This paper studies the following discrete systems of the complex Ginzburg-Landau equation: iu˙m-(α-iε)(2um-um+1-um-1)+iκum+βum2σum=gm, m∈Z. Under some conditions on the parameters α, ε, κ, β, and σ, we prove the existence of exponential attractor for the semigroup associated with these discrete systems.

1. Introduction

In the study of infinite dynamical systems, attractors occupy a central position (see, e.g., Chepyzhov and Vishik [1], Hale [2], Ladyzhenskaya [3], and Temam [4]). Exponential attractors are realistic objects intermediate between the global attractors and the inertial manifolds. There are several approaches for proving the existence of exponential attractors for parabolic and hyperbolic partial differential equations (PDEs) arising from mathematical physics. For example, we can refer to [5–7] for the existence of the exponential attractors for general evolution equations in Banach spaces, to [8] for the exponential attractors for reaction diffusion equations in unbounded domains, to [9] for the exponential attractors of the nonlinear wave equations, and to [10] for the exponential attractor for the generalized 2D Ginzburg-Landau equations. Also there are some references investigating the exponential attractors for lattice dynamical systems (LDSs). We can see [11–13] for the exponential attractors for first-order LDSs; see [14, 15] for the pullback exponential attractors for first- and second-order LDSs; see [16, 17] for second-order nonautonomous LDSs and discrete Zakharov equations for the uniform exponential attractors.

Lattice dynamical systems (LDSs) are currently under active investigation for their wide applications in electrical engineering [18], chemical reaction theory [19, 20], laser systems [21], and biology [22]. There are many references studying the asymptotic behavior of general LDSs. For instance, we can refer to [23–25] for the existence of global attractor, to [26–28] for the uniform attractor, to [11, 14, 15] for the exponential and pullback exponential attractor, and to [29, 30] for the random attractor. Also, there are some concrete applications of the above theory to the discrete PDEs. We can refer to [31–33] for discrete Klein-Gordon-Schrödinger equations, [34] for discrete three-component reversible Gray-Scott model, [35] for discrete coupled nonlinear Schrödinger-Boussinesq equations, [36] for discrete long-wave-short-wave resonance equations, and [37] for the discrete complex Ginzburg-Landau equation.

Lattice systems including coupled ordinary differential equations, coupled map lattices, and cellular automata are spatiotemporal systems with discretization in some variables. In some cases, lattice systems arise as the spatial discretization of partial differential equations on unbounded or bounded domains.

This paper will study the following discrete systems (lattice systems):(1)iu˙m-α-iɛ2um-um+1-um-1+iκum+βum2σum=gm,(2)um0=u0,m,m∈Z,where i is the unit of imaginary numbers and α,ɛ,κ,β,σ are parameters. Equation (1) can be regarded as a discrete analogue of the following complex Ginzburg-Landau equation on the real line: (3)iu˙+α-iɛuxx+iκu+βu2σu=g,x∈R.The complex Ginzburg-Landau equation is a simplified mathematical model for various pattern formation systems in mechanics, physics, and chemistry. We can refer to [10, 38, 39] for the detailed significations of the complex Ginzburg-Landau equation.

The existence of the exponential attractors for continuous complex Ginzburg-Landau equation in two-dimensional space was proved in [10]. Later, under some conditions on α,ɛ,κ,β,σ, and gm, [37] established the existence of global attractor for the semigroup associated with discrete systems (1)-(2). The aim of this paper is to prove the existence of exponential attractors for discrete systems (1)-(2). To this end, we will establish the following three items:

The solution operators associated with (1)-(2) generate a continuous semigroup {S(t)}t⩾0 in the phase space l2 and {S(t)}t⩾0 possesses a bounded and closed positively invariant set B⊂l2. Moreover, for any T>0, the map S(t) is Lipschitz continuous from [0,T]×B into B.

There exists a time T∗ such that the map S(T∗):=S∗:B↦B is an α-contraction on B.

The map S∗ satisfies the discrete squeezing property on B.

Compared with previous works such as [9], here we no longer require the compactness of the invariant set B (this fact was first noted by Babin and Nicolaenko [8] and then by Eden et al. [6]), which can usually be obtained by the compact embedding between Sobolev spaces when studying PDEs. Note that the compact embedding theorem of Sobolev spaces seems difficult to be applicable when studying LDSs. This is caused by the discrete characteristics of LDSs which restrict us to choose the phase spaces. Fortunately, the intrinsic characteristics of LDSs enable us to use the α-contraction property to compensate the compactness of the invariant set.

2. Positively Invariant Set and Lipschitz Continuity

Set(4)l2=u=umm∈Z,um∈C:∑m∈Zum2<+∞ and equip it with the inner product and norm as (5)u,v=∑m∈Zumv¯m,u2=u,u,u=umm∈Z,v=vmm∈Z∈l2,where v¯m denotes the conjugate of vm. Then l2,·,·,· is a separable Hilbert space. We now introduce the operators A, B, and B∗ on l2 as follows: (6)Aum=2um-um+1-um-1,∀m∈Z,Bum=um+1-um,B∗um=um-1-um,∀m∈Z. In fact, B∗ is the adjoint operator of B and one can easily check that (7)Au,v=B∗Bu,v=Bu,Bv,Bu,v=u,B∗v,∀u,v∈l2,Au2⩽16u2,Bu2⩽4u2,B∗u2⩽4u2,∀u∈l2. Using the notations introduced above, we can write problem (1)-(2) as(8)iu˙-α-iɛAu+iκu+βu2σu=g,(9)u0=u0,where u=(um)m∈Z, u2σu=um2σumm∈Z, g=gmm∈Z, and u0=um0m∈Z.

For the well-posedness of problem (1)-(2), we have the following.

Lemma 1 (see [<xref ref-type="bibr" rid="B39">37</xref>]).

Let α,ɛ,κ,β,σ>0 and g=(gm)m∈Z∈l2:

For any u0∈l2, problem (8)-(9) has a unique solution u∈C1([0,T0);l2) for some T0>0. Moreover, if T0<+∞, then limt→+T0-ut=+∞.

For any u0∈l2, the solution of problem (8)-(9) satisfies(10)ut2⩽u02e-κt+g2κ2,∀t⩾0.

Lemma 1(i) shows that, for each initial value u0∈l2, problem (8)-(9) possesses a unique solution. Letting t→+∞, we see from (10) that, for any u0∈l2, the corresponding solution u(t)∈l2 of problem (8)-(9) is uniformly (with respect to t) bounded for all t∈[0,+∞). Again, by Lemma 1(i), the solution exists globally; that is, problem (8)-(9) is globally well-posed. The above analysis implies that the solution operators (11)St:l2∋u0⟼Stu0=ut∈l2generate a continuous semigroup Stt⩾0 on l2. We next investigate the existence of the bounded and closed positively invariant set, as well as the Lipschitz property for the semigroup {St}t⩾0.

Lemma 2.

Let α,ɛ,κ,β,σ>0 and g=(gm)m∈Z∈l2. Then the semigroup {S(t)}t⩾0 possesses a bounded and closed positively invariant set B⊂l2.

Proof.

By (10) we see that the set(12)B≔u∈l2:u⩽2gκis a bounded and closed absorbing set for {S(t)}t⩾0. Thus, there is a time t∗:=t∗(B) such that S(t)B⊆B for any t⩾t∗. Set (13)B≔⋃τ⩾t∗SτB¯.Then B is the bounded and closed positively invariant set for {S(t)}t⩾0. The proof is complete.

The positively invariant property of B implies that(14)StB⊆B⊆B,∀t⩾0.

Lemma 3.

Let α,ɛ,κ,δ>0, σ⩾1/2, and g=gmm∈Z∈l2. Then the semigroup Stt⩾0 is Lipschitz continuous from 0,T×B into B for each T>0.

Proof.

Let u0,v0∈B, S(t)u0=u(t)=(um(t))m∈Z, S(t)v0=v(t)=(vm(t))m∈Z, and w(t)=u(t)-v(t). By (8),(15)iw˙-α-iɛAw+iκw+βu2σu-βv2σv=0,(16)w0=u0-v0.Using iw(t) to take inner product ·,· with both sides of (15) and then taking the real part, we obtain(17)12ddtwt2+ɛBu2+κwt2-Imβ∑m∈Zum2σum-vm2σvmw¯m=0.Now set f(x)=x2σ,x∈R+. Since σ⩾1/2, f′(x)=2σx2σ-1 is continuous and increasing on R+. By Cauchy inequality,(18)-Imβ∑m∈Zum2σum-vm2σvmw¯m⩽κ2wt2+β22κ∑m∈Zfumum-fvmvm2.Using mean value theorem, (12), and (14), we get(19)∑m∈Zfumum-fvmvm2⩽2+8σ22g2κ22σw2.It then follows from (17)–(19) that(20)ddtwt2+κ-δwt2⩽0,where(21)δ=δκ,β,σ,g≔2β2κ·1+4σ22g2κ22σ.Applying Gronwall inequality to (20) yields(22)wt2=Stu0-Stv02⩽w02eδ-κt,∀t⩾0,and, for any T>0, (23)Stu0-Stv0⩽u0-v0eδ-κT/2,∀t∈0,T. The proof is complete.

3. Existence of Exponential Attractor

For each positive number M, we define the orthogonal projection PM:l2↦l2 as (24)PMum=um,m⩽M;0,m>M and set QM=I-PM, where I is the identity operator on l2.

We next make some assumptions on the numbers α,κ,β,σ, and gm:

(H) Assume g=(gm)m∈Z∈l2, α,κ,β are positive, σ⩾1/2, and(25)κ>δ,

where δ is defined by (21).

The definitions of α-contraction and discrete squeezing property can be found in [2, 6].

Lemma 4.

Let assumption (H) hold. Then there exists a time T∗ such that the operator S(T∗):=S∗:B↦B is an α-contraction on B.

Proof.

Let u0,v0∈B, S(t)u0=u(t)=(um(t))m∈Z, S(t)v0=u(t)=(vm(t))m∈Z, and w(t)=u(t)-v(t). By (20), we have for any M∈N that (26)ddtwt2+κwt2⩽δwt2=δPMwt2+QMwt2, which, together with (25), gives (27)ddtwt2+κ-δwt2⩽δPMwt2,t⩾t∗.Thus we have(28)ddteκ-δtwt2⩽δeκ-δtPMwt2,t⩾t∗.Integrating both sides of (28) over [t∗,T] with T>t∗ and then using (22), we obtain(29)wT2⩽e-κ-δTw02+δκ-δmaxs∈t∗,TPMws2,T⩾t∗.Now we choose(30)T∗=maxt∗+ln256κ-δ+κ+δt∗+ln2562κ,lnκ+δ/2048βσκ/2g2σκ-δand it follows from (29) that(31)wT∗2⩽w02256+δκ-δmaxs∈t∗,T∗PMws2.Proceeding as that as [11] did, we can show δ/κ-δmaxs∈[t∗,T∗]PMw(s) is a precompact pseudometric on B, which, together with (31) and [2, Lemma 2.3.6], gives the desired result.

Remark 5.

Since Lemma 4 holds for any M∈N, we can specify some M∗ (see (40)). Then T∗ is chosen such that both e(δ-κ)T∗⩽1/256 and (41) hold.

Lemma 6.

Let assumption (H) hold. Then the operator S∗:B↦B satisfies the discrete squeezing property on B.

Proof.

Define a smooth function χx∈CR+,0,1 (see, e.g., [33]) such that (32)χx=0,0⩽x⩽1;1,x⩾2,χ′x⩽χ0constant,∀x∈R+. Let u0,v0∈B, S(t)u0=u(t)=(um(t))m∈Z, S(t)v0=v(t)=(vm(t))m∈Z, and w(t)=u(t)-v(t). Set ym=χm/M∗wm for each m∈Z and y=(ym)m∈Z, where M∗ is a positive integer that will be specified later. Using iy(t) to take inner product (·,·) with both sides of (15) and then taking the real part, we obtain (33)12ddt∑m∈ZχmM∗wm2+Reiα+ɛ∑m∈ZBwmBy¯m-Imβ∑m∈ZχmM∗um2σum-vm2σvmw¯m+κ∑m∈ZχmM∗wm2=0.By (31) and [37, (4.8)], we have for any t⩾t∗ that (34)Reiα+ɛ∑m∈ZBwtmBy¯tm⩾-2αχ0M∗w2,-Imβ∑m∈ZχmM∗fumum-fvmvmw¯m⩾-2βσ2gκ2σw2.Taking (33)-(34) into account, we obtain(35)ddt∑m∈ZχmM∗wm2+2κ∑m∈ZχmM∗wm2⩽Cw2,∀t⩾t∗,where C:=4χ0α/M∗+4βσ(2g/κ)2σ. By (22) and (35), we have for any t⩾t∗ that (36)ddt∑m∈ZχmM∗wm2+2κ∑m∈ZχmM∗wm2⩽Ceδ-κtw02,(37)ddt∑m∈ZχmM∗e2κtwm2⩽Ceδ+κtw02.Integrating both sides of (37) over [t∗,T] with T⩾t∗, we then get(38)∑m∈ZχmM∗e2κTwmT2⩽e2κt∗wt∗2+Cκ+δeδ+κTw02.Again from (22), we obtain that(39)QM∗wT2⩽eδ+κt∗-2κT+Cκ+δeδ-κTw02,T⩾t∗.We now take(40)M∗=2048χ0ακ+δ,and then from (25), (30), and (39), we have(41)eδ+κt∗-2κT∗⩽1256,4χ0ακ+δM∗eδ-κT∗⩽1512,4σβκ+δ2gκ2σeδ-κT∗⩽1512.Thus Q2M∗wT∗2=Q2M∗S∗u0-S∗v02⩽1/128w02=1/128u0-v02. Therefore, we can claim that if P2M∗(S∗u0-S∗v0)⩽Q2M∗(S∗u0-S∗v0), then(42)S∗u0-S∗v02=∑m≤2M∗wmT∗2+∑m≥2M∗wmT∗2=P2M∗S∗u0-S∗v02+Q2M∗S∗u0-S∗v02⩽2Q2M∗S∗u0-S∗v02⩽164u0-v02.The proof is complete.

Taking Lemmas 2, 3, 4, and 6 and [7, Theorem 3.1] into account, we now can state the main result of this paper as follows.

Theorem 7.

Let assumption H hold. Then, one has the following:

S∗ has an exponential attractor A∗ on B which satisfies the following:

M⊆A∗⊆B, where M is the global attractor of Stt⩾0;

S∗A∗⊆A∗; that is, A∗ is positively invariant under S∗;

A∗ has finite fractal dimension Dimf(A∗);

there exist two constants c1 and c2 such that, for each u∈B and every positive integer k, Dist(S∗ku,A)⩽c1e-c2k;

A=⋃0⩽t⩽T∗S(t)A∗ is an exponential attractor for Stt⩾0 on B and Dimf(A)⩽Dimf(A∗)+1.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This work is supported by NSFC with Grants nos. 41372264, 51279202, 41372308, and 91215301.

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