# Exponential Attractor for a  First-Order Dissipative Lattice Dynamical System

Authors: Xiaoming Fan
Journal: Journal of Applied Mathematics
 (2008)

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## Abstract

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


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## Body

## 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 system (LDS) is
an infinite system of ordinary differential equations (lattice ODEs) or of
difference equations. In some cases, they arise from spatial discretizations of
partial differential equations (PDEs), but they possess their own form.Letk∈ℕ be a fixed positive integer. Denote(1.1)ℓ2={u∣u=(ui)i∈ℤk,i=(i1,i2,…,ik)∈ℤk,ui∈ℝ,∑i∈ℤkui2<+∞},where ℤ is the set of integers. Define a linear
operator A acting on ℓ2 in the following way: for any u=(ui)i∈ℤk∈ℓ2,  i=(i1,i2,…,ik)∈ℤk,(1.2)(Aui)i∈ℤk=2ku(i1,i2,…,ik)−u(i1−1,i2,…,ik)−u(i1,i2−1,…,ik)−…−u(i1,i2,…,ik−1)−u(i1+1,i2,…,ik)−u(i1,i2+1,…,ik)−…−u(i1,i2,…,ik+1).In this paper, we will consider the following
first-order lattice dynamical system:(1.3)u˙+Au+λu+g¯(u)=q¯,t>0,u(0)=(u0i)i∈ℤk=u0,where λ>0, u=(ui)i∈ℤk,  Au=(Aui)i∈ℤk,  u˙=(u˙i)i∈ℤk denote the first-order derivative, and g¯(u)=(g(ui))i∈ℤk, q¯=(qi)i∈ℤkℓ2.
Then, problem (1.3) can be regarded as a discrete analogue of the following
reaction diffusion equation in ℝk:(1.4)∂tu−Δu+λu+g(u)=q(x),t≥0.One example is the
Chafee-Infante equation.Bates [1] and his collaborators made some results on a global attractor for
lattice dynamical system (LDS). Zhou [2] applied them to a first-order dissipative lattice dynamical systems
analogue to problem (1.3), 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
invariant M set is needed for the semigroup S(t),
and the system (S(t),M) possesses a global attractor 𝒜.
More specifically, the semigroup S(t) is not compact for all positive t.
So, it is difficult to find a compact and positively invariant M which is not the attractor 𝒜.
The first-order dissipative lattice dynamical systems analogue to problem (1.3)
is such an example. Babin and Nicolaenko [5] consider reaction-diffusion systems in unbounded domains, prove the
existence of exponential attractors for such systems, and estimate their
fractal dimension. In [5], the compactness assumption plays a relatively minor
role in the whole construction. In [6],  Eden et al. provide constructions of exponential
attractor for a Lipschitz α-contraction S on a closed bounded B that satisfies the discrete squeezing
property, where B is 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 in
[6] that if a map S is 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

LetE be a separable Hilbert space with the norm ∥⋅∥, B⊆E be nonempty closed bounded set, and S:B→B be a Lipschitz continuous map with Lipschitz
constant L.
In this paper, we will always denote dist the Hausdorff semi-distance of sets as
follows:(2.1)dist(B,C)=supx∈Binfy∈C∥x−y∥,for anyB,C⊂E.Definition 2.1.
S is asymptotically compact on B if for any {xn}n≥1⊆B,
there is a convergent subsequence of {Snxn} in E.Remark 2.2.
IfS is an α-contraction on B,
then S is asymptotically compact on B.Definition 2.3.
S is said to satisfy the discrete squeezing property on B if there exists an orthogonal projection PN of rank N such that for every u and v in B,(2.2)∥PN(Su−Sv)∥≤∥(I−PN)(Su−Sv)∥⇒∥Su−Sv∥≤18∥u−v∥.Definition 2.4.
A compact setM is called as an exponential attractor for (S,B) if (i)
A⊆M⊆B,
where A is the global attractor;(ii)
SM⊆M,
that M is positively invariant under S;(iii)
M has finite fractal dimension; and(iv)
there exist universal constantsc1, c2 such that for every u∈B,
for every natural number n, dist(Snu,M)≤c1e−c2n.LetP=PN be the orthogonal projection chosen as in the
definition of the squeezing property. Denote(2.3)ℱ=max{F∣allu,v∈Fsatisfying∥u−v∥≤2∥Pu−Pv∥}for the inclusion relation. From
the definition of ℱ,
we know P|ℱ is one-to-one on ℱ.
Clearly, Pℱ is a bounded closed set of a finite
dimensional vector space, and therefore, it is compact. So, ℱ as the preimage under the continuous map P|ℱ must also be compact.LetE(k) be a subset of the set Sk+1B,
which is formed by a finite union of exceptional sets of the form ℱ,
which is described above, hence all E(k) are compact.Lemma 2.5.
IfS is asymptotically compact on B,
then (2.4)M=⋃j,k=1+∞Sj(E(k))is relatively compact.Proof.
Let{yn}n≥1 be a sequence in M.
Then, two cases will appear as follows.
Case 1.
There exists a natural numberN0 such  that all yn are in ⋃j,k=1N0Sj(E(k));Case 2.
There exists a subsequence (still denoted by{yn}n≥1) satisfying for every yn,
there exists xn∈B such that yn=Snxn.
In Case 1, since E(k) is compact and S is continuous, there exists a convergent
subsequence of {yn}n≥1 that converges in M.
In Case 2, since S is asymptotically compact on B,
it is immediate that we can extract from {yn}n≥1 a subsequence that converges in H.
So, M is relatively compact.Theorem 2.6.
LetH be a separable Hilbert space and let B be a nonempty closed bounded subset of E.
Assume that (i)
Sis a Lipschitz continuous map with Lipschitz
constantLonB;(ii)
Sis asymptotically compact onB;(iii)
Ssatisfies the discrete squeezing property onB(with rankN0),
thenShas an exponential attractor onB:(2.5)ℳ=𝒜∪⋃j,k=1+∞Sj(E(k)),where𝒜is a global attractor forSonB, E(k)is as the above-mentioned. Moreover, the
fractal dimension ofℳsatisfies(2.6)dF(ℳ)≤N0max{1,Log(16L+1)2Log2}.Proof.
Note that all the limits point of⋃j,k=1+∞Sj(E(k)) belong to 𝒜.
Together with Lemma 2.5, the proof follows exactly in the same way as the proof
of Theorem 2.1 in [7].Remark 2.7.
In Theorem2.6, there are two advantages than all the
previous results on the existence of exponential attractor for S:(i)
B is not assumed to be compact;(ii)
ifS possesses a global attractor, then S is at least asymptotically compact. So, we
only check that if S satisfies the Lipschitz property and the
discrete squeezing property to obtain the existence of an exponential attractor
for S.


## 3. Exponential Attractor

Fori=(i1,i2,…,ik)∈ℤk,
we will always denote ∥i∥=max1≤j≤k|ij| in the following discussion. For any u=(ui)i∈ℤk∈ℓ2, i=(i1,i2,…,ik)∈ℤk,
define the operators Bj, B¯j,
and Aj, j∈{1,2,…,k} from ℓ2 to itself as follows: ui=u(i1,i2,…,ik)∈ℓ2, j=1,2,…,k,(3.1)(Bju)i=u(i1,…,ij+1,…,ik)−u(i1,…,ij,…,ik),(B¯ju)i=u(i1,…,ij,…,ik)−u(i1,…,ij−1,…,ik),(Aju)i=2u(i1,…,ij,…,ik)−u(i1,…,ij+1,…,ik)−u(i1,…,ij−1,…,ik).Then, we have(3.2)A=A1+A2+…+Ak,Aj=BjB¯j=B¯jBj,j=1,2,…,k.For anyu=(ui)i∈ℤk, v=(vi)i∈ℤk∈ℓ2,
we define inner product and norm of ℓ2 as follows:(3.3)(u,v)=∑i∈ℤkuivi,∥u∥=(∑i∈ℤk|ui|2)1/2,then ℓ2=(ℓ2,(⋅,⋅),∥⋅∥) is a Hilbert space. It is obvious that any u=(ui)i∈ℤk, v=(vi)i∈ℤk∈ℓ2,(3.4)(Au,v)=∑j=1k(Bju,Bjv)=∑j=1k(B¯ju,B¯jv),∑j=1k∥Bju∥2≤4k∥u∥2.We always make the following assumptions ong(s)∈C(ℝ):(H1)
g(0)≡0 and g(s)s≥0.(H2)
There exists an increasing continuous functionK(r):ℝ+→ℝ+ with K(0)=0 such that(3.5)sup|s|≤r|g′(s)|≤K(r2),where ℝ+=[0,+∞).Similar to [2, Theorem 1], we have.Theorem 3.1.
For any
initial datau0∈ℓ2,
there exists a unique local solution u(t) of problem (1.3) with u(0)=u0 such that u(t)∈C1([0,T],ℓ2) for any T>0.In fact, it will be showed in Lemma3.2 below
that the local solution u(t) of problem (1.3) exists globally, that is, u(t)∈C1([0,+∞),ℓ2).
It implies that the map(3.6)S(t):u(0)=u0↦u(t),ℓ2↦ℓ2,generates a continuous semigroup
from ℓ2 to itself.Lemma 3.2.
LetB0=B(0,r0) be a closed bounded ball of ℓ2,
centered at 0 with radius r0 where (3.7)r02=1λ2∥q¯∥2.For any bounded set B of ℓ2,
there exists T(B)≥0 such that(3.8)S(t)B⊆B0,∀t≥T(B).Proof.
The proof is easily obtained.Corollary 3.3.
For anyt≥0, S(t)B0⊆B0.We obtain the following lemma after some simple computation.Lemma 3.4.
Letu(t)=(ui)i∈ℤk∈ℓ2 be a solution of problem (1.3) with initial data u0=(u0i)i∈ℤk∈B0.
Then, for any t≥0, (3.9)∑∥i∥>N|ui(t)|2≤r02e−λt+8C0kr02λN+1λ2∑∥i∥≥N/2|qi|2.From Lemmas3.2 and 3.4, we have the following.Theorem 3.5.
The
semigroup{S(t)}t≥0 is asymptotically compact in ℓ2 and possesses a nonempty compact global
attractor 𝒜.
Furthermore, 𝒜⊆B0.
LetS(t)u0=U(t) and S(t)w0=W(t).
Since u0, w0∈B0,
by Corollary 3.3, U(t), W(t)∈B0,
for t≥0.
Let Z(t)=S(t)u0−S(t)w0=U(t)−W(t).
Then, Z(t) satisfies(3.10)Z˙+AZ+λZ+g¯(U)−g¯(W)=0,Z(0)=u0−w0.After some simple computation, we obtain the following.Lemma 3.6 (Lipschitz Property).
For anyu0, w0∈B0 and any T>0, (3.11)∥S(T)u0−S(T)w0∥≤e(K(r02)−λ)T∥u0−w0∥.Letn∈ℕ be a positive integer. Set(3.12)ω=(ω(−n,−n,…,−n,−n)ω(−n,−n,…,−n,−n+1)…ω(−n,−n,…,−n,n)ω(−n,−n,…,−n+1,−n)ω(−n,−n,…,−n+1,−n+1)…ω(−n,−n,…,−n+1,n)…………ω(n,n,…,n,−n)ω(n,n,…,n,−n+1)…ω(n,n,…,n,n)).For convenience, we  always
denote(3.13)En={ω=(ωi)i∈ℤk∈ℓ2∣ωiwith subscripts of componentsofωwhichare ordered as in(3.12)andωi=0,∥i∥>n},with the same inner product and
norm as those of ℓ2.Let𝒦(x) be the inverse function of K(x) in (H2). Set(3.14)T0=max{4λLog2,1λ(Log2+2Log∥q¯∥−2Logλ−Log𝒦(λ2))},(3.15)N0=min{N∈ℕ:8C0k∥q¯∥2λN+1λ2∑∥i∥≥N/2|qi|2≤12𝒦(λ2)}.SupposePN is an orthogonal projection of rank (2N+1)k on ℓ2 such that PNℓ2=EN.Lemma 3.7 (Discrete Squeezing Property).
For anyu0, w0∈B0,
if (3.16)∥PN0(S(T0)u0−S(T0)w0)∥≤∥(I−PN0)(S(T0)u0−S(T0)w0)∥,then(3.17)∥S(T0)u0−S(T0)w0∥≤18∥u0−w0∥.Proof.
DenoteUN0(t)=(I−PN0)u(t), WN0(t)=(I−PN0)w(t) and ZN0(t)=(I−PN0)(u(t)−w(t))=(I−PN0)Z(t).
Taking the inner product (⋅,⋅) in (3.10) with ZN0,
we have(3.18)ddt∥ZN0∥2+2λ∥ZN0∥2+2(g¯(U)−g¯(W),ZN0)≤0,where ∥ZN0∥2=∑∥i∥>N0|Zi|2.
By the mean value theorem,(3.19)|(g¯(U)−g¯(W),ZN0)|≤∑∥i∥>N0|g′(Ui+θi(Wi−Ui))|Zi|2,where θi∈(0,1), ∥i∥>N0.
By (H2) and Lemma 3.4, for t≥T0,(3.20)|g′(Ui+θi(Wi−Ui))|≤λ2,which implies(3.21)|(g¯(U)−g¯(W),ZN0)|≤λ2∑∥i∥>N0|Zi|2.By (3.18), (3.21), and the Gronwall
inequality, we have(3.22)∥ZN0(t)∥2≤e−λ(t−T0)∥u0−w0∥2,for all t≥T0.
So, for any u0, w0∈B0,
if(3.23)∥PN0(S(T0)u0−S(T0)w0)∥≤∥(I−PN0)(S(T0)u0−S(T0)w0)∥,then(3.24)∥S(T0)u0−S(T0)w0∥≤2∥(I−PN0)(S(T0)u0−S(T0)w0)∥≤18∥u0−w0∥.From Theorems2.6 and 3.5, Lemmas 3.6 and 3.7 in
this article, and [7, Theorem 3.1], we obtain.Theorem 3.8.
The
semigroupS(t) determined by problem (1.3) with (H1)-(H2) possesses anexponential attractor on B0: (3.25)𝔐=⋃0≤t≤T0S(t)(𝒜∪⋃j,k=1+∞S0j(E(k))),whose  fractal dimension
satisfies(3.26)dF(𝔐)≤c0(2N0+1)k+1,where T0 is as (3.14), S0=S(T0), E(k) is defined as in Section 2 and N0 is as (3.15), c0=max{1,Log(16e(K(r0)−λ)T0+1)/2Log2}.Remark 3.9.
Indeed, whenK(r02)<λ,
by Lemma 3.6, we easily know that S(t) has an exponential attractor of dimension zero
on B0,
which is an equilibrium point of problem (1.3) (the global attractor for S(t)).



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