(abdallah)(2004) upper semicontinuity of the attractor for lattice dynamical systems of partly...

7/30/2019 (Abdallah)(2004) Upper Semicontinuity of the Attractor for Lattice Dynamical Systems of Partly Dissipative Reactio

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UPPER SEMICONTINUITY OF THE ATTRACTOR FORLATTICE DYNAMICAL SYSTEMS OF PARTLYDISSIPATIVE REACTION-DIFFUSION SYSTEMS

AHMED Y. ABDALLAH

Received 12 July 2004 and in revised form 7 December 2004

We investigate the existence of a global attractor and its upper semicontinuity for the

infinite-dimensional lattice dynamical system of a partly dissipative reaction-diffusionsystem in the Hilbert space l2 l2. Such a system is similar to the discretized FitzHugh-Nagumo system in neurobiology, which is an adequate justification for its study.

1. Introduction

Lattice dynamical systems arise in various application fields, for instance, in chemical

reaction theory, material science, biology, laser systems, image processing and pattern

recognition, and electrical engineering (cf. [6, 7, 13]). In each field, they have their own

forms, but in some other cases, they appear as spatial discretizations of partial differential

equations (PDEs). Recently, many authors studied various properties of the solutions for

several lattice dynamical systems. For instance, the chaotic properties have been investi-

gated in [1, 7, 8, 10, 11, 17], and the travelling solutions have been carefully studied in

[2, 3, 7, 8, 9, 22].

From [18], we know that it is difficult to estimate the attractor of the solution semi-

flow generated by the initial value problem of dissipative PDEs on unbounded domains

because, in general, it is infinite dimensional. Therefore, it is significant to study the lat-

tice dynamical systems corresponding to the initial value problem of PDEs on unbounded

domains because of the importance of such systems and they can be regarded as an ap-

proximation to the corresponding continuous PDEs if they arise as spatial discretizationsof PDEs.

The main idea of this work is originated from [4, 16]. In [4, 19, 20], the researchers

proved the existence of global attractors for different lattice dynamical systems and they

investigated the finite-dimensional approximations of these global attractors. In fact,

Bates et al. [4] studied first-order lattice dynamical systems, and Zhou [20] gave a gen-

eralization of the result given by [4]. Zhou [19] studied a second-order lattice dynamical

system and investigated the upper semicontinuity of the global attractor.

Copyright 2005 Hindawi Publishing CorporationJournal of Applied Mathematics 2005:3 (2005) 273288

DOI: 10.1155/JAM.2005.273
http://dx.doi.org/10.1155/S1110757X04407049http://dx.doi.org/10.1155/S1110757X04407049


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274 Attractor for LDSs of PDRDSs

For a positive integer r, consider the Hilbert space

l2 =u= uiiZr : i= i1, i2, . . . , ir Zr, ui R, iZru2i 0, with the initial conditions

ui(0)= ui,0, vi(0)= vi,0, (1.4)

where is a positive constant, and are real constants such that

> 0, (1.5)

the operator A : l2 l2 is defined by, for all u= (ui)iZr l2, i= (i1, i2, . . . , ir),

(Au)i = 2ru(i1,i2,...,ir) u(i11,i2,...,ir) u(i1,i21,...,ir) u(i1,i2,...,ir1)

u(i1+1,i2,...,ir) u(i1,i2+1,...,ir) u(i1,i2,...,ir+1),(1.6)

and for j = 1,2, s,s1,s2 R,

qj =

qj,i

iZr l2, (1.7)

fj (s),gj (s) C1(R,R), fj (0)=gj (0)= 0, (1.8)

hj

s1,s2 C1

R

2,R

, hj (0,0)= 0, (1.9)

j f

j (s), (1.10)

gjss 0, (1.11)h1s1, s2s1 = h2s1, s2s2, (1.12)where j is a positive constant.


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Ahmed Y. Abdallah 275

For example, one can consider fj (s) = j s, where j is a positive constant, gj (s) =nji=0 i j s

2i+1, where nj is a nonnegative integer and i j is a nonnegative constant for each

i = 0,1, . . . , nj , h1(s1, s2) = sm111 s

m22 , and h2(s1,s2) = s

m11 s

m212 , where m1,m2 2 are con-

stants. In fact, for f1(s)=s, f2(s)= s, g2(s)= 0, h1(s1, s2)= s2, and h2(s1,s2)= s1, where and are positive constants, the system given by (1.3) and (1.4) can be regarded asa spatial discretization of the following partly dissipative reaction-diffusion system with

continuous spatial variable xRr and tR+:

ut u +u +g1(u) + v= q1, vt + vu = q2. (1.13)

The system (1.13) describes the signal transmission across axons and is a model of

FitzHugh-Nagumo equations in neurobiology, (cf. [5, 12, 15] and the references therein).

The existence of global attractors of the system given by (1.13) has been proved in a

bounded domain (cf. [14]) and in Rr (cf. [16]).

2. Preliminaries

We can write the operator A in the following form:

A=A1 +A2 + +Ar (2.1)

such that for j = 1,2, . . . ,r, and u= (ui)iZr l2,

Aj ui = 2u(i1,i2,...,ir) u(i1,i2,...,ij1,ij1,ij+1,...,ir) u(i1,i2,...,ij1,ij +1,ij+1,...,ir). (2.2)For j = 1,2, . . . ,r, define the operators Bj , B

j : l

2 l2 as follows: for u= (ui)iZr l2,Bj u

i = u(i1,i2,...,ij1,ij +1,ij+1,...,ir) u(i1,i2,...,ir),

Bj u

i = u(i1,i2,...,ij1,ij1,ij+1,...,ir) u(i1,i2,...,ir).(2.3)

Then, it follows that for j = 1,2, . . . , r,

Aj = Bj Bj = Bj B

j , (2.4)

there exists a constant C0 = C0(r) such that

Au2 C0u2,

Bj u2 = Bj u2 4u2, u l2, (2.5)Bj u, v

=

u,Bj v

, u,v l2. (2.6)

It is clear that A, Aj , Bj , and Bj , j = 1,2, . . . ,r, are bounded linear operators from l

2

into l2.We can represent the initial value problem, (1.3) and (1.4), in the following form:

u +

Au + f1(u) +g1(u) + h1(u, v)= q1,v+ f2(v) +g2(v)h2(u,v)= q2,

u(0)=

ui,0

iZr, v(0)=

vi,0

iZr,

(2.7)


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where u = (ui)iZr, v= (vi)iZr, Au = (Aui)iZr, and for j = 1,2, fj (u) = (fj (ui))iZr,gj (u)= (gj (ui))iZr, hj (u,v)= (hj (ui,vi))iZr, qj = (qj,i)iZr.

Consider the Hilbert space E := l2 l2, endowed with the inner product and norm as

follows: for j = (u(j)

,v(j)

)T

= ((u(j)

i ),(v(j)

i ))TiZr E, j = 1,2,

1,2

E =

u(1),u(2)

+

v(1), v(2)

,

E = ,1/2E , E.

(2.8)

Now, the system given by (2.7) is equivalent to the following initial value problem in

the Hilbert space E := l2 l2:

+ C() = F(), (0)=

u0, v0T

, (2.9)

where = (u,v)T, C()= (Au,0)T, F()= (G1(),G2())T,

G1()= q1 f1(u)g1(u)h1(u, v),

G2()= q2 f2(v)g2(v) +h2(u,v).(2.10)

For a given function f(s1,s2) C1(R2,R), let D1 f(a, b) represent the partial derivativeof f with respect to the first independent variable, s1, at (s1, s2)= (a, b), and let D2 f(a,b)represent the partial derivative of f with respect to the second independent variable, s2, at(s1, s2)= (a,b). From (1.9) and the mean value theorem, it follows that given u= (ui)iZr,v= (vi)iZr l2, there exist 1i, 2i (0,1), for each i Zr, such that

h1(u, v)

2=

iZr

h1

ui,vi2

=

iZr

D1h1

1iui,vi

ui + D2h1

0,2ivi

vi2

2 max|a|u

max|b|v

D1h1(a,b)2u2+ 2

max|b|v

D2h1(0,b)

2v2.

(2.11)

Thus, for u,v l2, we have h1(u,v) l2. Similarly one can show that for u, v l2, h2(u,v) l2. By using (1.8) and the mean value theorem, it is easy to show that f1, f2, g1, and g2map l2 into l2. From the above discussion, it is obvious that Fmaps E into E.

3. The existence of an absorbing set

First we will prove that there exists a unique local solution of the system given by (2.9)

in E. Suppose that (1.7), (1.8), and (1.9) are satisfied. Let G be a bounded set in E, and


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j = (u(j),v(j))= ((u(j)i ),(v

(j)i ))iZr G, j = 1,2, then

F

1

F

2

2E

= G11G122 +G21G222 2

f1u(1) f1u(2)2 + 4g1u(1)g1u(2)2+ 42

h1u(1),v(1)h1u(2), v(2)2 + 2f2v(1) f2v(2)2+ 4

g2v(1)g2v(2)2 + 42h2u(1),v(1)h2u(2),v(2)2.(3.1)

Using (1.9) and the mean value theorem, it follows that there exist 3i,4i (0,1), for eachi Zr, and L1 = L1(G) such that

h1u(1),v(1)h1u(2),v(2)2=

iZr

h1

u(1)i ,v(1)i

h1

u(2)i ,v

(2)i

2

=

iZr

D1h1

u(1)i + 3i

u(2)i u(1)i

, v(1)i

u(1)i u

(2)i

+D2h1

u(2)i , v

(1)i + 4i

v(2)i v

(1)i

v(1)i v

(2)i

2

2

max

|a|u(1)+u(2),|b|v(1)

D1h1(a, b)

2

u(1)u(2)

2

+ 2 max|a|u(2),|b|v(1)+v(2)

D2h1(a,b)2v(1) v(2)2 L1

u(1)u(2)2 +v(1) v(2)2.

(3.2)

Thus

h1u(1),v(1)h1u(2),v(2)2 L1122. (3.3)We can obtain similar results, as (3.3), for f1, f2, g1, g2, and h2 by using (1.8), (1.9), and

the mean value theorem. In such a case, using (3.1), there exists L2 = L2(G) such thatF1F22E L21 22. (3.4)Thus F is locally Lipschitz from E into E. In such a case from the standard theory ofordinary differential equations, we get the following lemma.

Lemma 3.1. If (1.7), (1.8), and (1.9) are satisfied, then for any initial data (0) = (u0,v0)T E, there exists a unique local solution (t) = (u(t),v(t))T of (2.9) such that C1([0,T),E) for some T > 0. IfT


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Let = (u,v)T E be a solution of (2.9). If we consider the inner product of (2.9) with in E, taking into account (2.4) and (2.6), we obtain

12

ddtu2 +

rj=1

Bj u2 + f1(u),u+ g1(u),u+ h1(u, v),u= q1,u, (3.5)1

2

d

dtv2 +

f2(v),v

+g2(v),v

h2(u,v),v

=

q2,v

, (3.6)

If we multiply (3.5) by||, (3.6) by||, and we sum up the results, taking into account(1.5) and (1.12), we find that

||2 ddtu2 + ||2 ddt

v2 + ||r

j=1

Bj u2 + ||f1(u),u+ ||f2(v),v+ ||

g1(u),u

+ ||

g2(v),v

= ||

q1,u

+ ||

q2,v

.

(3.7)

By using (1.8), (1.10), and the mean value theorem, for each i Zr, there exists a constant5i (0,1) such that

f1(u),u

=

iZrf1

ui

ui

=

iZrf1

5iui

u2i

1u

2. (3.8)

Thus we have

f1(u),u

1u

2,

f2(v),v 2v

2. (3.9)

From (1.11), we obtain

g1(u),u= iZrg1uiui 0, g2(v),v 0. (3.10)Now if we substitute (3.9) and (3.10) into (3.7), we find that

||

2

d

dtu2 +

||

2

d

dtv2 + 1||u

2 + 2||v2

||

q1,u

+ ||

q2,v||

21

q12 + 1||2

u2 +||

22

q22 + 2||2

v2.(3.11)

Thus if we choose

=min||,||,1||,2||

, (3.12)


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then it follows that

d

dt2E +

2E

q1

2

+

q2

2

. (3.13)

From the Gronwall lemma, we obtain

2E et02E + 1 et2

q12 + q22, (3.14)

limt

2E

q12 +

q22. (3.15)Inequality (3.15) implies that the solution semigroup {S(t)}t0 of (2.9) exists globally andpossesses a bounded absorbing set in E. In such a case, the maps

S(t) : (0) E S(t)(0)= (t) E, t 0, (3.16)

generate a continuous semigroup {S(t)}t0 on E. Now from Lemma 3.1 and (3.15), weare ready to present the following lemma.

Lemma 3.2. If (1.5), (1.7), (1.8), (1.9), (1.10), (1.11), and (1.12) are satisfied, then for

any initial data in E, the solution (t) of (2.9) exists globally for all t 0. That is, C1([0,),E). Moreover, there exists a bounded ballO = OE(0,r0) in E, centered at0 withradius r0, such that for every bounded setG ofE, there exists T(G) 0 such that

S(t)GO, t T(G), (3.17)

where r20 > ((/)q1)2 + ((/)q2)2. Therefore, there exists a constantT0 0 depend-

ing on O such that

S(t)O O, t T0. (3.18)

4. The existence of the global attractor

To prove the existence of the global attractor for the solution semigroup {S(t)}t0 of (2.9),we need to prove the asymptotic compactness of {S(t)}t0. Along the lines of [4], thekey idea of showing the asymptotic compactness for such a lattice system is to establish

uniform estimates on Tail Ends of solutions.

Lemma 4.1. If (1.5), (1.7), (1.8), (1.9), (1.10), (1.11), and (1.12) are satisfied and(0)=(u0,v0)O, where O is the bounded absorbing ball given byLemma 3.2, then for any > 0,there exist positive constants T() and K() such that the solution (t) = (u(t),v(t))T=(i(t))iZr = ((ui(t)),(vi(t)))

TiZr E of (2.9) satisfies

imK()

i(t)2E = imK()

u2i (t) + v2i (t) (4.1)for allt T(), where im =max1jr|ij|fori= (i1, i2, . . . , ir) Zr.


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Proof. Consider a smooth increasing function C1(R+,R) such that

(s)= 0, 0 s < 1,

0 (s) 1, 1 s < 2,

(s)= 1, s 2,

(4.2)

and there exists a constant M0 such that (s)M0 for all sR+.Let L be an arbitrary positive integer. Set wi = (im/L)ui, zi = (im/L)vi, w =

(wi)iZr, z= (zi)iZr, and = (w,z)T. Following [4], we take the inner product of (2.9)with in E, then it follows that

iZr

imL

12

ddt

u2i + f1uiui +g1uiui + h1ui,viui+

iZr

rj=1

Bj u

i

Bj w

i =

iZr

im

L

q1,iui,

(4.3)

iZr

im

L

1

2

d

dtv2i + f2

vi

vi +g2

vi

vi h2

ui,vi

vi

=

iZrim

L

q2,ivi. (4.4)

If we multiply (4.3) by||, (4.4) by||, and we sum up the results, taking into account

(1.5) and (1.12), we get

iZr

im

L

||2 ddtu2i + ||2 ddtv2i + ||f1uiui+||f2

vi

vi + ||g1

ui

ui + ||g2

vi

vi

+ ||

iZr

rj=1

Bj u

i

Bj w

i =

iZr

im

L

||q1,iui + ||q2,ivi

.

(4.5)

Using (1.8), (1.10), and the mean value theorem, it follows that for each i Zr

,

f1

ui

ui 1u2i , f2

vi

vi 2v2i . (4.6)

From (1.11), we obtain

g1

ui

ui 0, g2

vi

vi 0. (4.7)

Recalling (69) of [21], taking into account that Bj 2, j = 1,2, . . . ,r, one can see that

||

iZr

rj=1

Bj u

i

Bj w

i

16||nM0L

r20 , t T0. (4.8)


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Now if we substitute (4.6), (4.7), and (4.8), into (4.5), we find that for t T0,

iZr

im

L ||

2

d

dtu2i +

||

2

d

dtv2i + 1||u

2i + 2||v

2i

iZrim

L

||q1,iui + ||q2,ivi

+

16||nM0L

r20

iZrim

L

||

1q21,i +

1||

4u2i +

||

2q22,i +

2||

4v2i

+

16||nM0L

r20 .

(4.9)

Thus if we choose

=min

||,||,1||,2||

, (4.10)

then it follows that for t T0,iZr

im

L

d

dt

i(t)2E +i(t)2E

2

iZrim

L

q1,i

2+

q2,i

2+

32||nM0L

r20

2

imL

q1,i

2+

q2,i

2+

32||nM0L

r20 .

(4.11)

Since q1, q2 l2, then for a given > 0, we can fixL such that

2

imL

q1,i

2+

q2,i

2+

32||nM0L

r20

2, (4.12)

and in such a case we get that

iZrim

L

d

dt

i(t)

2E +

i(t)

2E

2, t T0. (4.13)

From the Gronwall lemma, we obtain

iZr

im

L

i(t)2E et iZr

im

L

i(0)2E+ 2 , t T0. (4.14)Since (0)= (u0,v0)TO, we have (0)E r0. (4.15)Therefore,

iZr

im

L

i(t)2E r20 et + 2 , t T0. (4.16)


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But again for > 0, there exists a constant T1 = T1() such that

r20 et

2, t T1. (4.17)

Using (4.16) and (4.17) with K()= 2L, T()=max{T0,T1}, we obtain

imK()

i2E = imK()

im

L

i2E

iZr

im

L

i2E , t T().(4.18)

The proof is completed.

Lemma 4.2. If (1.5), (1.7), (1.8), (1.9), (1.10), (1.11), and (1.12) are satisfied, then the solu-tion semigroup {S(t)}t0 of (2.9) is asymptotically compact in E, that is, if{n} is boundedin E andtn , then {S(tn)n} is precompact in E.

Proof. By using Lemmas 3.2 and 4.1, above, the proof of this lemma will be similar to

that of [19, Lemma 3.2].

Theorem 4.3. If (1.5), (1.7), (1.8), (1.9), (1.10), (1.11), and (1.12) are satisfied, then the

solution semigroup {S(t)}t0 of (2.9) possesses a global attractor in E.

Proof. From the existence theorem of global attractors, (cf. [18, Lemmas 2 and 4]), we

get the result.

5. Upper semicontinuity of the global attractor

Here we will study the upper semicontinuity of the global attractor of the solution

semigroup {S(t)}t0 of (2.9), in the sense that is approximated by the global attractorsof finite-dimensional versions of (2.9), as was done in [4] for a simpler system.

Let n be a positive integer, and

Zrn = i Z

r : im n. (5.1)For i= (i1, i2, . . . , ir) Zrn, consider w = (wi)imn R

(2n+1)r. For convenience, we reorder

the subscripts of components ofw as follows:

w =

w(n,n,...,n,n), w(n,n,...,n,n+1), . . . , w(n,n,...,n,n),

w(n,n,...,n+1,n),w(n,n,...,n+1,n+1), . . . ,

w(n,n,...,n+1,n), . . . ,w(n,n,...,n,n),w(n,n,...,n,n+1), . . . ,w(n,n,...,n,n)T

.

(5.2)

Let

X=w =

wiimn

: w R(2n+1)r, (5.3)

where subscripts of components ofw are ordered as in (5.2).


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Consider the (2n + 1)r-dimensional ordinary differential equations with the initialdata in X:

wi + (Aw)i + f1wi+g1wi+ h1wi, zi= q1,i,zi + f2zi+g2zih2wi,zi= q2,i,wi(0)=wi,0, zi(0)= zi,0

i Zrn, t > 0.

(5.4)

The system (5.4) can be written as

w +

Aw +

f1(w) +

g1(w) +

h1(w,z)=

q1,

z+ f2(z) + g2(z)h2(w,z)= q2,w(0)=

wi,0

imn

, z(0)=

zi,0imn

,

(5.5)

where w = (wi)imn, z= (zi)imn X, for k = 1,2, . . . ,r,

w(i1,...,n,ik+1,...,ir) =w(i1,...,n+1,ik+1,...,ir),

w(i1,...,n,ik +1,...,ir) =w(i1,...,n1,ik+1,...,ir),

(

Aw)(i1,i2,...,ir) = 2rw(i1,i2,...,ir) w(i11,i2,...,ir) w(i1,i21,...,ir) w(i1,i2,...,ir1)

w(i1+1,i2,...,ir) w(i1,i2+1,...,ir) w(i1,i2,...,ir+1),

(5.6)

and for j = 1,2,

fj (w)= fjwiimn, gj (w)= gjwiimn,hj (w,z)= hjwi,ziimn, qj = qj,iimn. (5.7)For w = (wi)imn X, define the linear operators Bj , Bj , Aj : XX, j = 1,2, . . . , r, by

Bj wi =w(i1,...,ij +1,...,ir) w(i1,...,ij ,...,ir),Bj wi =w(i1,...,ij1,...,ir) w(i1,...,ij ,...,ir), Aj wi = 2w(i1,...,ij ,...,ir) w(i1,...,ij1,...,ir) w(i1,...,ij +1,...,ir),(5.8)

then

A= A1 + A2 + + Ar, Aj = Bj Bj = Bj Bj , j = 1,2, . . . , r. (5.9)For w(j) = (w

(j)i )imn X, j = 1,2, i= (i1, i2, . . . , ir) Zrn, define

w(1), w(2)

X=

imn

w(1)i w(2)i ,

w(1)X= w(1),w(1)1/2X . (5.10)


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In such a case, it is clear that X= (X, X) is a Hilbert space, and E =XX is aHilbert space with the following inner product and norm: for Yj = (w(j), z(j))T= ((w

(j)i ),

(z(j)i ))

Timn

E, j = 1,2,

Y1,Y2

E = w(1), w(2)X + z(1), z(2)X, Y1 E = w(1)2X +z(1)2X1/2. (5.11)It is easy to check that problem (5.5) can be formulated to the following first-order

system in the Hilbert space E:Y+ C(Y)= F(Y), Y(0)= w(0), z(0)T E, (5.12)

where

Y= (w,z)T, C(Y) = ( Aw,0)T, F(Y)= G1(Y), G2(Y)T,G1(Y)= q1 f1(w) g1(w)h1(w, z),G2(Y)= q2 f2(z) g2(z) +h2(w, z).(5.13)

Similar to Section 2, one can see that, if (1.5), (1.7), (1.8), (1.9), (1.10), (1.11), and

(1.12) are satisfied, then (5.12) is a well-posed problem in

E. Thus for anyY(0)

E, there

exists a unique solution Y C([0,+), E)C1((0,+), E), see Lemmas 3.1 and 3.2, alsothere exist maps of solutions Sn(t) : Y(0) E Y(t)= Sn(t)Y(0) E, t 0, generating a

continuous semigroup {Sn(t)}t0 on E.Similar to Lemma 3.2 and Theorem 4.3, we can prove the following Lemma.

Lemma 5.1. If (1.5), (1.7), (1.8), (1.9), (1.10), (1.11), and (1.12) are satisfied, then there

exists a bounded ball O = O E(0,r0) in E, centered at 0 with radius r0 such that for everybounded set G of E, there exists T( G) 0 such that

Sn(t)

G

O, t T(

G), n= 1,2, . . . , (5.14)

where r0 is the same constant given byLemma 3.2, and it is independent ofn. Moreover, the

semigroup {Sn(t)}t0 possesses a global attractorn,n O E.Here we prove the upper semicontinuity of the global attractor of the solution semi-

group {S(t)}t0 of (2.9). In such a case, we should extend the element u= (ui)imn Xto an element ofl2 such that ui = 0 for im > n, still denoted byu.

Lemma 5.2. If (1.5), (1.7), (1.8), (1.9), (1.10), (1.11), and (1.12) are satisfied, andn(0)n, then there exists a subsequence {nk (0)} of{n(0)} and0 such thatnk (0) con-verges to 0 in E.

Proof. Consider n(t) = Sn(t)n(0)= (un(t),vn(t))T E to be a solution of (5.12). Sincen(0) n, n(t) n O for all t R+, and again the element n = (n,i)imn Ecan be extended to the element n = (n,i)iZr E, where n,i = (0,0)T for im > n, it


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follows that

n(t)

E =

n(t)

E =

un

2+

vn

21/2

r0,

tR+

, n=

1,2, . . . .

(5.15)

From (2.5) and (5.15), we get that Cn(t)2E Cn(t)2E = Aun2 C0un2 C0r20 , (5.16)for all tR+, n= 1,2, . . . .

Similarly, for j = 1,2, the element qj = (qj,i)imn R(2n+1)r can be extended to theelement qn,j = (qn,j,i)iZr l2, where qn,j,i = qj,i for im n and qn,j,i = 0 for im > n.From (1.8) and (1.9), we know that for j = 1,2,

fj (0)=gj (0)= hj (0,0)= 0. (5.17)

Therefore, Fn(t)2E = Fn(t)2E = qn,1 f1ung1unh1un, vn2+qn,2 f2vng2vn+h2un, vn2

2q12 + 4f1un2 + 8g1un2 + 82h1un,vn2

+ 2

q2

2

+ 4

f2

vn

2

+ 8

g2

vn

2

+ 82

h2

un,vn

2.

(5.18)

Using (1.9), (5.15), and the mean value theorem, there exist 6i, 7i (0,1) for each i Zr

such thath1un,vn2 = iZr

h1

un,i,vn,i2

=

iZr

D1h1

6iun,i,vn,i

un,i + D2h1

0,7ivn,i

vn,i

2 2

max

a,b[r0,r0]

D1h1(a,b)

2r20 + 2

max

b[r0,r0]

D2h1(0,b)

2r20

(5.19)

for all t R+, n = 1,2, . . . . Again using (1.8), (1.9), (5.15), and the mean value theorem,we can get similar results, as (5.19), for f1, f2,g1,g2,and h2. Thus from (5.18), it is obviousthat there exists a constant C1 = C1(r0, q1,q2, f1, f2,g1,g2,h1,h2) such that Fn(t)2E = Fn(t)2E C1, tR+, n= 1,2, . . . . (5.20)Now, by using (5.12), we obtain

n(t)

2E 2

C

n(t)

2E + 2

F

n(t)

2E. (5.21)

Hence, from (5.16) and (5.20), it follows that there exists a constant C2 = C2(r0, C0,C1)such that n(t)E C2, tR+, n= 1,2, . . . . (5.22)


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Therefore, from (5.28), (5.30), and (5.31), it is clear that

Jf1

un,i

(t)dt

Jf1

ui

(t)dt

0, as n. (5.32)

Similarly, one can get the same result, as (5.32), for f2, g1, g2, h1, and h2. In such a casefrom (5.24), (5.26), and (5.27), it follows that

ui + (Au)i + f1

ui

+g1

ui

+ h1

ui,vi= q1,i,

vi + f2

vi

+g2

vih2

ui, vi

= q2,i.

(5.33)

But J is arbitrary, thus (5.33) holds for all tR+, which means that (t) is a solution of(2.9). From (5.25), it follows that (t) is bounded for tR+, that is, (t) , therefore

n(0) (0)

, and the proof is completed.

Now we are ready to represent the main result of this section. In fact, the following

theorem shows that the global attractor of the lattice dynamical system (2.9) is upper

semicontinuous with respect to the (cutoff) approximate finite-dimensional dynamical

system (5.12).

Theorem 5.3. If (1.5), (1.7), (1.8), (1.9), (1.10), (1.11), and (1.12) are satisfied, then

limn

dEn,)= 0, (5.34)

where dE(n,)= supan infb a bE.

Proof. We argue by contradiction. If the conclusion is not true, then there exists a se-

quence nk nk and a constant K > 0 such that

dE

nk ,) K > 0. (5.35)

However, byLemma 5.2, we know that there exists a subsequence nkm ofnk such that

dEnkm ,) 0, (5.36)

which contradicts (5.35). The proof is completed.

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Ahmed Y. Abdallah: Department of Mathematics, University of Jordan, Amman 11942, JordanE-mail address: [email protected]
mailto:[email protected]:[email protected]

(abdallah)(2004) upper semicontinuity of the attractor for lattice dynamical systems of partly...

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