Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2014, Article ID 315318, 11 pages http://dx.doi.org/10.1155/2014/315318

Research Article

Pullback D-Attractor of Coupled Rod Equations with Nonlinear Moving Heat Source

Danxia Wang, Jianwen Zhang, and Yinzhu Wang

Department of Mathematics, Taiyuan University of Technology, Taiyuan 030024, China Correspondence should be addressed to Jianwen Zhang; jianwen.z2008@163.com Received 12 February 2014; Accepted 9 May 2014; Published 3 July 2014 Academic Editor: Bo-Qing Dong

Copyright © 2014 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.

We consider the pullback D-attractor for the nonautonomous nonlinear equations of thermoelastic coupled rod with a nonlinear moving heat source. By Galerkin method, the existence and uniqueness of global solutions are proved under homogeneous boundary conditions and initial conditions. By prior estimates combined with some inequality skills, the existence of the pullback D-absorbing set is obtained. By proving the properties of compactness about the nonlinear operator g1(-), g2(), and then proving the pullback D-condition (C), the existence of the pullback D-attractor of the equations previously mentioned is given.

1. Introduction

In this paper, we consider a thermoelastic coupled rod system: utt - pAu + yut + V9 + g1 (u) = f (x, t), (1)

dt - kAd + Vut = g2 (d) + Q (x, t), (2)

V0L = °

x e dû,

u (x, t) = uQ (x), ut (x, t) = pQ (x), d (x, t) = 90 (x) ,

x eQ. (4)

with an external force function and a nonlinear moving heat source function. Here u(x, t) is the rod elastic displacement. d(x, t) is the dimensionless temperature. p, y, k are all positive constants, where p is the square of wave velocity, y is the damping coefficient, and k is the thermal diffusivity. Q c R is abounded smooth domain. f(x, t) is the external force and f(x, t) is locally square integrable with respect to time for t e R, x e Q; that is, f(x,t) e L2loc(R,L2(Q)). Q(x,t) is the moving heat source and Q(x, t) is locally square integrable in time for t e R, x e Q; that is, Q(x,t) e L2loc(R,L2(Q)). gl (u) and g2(d) are all the nonlinear function, and gl (u) and

g2(0) are continuous on R, respectively. We give the pullback D-attractor for the nonautonomous nonlinear equations of thermoelastic coupled rod in space E1 = D(A) xVxV, where A = -A, V = H1(Q.).

Recently the research of the nonautonomous infinite dimensional dynamical system has been paid much attention and developed fast as evidence by the references cited in [17]. Chepyzhov and Vishik [1] firstly extend the notion of global attractor in the autonomous case to the concept of the uniform attractor for the nonautonomous case. But the uniform attractor is not applicable to the nonautonomous systems in which the trajectories can be unbounded as time increases to infinity. Therefore some new concepts and theories must be brought up for such nonautonomous case, where the concepts and the theorem of existence of the pullback D-attractor were advanced in [2-8] and so on.

Caraballo et al. [2] and so forth gave the existence of the pullback D-attractor for a nonautonomous N-S equation under the assumptions of asymptotic compactness and existence of a family of absorbing sets. Wang and Zhong [3] advanced the existence of the pullback D-attractor for the dissipative Sine-Gordon wave equation in an unbounded domain in which the external force did not need to be bounded. In [4, 5], The author studied the pullback attrac-tor of the reaction-diffusion equation and the generalized Korteweg-de Vries-Burgers equation, respectively. S. H. Park

and J. Y. Park [6] considered the nonautonomous modified Swift-Hohenberg equation

ut + A2u + 2Au + au + b\Wu\2 + u3 = g (x, t)

and proved the existence of the pullback attractor when its external force has exponential growth. The abovementioned systems are all specific systems. For the widespread used nonautonomous structural system in engineering, the study has been paid less attention. Park and Kang [7] studied the existence of the pullback D-attractor for nonautonomous suspension bridge equation because of being motivated by Ma et al. [8, 9]:

+ A u + put + ku+ + g (u) = f (x, t).

In this paper, based on Al-Huniti et al. [10] as the relaxation time f is not considered and Carlson [11], we study a more general nonlinear thermoelastic coupled system (1)-(4) of a rod due to a nonlinear moving heat source Q(x, t). We give the existence of a pullback D-attractor of above system by proving the existence of a pullback D-absorbing set and pullback condition (C) for the external force f(x, t) unnecessarily bounded.

In fact, we assume that the external forces f(x, t) and Q(x,t) satisfy f(x,t) e L2loc(R, L2(ty), f'(x,t) e L2loc(R, L2(Q)), and Q(x, t) e l2oc(£,L2(Q)),respectively,and for any t e R

f e& (|/(s)|2 + |/ (s)\2 + \Q(s)\2]ds<™,

where S > 0 is a small real number which will be characterized later.

On the assumptions of the nonlinear function g1(-), Park and Kang gave the assumption

lim sup

jsj ^ OT

\g[ (s)|

(where 0 < y < to) for nonautonomous suspension bridge equations in [7]. At present, we remove the assumption of [7] and we assume that the nonlinear function g1(-) e C2(R, R) satisfies the following assumptions:

(H1) we denote by G(s) the primitive of ^j(s);thatis,G(s) = f g1 (r)dr, and then

lim inf

jsj ^ OT

\G(S)\

(H2) g1(u) < 1 + \u\p+1 for some 0 < p < <m; (H3) g[(u) < C'(1 + \u\p) for some 0 < p < <m; (H4) there exists a constant C0 > 0 such that

lim inf S01 (s)-C0G(s) ^ 0;

(H5) gi(0) = 0.

We also assume that the nonlinear function g2(-) e C1(R,R) satisfies the following assumptions: there exists a constant a1 such that

g2 (0) = 0, \g'2 (s)I < a^, Vs e R.

Throughout this paper, we introduce the spaces H = L2(Q.) and V = Hq(Q.) and endow these spaces with the usual scalar products and norms (•, ■), \ ■ \, ((■, ■)), || • ||, where (u, v) = Jn uv dx, ((u, v)) = f VuVvdx. Because of defining A = -A, with reference to [7] we have the scalar products (Au, Au) and norm \Au\2 in the space D(A). By the Poincare inequality, there exists a proper constant A1, A2 > 0 such that

\\u\\2 > X 1\u\2, Vu e V, \Au\2 > X2\\u\\2, VueD(A).

2. Pullback DSEi-Attracting Set

By normal Galerkin method (see [1, 12-14]), we have the following theorem of existence and uniqueness of solutions to problems (1)-(4).

Theorem 1. Assume that p, y, k > 0, f(x,t), Q(x,t) e L2loc(R,H) and the assumptions of the functions g1 (■), g2() hold; then, for all T > 0 and any given u0 eV, p0 e H, 90 e H, problems (1)-(4) have a unique solution (u, Q) such that

u e C0 (RT; V) n C1 (RT; H), d e C0 (RT; H) , (13)

where Rr = [t, to}.

Moreover f'(x,t) e L2loc(r,T;H),forallT > 0, u0 e D(A), p0 e V, 90 e V; then

ueC0 (Rr;D(A))nC1 (Rr;V), 0 e C0 (Rr;V). (14)

For simplicity, we write y(r) = (u(r),dru(r),0(r)) = (u(r),p(r),d(r)), y0 = (u0,p0,90). We denote by E0 = Fx H x H the space of vector functions y(r) = (u(r), p(r), 9(r))

with the norm

= \\u\\2 + \p\2 +

in E0 and denote

by Ex = D(A) xV xV the space of vector functions y(r) = (u(r),p(r),G(r)) with the norm \\y\\Ei = \ Au\2 + \\p\\2 + ||6>||2 in E1. We can construct the nonautonomous dynamical system generated by problems (1)-(4) in E0 or E1. We consider Q = R, dtt = t + t, and then we define

T,y0) = y (t + T, T,y0)

= (u(t + T),p(t + T),G(t + T)], (15) t e R, t>0, y0e E0,E1. The uniqueness of solutions to problems (1)-(4) implies that ®(t,T,yQ) = ®(t,s + T,®(s,T,yQ)),

t eR, t>0, y0 e E0,E1.

And, for all t e R, t > 0, the mapping 0(t, t, ■) : E0 ^ E0 (or E1 ^ E1) defined by (15) is continuous. Consequently, the mapping 0(t, t, ■) defined by (15) is a continuous cocycle on E0 or E1.

Let R be the set of all functions r : R ^ (0, such

lim eStr2 (t) = 0,

where 0 < S < 2a1 and a1 = min{3al16,yl2,kAl4,aC0l2] and DS Eo denotes the class of all families D = {D(t); t e R] с P(E0) such that D(t) с B(0,rß(t)) for some rß e R, where B(0,rß(t)) denotes the closed ball in E0 centered on 0 with radius rß(t).

Theorem 2. Assume that ß, y, k > 0 and the assumptions of the functions g1(-), g2(-) hold. Suppose that f(x,t) e L2loc(R,H) and Q(x,t) e L2loc(R,H) satisfy (7). Then there exists a pullback DSEo -attracting set in E0 for the nonautonomous dynamical system (в, Ф) defined by (15).

Proof. Let t e R, t > 0, and y0 = (u0, p0, в0) e E0 be fixed. Define

u(r) = u (r, t -t, u0), p (r) = u (r, t - t, P0) , ef(r) = e(r,t-T,e0), for r > t - T, (u (r), p (r) , e(r)) = Ф (r - t + T,t - T, y0)

for r > t - T.

Taking the scalar product in H of (1) with v = u + au and taking the scalar product in H of (2) with Q, after a computation of addition, we obtain

u\\2 + |v|2 + \в\~ ) + y\v\'- a\v\

+ aß\\u\\2 + к||в|| + a2 (u, v) - ya (u, v) + a (ve, u) + (g1 (u), v)

= (f, v) + (g2 (в), в) + (q, в).

For simplicity, define <(u) = Jn G(u)dO.. By the assumption (H1), it is obvious that <(u) > 0. By the assumption (H4) of g1 (■), we have

(g1 (u),u)-C0 I G(u)dn + —\u\2 >-M, (20) Jn 16

( g1 (u) , v)

= (gi (u), u) + a (gi (u),u)

> [ —G(u)dO. + aC0 [ G(u)dO. Jn dr Jn

A 2 -a—\u\ - aM 16

> —ф (u) + aC0(^ (u) - a—\u\ - aM.

Considering assumption (11) of g2(-), we have

\(g2 (Q),Q)\^\g2 (Q)||Q i^Q2-

By the Young inequality and (12), we have

U V)|S;I/I" + llv\2;

(21) (22)

22 2 a 2 a 2

a (u, v) > - — \u\ - —\v\ ;

I w 2У2<x\ |2 aA 2 - ya (u, v) >--— \v\ - — M ;

/ ^r \ a \x\2 aß ,,2

a(ve, u) >--\в\ --!-\\u\\2.

v ' 2ß\ \ 2 11 11

Letting 0 < a < min{^—/2 + A/4,-(1 + 2y2/—) +

^(2y2/— + 1)2 + y/2,k—p/2,1} and taking a1/k < A/4 and fi > 1, we infer from (19) that

\-(ß\\u\\2 + M2 + \в\2 + 2фЫ)]

+ ^ + 2м2 + Т^ + a^(u) (24)

<~\f\2 +aM+±\Q\2.

1\f\~ +aM + y \ A k

Also taking a1 = min{3a|16, y/2, kA/4, aC0/2], we have

(ß\\u\\2 + \v\2 + \в\2 +2ф (u)) \2

+ 2a1 (ß\\u\\2 + \v\2 + \в\ + 2ф (u)) (25)

< -\f\2 + 2aM+—\Q\2-

Note that

dd~eSr (ß\\u\\2 + \v\2 + \в\2 + 2ф (u))

SeSr (ß\\u\\2 + \v\2 + \в\ +2ФМ)

+ eSrddr(ß\\u\\2 + M2 + \в\2 + 2ф (u

and by (25), we have

^ (flmI2 + |v|2 + |0|2 + 20(«))

< (5 - 2ax) e5r (^||w||2 + |v|2 + |0|2 + 20 («)) (27)

+^ + 2^+,|lQI2).

By integrating (27) over the interval [i - r, i], we obtain e5t (^|M(i)|2 + |v(i)|2 + |0(i)|2 + 20(M)) <e5(t-r) (^|M(i-r)|2 + |v(i-r)|2

+ |0(i-r)|2 + 20(w(i-r)))

(* Ss ( 2 | r|2 2 ,2\ , 2«M / St S(t-r)\

+ L6 (y|y| -e )

+ J^ (5-2ai)e5s (£||M(S)||2 + |v (S)|2

+ 20 (M (S))) ds.

Since 5 < 2a1, we have 0||M(f)||2 + |v (i)|2 + |0(i)|2 + 20(«(i)) <e-5r (^|M(i-r)||2 + |v(i-r)|2

+ |0(i-r)|2 +20(w(i-r))) -St f s^2| p 2 2aM / -5tn

Note that

||"||2 + |pr + |0|2 +

< (l + f)(№l|2) + 2|v|2 + |0|2 + 20(M).

If we take = max{2,1 + 2a2/A^|, we infer from (29) that ||w||2 + |p|2 + |flf + 20(M)

<Ci (^||M||2 + |v|2 + |0|2 + 20(«)) <Cie-5r (^|M(i-r)|2 + |v(i-r)|2 + |0(i-r)|2 +20(w(i-T

+ 2C^iM (i - e-är)

<CiC2e-dT [||w(i-T)||2 + |p(i-r)r

+ |0(i-r)|2 + 20(M(i-r))]

Let DS>B be given. For all - r) = 6 ^ - T), * 6 R and r > 0, from the assumption (H4) of ^1(-), we know that 0(w(i - r)) is bounded. So we easily obtain from (31)

II® (^-*,*>)£,

<CiC2e-äT [||w(i-T)||2 + | p(i-r)| 2

+ |0(i-r)|2 + 20(w(i-r))]

for all e D(i - r), (6 .R, and r > 0. Set

fo (i))2 = 2Cie-St J'^ e& (^|/|2 + ¿|Q|2) ds

and consider the family £S)E of closed balls in £0 defined by Bs(i) = jy 6 £0, ||y||£ < £s(i)}. It is easy to checkthat 6 Ds E and is pullback Ds E -absorbing for the cocycle O by

(15).° ' ° □

In order to prove the pullback -DSEi-attractor, let .Rs be the set of all functions r : R ^ (0, +ot) which satisfies (17) and -Ds>Ei denotes the class of all families D = j-D(i); ( 6 i} c P(£1) such that D(i) c 1(0, r6(f)) for some r5 6 R, where 5(0, rg(i)) denotes the closed ball in £1 centered on 0 with radius rg(i).

Theorem 3. Assume that p, y, k > 0 and the assumptions of 9\(•), 02(0 hold. /(x,f) 6 L]oc(R,L2m,f'(x,t) 6 i20C(^,L2(Q)), and Q(x,i) 6 l20C(£, L2(Q)) satisfy (7). Then there exists a pullback -DSEi -attracting set in £1 for the nonautonomous dynamical system (0, O) defined by (15).

Proof. Let t e R, t > 0 and y0 = (u0, p0,d0) e E1 be fixed. Take the scalar product in H of (1) with Av = Au' + aAu, and take the scalar product in H of (2) with A0; then make summation to get

1-j^(ß\Au\2 + \\V\\2 + \\e() + YM2 -a\\y\\2 + aß\Au\2 + k\Ad\2 + a2 ((u, v))

- ya (Au, v) + a (a1/2Q, Au) + (g1 (u), Av) = (g2 (d),A0) + (f,Av) + (Q,Ad).

(0i (u), Av)

= (g1 (u), Au') + a (g1 (u), Au)

= -—_(91 (u), Au) - (g1 (u) u , Au) + a (g1 (u), Au);

(f,Av)

(f, Au) + a (f, Au) - (f', Au) ,

we infer from (34) that

1- (ß\Au\2 + \\V\\2 + ||e||2 + 2 (gi (u) ,Au)-2 (f, Au))

+ y\\v\\2 - a\\v\\2 + aß\Au\2 + k\Ad\2 + a2 (Au, v)

- ya (Au, v) + a (a1/2q, Au) + a (g1 (u), Au) - a (f, Au)

< - (f', Au) + (Q, Ad) + (g2 (d) , Ad) + (g'1 (u) u, Au) .

2 aß 2 4a3 2

a2 (Au, v)>--6\Au\2 - ~^\v\ ;

aß 2 4y2a 2 - ya (Au, v)>--L\Au\2 -^M2;

a(A1/2 (Q),Au)>-4a\A1/2d\2 -a6\Au\2

-(f'>Au)<a-ß\f'\2 + aß\Au\2;

(Q,A8)<2k\Q\2 + k\A8\\ (g2 (d),Ad)<k\g2 (0)|2 + k-\Ae\2

and consider the assumption (H3) of g1(-) combined with Sobolev-embed theorem

\(g1 (u)u,Au)\

= | g1 (u)u Audx Jn

< \C'\ I 1(1 + \u\p)u'Au\dx

8C ~äß

\pl2 + aß\Au\2 + \u\2»(u')"dx

16 aß Jn v 7

8C'2. ,2 aß 2 (4C'2\

+T6\Au\ +y\0ß)

\\2 + l\\p\\2

8C21 ,2 aß 2 (4C'2\ -Oß\p\ +T6\M +y(-aß)

4ci2\2„ ..2

+ jM2 + a2T\\u\\2,

and then we infer from (36) that IA (p\Au\2 + \\v\\2 + ||0||2 + 2 (gi (u), Au) - 2 (f, Au))

+ y\\v\\2 - a\\v\\2 + ^\Au\2 + 4\A£\

11 aß

3k\ ,~\2

- jaWdf +a(g1 (u), Au) - a (f, Au)

<i\fr+kkQ+> ß

2 4a3 + 4ay2 2 ' \v\2

a2 (4C'2^2 — +

2 \ aß

2 8C'2 \P\2-

Let 0 < a < min{3kXß/(16 + 2ß),3y/4}. By the Gronwall lemma we have from (39)

ß\Au\2 + \\v\\2 + llell2 + 2 (g1 (u), Au) - 2 (f, Au)

< e~m ( ß\Au0\2 + ||Vo||2 + ki

+ 2 (g1 (uo), Auo) -2(f(t-r), Auo) )

-a(t-s)

a-ßifi2+44 pi"

8 (a3 + ya2)

-a(t-s)

a2 4C'2"" — +

" 8C \P\"

ds. (40)

Considering that

2 (M) , AM) > -1 |Au|2 - 4 [ (w)|2dx; 4 Jn

2 (/, AM) > -^Awl2 -4|/|2

2C1C2aM a<?

[ (w)|2dx< [ (l + |w|(p+1))2dx<2|Q|2 + 2||w||2, andthenwehavefrom(42)

by the assumption (H2) of ^1(-), we have

|Aw|2 + ||v||2 +

< 2e^T ( ^|AM0|2 + ||Vo||2 + ||0Q||2

+ 2 (^i (mq) , Amq) - 2 (/ (i - r), Amq) )

+ 4 | e Jt_r

-a(t_s)

+ ^ + T A'

4 (a3 + ay2)/ . 2 2a:

2|p|2 + ^N1'

a2 4C'' +

8C'2, ,2

2 . ol r|2

■ ds

16|Q|2 + 16||w||2 + 8|/|2.

C3 = max

ßA + 2 + ( aß 1 y'

|A«|2 + b||2 + K

<^((2^+4^ + 2)C6|AMQ|2

+ 2||^o||2 + |0q|2

+ 2C6e-ar (|Q|2 + |/(i-r)|2) C1C2C3C6 i 12 , 11 ||2 , ||K ||2

2C1C2C3Cg _är

e ( |AMq| + ||po|| + |ro

C1C3 C6 |

a - <5

2MC1C3C6

Le-i (,-<>2+;l|Q|,)di

+ C4C6

e_a(t_s) (|^|2 + |Q|2)d,+ i6|Q|2

2 . ol r|2

+ 16||M||2 +8|/| C1C2C3 Cfi _gr

e + CrQe

x (|AMq|2+ipoi2+|0Q|2;

8a4 + 8a2y2 + 8C'2 2flx2

C4 = max 1 , 1 t fc

C5 = 2ß +

4a2 +2

C6 = max -

, 2a2 „

1 + -T'2

Since S < a,we have from (32)

C3 | e_a(t_s) J„M(S)„2 + |jP(,)|2 + 10

e_Sr (|AMqI2 + W2 + l-12)

2C1C2C3 _gr

■0(t

+ 2C6e_"r (|Q|2 + |/(i-T)|2) + ^3^. f* / 21 -,2 2 2\ , 2MC,C3C6

e_a(t_s) (^ + |Q| 2)ds+i6C6 |Q|2

+ 16C6||w||2 + 8C6I/I2.

Let D e Ds>Bi be given. For all y(i - r) = y0 6 ^ - T)> i e -R and r > 0, from the assumption (H:) of we know that 0(w(i - r)) is bounded and positive. So we easily obtain from (45)

C1C2C3C6 _gr

e + CQe

x ( | Aw0 | 2 + ||po|| 2 + ||0o||2) + 2ClC2^e^(«0)

+ 2C6e-aT (|Q|2 + |/(i-r)|2}

CiC3C6 f (2 2 + 2 2)ds,2MCiC3C6

J-TO V V Afc )

+ C4C6 f e-a(t-s) (|/|2 + |Q|2) ds + 16C6|Q|2

+ 16C6||w||2 + 8C6|/|2

for all y0 6 D(i - r), (6 P, and r > 0. Set

=2{%-r L

2MC1C3C6

e-a(t-s) (|/|2 + |Q|2)ds

16C6|Q|2 + 16C6(^5>£o (i))2 + 8C6|/|2

(48) □

The family ßäEi of closed balls in £1

fi5>Ei (i) = {7e£i,||7||El (*)}

is pullback -Dg>Ei-absorbing for the cocycle O in £1.

3. The Pullback D5>£i-Attractor in £1

In order to get the existence of the pullback Dg>Bi-attractor, we first introduce the following Lemma.

Lemma 4. Let H be an infinite dimensional Hilbert space and let the family {«¡jigN be an orthonormal basis of H. Suppose that /(%, t), /'(%, t), Q(x, i) 6 l20C(P, H) and for any t 6 R,

effs (|/ (x, s)|H + |/' (x, i)|2 + |Q (x, s)H ds < to,

for any a > 0.

n — TO

7-PJ/(*,*)|H +|(/-Pn)/' (*,s)|

+ |(7-Pn)Q(x,S)|^)d5 = 0, Vi 6 P,

where Pn : H ^ span{w1, w2,..., «n| is the orthogonal projector.

Proof. Let ^(f) = (/(x,i),«;)H, £ = (/'(x,i),«j)H, and C;(i) = (Q(X,i),«;),so

/ (x, i) = (i) /' (x, i) = I^ (i)

>=i >=i

Q(*,i) = I<i (¿K-

For any i e P and any e > 0,

f effs (|/ (x, s)£ + |/' (x, s)£ + |Q (*, s)&) ds

= I I (to(*)|2 + (*)|2 + K>W|2)^<TO >=ij-m

we can choose N1, N"2, N"3 large enough so that

m ft P If ^ (s)|2ds < 3,

m ft P If (S)|2dS<3,

m ft c

If (S)|2dS<3.

Then for any t 6 P and any e > 0, we put N0 = max{N1, N"2, N3| to get

I I ^ (k, (^)l2 + (^)l2 + K, (S)|2)dS<£. (54)

>=N0 J-TO

That is, for any t 6 P, any e > 0, and n > N"0,

ft (|(J-PJ/(*,S)|h +|(i-Pn)/' (*,*)

+ |(7-Pn)Q(x,S)|^)di<e.

lim I e~

n —> TOTO

J-Pn)/(*,S)|H +|(J-^n)/' (X,5)|H

+ |(7-Pn)Q(x,S)|H)di = 0.

In order to obtain the pullback -DSEi-attractor in £1, we also need the following Lemmas of the properties of compactness about the nonlinear operator ^1(-), (•).

Lemma 5. Let be a C2(P, P) function from P into P satisfying (H2); then : -D(A) ^ -H0(Q) is continuously compact; that is, (•) is continuous and maps a bounded subset of -D(A) into a precompact subset of H0(Q).

Proof. Let B = BD(A) be a bounded set in D(A). Assume that {un} is a bounded sequence in B. From Sobolev embedding Theorem, the embeddings D(A) ^ Lp, Vp > 1 and D(A) ^ u>1'p(Vp > 1) are compact. We assume that {un} is bounded and converges to u0 in Lp and W1'?, respectively By Minkowski inequality, we see that

Iv(#i (un) - 01 (uo))\2dxj

£ [9l (u„)-01 (Uo)Vunfdx £ [9i (uo)^(u„ -Uo)]2dx

By Holder inequality, we have

[g[ (un)-g[ ("o)] dx

< (£ \g[ (u„)- g'i (uo)l\2pdx) (£\vun\2qdx

< c(rs,Ei (t)f\g[ (un)- g'i (uo)\L2P,

where 1/p' + 1/q' + 1/2 = 1 and C(R2SE (t)) is a constant depending on R^E (t) and the embedding constant. Due to the assumption (H3) of g1 (■) and a classical continuity result, it follows that

\g[ (u„)-g[ (uo)\L2P —> (59)

Also by the Holder inequality

2 1 1/2

[g[ (uo)V(un -Uo)]dx} > 0. (60)

The proof is completed. □

Lemma 6. Let g2() be a C1(R,R) function from R into R satisfying (11); then g2 : H0(Q) ^ L (Q) is continuously compact.

Proof. Let B = BHi(n) be a bounded set in H^ (Q) and assume {0} to be a bounded sequence in B. From Sobolev embedding theorem, the embedding H0 (Q) ^ Lp, Vp >1 is compact, so we assume that 9n is bounded and converges to 9o in Lp. Let dn - do = wn; then there exists d = d(x) e [0,1] such that

2 \i/2 \{g2 (dn)-g2 faffl dx.

£ [g2 {do + 0un)un]2dx

By Holder inequality, we have

£ I(g2 (e„)-02 (0o))\2 dx

< \g2 (do + dœ^1^1^

where q is the conjugate of p' (i.e., 1/p' + 1/q' = 1). Combined with the assumption (11), the proof is completed. □

Lemma 7. Let g1(■) be a C2(R,R) function from R into R satisfying (H2); moreover, g1(0) = 0. LetB be a bounded subset ofD(A). Thenforanye > 0, there exists some no such that when n>nn

\\(l-pn)gi (w)||<£, VueB,

where Pn : V ^ span[w1, w2,..., wn} is the orthogonal projection.

Proof. Note that g1(u) e L2(O) for u e D(A). By Lemma 5, we see that g1(-) maps bounded subsets of D(A) into precompact subsets of H0)(O). Let B be a bounded subset of D(A) and let e > 0 be given arbitrarily. Since g1(B) is precompact in H0)(O), there is a finite number of elements v1, v2,..., vk e g1(B) such that

gi (B)c \JB(v,,2).

We take no > 0 sufficiently large so that

\\(I-Pn)g1 (u)\\<e, VueB

for all 1 < i < k, when n> no. Then by g1(B)c1<i<kB(vi, e/2), we have

\\(i-Pn)g1

Vu e V,

where Pn : D(A) ^ span{w1,..., wn} is the orthogonal projection. □

Lemma 8. Let g2() be a C2(R,R) function from R into R satisfying (11). Let B be a bounded subset of H^(Q). Then for any e > 0, there exists some no such that when n> no

p-pn)g2 VdeB, (67)

where Pn : H^(Q) ^ span{w1,w2,..., wn} is the orthogonal projection.

Theorem 9. Assume that p, y, k > 0 and the assumptions of 9X() g2(■), K) hold. f(x,t) e L2l0C(R,L2(Q)), f'(x,t) e L2loc(R,L2(Q)), and Q(x,t) e L2l0C(R, L2(Q)) satisfy (7); then there exists a pullback DSEi-attractor in E1 for the nonautonomous dynamical system (d, O) defined by (15).

Proof. In order to prove the result of the theorem, we only need to check the pullback DS E -condition (C).

Let {wk}'jjj=1 be an orthonormal basis of H which consists of eigenvectors of A. The corresponding eigenvalues are denoted by Xk, k = 1,2,... and 0 < X1 < X2 < X3 < ..., Xj ^ >x>, as j ^ >x>. Then {wk}jj=1 is also an orthonormal basis of V and D(A). We write Vn = {m1,w2 ,...,wn} and Pn : V ^ Vn is an orthogonal projector. For any u e V, we write

n\L2q'

u = Pnu + (I - Pn) u = u1 + u2.

Take the scalar product in H of (1) with Av2 = A«2 + aAw2, and take the scalar product in H of (2) with A02; then make summation to get

(^lA"2'2 + I I V2 I I 2 + I I 02 I I 2) + y I I V2 I I 2 - a ' 1 ' 2 2

+ aß|A«2|2 + fc|A02| + a2 ((«2, V2)) - ya (Am2, v2) + a (a1/202, Aw2) + ((('-^>0i («)> V2))

= (a - Pj ^2 (0) , A02) + ((/ - P„) /, AV2) + ((7-P„)Q,A02).

22 a II 112 a I 112

2 a \ \ ^ № I 112 ^ II 112

a ((«2, v2)) > 2 ' ' ^2 I I - 1 I V2 I I ;

fA \ ay2| ^ | 2

- ya (A«2, v2)>- — | V2 | - — | Aw 2 h

a(A1/20-;AM2)>-^|02|2 -^f|A«2|2,

!((/ - P„) ^ («), V2) | < ^(J - P„) ^ («)I2 + ^IHI2;

|((i-P„>02 (0)>A02)|<i||(/-P„)^2

2 fc" |2

|((7-P„)/,Av2)|

= -TT (/2> A«2) + a (/2, A«2) - (/', A«2)

aß, 12 2a^ |2 2 | r/|2 — |am9| + — | ^ | 1 1 f 1 4 | 2| ß

< ^7(/2,AM2) + — | A«2 | +—| /2 | + 2

here we set /2 = (7 - P„)/, and

<-"(J-P„^i (M)||2 + fc|(7-P„)^i

+ )Q|2 + 2?|(J-P")^2|2

(69) By the Gronwall lemma, we obtain

Am2 - -¡5/2

+ II v2 I +

I 2 III III 2

2" T 21|

A«2 (i - T) - -/2 (i - T)

+ ||V2 (i-r)||2 +||02 (i-T)"2

+ 1 e-2"i(t-s) (-''(/- P„) (M)"2

+ fc|(7-P„(0)|2 + ||(/-P„)Q|2

2 In "2

|A«2|2 +IIP2"2 +"02

< 2e-2"ir

2U2 + — )|AM2 (i-r)|2 + 2"^ (i-r)

"2 2 |

+ "02 (i-r)|f+^|/2 (i-r)|2

|((7-P„)Q;A0-2)| < fc|(/-P„)Q|2 + fc|A02|2, (71) +2[-re-2^ (

+ 2|f e-2^ (2"(/-P„)^i (M)"2

and letting 0 < a < minjßA/4, ((-A + y2/ß) +

, + y2/^)2 + yA2/2)/A,fcA^/2|, setting ax = minja/8, yA/2, fcA/4|, we have, from (69),

A"2 - ^/2

+ 2a1 ( ß

+ "v2I +

I 2 II II 2

A"2 - ^/2

+ "Vo" +

2 II II 2

+ 2a2||M2||2 + 2/2|2-

Then given any D 6 , we have

11^2 (T,i-r,7o)||

2 (^2 + (i-T)|2 + 2||^2 (i-r)

M- II2 2'

+ ||02 (f-T)|| +^|/2 (i-T)

2 i' g-2"1 (t-s) (^(/-Pj^ Jt-T V y

(0)'2)ds

+ 2il (||(J-P")Q|2 + |(/-P.)/2i

o 2II 112 2 | , a

+ 2a I I «2 I I + — | ^2 |

:= 7i + /2 + /3 +

for any y(f - r) = 6 ^ - T) and i e i, r > 0. Now we estimate 7j, 72, i3, X4 one by one. Given any e > 0 and any t 6 .R,it is easy to see that

^l/2|2 "^0, (76)

so there exists >0 such that

A (77)

for all r > Tj, y0 6 D(i - r).

By Lemmas 7-8, we can choose Mj 6 N such that

*2 <7 (78)

for any n > Mj, r > r2.

By Lemma 4, we can choose n2 large enough such that

for n > n2.

By (32), there exists r3 >0 such that, for r > r3, < to and the embedding from -D(A) into V is

compact combined with (2/^)|/2|2 ^ 0, so we can choose n3 large enough such that

for n > n3, r > r3.

By above analysis, if we choose r0 = max{r1; r2, r3|, n0 = max|n1; n2, n3|, then

11^2 (r, i-r, 70 ^

for any r > r0, n > n0, y0 eD(i-r).

This implies pullback -DSEi -condition (C). □

Conflict of Interests

The authors declare that there is no conflict of interests

regarding the publication of this paper.

Acknowledgments

The project is supported by the National Natural Science

Foundation of China (Grant no. 10772131), the Natural

Science Foundation of Shanxi Province, China (Grant no.

2010011008), and the Natural Science Foundation for Young

Scientists of Shanxi Province, China (Grant no. 20110210022).

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