# Exponential Attractor for Lattice System of Nonlinear Boussinesq Equation

Authors: Min Zhao; Shengfan Zhou
Journal: Discrete Dynamics in Nature and Society
 (2013)

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## Abstract

We study the lattice dynamical system of a nonlinear Boussinesq equation. We first verify the Lipschitz continuity of the continuous semigroup associated with the system. Then, we provide an estimation of the tail of the difference between two solutions of the system. Finally, we obtain the existence of an exponential attractor of the system.


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## Body

## 1. Introduction

Lattice dynamical systems (LDSs) have a wide range of applications in many areas such as electrical engineering, chemical reaction theory, laser systems, material science, and biology [1, 2]. In recent years, many works about the asymptotic behavior of LDSs have been done, which include the global attractor, see [3–11] and the references therein. However, the global attractor sometimes attracts orbits at a relatively slow speed and it might take an unexpected long time to be reached. For this reason, the exponential attractor having finite fractal dimension and attracting all bounded sets exponentially was introduced, and it has been studied for a large class of LDSs, see [12–15] and the references therein. Han presented in [13] some sufficient conditions for the existence of exponential attractor for LDSs in the weighted space of infinite sequences and applied the result to obtain the existence of exponential attractors for some LDSs. Zhou and Han in [15] presented some sufficient conditions for the existence of uniform exponential attractor for LDSs, which is easier to verify the existence of exponential attractor for some LDSs. Abdallah in [3] considered the following initial problem of lattice system of nonlinear Boussinesq equation:
(1)u¨i+δu˙i+α(Au)i+β(Bu)i+λui0000-13k(D(D*u)3)i=fi,i∈ℤ,(2)ui(0)=ui,0,u˙i(0)=u1i,0,i∈ℤ,
where δ, α, λ, and  k are positive constants, β is a real constant; for i∈ℤ, ui,fi∈ℝ; u=(ui)i∈ℤ and A, B, D, and  D* are linear operators (see Section 3 for details). Equation (1) can be regarded as a spatial discretization of the following nonlinear damped Boussinesq equation on ℝ:
(3)utt+δut+αuxxxx+βuxx+λu-kux2uxx=f(x),
which appears in many fields of physics and mechanics, for example, long waves in shallow water, nonlinear elastic beam systems, thermomechanical phase transitions, and some Hamiltonian mechanics. Abdallah has in [3] investigated the existence and finite-dimensional approximation of the global attractor for (1) under the following conditions:
(4)f=(fi)i∈ℤ∈l2,λ>4|β|.
In this paper, motivated by the ideas of [13, 15], we will further prove the existence of an exponential attractor for the system (1) under the condition (4).The paper is organized as follows. In Section2, we present some preliminaries. Section 3 is devoted to the existence of an exponential attractor for (1).


## 2. Preliminaries

In this section, we present the definition of an exponential attractor and some sufficient conditions for the existence of an exponential attractor for a semigroup in a separable Hilbert space from [13, 15].LetEs be a separable Hilbert space, let  𝒪s be a bounded subset of Es, and let  {S(t)}t≥0 be a semigroup acting on 𝒪s which satisfy: S(t)S(s)=S(t+s), S(0)=IEs, for all t, s≥0, and S(t)𝒪s⊆𝒪s for t≥0, where IEs is the identity operator on Es.Definition 1.
A setℳs is called an exponential attractor for the semigroup {S(t)}t≥0 on 𝒪s, if(i)
ℳs is compact;(ii)
ℬs⊆ℳs⊆𝒪s, where ℬs is the global attractor;(iii)
S(t)ℳs⊆ℳs, t>0;(iv)
ℳs has a finite fractal dimension;(v)
there exist two positive constantsa1 and a2 such that dist (S(t)u,ℳs)≤a1e-a2t for all u∈𝒪s, t≥0.LetEsN be a N-dimensional subspace of Es,N∈ℕ. We define the bounded N-dimensional orthogonal projection PN:Es→EsN from Es into EsN and QN=IEs-PN.As a direct consequence of [13, Theorem 2.5] and [15, Theorem 2.1], we have the following theorem.Theorem 2.
Let{S(t)}t≥0 be a continuous semigroup on Es and let  𝒪s be a closed bounded subset of Es such that S(t)𝒪s⊆𝒪s, for t≥0. If there exist t*>0, a constant L=L(t*)>0 and a N-dimensional subspace EsN of Es(N=N(t*)∈ℕ) such that for any u1,u2∈𝒪s,
(5)∥S(t)u1-S(t)u2∥Es≤L∥u1-u2∥Es,∀t∈[0,t*],∥QN(S(t)u1-S(t)u2)∥Es2≤1128∥u1-u2∥Es2,
Then, (i)
S(t*)=S* has an exponential attractor ℳs* on 𝒪s with dimf(ℳs*)≤K0NlnL2+1, where K0 is a constant;(ii)
ℳs=⋃0≤t≤t*S(t)ℳs* is an exponential attractor for {S(t)}t≥0 on 𝒪s that dimf(ℳs)≤dimf(ℳs*)+1, and there exist two positive constants a1 and a2 such that dist(S(t)u,ℳs)≤a1e-a2t for all u∈𝒪s, t≥0.


## 3. Exponential Attractor for System (1)

Letl2={u=(ui)i∈ℤ:ui∈ℝ,∑i∈ℤui2<+∞} and equip it with the inner product and norm as
(6)(u,v)=∑i∈ℤ‍uivi,∥u∥2=(u,u),u=(ui)i∈ℤ,v=(vi)i∈ℤ∈l2.
Then, (l2,∥·∥,(·,·)) is a separable Hilbert space. The linear operators A, B, D, and  D* are defined from l2 into l2 as follows: for any u=(ui)i∈ℤ∈l2,
(7)(Au)i=ui+2-4ui+1+6ui-4ui-1+ui-2,(Bu)i=ui+1-2ui+ui-1,(Du)i=ui+1-ui,(D*u)i=ui-ui-1,∀i∈ℤ,
then, A=B2, B=D*D=DD*.The system (1) with initial data (2) is equivalent to the following vector form:
(8)u¨+δu˙+α(Au)+β(Bu)+λu-13kD(D*u)3=f,∀t>0,u(0)=(ui,0)i∈ℤ,u˙(0)=(u1i,0)i∈ℤ,
where u=(ui)i∈ℤ, Au=((Au)i)i∈ℤ, Bu=((Bu)i)i∈ℤ,  D(D*u)3=((D(D*u)3)i)i∈ℤ, f=(fi)i∈ℤ.Letting(9)v=u˙+εu,φ=(uv),whereε>0,
then, the system (8) can be written as the following initial value problem:
(10)φ˙+C(φ)=F(φ),φ(0)=(u(0),v(0))T=(u(0),u˙(0)+εu(0))T,
where
(11)C(φ)=(εu-vαAu+λu+(δ-ε)(v-εu)),F(φ)=(0-βBu+13kD(D*u)3+f).
We define
(12)(u,v)λ=(Bu,Bv)+λ(u,v),∥u∥λ=(∥Bu∥2+λ∥u∥2)1/2,∀u=(ui)i∈ℤ,00000000000000000v=(vi)i∈ℤ∈l2.
Then, the bilinear form (·,·)λ is an inner product on l2 and the induced norm ∥·∥λ is equivalent to ∥·∥. Let lλ2=(l2,(·,·)λ,∥·∥λ) and let H=lλ2×l2, then, H is a separable Hilbert space with the following norm:
(13)∥φ∥H=(∑i∈ℤ‍((Bu)i2+λui2+vi2))1/2,0000000∀φ=(ui,vi)i∈ℤ∈H.In this section, we will study the existence of an exponential attractor of (10) in the space H.Lemma 3 (see [3]).
Assume (4) holds. Then, there exist small ε>0 and M1>0, such that
(14)ε2+3ε≤2δ,δ≥4ε,ε(1+δ)+4|β|≤λ,λ(1-M14λ-2M1)≥4|β|.
Moreover,(1)
for any initial dataφ(0)=(u(0),v(0))T∈H, there exists a unique solution φ(t)=(u(t),v(t))T of (10), such that φ(·)∈𝒞([0,∞),H)∩𝒞1((0,∞),H), and the solution map
(15)Sε(t):φ(0)=(u(0),v(0))T∈H↦φ(t)=(u(t),v(t))T∈H,t>0,generates a continuous semigroup{Sε(t)}t≥0 on H.(2)
The semigroup{Sε(t)}t≥0 possesses a closed bounded absorbing ball 𝒪=B(0,r0)={φ∈H:∥φ∥H≤r0}⊂H, where r0=2∥f∥/M3, M3=M1M2, M2=min{1/8,δ/8ε,   α/2,M1}. Therefore, there exists a constant T0=T0(𝒪) such that Sε(t)𝒪⊆𝒪,  for t≥T0.(3)
For anyη>0, there exist K(η)∈ℕ and T(η)≥0 such that the solution φ(t)=((ui(t),vi(t)))i∈ℤ of (10) with φ(0)∈𝒪 satisfies
(16)∑|i|≥K(η)‍∥φi(t)∥H200=∑|i|≥K(η)‍((Bu(t))i2+λ(ui(t))2+(vi(t))2)00≤η,∀t≥T(η).(4)
The semigroup{Sε(t)}t≥0 of (10) possesses a global attractor ℬ⊂𝒪⊂H.In the following, we first verify the Lipschitz continuity of{Sε(t)}t≥0 and provide an estimation of the tail of the difference between two solutions of (10). Then, we obtain the existence of an exponential attractor of (10) by Theorem 2.Forj=1,2, φ(j0)∈𝒪, t≥0, let φ(j)(t)=Sε(t)φ(j0)=(u(j)(t),v(j)(t)) be the solutions of (10). Set Φ(t)=φ(1)(t)-φ(2)(t) = Sε(t)φ(10)-Sε(t)φ(20) = (ω(t),ζ(t)) = ((ωi(t))i∈ℤ,(ζi(t))i∈ℤ), we have by (10) that
(17)Φ˙+C(Φ)=F(φ(1))-F(φ(2)),Φ(0)=φ(10)-φ(20).Lemma 4.
Assume that (4) and (14) hold. Let
(18)M4=max{2+4ε2λ-4|β|+εδ-2ε2,8},M5=min{18,α2,λ-4|β|2λ,δ-2ε4ε},M6=max{α+12,2λ+4|β|+εδ-ε22λ},L(t)=M6M5e|-ε+8kr02M4|t,
Then, (1)
(19)∥Sε(t)φ(10)-Sε(t)φ(20)∥H2≤L(t)∥φ(10)-φ(20)∥H2,∀t≥0.(2)
There existT*>0 and M*∈ℕ, such that
(20)∑|i|>M*‍∥(Sε(T*)φ(10)-Sε(T*)φ(20))i∥H2≤1128∥φ(10)-φ(20)∥H2,where
(21)∥Φi(t)∥H2=∥(ωi(t),ζi(t))∥H2=(Bω(t))i2+λ(ωi(t))2+(ζi(t))2,00000000000000000000∀i∈ℤ.Proof.
(1) Taking the inner product(·,·)H of (17) with Φ(t), we obtain
(22)((13kD(D*u(1))3-13kD(D*u(2))3)ω¨+δω˙+α(Aω)+β(Bω)+λω0-(13kD(D*u(1))3-13kD(D*u(2))3),ω˙+εω)=0.
We can write (22) into the following form:
(23)ddtP(t)+N(t)=0,
where
(24)P(t)=12∥ω˙∥2+α2∥Bω∥2-β2∥Dω∥2+λ2∥ω∥2+ε(ω˙,ω)+εδ2∥ω∥2,N(t)=(δ-ε)∥ω˙∥2+εα∥Bω∥2-εβ∥Dω∥2+ελ∥ω∥2-13k(D((D*u(1))3-(D*u(2))3),ω˙+εω).
Then,
(25)εP(t)-N(t)00=3ε-2δ2∥ω˙∥2-εα2∥Bω∥2+εβ2∥Dω∥20000+ε(εδ-λ)2∥ω∥2+ε2(ω˙,ω)0000+13k(D((D*u(1))3-(D*u(2))3),ω˙+εω)00≤ε2+3ε-2δ2∥ω˙∥20000+ε(ε(1+δ)+4|β|-λ)2∥ω∥20000+13k(D((D*u(1))3-(D*u(2))3),ω˙+εω)00≤13k(D((D*u(1))3-(D*u(2))3),ω˙+εω).
Since
(26)P(t)≥14∥ω˙∥2+α2∥Bω∥2+(λ-4|β|+εδ-2ε22)∥ω∥2,
thus,
(27)13k(D((D*u(1))3-(D*u(2))3),ω˙+εω)00=13k((D*u(1))3-(D*u(2))3,D*(ω˙+εω))00≤8kr02(2∥ω˙∥2+(1+2ε2)∥ω∥2)00≤8kr02M4P(t).
From (23), (25), and (27), it follows that for t>0,
(28)ddtP(t)≤(-ε+8kr02M4)P(t).
Applying Gronwall’s inequality to (28), we obtain
(29)P(t)≤e(-ε+8kr02M4)tP(0).
Since
(30)P(t)≥14∥ω˙∥2+α2∥Bω∥2+λ-4β2∥ω∥2+εδ-2ε22∥ω∥2≥M5∥Φ(t)∥H2,(31)P(t)≤12∥ω˙∥2+α2∥Bω∥2+2|β|∥ω∥2+λ2∥ω∥2+ε(ω˙,ω)+εδ2∥ω∥2≤M6∥Φ(t)∥H2.
From (29) to (31), it follows that for t>0,
(32)∥Φ(t)∥H2≤M6M5e(-ε+8kr02M4)t∥Φ(0)∥H2.
(2) Choosing a smooth increasing functionξ∈𝒞1(ℝ+,ℝ) satisfies
(33)ξ(s)=0,0⩽s<1,0⩽ξ(s)⩽1,1⩽s<2,ξ(s)=1,s⩾2,|ξ′(s)|≤C0,s∈ℝ+,
where C0 is a positive constant. For t≥0, let Ψi=ξ(|i|/M)Φi, y=(ξ(|i|/M)ωi(t))i∈ℤ, z=(ξ(|i|/M)ζi(t))i∈ℤ, where M∈ℕ. Taking the inner product (·,·)H of (17) with Ψ={Ψi(t)}i∈ℤ, we obtain
(34)((13kD(D*u(1))3-13kD(D*u(2))3)ω¨+δω˙+α(Aω)+β(Bω)+λω0-(13kD(D*u(1))3-13kD(D*u(2))3),z)=0.
Similar to (4.3)–(4.5) in [3], we can get
(35)ddtP1(t)+N1(t)=0,
where
(36)P1(t)=∑i∈ℤ‍ξ(|i|M)(12ω˙i2+α2(Bω)i2-β2(Dω)i200000000000000000+λ2ωi2+εω˙iωi+εδ2ωi2),N1(t)=∑i∈ℤ‍(ξ(|i|M)((δ-ε)ω˙i2+εα(Bω)i200000000000-εβ(Dω)i2+ελωi2)(|i|M))000000+13k((ξ(|i|M)(D*ζ)i)i∈ℤ(D*u(1))3-(D*u(2))3,000000000000(ξ(|i|M)(D*ζ)i)i∈ℤ)000000+∑i∈ℤ‍(α(Bω)i((Bz)i-ξ(|i|M)(Bζ)i)000000000000+β(Dω)i((Dz)i-ξ(|i|M)(Dζ)i)000000000000+13k((D*u(1))i3-(D*u(2))i3)000000000000×((Dz)i-ξ(|i|M)(Dζ)i)).
Then,
(37)εP1(t)-N1(t)00≤-13k((D*u(1))3-(D*u(2))3,(ξ(|i|M)(D*ζ)i)i∈ℤ)0000-∑i∈ℤ‍(α(Bω)i((Bz)i-ξ(|i|M)(Bζ)i)0000000000+β(Dω)i((Dz)i-ξ(|i|M)(Dζ)i)0000000000+13k((D*u(1))i3-(D*u(2))i3)0000000000×((Dz)i-ξ(|i|M)(Dζ)i)).
By (3) of Lemma 3, there exist K1=K(λε/16kM4), T1=T(λε/16kM4), such that
(38)∑|i|≥K1‍((Bu(j)(t))i2+λ(ui(j)(t))2+(vi(j)(t))2)≤λε16kM4,j=1,2,∀t≥T1.
This implies that
(39)(ui(1)(t))2+(ui(2)(t))2≤ε8kM4,∀|i|>K1,t≥T1.
Then, for M>K1+1, t≥T1,
(40)-13k((D*u(1))3-(D*u(2))3,(ξ(|i|M)(D*ζ)i)i∈ℤ)00≤13k∑i∈ℤ‍ξ(|i|M)|(D*ω)i(D*ζ)i|0000×((D*u(1))i2+(D*u(1))i(D*u(2))i+(D*u(2))i2)00≤k∑i∈ℤ‍ξ(|i|M)|(D*ω)i(D*ζ)i|0000×((u(1))i2+(u(2))i2+(u(1))i-12+(u(2))i-12)00≤ε2P1(t)+C0εM64M∥Φ∥H2.
Since
(41)-∑i∈ℤ‍(α(Bω)i((Bz)i-ξ(|i|M)(Bζ)i)00000+β(Dω)i((Dz)i-ξ(|i|M)(Dζ)i)00000+13k((D*u(1))i3-(D*u(2))i3)00000×((Dz)i-ξ(|i|M)(Dζ)i))00≤C0M((6α+3|β|+k6)α∥Bω∥2+(2|β|+384kr04)∥ω∥200000000+(6α+3|β|+k6)∥ζ∥2)00≤C0M7M∥Φ∥H2,
where M7=max{α,(2|β|+384kr04)/λ,(6α+3|β|+k)/6}. From (32), (35), (37), and (40)-(41), it follows that for M>K1+1, t≥T1,
(42)ddtP1(t)≤-ε2P1(t)+C0M6M8M5Me(-ε+8kr02M4)t∥Φ(0)∥H2,
where M8=εM6/4+M7. Applying Gronwall’s inequality to (42) from T2 to t, where T2=max{T0,T1}, we obtain that for M>K1+1,
(43)P1(t)≤e-(ε/2)(t-T2)P1(T2)+2C0M6M8|-ε+16kr02M4|M5Me(-ε+8kr02M4)t×∥Φ(0)∥H2.
Similar to (30), we can get
(44)M5∑i∈ℤ‍ξ(|i|M)∥Φi(t)∥H2≤P1(t).
Since
(45)P1(T2)≤P(T2).
By (31)-(32), we obtain
(46)P(T2)≤M62M5e(-ε+8kr02M4)T2∥Φ(0)∥H2.
From (43) to (46), it follows that for t≥T2, M>K1+1,
(47)∑i∈ℤ‍ξ(|i|M)∥Φi(t)∥H200≤(M62M52e-(εt+εT2-16kr02M4T2)/200000+2C0M6M8|-ε+16kr02M4|M52Me(-ε+8kr02M4)t)∥Φ(0)∥H2.
Letting
(48)T*=max{1ε(2(ln(256M62)-lnM52)0T*=max00+16kr02M4T2-εT2(ln(256M62)-lnM52)),T21ε(2(ln(256M62)-lnM52)};M*=max{1024C0M6M8|-ε+16kr02M4|M52e(-ε+8kr02M4)T*,2K1+3},
we then have
(49)M62M52e-(εT*+εT2-16kr02M4T2)/20000+2C0M6M8|-ε+16kr02M4|M52M*e(-ε+8kr02M4)T*≤1128,∑|i|>M*‍∥Φi(T*)∥2≤1128∥Φ(0)∥2.As a direct consequence of (1)-(2), (4) of Lemma3, (1)-(2) of Lemma 4 and Theorem 2, we have our main result.Theorem 5.
Assume that (4) and (14) hold. Then, the semigroup {Sε(t)}t≥0 of (10) possesses an exponential attractor ℳ on 𝒜=⋃t≥T0‍Sε(t)𝒪¯ with (i) ℳ is compact; (ii) ℬ⊂ℳ⊂𝒪, where ℬ is the global attractor; (iii) ℳ has a finite fractal dimension dimf(ℳ)≤2K0(2M*+1)lnL(T*)+1+1, where K0 is a constant and T* and M* are as in (48); and (iv) there exist two positive constants k1 and k2 such that dist(Sε(t)u,ℳ)≤k1e-k2t for all u∈𝒪,t≥0.



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