Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2013, Article ID 769514, 8 pages http://dx.doi.org/10.1155/2013/769514

Research Article

Strong Attractor of Beam Equation with Structural Damping and Nonlinear Damping

Danxia Wang,1 Jianwen Zhang,2 and Yinzhu Wang3

1 School of Mathematics, Taiyuan University of Technology, Taiyuan 030024, China

2 Department of Mathematics, Taiyuan University of Technology, Taiyuan 030024, China

3 Department of Mathematics, Taiyuan University of Science and Technology, Taiyuan 030024, China

Correspondence should be addressed to Jianwen Zhang; jianwen.z2008@163.com.cn Received 4 December 2012; Revised 18 February 2013; Accepted 27 February 2013 Academic Editor: Shueei M. Lin

Copyright © 2013 Danxia Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This paper is mainly concerned with the existence of a global strong attractor for the nonlinear extensible beam equation with structural damping and nonlinear external damping. This kind of problem arises from the model of an extensible vibration beam. By the asymptotic compactness of the related continuous semigroup, we prove the existence of a strong global attractor which is connected with phase space

1. Introduction

Global attractor is a basic concept in the study of long-time behavior of nonlinear dissipative evolution equations with various dissipation. There have been many methods to prove the existence of the global attractor. It can be proved by the theory of a-contractions of the solution semigroup S(t), such as [1-3] and the reference therein. It can also be proved by the decomposition of the solution semigroup S(t) (see Hale [4], Temam [5], etc.).

In this paper, we use the method of the asymptotically compact property of the solution semigroup S(t) which is different from the method of [1-5] to prove the existence of a strong global attractor for the Kirchhoff type equations with structural damping and nonlinear external damping which arises from the model of the nonlinear vibration beam

2 2 utt + ah u + y A ut

+ \Vu\2dx) + N(J VuVut dx))hu (1)

+ g(u) + f (ut) = h(x), in Ox R+,

u = Au = 0 on dOx R+, (2)

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

ut (x, 0) = u1 (x) in Q,

where a, y, and p are all positive constants, Q is a bounded domain of RN with smooth boundary r = dQ, M(s), N(s), g(s), and f(s) are nonlinear functions specified later, and h e L2(Q) is an external force term. u(t) represents the vertical deflection of the beam, and u = u(x, t) is areal-valued function on Qx [0, +rc>).

In this context of problem (1), based on the vibrating beams equation

\ux (s,t)\2ds)uxx = 0

which is proposed by Woinowsky-krieger [6]; Ma and Narciso [7] considered problem (1) without structural damping and posed a weak global attractor in weak phase space H¡2(0)xL (O). Eden and Milani [8] considered the existence of exponential attractor for problem (1) with f(u) = 0 and a linear weak damping g(ut) = ut, M(-) being a nonlinear function and without structural damping. Ball [9] presented the existence and uniqueness of global solutions for problem (1) with/ = g = h = 0, M(-), N(-) are all linear functions.

On the other hand, the existence of the attractor for a related problem, with the boundary conditions u = Au = 0

of (2) replaced with u = Vu = 0, was considered by Ma and Narciso [7], Eden and Milani [8] with a linear damping ut or nonlinear damping f(ut) without structural damping, respectively. Chueshov and Lasiecka [10] considered a kind of boundary condition which is u = Au = 0 but without structural damping.

Generally speaking, there have been many works on the long-time behavior for nonlinear beam equations [6-10]. But for the beam equation (1) with structural damping, in strong phase space D(A2) x H^(O) n H2(O), the global solutions and the strong global attractor have not still been proved until now.

The outline of this paper is arranged as follows: in Section 2 we give the existence and uniqueness of global solutions in space C(R+;D(A2) x H^(O) n H2(O)), in Section 3 we give the boundedness of solutions in phase space D(A2) x H^ (O) n H2(O), and finally in Section 4, we give

where <p(u) = Jn G(u)dx, G(u) = Jn g(u)du, and k1, k2 are all constants, C1 >1.

Our analysis is based on the following Sobolev spaces: H = L2(O), V = Hl(O) n H2(O), with the usual inner

u\ = (u, u)1'2, Vu, v e L2 (Q),

products and norms as follows, respectively:

(u, v) = I uv dx Jn

(Au,Av)= I AuAv dx, \\u\\ = (Au,Au)1/2, Jn

Vu, v e Hl (Q) n H2 (Q).

Consider D(A2) = {u | u e V,u e H4(O),A2u e H, Auldn = 0} with the inner products (A2u,A2u) and the norms lA2ul2 = (A2u, A2u).

Take E0 = Hi (O) n H2(O) x L2(O) and E = D(A2) x

the proof of the existence of a strong global attractor in phase H2(O) n H^(O) with the inner products and norms as

follows, respectively:

(yi,y2)E0 = (A^1,A^2) + (VV , \y\Eo = (y,y)E02,

Vyi = (uj, Vj)T, y = (u, v)t e E0, i = 1,2, (yv y2)E = (A2U, a2^2) + (AV1,AV2) , \y\E = ^ y)f,

space D(A2) x H^(O) n H2(O).

2. Some Assumptions and Existence of Global Solution

In (1), we assume that damping term and the source term are in the form of

f(ut) = \ut\rut> g(u) = \u\pu

Vyt = (ui> Vj) , y = (u, v)T e E, i = 1,2.

0 < p,

if N>3,

N-2 p,r > 0 if N=1,2.

We assume that the nonlinear functions M,N : R+ ^

Note that assumption (6) implies that H^(O)nH2(O) ^ Hl(O) ^ L2(p+1\Q.), with p = p or p = r.

Finally, we assume that X, a are the first eigenvalue of A2 and A, respectively; then we have

R+ are all class C1 ,and satisfying M(0) = 0, N(0) = 0 and

M(s)s> M (s), where M (s) = I M (z) dz,

M (s) > 2s; N (s) >s, Vs e R.

\\u\\2 > a\u\2, Vu e V,

\A2u\ >X\\v\\2, VueD(A2).

The functions f, g : R ^ R are also class C1,with f(0) = g(0) = 0, a1 < f'(v) < a2, and \g'(u)\ < k0(l + \u\p) for all u, v e R, where a1, a2, and k0 are all constants. There also exists constants k5, k6 such that

\f(u)-f(v)\ <k5 (l + \u\r + \v\r)\u- v\, Vu, v eR,

\g (u)-g(v)\<k6 (1 + \u\p + \v\p) \u- v\, Vu, v e R.

In addition, nonlinear function g(-) also satisfies

, . tt-£Y„ ll2 ,

(p(u)+--\\u\\ >-k1,

f & 2 I g(u)udx-C1 <p(u)+—\\u\\ >-k2, Jn 4

In the following, we state the result of the existence and uniqueness of the solutions for systems (1)-(3).

Theorem 1. Assume that (u0 ,u1) e E, he L2(O), and the assumptions of these functions M(-), N(-), f(-), and g(-) hold; then problems (1)-(3) have unique solutions (u,ut) e C([0, T];D(A2)) xC([0, T];H^(O)nH2(O)) depending continuously on initial data in E.

By virtue of Galerkin method, we may prove Theorem 1 combined with the priori estimates of Section 3.

According by Theorem 1, for any t > 0,we may introduce the mapping

[S(t), t > 0} : {u0,u1} —> [u(t),ut (i)|.

It maps E into itself, and it enjoys the usual semigroup properties as follows:

S(0) = I, S(t + r) = S(t)S (t) , Vt>0.

And it is obvious that the map |S(i), t > 0}, for all t e R, is continuous in space E. In the following, we will introduce the existence of bounded absorbing set and global attractor in space E for map |S(i), t > 0}.

3. The Existence of Bounded Absorbing Set in Space E

In this section, we will show boundedness of the solutions for systems (1)-(3).

Theorem 2. Assume that these assumptions of Theorem 1 hold then for the dynamic system determined by problems (1)-(3), there exists the boundary absorbing set in space E.

Proof. Taking the inner products of v = ut + eu with both sides of (1) and then making summation, we have

^Ш^(^) - zlvl2 + £(a -+ £P\^U\2 + y!v\2

+ e2 (u, v) + sM (z) + N(z)z + (g (u), v) (15)

+ (f (ut), v) = (h, V),

where M(z) = j M(z)dz, z(t) = \Vu\2 and e is fixed at arbitrary time, and here the energy function E(t) is defined on E0 by

E (t) = \v\2 + (a- ey) \\u\\2 + p\Vu\2 + M (z) + \Уи\4. (16)

where % among 0 and v - eu. Set

E(t) = \v\2 + (a - ey)\\u\\2 + p\Vu\2 + M(z) + \Vu\A +2<p(u) +2k1.

Consider

2 e2 ea га2 \ 2

2 T 2a2

Y (t) = ( ea- e у

I ,2 £2 У°2 ш2\ , ,2

+ (yA2-e-4+ai )\v\2

+ eß\4u\ + eM (z) + N(z)z + eC1f (u) + ek1.

So (15) is transformed into

1-^-E(t) + Y(t)<^1 \h\2 + ek2 + eki. (22) 2 dt yo2

Considering the assumptions M(s)s > M(s), M(s) > 2s, N(s) > s, \\uf > a2\u\2, and \A2u\2 > X2\\u\\2, C1 > 1 and letting 0 < e < min|(aa2 + 2a2)/(2ya2 + 4),-(3 + a2) +

^(3 + a2)2 + (4a1 + 3ya2)} = e0,we have

Y(t)-E(t) > 0.

Considering the assumption Jn ug(u)dx - Cicp(u) + Substituting (23)into (22), we have

(а/4)||м|| >-fc2,wehave

(g (u), v) = (и) + e(g (и), u) d

> —ф (и) + еС1Ф (и) dt

Ш11 Ii 2 г

--ЦиЦ - ек2.

With \е2(и, v)\ < (е2/а2)ЦиЦ2 + (e2/4)\v\2,wehave - e\v\2 + е (а - еу) ||м||2 + eß\Vu\2 + y\v\2 + е2 (и, V)

1^E(t) + -E(t)<-^r\h\2 + гк2 + гк1. (24)

о Ji о у02

2 dt 2

>(еа-е2у-^ ) Ци\\2 + ( уХ2 - е - ^ ) \ v\2 (18)

+ £p\Vu\2.

With the assumptions f(0) = 0, f e C1(R,R), and a1 < f'(v) < a2 and by using Mean Value Theorem and Mean Value inequality, we have

(f(ut), v)= | f' (0 v2dx-e^f (Z)uv dx that is,

On the one hand, applying the Gronwall inequality to (24), we get

E(t)<E(0)e-£t + -(-^r\h\2 + ek2 + ekA , t > 0. (25) e \ya2 /

Note that \\w(0)\\ and \wt(0)\ are bounded; then there exists a positive constant R> 0 such that E(0) < R2 is bounded; so

limsupE(t) < pi = - \h\2 + ek2 + ek1). (26)

t^rn £ \ytJ )

On the other hand, considering that (p(u) + ((a-ey)/8)\\u\\2 > -k1, fixing ^0 > p0, and assuming that E(0) < R2, then as t>to = to(R,po) = (1/£o) logW^2 - Po)), we have

> [a, -

2 \ I 12 ea2 и i|2 V M

E(t) <

, l2 a-ev 2 2

\v\ + m <p0.

Take the inner products by A2v in both sides of (1); then make summation to get

IjAWvf + (a-£Y)\A2u\2)

2u\2)-£\\••'|2

-(¡3 + M(z (t)) + N(Z (t))) (Au, A2v)

+ y\A2v\ + e2 (a2u, v) + eM (z) + N(Z)Z

+ (g(u),A2v) + (f (v - eu), A2v) = (h, A2v) .

Considering the continuity of the functions M'(-) and N' (■), we have

-(¡3 + M(z (t)) + N (Z (t))) (Au, A2v) >-(p + C2V0 + C3^20)lAul\A2v\

{ß + c2^l + c3h0,)h0, y

_^\A2v\

where C2, C3 are all positive constants. Also

£2 (a2u, v) > -^\a2u\2 _—\\v\\2, y ) q2 114

(h, A2v) = ^ (h, A2u) + £ (h, A2u) .

In addition, with lg'(u)l < k0(1 + lulp), there exists a constant k3 such that lg(u)lL™ < k3, lg'(u)lL™ < k3; so

(g(u),A2v)

= It (0 (U), a2v) - (9' (U) Ut, A2u) + e(d (u), A2u)

> dt(0 (U), A2u) + £(0 (U), A2u) _ ~ I A'

^ 1 . 2 12 _ 2k3Vo

Also by using Schwarz and Mean Value inequalities and Mean Value Theorem, we have

(f (v _ £u), A2v)

y\ A 2 ¡2 1 < l\a2v\ + -

41 1 y

(f (V)2 (v _£u)dx

<|Ia2v|2+ j ,

where V among 0 and v _ £u. Set

22 2 £ £

Yi (t) = [£(X_£ y_ — _ — ) \ A u\

■ e(g(u), A2u)

(h, A2u),

andwriteM = (£2/y)(1 + 3£2)^2 + ((ß + C2Po + C3^o)Po)ly + K2..2 1 2

2k3^2l£2; then (29) is transformed into

--Ei (t) + Yi (t)<M. 2 at

Here the function

E1 (t) = \\v\\2 + (a- ey)\a2u\ + 2(g (u), A2u) + 2 (h, A2u)

is obtained by the energy function being changed slightly. Let 0 < e < min{e0,2a/(y + (2/a2) + (1/4)),-3 +

^9 + 2yX2, a/(y + (1/8))}, we have Y1(t) > (e/2)E1 (t), and so

--E, (t) + -Ex (t)<M, Vt>tQ(B). (37)

Then an application of the Gronwall inequality leads to

E1 (t) < E1 (0) el-£(t-t°)]

Vt > t0 (ß).

If B c BE(0, p), there exists a positive constant R1 > 0 such that E1(t0) < R2.

Putting t1 satisfing t1 - t0 > (1/e) log .Rp then as t > t1, we get

E1 (t) < R2e

2e-ex(1/e) log $ + 2M = 1 + 2M

£ \ I 2 ¡2 2 K_£y_8)\Au\ + \\v\

16n ,2 2,™ , 2M < —\h\2 + —ki \Q\ + 1 +-.

£ £ 3 £

The global estimate (40) shows the existence of an absorbing set of S(t). □

4. The Existence of Global Attractor in Space E

The general theory [11] indicates that the continuous semigroup S(t) defined on a Banach space X has a global attractor which is connected when the following conditions are satisfied.

(i) There exists a bounded absorbing set B c X such that for any bounded set B0 c X,

dist (S (t) B0, B) —> 0, as t —>

(ii) S(t) is asymptotically compact; that is, for any bounded sequence {un} in X and {tn} tending to >x>, there exists a subsequence {n} such that {S(tni )uni} is convergent as n ^ <m.

Theorem 3. Under the assumptions of Theorem 1, the continuous semigroup S(t) has a global attractor which is connected to E.

Proof. Let u, v be two solutions of Problems (1)-(3) in space C(R+;E) as shown above corresponding to the initial data (u0,u1) and (v0, v1) with \(u0,u1)\2E + |v0, v1\2E < R2, respectively. Then w = u - v satisfies

wtt + aA2w + yA2wt - pAw

= (M (\VU\2) AU-M (\Vv\2) Av) + ( N (J VuVut dx)Au-NQ VvVvt dx) Av

-(g (u)-g(v))-(f(ut )-f(vt)),

\(w,wt)\2Eo <C(y2)

Taking the inner products in both sides of (42) by wt, Aw, and Awt, respectively, we have

\jt(\™\ + «\Aw\2 + p\Vw\2) + y\Awt\2 = (M (\Vu\2) Au-M (\Vv\2) Av, wt)

+ (n(J VuVut dx)Au-N(J VvVvt dx)Av,wt + (g (u) - g (v) + f (ut) - f (vt), wt),

(y\A2^\2 + 2 (Aw, Awt)) + a|A2w|2

- p(Aw,A2w) + \Awt\2 = (M (\VU\2) AU-M (\Vv\2) Av, A2W)

+ (n (J VuVut dx)Au-N(| VvVvt dx) Av, A2w

+ (g (u) - g (v) + f (ut) - f (vt), a2w) ,

±- (\Awt\2 + a\A2w\2) + y\A2wt\2 - p (a2w, Awt) = (M (\Vu\2) Au-M (\Vv\2) Av, A2wt) + (n (J VuVut dx)Au-N(J VvVvt dx) Av, A2wt) + (g (u) - g (v) + f (ut) - f (vt), A2wt) .

Equation (46)+ k x (45) +kx (44) yields

1 d ( \ a \2 \A 2 \2

— — ( \Au>J + a\A w\ 2dt\ \ ^ \ \

+ fcy\A2w\ +2k(A2w,wt) +k\wt\2 +ka\Aw\2 + kp\Vw[

+ y \ A2wt \ +fca\A2w\ +fc\Awt\2 + ky\Awt

= (g (u) -g (v) +f (ut) -f (vt), A wt +kA w +kwt

+ (m (\Vu\2) Vu - M(\Vv\2) Av

+ n(J VuVut dx) Au

-n(j VvVvt dx) Av, A2wt)

+ k(M (\VU\2) AU-M (\Vv\2) Av

+ N (J VuVut dx) Au

-N(J VvVvt dx) Av, A

+ k (M (\VU\2) AU-M (\Vv\2) Av

+ n(j VuVut dx) Au

-n(j VvVvt dx)Av,wt + (¡3Aw, A2wt) + k$ (Aw, A2w) .

Consider that

(M (\Vu\2) Au - M(\Vv\2) Av, A2wt)

= M' (%) \Vu\2 J AwA2wtdx + M' (%) Jn

x I Vw (Au + Vv) dx I AvA2wtdx Jn Jn

< C2V0 ^\A2wt\ + 2C2^2 \V^\\A2wt\

('Cjtà + fatifa,2 y \ * 2 f

< 2-\Aw\ + -\A wt\ ;

y 1 1 8\ t\ '

k (M (\Vu\2) Au_M (\Vv\2) Av, A2w)

= kM' (q0)\Vu\2 I AwA2wdx + kM' fa)

x I Vw (Au + Vv) dx I AvA2wdx

< kC2 \Aw\\A2w\ + \Aw\ \A2w\) [C2k^02 (1 + (2la))]

n a ka\.2 ¡2 \Aw\ + — \A w\ ;

k(M(\Vu\2)Au_M(\Vv\2) Av, wt)

= Um' (n0)\Vu\2 { Awwtdx + kM' fa)

x I Vw (Au + Vv) dx I Avwtdx

< lC2 (rf, \Aw\ \wt\ + \Aw\ \w,

C^l (1 + (2/a))

2 I \2

-\Aw\ + \wt\ ,

where q0 is among 0 and lVul2, is among lVul2 and |Vv|2, and

N( I VuVutdx)Au_ N( I VvVvt dx) Av, A2w

N' (V0) \ VuVut dx \ AwA2wt dx + N' fa)

x (J VvVwt dx + J VwVut dx

< C3 [d \Aw\ \A2wt\ + (^0\wt\ + d0 \Aw\)^0 \A2wt\]

x I AvA wtdx

-3 I 1

y V I y

k(N(] VuVutdx)Au_ N( I VvVvtdx) Av, A2w

kN' fa)[ VuVutdx[ AwA2wdx + kN' fa)

x (I VvVwt dx + I VwVut dx

<kC3 \Aw\ |A2w| + fa \wt\+^0 \Aw\)d0 |A2w|]

x I AvA wdx

(4C3^)k . la 2 .2 (4C3H°)k

-\Aw\ +--\A w\ +--1

a 8 1 1 a

k^Q VuVutdx)Au_N(I VvVvtdx) Av,wt

= kN' fa) I VuVut dx I Awwt dx + kN' fa) Jn Jn

VvVwt dx + I VwVut dx Jn

x I Avwtdx

< №3 \Aw\ \wt\ + fa\wt\ +^0 \Aw\)d0 M]

< kC3d0\Aw\2 + kC^^w^ + kC^^w^,

where ^0 is among 0 and Jn VuVutdx, ^ is among Jn VuVutdx and Jn VvVvtdx.

Also considering lg(u) - g(v)l < k6(1 + lulp + |v|p)|w -v| for all u, v e R, lf(u) - f(v)l < k5(1 + H + lvlr)lu - v|, for all u, v e R, and p/(2(p + 1)) + 1/(2(p + 1)) + (1/2) = 1, r/(2(r+1)) + 1/(2(r+1)) + (1/2) = 1,byHolder inequality we have

pl(2(p+1))

_ (g (u) _g(v)+f (ut) _ f (vt), A2wt) <k6 I (1 + \u\p + \v\p)\w\\A2wt\dx

+k5 L (i+^r+\vt\r)\wMA2wt

<k6\In (1 + \u\p + \v\p)2(p+1)lpdXj x \w\2(p+1) \A2wt\ +ML (1+^+v\T+1)lr

x \wt\2(r+1) \A2wt\

< C fa) \Vw\\A2wt\ + C fa)\VwtI\A2wt\ _ (g (u) _g(v)+f (ut) _ f (vt), kA2w)

rl(2(r+1))

+ \u\p + \v\p)\w\\A2w\ dx

1 + \uf\r + D \wt\ \A2w\ dx

+ \u\p + \v\pfp+1)lpdx

pl(2(p+1))

x \w\2(p+1) \A + kk

[In (1 + ^ + \vt\r)2(r+11'r dx

x \wt\2(r+1) \A2w\

rl(2(r+1))

< kC (d0) \w\2(p+1) \a2w\ + kC (d0) \wt\2(r+1) \a2w\

< kC (to) \Vw\\A2w\ + kC (to) |Vwt|\A2w\;

- (g (u) - g (v) + f (ut) - f (vt), kwt

<kk6 L (1 + \u\p + \v\p)\w\\wt\dx Jn

+ kk5 L (1 + I ut \r + \ vt \r) \ wt \ \ wt \ dx

L (l + \u\p + \y\pfp+1)lp dx Jn

pl(2(p+l))

\l(p+l)

+ kk„

. ,r . ,r\2(r+l)lr 1 + \Ut\ + \Vt\ )

rl(2(r+l))

X \Wt\2(r+l) \Wt\

< kC (ft0)\w\2(p+1) \wt\ + kC (ft0)\wt\2(r+1) \wt\

< kC (ft,) \Vw\ \wt\ + kC (ft)\Vwt\ \wt\. Setting

E2 (t) = \Awt\2 + a J a2w J 2 + ky\a2w\2 + 2k (a2w, wt)

+ k\wt\2 + ka\Aw\2 + k.ß\Vw\2 + ß\VAw\2, Y2 (t)=2\A2Wt\2 + ^\a2u,\2 +\\Awt\2

+ Y\Awt\2 + kp\VAw\2,

then substituting (48)-(56) into (47), by Schwarz inequality and Young inequality, and taking k > (4C2(ft))/ya2 and k > (8kC2(ft))/aa2y, we have

(t) + Y2 (t)<c(\Aw\2 + \wt\2).

Again setting £ = max{4/a + 4/k + 4y/a,2/k,(4/X2 + 4/a\4)(k/k),2/yX2 + (2/y\2)(k/k)}, and considering that -2k(A2w,wt) > -k\A2w\2-k\wt\2,wehave%Y2(t)-E2(t) > 0. On the one hand, from (58) we have

l^ + h (*) < c(\Aw\2 + \Wt\2). Applying the Gronwall inequality to (59), we get

E2 (t) < E2 (0) e

<2/Ç)t

cfo e-(2lOt (\\w(t)\\2 + \wt (r)\2)dr.

On the other hand, with (2kwt,A2w) > -(ky/2)\A2w\2 -(4k/y)\wt\2 and setting k > 4k/y, we get

E2 (t) > |Au>t| + rnA w\ .

\w,wt\2E < CE2 (0)e-2m

e-m)t (\Aw (t)\2 + \wt (T)\2)dr.

Now, let {(u0m, ulm)} be a bound sequence in B0 c E, and [um(t),umt(t)} the corresponding solutions of problems (1)-(3) in C(R+, E). We assume tn > tm. Let T > 0 and tn, tm > T. Then, applying estimate (62) to wm'n = un(t + tn -T)- um(t + tm -T), t > 0, we have

\( m,n m.n\ \(W ,Wt )e

<CC(ft0

-(2/Qt

X sup \(Un (tn -T + S)- Um {tm -T + S)) ,

(unt (tn -T + S)- Umt (tm -T + S^^ By taking t = T in the above, we have

\(Un (tn)-Um (tm),Unt { n

<CC(ft)e-(2/VT + C (ft)

X sup \(un (tn + s)-Um (tm +s)),

{unt {tn + s)-Umt {trn + S))\e0.

By Sobolev embedding Theorem, for any T > 0, we can extract a subsequence {(uni,un/t}which is convergent in C([0, T];Eo) for any T > 0. For any e > 0,we first fix T > 0 such that

CC (fto

e-(2K,)T < £ 2'

And, next, taking large m ,n , we have

C (ft0) sup \(Un (tn + S)-Um (tm + S) ,

0< s< T

Unt (tn + s)-Umt (tm + s))Ie0 < 2.

(59) Then by (62) we have that

\ {un' {tn') - Um> {tm' ) > Un't {tn') - Urn't {trn' < £. (67)

We conclude that S(t) is asymptotically compact on E. The theorem is now proved. □

Acknowledgments

This work is partially supported by the Natural Science Foundation of China (11172194) and Shanxi Province (2011021002-2 and 2010011008). The authors also wish to give their thanks to the referees for their comments to improve the presentation of this paper.

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