# Pullback Exponential Attractors for Nonautonomous Klein-Gordon-Schrödinger Equations on Infinite Lattices

Authors: Chunqiu Li; Min Zhao; Caidi Zhao
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

This paper proves the existence of the pullback exponential attractor for the process associated to the nonautonomous Klein-Gordon-Schrödinger equations on infinite lattices.


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## Body

## 1. Introduction

Lattice dynamical systems (LDSs for short), including coupled ordinary differential equations (ODEs), coupled map lattices, and cellular automata [1], are spatiotemporal systems with discretization in some variables. In some cases, LDSs arise as the spatial discretization of partial differential equations (PDEs) on unbounded or bounded domains. LDSs occur in a wide variety of applications, ranging from image processing and pattern recognition [2–4] to electrical engineering [5], chemical reaction theory [6, 7], laser systems [8], material science [9], biology [10], and so forth.Nowadays, LDSs have drawn much attention from mathematicians and physicists [1]. Various properties of solutions for LDSs have been widely studied. For example, the stochastic LDSs were investigated in [11, 12]. The global and uniform attractors of LDSs were examined in [13–19]. The exponential and uniform exponential attractors of LDSs were investigated by [20–24].At the same time, the asymptotic theory of LDSs has been widely used on many concrete lattice equations from mathematical physics. For example, lattice reaction-diffusion equations [25], discrete nonlinear Schrödinger equations [26], lattice FitzHugh-Nagumo systems [27], lattice Klein-Gordon-Schrödinger (KGS) equations [28], and lattice three component reversible Gray-Scott equations [29].Very recently, Zhou and Han [30] presented some sufficient conditions for the existence of the pullback exponential attractor for the continuous process on Banach spaces and weighted spaces of infinite sequences. Also, they applied their results to study the existence of pullback exponential attractors for first-order nonautonomous differential equations and partly dissipative differential equations on infinite lattices with time-dependent coupled coefficients and time-dependent external terms in weighted spaces.In this paper, we will use the abstract theory of [30] to study the pullback exponential behavior of solutions for the following nonautonomous lattice systems:
(1)iz˙m-(Az)m+iαzm+zmum=fm(t),m∈ℤ,t>τ,(2)u¨m+νu˙m+(Au)m+μum-β|zm|2=gm(t),m∈ℤ,t>τ,
with initial conditions:
(3)zm(τ)=zm,τ,um(τ)=um,τ,u˙m(τ)=u1m,τ,m∈ℤ,τ∈ℝ,
where A is a linear operator defined as
(4)(Au)m=2um-um+1-um-1,∀u=(um)m∈ℤ,zm(t)∈ℂ,um(t)∈ℝ, ℂ and ℝ are the sets of complex and real numbers, respectively, ℤ is the set of integer numbers, i=-1 is the unit of the imaginary numbers, and α,β,ν, and μ are positive constants.Equations (1)-(2) can be regarded as a discrete analogue of the following nonautonomous KGS equations on ℝ:
(5)izt+zxx+iαz+zu=f(x,t),utt+νut-uxx+μu-β|z|2=g(x,t).
Equations (5) describe the interaction of a scalar nucleon interacting with a neutral scalar meson through Yukawa coupling [31], where z and u represent the complex scalar nucleon field and the real meson field, respectively, and the complex-valued function f(x,t) and the real-valued function g(x,t) are the time-dependent external sources. There are many works concerning the Cauchy problem and the initial boundary value problem of the continuous model of KGS equations or its related version, see [32–36] and references therein.We want to mention that the lattice KGS equations have been studied by [28, 37]. In [37], the authors first presented some sufficient conditions for the existence of the uniform exponential attractor for a family of continuous processes on separable Hilbert spaces and the space of infinite sequences. Then, they studied the existence of uniform exponential attractors for the dissipative nonautonomous KGS lattice system and the Zakharov lattice system driven by quasi-periodic external forces. In [28], the authors first proved the existence of compact kernel sections and gave an upper bound of the Kolmogorov ε-entropy for these kernel sections. Also they verified the upper semicontinuity of the kernel sections. Articles in [28, 37] use the same transformation of the variable u=(um)m∈ℤ.The aim of the present paper is to use the abstract result of [30] to prove the existence of pullback exponential attractors for the LDSs (1)–(3). When verifying the discrete squeezing property (see Lemma 5(II)) of the generated process, we encounter the difficulty coming from the nonlinear terms umzm and |zm|2 in the coupled lattice equations. To overcome this difficulty, we make a proper transformation of the variable u=(um)m∈ℤ and use the technique of cutoff functions. We want to remark that the idea concerning the transformation of the variable u originates from articles in [28, 37], but our transformation is other than that of [28, 37].The rest of the paper is organized as follows. In Section2, we first introduce some spaces and operators. Then, we recall some results on the existence, uniqueness, and some estimates of solutions. Section 3 is devoted to proving the existence of the pullback exponential attractor for the process associated to the lattice KGS equations.


## 2.  Preliminaries

We first introduce the mathematical setting of our problem. Set(6)ℓ2={u=(um)m∈ℤ,um∈ℝ:∑m∈ℤ‍um2<+∞},L2={u=(um)m∈ℤ,um∈ℂ:∑m∈ℤ‍|um|2<+∞}.For brevity, we useX to denote ℓ2 or L2, and equip X with the inner product and norm as
(7)(u,v)=∑m∈ℤ‍umv-m,∥u∥2=(u,u),∀u=(um)m∈ℤ,v=(vm)m∈ℤ∈X,
where v-m denotes the conjugate of vm. For any two elements u,v∈X, we define a bilinear form on X by
(8)(u,v)μ=(Bu,Bv)+μ(u,v),
where μ is the constant in (2) and B is a linear operator defined as
(9)(Bu)m=um+1-um,∀m∈ℤ,∀u=(um)m∈ℤ∈X.We also define a linear operatorB* from X to X via
(10)(B*u)m=um-1-um,∀m∈ℤ,∀u=(um)m∈ℤ∈X.In fact,B* is the adjoint operator of B and one can check that
(11)(Au,v)=(B*Bu,v)=(Bu,Bv),(Bu,v)=(u,B*v),∀u,v∈X,∥Bu∥2=∥B*u∥2≤4∥u∥2,∥Au∥2≤16∥u∥2,∀u∈X.Clearly, the bilinear form(·,·)μ defined by (8) is also an inner product in X. Since
(12)μ∥u∥2≤μ∥u∥2+∥Bu∥2=∥u∥μ2≤(μ+4)∥u∥2,∀u∈X,
the norm ∥·∥μ induced by (·,·)μ is equivalent to the norm ∥·∥. Write
(13)ℓ2=(ℓ2,(·,·),∥·∥),ℓμ2=(ℓ2,(·,·)μ,∥·∥μ),L2=(L2,(·,·),∥·∥);
then ℓ2, ℓμ2, and L2 are all Hilbert spaces. Set
(14)Eμ=ℓμ2×ℓ2×L2.For any two elementsψ(j)=(u(j),v(j),z(j))T∈Eμ, j=1,2, the inner product and norm of Eμ are defined as
(15)(ψ(1),ψ(2))Eμ=(u(1),u(2))μ+(v(1),v(2))+(z(1),z(2))=∑m∈ℤ‍({zm(1)zm(2)¯}(Bu(1))m(Bu(2))m+μum(1)um(2)+vm(1)vm(2)+zm(1)zm(2)¯),∥ψ∥Eμ2=(ψ,ψ)Eμ,∀ψ∈Eμ,
where zm(2)¯ stands for the conjugate of zm(2).For convenience, we will express (1)–(3) as an abstract Cauchy problem of first-order ODE with respect to time t in Eμ. To this end, we put u=(um)m∈ℤ, z=(zm)m∈ℤ, zu=(zmum)m∈ℤ, |z|2=(|zm|2)m∈ℤ, f(t)=(fm(t))m∈ℤ, g(t)=(gm(t))m∈ℤ, zτ=(zm,τ)m∈ℤ, and uτ=(um,τ)m∈ℤ, u1τ=(u1m,τ)m∈ℤ. Then, we rewrite (1)–(3) in a vector form as
(16)iz˙-Az+iαz+zu=f(t),t>τ,(17)u¨+νu˙+Au+μu-β|z|2=g(t),t>τ,(18)z(τ)=zτ,u(τ)=uτ,u˙(τ)=u1τ,τ∈ℝ.
Set
(19)v=u˙+δu,whereδ=μν2(μ+ν2)>0,ψ=(u,v,z)T, F(ψ,t)=(0,β|z|2+g(t),izu-if(t))T,
(20)Θ=(δI-I0A+μI+δ(δ-ν)I(ν-δ)I000iA+αI).Then, (16)–(18) can be written as
(21)ψ˙+Θψ=F(ψ,t),t>τ,(22)ψ(τ)=ψτ=(uτ,vτ,zτ)T=(uτ,u1τ+δuτ,zτ)T,τ∈ℝ.Let𝒞b(ℝ,X) be the set of continuous and bounded functions from ℝ into X, then for each f(t)∈𝒞b(ℝ,X), we have supt∈ℝ∑m∈ℤ|fm(t)|2<+∞. Write
(23)ℋ={{(fm(t))m∈ℤ}{∑|m|≥M(ε,τ)‍}f(t)=(fm(t))m∈ℤ∈𝒞b(ℝ,X):foreachτ∈ℝ,∀ε>0,∃M(ε,τ)∈ℕ,suchthat∑|m|≥M(ε,τ)‍|fm(s)|2≤εforanys≤τ{(fm(t))m∈ℤ}{∑|m|≥M(ε,τ)‍}}.
We next recall some results of solutions to (21)-(22).Lemma 1 (see [28]).
Letf(t)=(fm(t))m∈ℤ∈𝒞b(ℝ,L2), g(t)=(gm(t))m∈ℤ∈𝒞b(ℝ,ℓ2). Then, for any initial data ψτ=(uτ,vτ,zτ)T∈Eμ, there exists a unique solution ψ(t)=(u(t),v(t),z(t))T∈Eμ of (21)-(22), such that ψ(t)∈𝒞([τ,+∞),Eμ)∩𝒞1((τ,+∞),Eμ). Moreover, the mapping
(24)U(t,τ):ψτ=(uτ,vτ,zτ)T∈Eμ↦ψ(t)=(u(t),v(t),z(t))T∈Eμ,∀t≥τ
generates a continuous process {U(t,τ)}t≥τ on Eμ, where vτ=u1τ+δuτ.Lemma 2 (see [28]).
Letf(t)=(fm(t))m∈ℤ∈𝒞b(ℝ,L2), g(t)=(gm(t))m∈ℤ∈𝒞b(ℝ,ℓ2). Then, the solution ψ(t)=(u(t),v(t),z(t))T∈Eμ of (21)-(22) corresponding to initial data ψτ=(uτ,vτ,zτ)T∈Eμ satisfies
(25)∥ψ(t)∥Eμ2≤C0e-2ϑ0(t-τ)+12R02,∀t≥τ,
where C0,ϑ0, and R0 are positive constants independent of  t and τ.Lemma 3 (see [28]).
Letf(t)=(fm(t))m∈ℤ∈𝒞b(ℝ,L2), g(t)=(gm(t))m∈ℤ∈𝒞b(ℝ,ℓ2). Then, the process {U(t,τ)}t≥τ corresponding to (21)-(22) possesses a uniformly bounded absorbing set ℬ0⊂Eμ, such that for any bounded set ℬ of Eμ, there exists a time t(τ,ℬ)≥τ yielding
(26)U(t,τ)ℬ⊂ℬ0,∀t≥t(τ,ℬ),
where ℬ0=ℬ(0,R0)⊂Eμ is a closed ball centered at 0 with radius R0.Lemma3 shows that there exists a time t0=˙t0(τ,ℬ0)≥τ, such that
(27)U(t,τ)ℬ0⊂ℬ0,∀t≥t0.Lemma 4 (see [28]).
Letf(t)=(fm(t))m∈ℤ∈ℋ with X=L2 and g(t)=(gm(t))m∈ℤ∈ℋ with X=ℓ2, respectively. Then, for any ε>0, there exist t*=˙t(ε,ℬ0)>t0 and M0(ε,τ,ℬ0)∈ℕ, such that when t≥t*, the solution U(t+τ,τ)ψτ=ψ(t+τ)=(ψm(t+τ))m∈ℤ∈Eμ of (21)-(22) with ψτ∈ℬ0 satisfies
(28)∑|m|>M0(ε,τ,ℬ0)‍|(U(t+τ,τ)ψτ)m|Eμ2=∑|m|>M0(ε,τ,ℬ0)‍|ψm(t+τ)|Eμ2≤ε2,
where |ψm|Eμ2=|(Bu)m|2+μum2+vm2+|zm|2.


## 3. Existence of the Pullback Exponential Attractors

In this section, we prove the existence of the pullback exponential attractor for the process{U(t,τ)}t≥τ defined by (24). Write
(29)Eμ(N)={ψ=(ψm)m∈ℤ∈Eμ∣ψm=(0,0,0)Tif|m|>N},
then Eμ(N) is a 4(2N+1)-dimensional subspace of Eμ. Define a bounded projection PN:Eμ↦Eμ(N)⊂Eμ by
(30)(PNψ)m={ψm,|m|≤N;0,|m|>N,ψ=(ψm)m∈ℤ∈Eμ.Lemma 5.
(I) For anyT>t0, there exists some LT>0 (independent of τ), such that for every τ∈ℝ and any t∈[t0,T],
(31)∥U(t+τ,τ)ψτ(1)-U(t+τ,τ)ψτ(2)∥Eμ≤LT∥ψτ(1)-ψτ(2)∥Eμ,∀ψτ(1),ψτ(2)∈ℬ0.
(II) There exist two positive constantsT*>t0 and γ∈(0,1/2), and a 4(2N*+1)-dimensional orthogonal projection PN*:Eμ↦Eμ(N*) for some N*∈ℕ, such that for every τ∈ℝ and ψτ(1),ψτ(2)∈ℬ0,
(32)∥(I-PN*)[U(T*+τ,τ)ψτ(1)-U(T*+τ,τ)ψτ(2)]∥Eμ≤γ∥ψτ(1)-ψτ(2)∥Eμ.Proof.
(I) For anyτ∈ℝ, let
(33)ψ(1)(t)=U(t,τ)ψτ(1),ψ(2)(t)=U(t,τ)ψτ(2),∀t≥τ
be two solutions of  (21)-(22) with initial conditions ψτ(1),ψτ(2)∈ℬ0, respectively. Set
(34)ud(t)=u(1)(t)-u(2)(t),vd(t)=v(1)(t)-v(2)(t),zd(t)=z(1)(t)-z(2)(t),ψd(t)=ψ(1)(t)-ψ(2)(t).
From (21)-(22), we get
(35)ψ˙d+Θψd=F(ψ(1),t)-F(ψ(2),t),∀t>τ.ψd(τ)=ψτ(1)-ψτ(2).
Taking the real part of the inner product of (35) with ψd in Eμ, we obtain
(36)12ddt∥ψd∥Eμ2+Re(Θψd+F(ψ(2),t)-F(ψ(1),t),ψd)Eμ=0,∀t≥τ.
SinceΘ:Eμ↦Eμ is a bounded linear operator, F:Eμ×ℝ↦Eμ is a locally Lipschitz continuous operator (see Lemma 2.2 in [28]), and ℬ0 is a bounded set in Eμ, we see that there exist two positive constants C1 and K1=K1(ℬ0), such that
(37)Re(Θψd+F(ψ(2),t)-F(ψ(1),t),ψd)Eμ≤(C1∥ψd∥Eμ+∥F(ψ(2),t)-F(ψ(1),t)∥Eμ)∥ψd∥Eμ≤(C1+K1)∥ψd∥Eμ2,∀t≥τ.
Combining (36) and (37), we get
(38)ddt∥ψd∥Eμ2-K2∥ψd∥Eμ2≤0,∀t≥τ,
where K2=2(C1+K1). Applying Gronwall inequality to (38) on [τ,τ+t] with t∈[t0,T], we obtain
(39)∥ψd(t+τ)∥Eμ2≤eK2t∥ψd(τ)∥Eμ2,∀t∈[t0,T].
Thus,
(40)∥ψd(t+τ)∥Eμ=∥ψ(1)(t+τ)-ψ(2)(t+τ)∥Eμ=∥U(t+τ,τ)ψτ(1)-U(t+τ,τ)ψτ(2)∥Eμ≤LT∥ψτ(1)-ψτ(2)∥Eμ,∀t∈[t0,T],
where LT=eK2T does not depend on τ.
(II) Define a smooth functionχ(x)∈𝒞1(ℝ+,[0,1]), such that
(41)χ(x)={0,0≤x≤1;1,x≥2,|χ′(x)|≤χ0(positiveconstant),∀x∈ℝ+.
Set(42)pd=(pdm)m∈ℤwithpdm=χ(|m|M)udm,qd=(qdm)m∈ℤwithqdm=χ(|m|M)vdm,wd=(wdm)m∈ℤwithwdm=χ(|m|M)zdm,ϕd=(ϕdm)m∈ℤwithϕdm=(pdm,qdm,wdm),
where M is a positive integer that will be specified later. From (16), we see that
(43)iz˙d-Azd+iαzd+z(1)u(1)-z(2)u(2)=0.
Taking the imaginary part of the inner product of (43) with wd in L2, we get
(44)12ddt∑m∈ℤ‍χ(|m|M)|zdm|2-Im(Azd,wd)+α∑m∈ℤ‍χ(|m|M)|zdm|2+Im(z(1)u(1)-z(2)u(2),wd)=0.
Now, we have(45)-Im(Azd,wd)=-Im(Bzd,Bwd)=-Im(∑m∈ℤ‍(zdm+1-zdm)×(χ(|m+1|M)z-dm+1-χ(|m|M)z-dm){∑m∈ℤ‍})=Im(∑m∈ℤ‍(χ(|m|M)zdm+1z-dm+χ(|m+1|M)z-dm+1zdm){∑m∈ℤ‍})=Im(∑m∈ℤ‍(χ(|m+1|M)-χ(|m|M))z-dm+1zdm)≥-∑m∈ℤ‍|χ′(m~M)|1M|zdm+1||zdm|≥-2χ0M∥ψd∥Eμ2,∀t≥τ,
where m~ locates between |m+1| and |m|. According to Lemma 4, we know that there exist t1(t1>t0) and M1(ε,τ,ℬ0)∈ℕ, such that when t≥t1 and M>M1(ε,τ,ℬ0), we obtain
(46)Im∑m∈ℤ‍(zm(1)um(1)-zm(2)um(2),χ(|m|M)(zm(1)-zm(2)))≤∑m∈ℤ‍χ(|m|M)|zm(1)um(1)-zm(2)um(2)||zm(1)-zm(2)|=∑m∈ℤ‍χ(|m|M)|zm(1)(um(1)-um(2))+um(2)(zm(1)-zm(2))|×|zm(1)-zm(2)|≤α4∑m∈ℤ‍χ(|m|M)|zm(1)-zm(2)|2+αμδ2×∑m∈ℤ‍χ(|m|M)|um(1)-um(2)||zm(1)-zm(2)|≤α2∑m∈ℤ‍χ(|m|M)|zm(1)-zm(2)|2+μδ4∑m∈ℤ‍χ(|m|M)|um(1)-um(2)|2,
which implies that when t≥t1 and M>M1(ε,τ,ℬ0)(47)Im(z(1)u(1)-z(2)u(2),wd)=Im∑m∈ℤ‍χ(|m|M)(zm(1)um(1)-zm(2)um(2),(zm(1)-zm(2)))≥-α2∑m∈ℤ‍χ(|m|M)|zm(1)-zm(2)|2-μδ4∑m∈ℤ‍χ(|m|M)|um(1)-um(2)|2.
Then, taking (44)–(47) into account, we obtain for every t≥t1 and M>M1(ε,τ,ℬ0) that
(48)ddt∑m∈ℤ‍χ(|m|M)|zdm|2+α∑m∈ℤ‍χ(|m|M)|zdm|2≤μδ2∑m∈ℤ‍χ(|m|M)udm2+4χ0M∥ψd∥Eμ2.
From (17) and (19), we obtain
(49)v˙d+(ν-δ)vd+(δ(δ-ν)+μ)ud+Aud-β(|z(1)|2-|z(2)|2)=0.
Taking the inner product of (49) with qd in ℓ2, we obtain
(50)12ddt∑m∈ℤ‍χ(|m|M)vdm2+(ν-δ)×∑m∈ℤ‍χ(|m|M)vdm2+((δ(δ-ν)+μ)ud,qd)+(Aud,qd)-β(|z(1)|2-|z(2)|2,qd)=0.
It is clear thatqd=p˙d+δpd. Then, (50) can be rewritten as
(51)ddt∑m∈ℤ‍χ(|m|M)(μudm2+vdm2)+2(ν-δ)∑m∈ℤ‍χ(|m|M)vdm2+2μδ∑m∈ℤ‍χ(|m|M)udm2+2δ(δ-ν)(ud,qd)+2(Aud,qd)-2β(|z(1)|2-|z(2)|2,qd)=0.
Also, we have(52)(Aud,qd)=(Aud,p˙d)+δ(Aud,pd)=(Bud,Bp˙d)+δ(Bud,Bpd).
By some computations, we get(53)(Bud,Bp˙d)=∑m∈ℤ‍(Bud)m(Bp˙d)m=∑m∈ℤ‍(Bud)m[χ(|m+1|M)u˙dm+1-χ(|m|M)u˙dm]=∑m∈ℤ‍(Bud)m[(Bu˙d)mχ(|m|M)+(χ(|m+1|M)-χ(|m|M))u˙dm+1]=∑m∈ℤ‍χ(|m|M)(Bud)m(Bu˙d)m+∑m∈ℤ‍χ′(m~M)1M(udm+1-udm)×(vdm+1-δudm+1)≥12ddt∑m∈ℤ‍χ(|m|M)|(Bud)m|2-2χ0(μ+δ+1)Mμ∥ψd∥Eμ2,∀t≥τ,δ(Bud,Bpd)=δ∑m∈ℤ‍(Bud)m(Bpd)m=δ∑m∈ℤ‍(Bud)m[χ(|m+1|M)udm+1-χ(|m|M)udm]=δ∑m∈ℤ‍(Bud)m[(Bud)mχ(|m|M)+(χ(|m+1|M)-χ(|m|M))udm+1]=δ∑m∈ℤ‍χ(|m|M)|(Bud)m|2+δ∑m∈ℤ‍χ′(m~M)1M(udm+1-udm)udm+1≥δ∑m∈ℤ‍χ(|m|M)|(Bud)m|2-2χ0δMμ∥ψd∥Eμ2,∀t≥τ.
Here m~ locates between |m+1| and |m|. Inserting (53) into (52), we get
(54)(Aud,qd)≥12ddt∑m∈ℤ‍χ(|m|M)|(Bud)m|2+δ∑m∈ℤ‍χ(|m|M)|(Bud)m|2-2χ0(μ+2δ+1)Mμ∥ψd∥Eμ2,∀t≥τ.
According to Lemma4, we can see that there exist t2>t1 and M2(ε,τ,ℬ0)>M1(ε,τ,ℬ0), such that when t≥t2 and M>M2(ε,τ,ℬ0),
(55)β(|z(1)|2-|z(2)|2,qd)=β∑m∈ℤ‍χ(|m|M)(|zm(1)|2-|zm(2)|2)vdm≤β∑m∈ℤ‍χ(|m|M)(|zm(1)|+|zm(2)|)|zdm||vdm|≤β2αν2β∑m∈ℤ‍χ(|m|M)|zdm||vdm|≤ν2∑m∈ℤ‍χ(|m|M)vdm2+α4∑m∈ℤ‍χ(|m|M)|zdm|2.
It follows from (51) and (54)-(55) that when M>M2(ε,τ,ℬ0) and t≥t2, we have
(56)ddt∑m∈ℤ‍χ(|m|M)[|(Bud)m|2+μudm2+vdm2]+2δ∑m∈ℤ‍χ(|m|M)|(Bud)m|2+2μδ∑m∈ℤ‍χ(|m|M)udm2+(ν-2δ)×∑m∈ℤ‍χ(|m|M)vdm2-2δ(ν-δ)(ud,qd)≤α2∑m∈ℤ‍χ(|m|M)|zdm|2+4χ0(μ+2δ+1)Mμ∥ψd∥Eμ2.
Combining (48) and (56), when t≥t2, we obtain
(57)ddt∑m∈ℤ‍χ(|m|M)[|(Bud)m|2+μudm2+vdm2+|zdm|2]+2δ∑m∈ℤ‍χ(|m|M)|(Bud)m|2+3μδ2∑m∈ℤ‍χ(|m|M)udm2+(ν-2δ)×∑m∈ℤ‍χ(|m|M)vdm2+α2∑m∈ℤ‍χ(|m|M)|zdm|2-2δ(ν-δ)(ud,qd)≤4χ0(2μ+2δ+1)Mμ×∥ψd∥Eμ2,∀M>M2(ε,τ,ℬ0).
Sinceδ2ν2=μδ(ν/2-δ), we get for any m∈ℤ that
(58)μδudm2+(ν2-δ)vdm2-2δ(ν-δ)udmvdm≥μδudm2+(ν2-δ)vdm2-2δν|udmvdm|≥0.
Thus,
(59)μδ∑m∈ℤ‍χ(|m|M)udm2+(ν2-δ)∑m∈ℤ‍χ(|m|M)vdm2-2δ(ν-δ)(ud,qd)≥0.
We then conclude from (57) and (59) that when M>M2(ε,τ,ℬ0) and t≥t2,
(60)ddt∑m∈ℤ‍χ(|m|M)[|(Bud)m|2+μudm2+vdm2+|zdm|2]+2δ∑m∈ℤ‍χ(|m|M)|(Bud)m|2+μδ2∑m∈ℤ‍χ(|m|M)udm2+(ν2-δ)∑m∈ℤ‍χ(|m|M)vdm2+α2∑m∈ℤ‍χ(|m|M)|zdm|2≤4χ0(2μ+2δ+1)Mμ∥ψd∥Eμ2.
Choosingϑ=min{δ/2,ν/2-δ,α/2}, we obtain
(61)ddt∑m∈ℤ‍χ(|m|M)|ψdm|Eμ2+ϑ∑m∈ℤ‍χ(|m|M)|ψdm|Eμ2≤4χ0(2μ+2δ+1)Mμ∥ψd∥Eμ2,∀t≥t2,∀M>M2(ε,τ,ℬ0).
Applying Gronwall inequality to (61) from τ+t2 to τ+t with t>t2, we get for every ψτ(1),ψτ(2)∈ℬ0,M>M2(ε,τ,ℬ0) and t≥t2 that
(62)∑m∈ℤ‍χ(|m|M)|ψdm(t+τ)|Eμ2≤e-ϑ(t-t2)∑m∈ℤ‍χ(|m|M)|ψdm(τ+t2)|Eμ2+4χ0(2μ+2δ+1)Mμ∫τ+t2τ+t∥ψd(s)∥Eμ2e-ϑ(t+τ-s)ds.
By (39), when M>M2(ε,τ,ℬ0) and t≥t2 we have
(63)e-ϑ(t-t2)∑m∈ℤ‍χ(|m|M)|ψdm(τ+t2)|Eμ2≤e-ϑ(t-t2)∥ψd(τ+t2)∥Eμ2≤e-ϑ(t-t2)+K2t2∥ψd(τ)∥Eμ2,4χ0(2μ+2δ+1)Mμ∫τ+t2τ+t∥ψd(s)∥Eμ2e-ϑ(t+τ-s)ds≤4χ0(2μ+2δ+1)Mμ∥ψd(τ)∥Eμ2×∫τ+t2τ+te-ϑ(t+τ)+(ϑ+K2)s-K2τds=4χ0(2μ+2δ+1)Mμ∥ψd(τ)∥Eμ2e-ϑ(t+τ)-K2τ×∫τ+t2τ+te(ϑ+K2)sds≤4χ0(2μ+2δ+1)Mμ(ϑ+K2)∥ψd(τ)∥Eμ2eK2t.
Thus, it follows from (62)-(63) that for any t≥t2 and M>M2(ε,τ,ℬ0),
(64)∑m∈ℤ‍χ(|m|M)|ψdm(t+τ)|Eμ2≤e-ϑ(t-t2)+K2t2∥ψd(τ)∥Eμ2+4χ0(2μ+2δ+1)Mμ(ϑ+K2)∥ψd(τ)∥Eμ2eK2t.
Pick two sufficient large numbersT*≥t2 and M*>M2(ε,τ,ℬ0) to satisfy
(65)e-ϑ(T*-t2)+K2t2+4χ0(2μ+2δ+1)M*μ(ϑ+K2)eK2T*=˙γ2<14.
Then, from (64), we have for N*>2M* that
(66)∑|m|>N*‍|ψdm(T*+τ)|Eμ2≤∑m∈ℤ‍χ(|m|M*)|ψdm(T*+τ)|Eμ2≤γ2∥ψd(τ)∥Eμ2;
that is,
(67)∥(I-PN*)[U(T*+τ,τ)ψτ(1)-U(T*+τ,τ)ψτ(2)]∥Eμ≤γ∥ψτ(1)-ψτ(2)∥Eμ,
where γ<1/2. The proof is complete.Now, we can state the main result of this paper.Theorem 6.
Let the conditions of Lemma4 hold. Then, the process {U(t,τ)}t≥τ associated with (21)-(22) possesses a pullback exponential attractor {𝒜(t)}t∈ℝ, satisfying(1)
(compactness and finiteness of dimension) for eacht∈ℝ, {𝒜(t)} is a compact set of Eμ, and the fractal dimension dimF𝒜(t) is finite and uniformly bounded in t; that is,
(68)supt∈ℝdimF𝒜(t)<∞;(2)
(positive invariant property)U(t,τ)𝒜(τ)⊂𝒜(t) for all t≥τ;(3)
(pullback exponential attractivity) there exist an exponentη>0 and two positive-valued functions Q,ℱ:ℝ+↦ℝ+, such that for any bounded set ℬ⊂Eμ,
(69)DistEμ(U(t,τ)ℬ,𝒜(t))≤Q(∥ℬ∥Eμ)e-η(t-τ),τ∈ℝ,τ+ℱ(∥ℬ∥Eμ)≤t<∞,where DistEμ(·,·) is the Hausdorff semidistance between two subsets of Eμ.Proof.
Using Lemmas 2.3 and 3.1 and Theorem 2 of [30], we obtain the result.Remark 7.
The spectrum of Lyapunov exponents is the most precise tool for identification of the character of motion of a dynamical system [38]. There are some works on the estimation of the dominant Lyapunov exponent of nonsmooth systems by means of synchronization method, one can refer to the articles of Stefański  et al. [38–40]. In [38], Stefański and Kapitaniak presented a method to estimate the value of largest Lyapunov exponent both for discrete dynamical systems of known difference equations and also for discrete maps reconstructed from the time evolution of the given system. Following this clue, we can ask naturally the problem that whether the method presented in [38] could be applied to estimate Lyapunov exponents for the trajectories on the pullback attractor {𝒜(t)}. It is an interesting and challenging issue for us to investigate.



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*Source: 809476-2013-07-01.xml*