Exponential Attractor for a First-Order Dissipative Lattice Dynamical System

Journal of Applied Mathematics Volume 2008, Article ID 354652,8pages doi:10.1155/2008/354652

Research Article

Exponential Attractor for a First-Order Dissipative Lattice Dynamical System

Xiaoming Fan

Department of Mathematics, Southwest Jiaotong University, Chengdu 610031, China

Correspondence should be addressed to Xiaoming Fan,[email protected] Received 22 November 2007; Accepted 2 March 2008

Recommended by Andrei Agrachev

We construct an exponential attractor for a first-order dissipative lattice dynamical system arising from spatial discretization of reaction-diffusion equations inR^k. And we obtain fractal dimension of the exponential attractor.

Copyrightq2008 Xiaoming Fan. 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.

1. Introduction

Lattice systems arise in many applications, for example, in chemical reaction theory, image processing, pattern recognition, material science, biology, electrical engineering, laser systems, and so forth. A lattice dynamical systemLDSis an infinite system of ordinary differential equations lattice ODEs or of difference equations. In some cases, they arise from spatial discretizations of partial differential equationsPDEs, but they possess their own form.

Letk∈Nbe a fixed positive integer. Denote ²

u|u u_i

i∈Z^k, i

i₁, i₂, . . . , i_k

∈Z^k, u_i∈R,

i∈Z^k

u²_i <∞

, 1.1

whereZis the set of integers. Define a linear operatorAacting on²in the following way: for anyu ui_i∈Zk ∈²,i i1, i2, . . . , ik∈Z^k,

Au_i

i∈Z^k2ku_i1,i2,...,ik−u_{i1−1,i2,...,ik}−u_{i1,i2−1,...,ik}− · · ·

−ui1,i2,...,ik−1−ui11,i2,...,ik−ui1,i21,...,ik− · · · −ui1,i2,...,ik1. 1.2 In this paper, we will consider the following first-order lattice dynamical system:

˙

uAuλugu q, t >0, u0

u0i

i∈Z^k u0, 1.3

whereλ > 0,u ui_i∈Z^k,Au Aui_i∈Z^k, ˙u u˙i_i∈Z^k denote the first-order derivative, and gu gui_i∈Z^k,q qi_i∈Z^k2. Then, problem1.3can be regarded as a discrete analogue of the following reaction diffusion equation inR^k:

∂tu−Δuλugu qx, t≥0. 1.4

One example is the Chafee-Infante equation.

Bates 1 and his collaborators made some results on a global attractor for lattice dynamical systemLDS. Zhou2 applied them to a first-order dissipative lattice dynamical systems analogue to problem1.3, proved the existence of the global attractor for the LDS, and considered the finite-dimensional approximation of the attractor. Wang 3 and Zhao and Zhou 4 studied asymptotic behavior of nonautonomous lattice systems. In standard definition of exponential attractor, a compact and positively invariantMset is needed for the semigroupSt, and the systemSt, Mpossesses a global attractorA. More specifically, the semigroupStis not compact for all positivet. So, it is difficult to find a compact and positively invariantMwhich is not the attractorA. The first-order dissipative lattice dynamical systems analogue to problem1.3 is such an example. Babin and Nicolaenko5 consider reaction- diffusion systems in unbounded domains, prove the existence of exponential attractors for such systems, and estimate their fractal dimension. In5 , the compactness assumption plays a relatively minor role in the whole construction. In6 , Eden et al. provide constructions of exponential attractor for a Lipschitzα-contractionSon a closed boundedBthat satisfies the discrete squeezing property, whereBis not assumed to be compact.

The main novelty of this work is that we make an improvement in the constructions of exponential attractors is indicated in6 that if a mapSis asymptotically compact on a closed bounded B that satisfies the discrete squeezing property, then S possesses an exponential attractor. S is not assumed to be α-contraction in the result. We apply the result to study an exponential attractor for a first-order dissipative lattice dynamical system. We not only construct an exponential attractor for the lattice dynamical system and consider its finite- dimensional approximation, but also obtain an upper bound of its fractal dimension.

2. A key theorem

LetEbe a separable Hilbert space with the norm·,B ⊆Ebe nonempty closed bounded set, andS:B →Bbe a Lipschitz continuous map with Lipschitz constantL. In this paper, we will always denote dist the Hausdorffsemi-distance of sets as follows:

distB, C sup

x∈Binf

y∈Cx−y, for anyB, C⊂E. 2.1

Definition 2.1.S is asymptotically compact onB if for any {xn}_n≥1 ⊆ B, there is a convergent subsequence of{Sⁿxn}inE.

Remark 2.2. IfSis anα-contraction onB, thenSis asymptotically compact onB.

Definition 2.3. Sis said to satisfy the discrete squeezing property onBif there exists an orthogonal projectionPNof rankNsuch that for everyuandvinB,

PNSu−Sv≤I−PN

Su−Sv⇒Su−Sv≤1

8u−v. 2.2

Definition 2.4. A compact setMis called as an exponential attractor forS, Bif iA⊆M⊆B, whereAis the global attractor;

iiSM⊆M, thatMis positively invariant underS;

iiiMhas finite fractal dimension; and

ivthere exist universal constants c₁, c₂ such that for every u ∈ B, for every natural numbern, distSⁿu, M≤c1e^−c2n.

LetP PN be the orthogonal projection chosen as in the definition of the squeezing property. Denote

Fmax

F|allu, v∈F satisfyingu−v ≤√

2P u−P v 2.3

for the inclusion relation. From the definition ofF, we knowP|Fis one-to-one onF. Clearly, PFis a bounded closed set of a finite dimensional vector space, and therefore, it is compact.

So,Fas the preimage under the continuous mapP|Fmust also be compact.

LetE^kbe a subset of the setS^k1B, which is formed by a finite union of exceptional sets of the formF, which is described above, hence allE^kare compact.

Lemma 2.5. IfSis asymptotically compact onB, then

M ^∞

j,k1

S^j E^k

2.4

is relatively compact.

Proof. Let{yn}_n≥1be a sequence inM. Then, two cases will appear as follows.

Case 1. There exists a natural numberN0such that allynare in_N0

j,k1S^jE^k;

Case 2. There exists a subsequencestill denoted by{yn}_n≥1satisfying for everyyn, there exists xn∈Bsuch thatynSⁿxn.

In Case 1, since E^k is compact and S is continuous, there exists a convergent subsequence of{yn}_n≥1 that converges inM. In Case2, sinceSis asymptotically compact on B, it is immediate that we can extract from{yn}_n≥1a subsequence that converges inH. So,M is relatively compact.

Theorem 2.6. LetHbe a separable Hilbert space and letBbe a nonempty closed bounded subset ofE.

Assume that

iSis a Lipschitz continuous map with Lipschitz constantLonB;

iiSis asymptotically compact onB;

iiiS satisfies the discrete squeezing property onB (with rankN₀), thenShas an exponential attractor onB:

MA ∪ ^∞

j,k1

S^j E^k

, 2.5

whereAis a global attractor forSonB,E^kis as the above-mentioned. Moreover, the fractal dimension ofMsatisfies

d_FM≤N₀max

1,Log16L1 2 Log 2

. 2.6

Proof. Note that all the limits point of_∞

j,k1S^jE^kbelong toA. Together withLemma 2.5, the proof follows exactly in the same way as the proof of Theorem 2.1 in7 .

Remark 2.7. In Theorem 2.6, there are two advantages than all the previous results on the existence of exponential attractor forS:

iBis not assumed to be compact;

iiifSpossesses a global attractor, thenSis at least asymptotically compact. So, we only check that ifSsatisfies the Lipschitz property and the discrete squeezing property to obtain the existence of an exponential attractor forS.

3. Exponential attractor

Fori i1, i₂, . . . , i_k∈Z^k, we will always denoteimax_1≤j≤k|ij|in the following discussion.

For anyu ui_i∈Z^k ∈ ²,i i1, i₂, . . . , i_k ∈ Z^k, define the operatorsB_j, B_j, and A_j,j ∈ {1,2, . . . , k}from²to itself as follows:uiui1,i2,...,ik∈²,j 1,2, . . . , k,

B_ju

iu_{i1,...,ij1,...,ik}−u_{i1,...,ij,...,ik}, Bju

iui1,...,ij,...,ik−ui1,...,ij−1,...,ik, A_ju

i2u_{i1,...,ij,...,ik}−u_{i1,...,ij1,...,ik}−u_{i1,...,ij−1,...,ik}.

3.1

Then, we have

AA₁A₂· · ·A_k, A_jB_jB_jB_jB_j, j1,2, . . . , k. 3.2

For anyu ui_i∈Zk,v vi_i∈Zk ∈², we define inner product and norm of²as follows:

u, v

i∈Z^k

uivi, u

i∈Z^k

ui²1/2

, 3.3

then^{2 2},·,·,·is a Hilbert space. It is obvious that anyu ui_i∈Z^k,v vi_i∈Z^k∈²,

Au, v ^k

j1

B_ju, B_jv ^k

j1

B_ju, B_jv ,

k j1

B_ju²≤4ku². 3.4

We always make the following assumptions ongs∈CR:

H1g0≡0 andgss≥0.

H2There exists an increasing continuous functionKr : R →R withK0 0 such that

sup

|s|≤r

gs≤K r²

, 3.5

whereR 0,∞.

Similar to2, Theorem 1 , we have.

Theorem 3.1. For any initial datau₀ ∈ ², there exists a unique local solutionutof problem1.3 withu0 u0such thatut∈C¹0, T , ²for anyT >0.

In fact, it will be showed inLemma 3.2below that the local solutionutof problem1.3 exists globally, that is,ut∈C¹0,∞, ². It implies that the map

St:u0 u0−→ut, ²−→², 3.6

generates a continuous semigroup from²to itself.

Lemma 3.2. LetB₀B0, r0be a closed bounded ball of², centered at 0 with radiusr₀where

r₀² 1

λ²q². 3.7

For any bounded setBof², there existsTB≥0 such that

StB⊆B₀, ∀t≥TB. 3.8

Proof. The proof is easily obtained.

Corollary 3.3. For anyt≥0,StB0⊆B0.

We obtain the following lemma after some simple computation.

Lemma 3.4. Letut ui_i∈Z^k ∈²be a solution of problem1.3with initial datau₀ u0i_i∈Z^k ∈ B0. Then, for anyt≥0,

i>N

u_it²≤r₀²e^−λt8C0kr₀²

λN 1

λ²

i≥N/2

q_i². 3.9

From Lemmas3.2and3.4, we have the following.

Theorem 3.5. The semigroup{St}_t≥0 is asymptotically compact in² and possesses a nonempty compact global attractorA. Furthermore,A ⊆B₀.

LetStu0 UtandStw0Wt. Sinceu0,w0∈B0, byCorollary 3.3,Ut,Wt∈B0, fort≥0. LetZt Stu0−Stw0Ut−Wt. Then,Ztsatisfies

Z˙ AZλZgU−gW 0, Z0 u₀−w₀. 3.10

After some simple computation, we obtain the following.

Lemma 3.6Lipschitz property. For anyu₀,w₀∈B₀and anyT >0,

STu0−STw0≤e^Kr02−λTu0−w0. 3.11 Letn∈Nbe a positive integer. Set

ω

⎛

⎜⎜

⎜⎝

ω−n,−n,...,−n,−n ω−n,−n,...,−n,−n1 . . . ω−n,−n,...,−n,n

ω−n,−n,...,−n1,−n ω−n,−n,...,−n1,−n1 . . . ω−n,−n,...,−n1,n

. . . . . . . . . . . . ωn,n,...,n,−n ωn,n,...,n,−n1 . . . ωn,n,...,n,n

⎞

⎟⎟

⎟⎠. 3.12

For convenience, we always denote E_n

ω ω_i

i∈Z^k∈²|ω_i with subscripts of components

ofωwhich are ordered as in3.12andω_i0, i> n , 3.13 with the same inner product and norm as those of².

LetKxbe the inverse function ofKxinH2. Set

T0max 4

λLog 2,1 λ

Log 22 Logq −2 Logλ−LogK λ

2

, 3.14

N0min

N∈N: 8C0kq²

λN 1

λ²

i≥N/2

qi²≤ 1 2K

λ 2

. 3.15

SupposeP_Nis an orthogonal projection of rank2N1^kon²such thatP_N²E_N. Lemma 3.7Discrete squeezing property. For anyu0,w0∈B0, if

P_N0 S

T₀ u₀−S

T₀

w₀≤I−P_N0 S

T₀ u₀−S

T₀

w₀, 3.16

then

S T₀

u₀−S T₀

w₀≤1

8u₀−w₀. 3.17

Proof. DenoteU_N0t I−PN0ut,W_N0t I−PN0wtandZ_N0t I−PN0ut−wt I−PN0Zt. Taking the inner product·,·in3.10withZN0, we have

d

dtZ_N0²2λZ_N0²2

gU−gW, ZN0

≤0, 3.18

whereZN0²

i>N0|Zi|². By the mean value theorem, gU−gW, ZN0≤

i>N0

g U_iθ_i

W_i−U_iZ_i², 3.19

whereθi∈0,1,i> N0. ByH2andLemma 3.4, fort≥T0, g

Uiθi

Wi−Ui≤λ

2, 3.20

which implies

gU−gW, ZN0≤λ 2

i>N0

Z_i². 3.21 By3.18,3.21, and the Gronwall inequality, we have

ZN0t²≤e^−λt−T0u0−w0², 3.22 for allt≥T₀. So, for anyu₀,w₀∈B₀, if

PN0

S T0

u0−S T0

w0≤I−PN0

S T0

u0−S T0

w0, 3.23 then

S T₀

u₀−S T₀

w₀≤2I−P_N0 S

T₀ u₀−S

T₀

w₀≤ 1

8u₀−w₀. 3.24 From Theorems2.6and3.5, Lemmas3.6and3.7in this article, and7, Theorem 3.1 , we obtain.

Theorem 3.8. The semigroupStdetermined by problem1.3with (H1)-(H2) possesses an exponen- tial attractor onB0:

M

0≤t≤T0

St

A ∪ ^∞

j,k1

S^j₀ E^k

, 3.25

whose fractal dimension satisfies

dFM≤c0

2N01_k

1, 3.26 whereT0is as3.14,S0 ST0,E^kis defined as inSection 2andN0 is as3.15,c0 max{1, Log16e^Kr0−λT01/2 Log 2}.

Remark 3.9. Indeed, whenKr₀² < λ, by Lemma 3.6, we easily know thatSthas an expo- nential attractor of dimension zero onB₀, which is an equilibrium point of problem1.3 the global attractor forSt.

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