The set A is said to be a global attractor of the m−semiflow G ifthe following conditions hold

8 C.T. ANH, N.M.CHUONG AND T.D. KE

• A is an attracting, i.e. dist(G(t, B),A) → 0 as t → ∞ for all boundedsubsets B ⊂ E;

• A is negatively semi-invariant, i.e. A ⊂ G(t,A) for arbitrary t > 0;• If B is an attracting of G then A ⊂ B.

The following theorem gives the sufficient conditions for the existence of a globalattractor for the m−semiflow G.

th2 Theorem 3.2 ([7]). Suppose that the m−semiflow G has the following properties

(1) G is pointwise dissipative, i.e., there exists K > 0 such that for u0 ∈E, u(t) ∈ G(t, u0) one has ‖u(t)‖E 6 K if t > t0(‖u0‖E);

(2) G(t, .) is a closed map for any t > 0, i.e., if ξn → ξ, ηn → η, ξn ∈ G(t, ηn)then ξ ∈ G(t, η);

(3) G is asymptotically upper semicompact, i.e. if B is a bounded set in E suchthat for some T (B), γ+

T (B)(B) is bounded, any sequence ξn ∈ G(tn, B) with

tn → ∞ is precompact in E. Here γ+T (B)(B) is the orbit after the time

T (B).

Then G has a compact global attractor in E. Moreover, if G is a strict m−semiflowthen A is invariant, i.e. G(t,A) = A for any t > 0.

Now we construct the multi-valued mapping

G(t, u0) = u(t) | u(.) is the solution of (1.1), u(0) = u0,

and check that G is a strict m−semiflow in the sense of Definition 3.1. Assumethat ξ ∈ G(t1 + t2, u0), then ξ = u(t1 + t2) where u(t) is a solution of (1.1).Denoting v(t) = u(t+ t2), we see that v(.) is also in the set of solution of (1.1) withrespect to initial condition that v(0) = u(t2). Therefore ξ = v(t1) ∈ G(t1, u(t2)) ⊂G(t2,G(t2, u0)). It is remain to show that G(t1,G(t2, u0)) ⊂ G(t1 + t2, u0). Ifξ ∈ G(t1,G(t2, u0)) then ξ = v(t1), where v(0) ∈ G(t2, u0). One can suppose thatv(0) = u(t2) where u(0) = u0. Set

w(τ) =

u(τ), 0 6 τ < t2,

v(τ − t2), τ > t2.

Since u and v are the solutions of (1.1), we obtain that w is a solution of (1.1) withw(0) = u(0) = u0. In addition, by the fact that ξ = v(t1) = w(t1 + t2), we haveξ ∈ G(t1 + t2, u0).

In order to show that G is pointwise dissipative, we need the following lemma(see [9]).

lm1 Lemma 3.3. Assume that y(t), t > 0, is an absolute continuous and nonnegativefunction. If there exist β > 0, γ > 0 such that

dy(t)

dt+ βyα(t) 6 γ

where α > 1, then

y(t) 6

(γ

β

) 1α

+1

[β(α − 1)t]1

α−1

.

GLOBAL ATTRACTOR FOR THE M-SEMIFLOW 9

Let u(t) ∈ G(t, u0), one gets

1

2

d

dt‖u(t)‖2

L2(Ω) + a(u, u) + C3‖u(t)‖q

Lq(Ω) 6 C4|Ω| + ‖g‖L2(Ω)‖u(t)‖L2(Ω).

Noting that

‖u‖L2(Ω) 6 C‖u‖Lq(Ω), C = C(q, Ω) > 0,

‖g‖L2(Ω)‖u‖L2(Ω) 6 ǫ‖u‖q

L2(Ω) + Cǫ‖g‖q′

L2(Ω) for ǫ > 0,

we have

1

2

d

dt‖u‖2

L2(Ω) + a(u, u) + (C3

C− ǫ)‖u‖q

L2(Ω) 6 C4|Ω| + Cǫ‖g‖q′

L2(Ω). (3.6) e12

Now removing a(u, u) in (3.6) and taking the result of Lemma 3.3 for

y(t) = ‖u‖2L2(Ω), α =

q

2, β =

C3

C− ǫ, γ = C4|Ω| + Cǫ‖g‖

q′

L2(Ω)

we deduce that G is pointwise dissipative.

The following result is the key in this section.

lm2 Lemma 3.4. G(t∗, .) is a compact mapping for each t∗ ∈ (0, T ].

Proof. Assume that B is a bounded set in L2(Ω) and ξn ∈ G(t∗, B). By the definitionof G, there exists a sequence un(t) such that un(t) is the solution of (1.1) withthe initial data belongs to B and un(t∗) = ξn.

Then we have

1

2

d

dt‖un(t)‖2

L2(Ω) + a(un, un) +

∫

Ω

f(un)un =

∫

Ω

gun. (3.7) e13

Using (1.3) and the Cauchy inequality, we get

d

dt‖un(t)‖2

L2(Ω) 6 ‖un(t)‖2L2(Ω) + ‖g‖2

L2(Ω) + 2C4|Ω|. (3.8) e14

Let dB be the diameter of B, we obtain that

‖un(t)‖2L2(Ω) 6 C = C(dB , ‖g‖L2(Ω), |Ω|, C4), for all t ∈ [0, T ], (3.9) e15

thanks to Gronwall inequality. Thus un(t) is bounded in L2(Ω) for all t ∈ [0, T ].Now integrating (3.7) by t and using (1.3) again, we arrive at

‖un(T )‖2L2(Ω) + 2

∫ T

0

a(un, un)dt + 2

∫

QT

|un|q 6 ‖un(0)‖2

L2(Ω)

+ ‖un‖2L2(QT ) + T ‖g‖2

L2(Ω) + 2TC4|Ω|. (3.10) e16

From the last inequality and (3.9), we deduce that∫ T

0

a(un, un)dt is a bounded sequence, (3.11) e17

∫

QT

|un|q

6 C = C(T, dB, ‖g‖L2(Ω), |Ω|, C4) (3.12)

or, equivalently, un is bounded in Lq(QT ). Then, using hypothesis (1.2), one can

state that f(un) is bounded in Lq′

(QT ).

10 C.T. ANH, N.M.CHUONG AND T.D. KE

Now rewriting equation (1.1) for un in dual space Lp′

([0, T ], V ∗) + Lq′

(QT ), wehave

u′n = a(un, .) − f(un) + g.

Arguing similarly as in the proof of theorem 3.1, we conclude that

u′n is bounded in Lp′

(0, T ; V ∗) + Lq′

(QT ). (3.13) e18

Putting claims (3.11) and (3.13) in proposition 2.4, we can state that un isprecompact in Lp(QT ). It follows that un(x, t) → u(x, t) a.e. (x, t) ∈ QT . Inaddition, Proposition 2.5 ensures that u ∈ C([0, T ]; L2(Ω)).

In case t = t∗, one has un(t∗) u(t∗) in L2(Ω). It remains to show that‖un(t∗)‖L2(Ω) → ‖u(t∗)‖L2(Ω).

Let us denote

Jn(t) = ‖un(t)‖L2(Ω) − 2

∫

Qt

gun − 2C4|Ω|t,

J(t) = ‖u(t)‖L2(Ω) − 2

∫

Qt

gu − 2C4|Ω|t.

Obviously, Jn and J are decreasing on [0, T ]. In addition, Jn(t) → J(t) for a.e.t ∈ [0, T ]. Suppose that tm is an increasing sequence in [0, T ], tm → t∗ as m → ∞.Then

• Jn(tm) → Jn(t∗) as m → ∞,• Jn(tm) → J(tm) as n → ∞.

So

Jn(t∗) − J(t∗) 6 Jn(tm) − J(t∗) = Jn(tm) − J(tm) + J(tm)− J(t∗) < ε, for ε > 0.

Similarly, J(t∗) − Jn(t∗) < ε. Therefore Jn(t∗) → J(t∗) and then ‖un(t∗)‖L2(Ω) →‖u(t∗)‖L2(Ω) as n → ∞.

We can now state the main result of the paper.

th3 Theorem 3.5. Under the conditions (1.2)-(1.3), the strict m−semiflow G gener-ated by (1.1) has an invariant compact global attractor in L2(Ω).

Proof. It suffices to check the hypotheses (2) and (3) in Theorem 3.2. Assume thatξn ∈ G(t, ηn), ξn → ξ, ηn → η in L2(Ω). Then there exists the sequence unsatisfying

un(t) = ξn, un(0) = ηn.

Using the same arguments as in the proof of Theorem 3.1, we have

• un → u in Lp(QT ),• un(t) u(t) in L2(Ω) for arbitrary t ∈ [0, T ] (and then u(0) = η),

• f(un) f(u) in Lq′

(QT ),

•dun

dt

du

dtin Lp′

(0, T ; V ∗) ∩ Lq′

(QT )

up to a subsequence. Hence, passing to the limit in the equality∫ T

0

〈u′n, v〉 +

∫ T

0

a(un, v) +

∫ T

0

dt

∫

Ω

f(un)v =

∫ T

0

dt

∫

Ω

gv

we conclude that u(t) is the solution of (1.1) with respect to initial condition u(0) =η. Thus, ξ ∈ G(t, η).

GLOBAL ATTRACTOR FOR THE M-SEMIFLOW 11

For the hypothesis (3), one observes that

G(tn, B) = G(t∗ + tn − t∗, B) ⊂ G(t∗,G(tn − t∗, B)) ⊂ G(t∗, B∗)

where t∗ > 0 and B∗ is a bounded set in L2(Ω). Using Lemma 3.4, we concludethat, if ξn ∈ G(tn, B) then ξn is precompact in L2(Ω).

4. Some further remarks

Using arguments as in [4], one can show that the global attractor depends upper-semicontinuously on the nonlinearity. Moreover, by arguments in [10], we can provethe Kneser property holds for the solutions to problem (1.1), which implies theconnectedness of the global attractor in L2(Ω).

The uniqueness for problem (1.1) in the case f ≡ 0 was proved in [8]. However,in our case, the conditions implied the uniqueness remain unknown. Let us take anote that, if the uniqueness of (1.1) is guaranteed by some additional hypotheses,then the m−semiflow G turn to be a single-valued one and the compact globalattractor A obtained in Theorem 3.5 is exactly the one for single-valued semiflowin the usual sense [9].

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Cung The Anh

Department of Mathematics, Hanoi National University of Education

136 Xuan Thuy, Cau Giay, Hanoi, Vietnam

E-mail address: anhctmath@hnue.edu.vn

Nguyen Minh Chuong

Institute of Mathematics, Vietnamese Academy of Science and Technology

18 Hoang Quoc Viet, 10307 Hanoi, Vietnam

E-mail address: nmchuong@math.ac.vn

12 C.T. ANH, N.M.CHUONG AND T.D. KE

Tran Dinh Ke

Department of Mathematics, Hanoi National University of Education

136 Xuan Thuy, Cau Giay, Hanoi, Vietnam

E-mail address: ketd@hn.vnn.vn