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

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.

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: 𝑖𝑧̇_𝑚−(𝐴𝑧)_𝑚+𝑖𝛼𝑧_𝑚+𝑧_𝑚𝑢_𝑚=𝑓_𝑚(𝑡)𝑢̈,𝑚∈ℤ,𝑡>𝜏,_𝑚+𝜈𝑢̇_𝑚+(𝐴𝑢)_𝑚+𝜇𝑢_𝑚−𝛽𝑧_𝑚²=𝑔_𝑚(𝑡),𝑚∈ℤ,𝑡>𝜏, (1)(2) with initial conditions: 𝑧_𝑚(𝜏)=𝑧_𝑚,𝜏,𝑢_𝑚(𝜏)=𝑢_𝑚,𝜏,𝑢̇_𝑚(𝜏)=𝑢_1𝑚,𝜏,𝑚∈ℤ,𝜏∈ℝ, (3) where 𝐴 is a linear operator defined as (𝐴𝑢)_𝑚=2𝑢_𝑚−𝑢_𝑚+1−𝑢_𝑚−1,∀𝑢=𝑢_{𝑚𝑚∈ℤ}, (4)𝑧_𝑚(𝑡)∈ℂ,𝑢_𝑚(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: 𝑖𝑧_𝑡+𝑧_𝑥𝑥(),𝑢+𝑖𝛼𝑧+𝑧𝑢=𝑓𝑥,𝑡_𝑡𝑡+𝜈𝑢_𝑡−𝑢_𝑥𝑥+𝜇𝑢−𝛽|𝑧|²=𝑔(𝑥,𝑡). (5) 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=(𝑢_𝑚)_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 𝑢_𝑚𝑧_𝑚 and |𝑧_𝑚|² in the coupled lattice equations. To overcome this difficulty, we make a proper transformation of the variable 𝑢=(𝑢_𝑚)_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 Section 2, 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 ℓ²=𝑢=𝑢_{𝑚𝑚∈ℤ},𝑢_𝑚∈ℝ∶_𝑚∈ℤ𝑢²_𝑚,𝐿<+∞₂=𝑢=𝑢_{𝑚𝑚∈ℤ},𝑢_𝑚∈ℂ∶_𝑚∈ℤ𝑢_𝑚²<+∞. (6)

For brevity, we use 𝑋 to denote ℓ² or 𝐿², and equip 𝑋 with the inner product and norm as (𝑢,𝑣)=_𝑚∈ℤ𝑢_𝑚𝑣_𝑚,‖𝑢‖²=(),𝑢𝑢,𝑢∀𝑢=_{𝑚𝑚∈ℤ},𝑣=𝑣_{𝑚𝑚∈ℤ}∈𝑋, (7) where 𝑣_𝑚 denotes the conjugate of 𝑣_𝑚. For any two elements 𝑢,𝑣∈𝑋, we define a bilinear form on 𝑋 by (𝑢,𝑣)_𝜇=(𝐵𝑢,𝐵𝑣)+𝜇(𝑢,𝑣), (8) where 𝜇 is the constant in (2) and 𝐵 is a linear operator defined as (𝐵𝑢)_𝑚=𝑢_𝑚+1−𝑢_𝑚,∀𝑚∈ℤ,∀𝑢=𝑢_{𝑚𝑚∈ℤ}∈𝑋. (9)

We also define a linear operator 𝐵^∗ from 𝑋 to 𝑋 via 𝐵^∗𝑢_𝑚=𝑢_𝑚−1−𝑢_𝑚,∀𝑚∈ℤ,∀𝑢=𝑢_{𝑚𝑚∈ℤ}∈𝑋. (10)

In fact, 𝐵^∗ is the adjoint operator of 𝐵 and one can check that (𝐴𝑢,𝑣)=𝐵^∗𝐵𝑢,𝑣=(𝐵𝑢,𝐵𝑣),(𝐵𝑢,𝑣)=𝑢,𝐵^∗𝑣,‖‖∀𝑢,𝑣∈𝑋,𝐵𝑢²=𝐵^∗𝑢²≤4‖𝑢‖²,‖𝐴𝑢‖²≤16‖𝑢‖²,∀𝑢∈𝑋. (11)

Clearly, the bilinear form (⋅,⋅)_𝜇 defined by (8) is also an inner product in 𝑋. Since 𝜇‖𝑢‖²≤𝜇‖𝑢‖²+‖𝐵𝑢‖²=‖𝑢‖²_𝜇≤𝜇+4‖𝑢‖²,∀𝑢∈𝑋, (12) the norm ‖⋅‖_𝜇 induced by (⋅,⋅)_𝜇 is equivalent to the norm ‖⋅‖. Write ℓ²=ℓ²,(),‖⋅,ℓ⋅,⋅‖_2𝜇=ℓ²,(⋅,⋅)_𝜇,‖⋅‖_𝜇,𝐿²=𝐿²,(⋅,⋅),‖⋅‖; (13) then ℓ², ℓ²_𝜇, and 𝐿² are all Hilbert spaces. Set 𝐸_𝜇=ℓ²_𝜇×ℓ²×𝐿². (14)

For any two elements 𝜓^(𝑗)=(𝑢^(𝑗),𝑣^(𝑗),𝑧^(𝑗))^𝑇∈𝐸_𝜇, 𝑗=1,2, the inner product and norm of 𝐸_𝜇 are defined as 𝜓⁽¹⁾,𝜓⁽²⁾_𝐸𝜇=𝑢⁽¹⁾,𝑢⁽²⁾_𝜇+𝑣⁽¹⁾,𝑣⁽²⁾+𝑧⁽¹⁾,𝑧⁽²⁾=_𝑚∈ℤ𝐵𝑢⁽¹⁾_𝑚𝐵𝑢⁽²⁾_𝑚+𝜇𝑢_𝑚⁽¹⁾𝑢_𝑚⁽²⁾+𝑣_𝑚⁽¹⁾𝑣_𝑚⁽²⁾+𝑧_𝑚⁽¹⁾𝑧_𝑚⁽²⁾,𝜓²_𝐸𝜇=𝜓,𝜓_𝐸𝜇,∀𝜓∈𝐸_𝜇, (15) where 𝑧_𝑚⁽²⁾ stands for the conjugate of 𝑧_𝑚⁽²⁾.

For convenience, we will express (1)–(3) as an abstract Cauchy problem of first-order ODE with respect to time 𝑡 in 𝐸_𝜇. To this end, we put 𝑢=(𝑢_𝑚)_𝑚∈ℤ, 𝑧=(𝑧_𝑚)_𝑚∈ℤ, 𝑧𝑢=(𝑧_𝑚𝑢_𝑚)_𝑚∈ℤ, |𝑧|²=(|𝑧_𝑚|²)_𝑚∈ℤ, 𝑓(𝑡)=(𝑓_𝑚(𝑡))_𝑚∈ℤ, 𝑔(𝑡)=(𝑔_𝑚(𝑡))_𝑚∈ℤ, 𝑧_𝜏=(𝑧_𝑚,𝜏)_𝑚∈ℤ, and 𝑢_𝜏=(𝑢_𝑚,𝜏)_𝑚∈ℤ, 𝑢_1𝜏=(𝑢_1𝑚,𝜏)_𝑚∈ℤ. Then, we rewrite (1)–(3) in a vector form as 𝑖̇(𝑡)̈̇|𝑧|𝑧−𝐴𝑧+𝑖𝛼𝑧+𝑧𝑢=𝑓,𝑡>𝜏,𝑢+𝜈𝑢+𝐴𝑢+𝜇𝑢−𝛽²(𝑡)𝑧(𝜏)=𝑔,𝑡>𝜏,=𝑧_𝜏,𝑢(𝜏)=𝑢_𝜏,𝑢̇(𝜏)=𝑢_1𝜏,𝜏∈ℝ. (16)(17)(18) Set 𝑣=𝑢̇+𝛿𝑢,where𝛿=𝜇𝜈2𝜇+𝜈²>0, (19)𝜓=(𝑢,𝑣,𝑧)^𝑇, 𝐹(𝜓,𝑡)=(0,𝛽|𝑧|²+𝑔(𝑡),𝑖𝑧𝑢−𝑖𝑓(𝑡))^𝑇, Θ=()𝐼()𝛿𝐼−𝐼0𝐴+𝜇𝐼+𝛿𝛿−𝜈𝜈−𝛿𝐼000𝑖𝐴+𝛼𝐼. (20)

Then, (16)–(18) can be written as ̇𝜓(𝜏)𝜓+Θ𝜓=𝐹𝜓,𝑡,𝑡>𝜏,=𝜓_𝜏=𝑢_𝜏,𝑣_𝜏,𝑧_𝜏^𝑇=𝑢_𝜏,𝑢_1𝜏+𝛿𝑢_𝜏,𝑧_𝜏^𝑇,𝜏∈ℝ. (21)(22)

Let 𝒞_𝑏(ℝ,𝑋) be the set of continuous and bounded functions from ℝ into 𝑋, then for each 𝑓(𝑡)∈𝒞_𝑏(ℝ,𝑋), we have sup_𝑡∈ℝ∑_𝑚∈ℤ|𝑓_𝑚(𝑡)|²<+∞. Write ℋ=𝑓(𝑡)=𝑓_𝑚(𝑡)_𝑚∈ℤ∈𝒞_𝑏()∶()ℝ,𝑋foreach𝜏∈ℝ,∀𝜀>0,∃𝑀𝜀,𝜏∈ℕ,suchthat_{|𝑚|()≥𝑀𝜀,𝜏}𝑓_𝑚(𝑠)².≤𝜀forany𝑠≤𝜏 (23) We next recall some results of solutions to (21)-(22).

Lemma 1 (see [28]). Let 𝑓(𝑡)=(𝑓_𝑚(𝑡))_𝑚∈ℤ∈𝒞_𝑏(ℝ,𝐿²), 𝑔(𝑡)=(𝑔_𝑚(𝑡))_𝑚∈ℤ∈𝒞_𝑏(ℝ,ℓ²). Then, for any initial data 𝜓_𝜏=(𝑢_𝜏,𝑣_𝜏,𝑧_𝜏)^𝑇∈𝐸_𝜇, there exists a unique solution 𝜓(𝑡)=(𝑢(𝑡),𝑣(𝑡),𝑧(𝑡))^𝑇∈𝐸_𝜇 of (21)-(22), such that 𝜓(𝑡)∈𝒞([𝜏,+∞),𝐸_𝜇)∩𝒞¹((𝜏,+∞),𝐸_𝜇). Moreover, the mapping 𝑈(𝑡,𝜏)∶𝜓_𝜏=𝑢_𝜏,𝑣_𝜏,𝑧_𝜏^𝑇∈𝐸_𝜇(𝑡)=(𝑢(𝑡)(𝑡)(𝑡↦𝜓,𝑣,𝑧))^𝑇∈𝐸_𝜇,∀𝑡≥𝜏 (24) generates a continuous process {𝑈(𝑡,𝜏)}_𝑡≥𝜏 on 𝐸_𝜇, where 𝑣_𝜏=𝑢_1𝜏+𝛿𝑢_𝜏.

Lemma 2 (see [28]). Let 𝑓(𝑡)=(𝑓_𝑚(𝑡))_𝑚∈ℤ∈𝒞_𝑏(ℝ,𝐿²), 𝑔(𝑡)=(𝑔_𝑚(𝑡))_𝑚∈ℤ∈𝒞_𝑏(ℝ,ℓ²). Then, the solution 𝜓(𝑡)=(𝑢(𝑡),𝑣(𝑡),𝑧(𝑡))^𝑇∈𝐸_𝜇 of (21)-(22) corresponding to initial data 𝜓_𝜏=(𝑢_𝜏,𝑣_𝜏,𝑧_𝜏)^𝑇∈𝐸_𝜇 satisfies 𝜓(𝑡)²_𝐸𝜇≤𝐶₀e^{−2𝜗0(𝑡−𝜏)}+12𝑅²₀,∀𝑡≥𝜏, (25) where 𝐶₀,𝜗₀, and 𝑅₀ are positive constants independent of 𝑡 and 𝜏.

Lemma 3 (see [28]). Let 𝑓(𝑡)=(𝑓_𝑚(𝑡))_𝑚∈ℤ∈𝒞_𝑏(ℝ,𝐿²), 𝑔(𝑡)=(𝑔_𝑚(𝑡))_𝑚∈ℤ∈𝒞_𝑏(ℝ,ℓ²). Then, the process {𝑈(𝑡,𝜏)}_𝑡≥𝜏 corresponding to (21)-(22) possesses a uniformly bounded absorbing set ℬ₀⊂𝐸_𝜇, such that for any bounded set ℬ of 𝐸_𝜇, there exists a time 𝑡(𝜏,ℬ)≥𝜏 yielding 𝑈(𝑡,𝜏)ℬ⊂ℬ₀,∀𝑡≥𝑡(𝜏,ℬ), (26) where ℬ₀=ℬ(0,𝑅₀)⊂𝐸_𝜇 is a closed ball centered at 0 with radius 𝑅₀.

Lemma 3 shows that there exists a time 𝑡₀=̇𝑡₀(𝜏,ℬ₀)≥𝜏, such that 𝑈(𝑡,𝜏)ℬ₀⊂ℬ₀,∀𝑡≥𝑡₀. (27)

Lemma 4 (see [28]). Let 𝑓(𝑡)=(𝑓_𝑚(𝑡))_𝑚∈ℤ∈ℋ with 𝑋=𝐿² and 𝑔(𝑡)=(𝑔_𝑚(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)𝜓_𝜏=𝜓(𝑡+𝜏)=(𝜓_𝑚(𝑡+𝜏))_𝑚∈ℤ∈𝐸_𝜇 of (21)-(22) with 𝜓_𝜏∈ℬ₀ satisfies _{|𝑚|>𝑀0(𝜀,𝜏,ℬ0)}𝑈(𝑡+𝜏,𝜏)𝜓_𝜏𝑚²_𝐸𝜇=_{|𝑚|>𝑀0(𝜀,𝜏,ℬ0)}𝜓_𝑚(𝑡+𝜏)²_𝐸𝜇≤𝜀², (28) where |𝜓_𝑚|²_𝐸𝜇=|(𝐵𝑢)_𝑚|²+𝜇𝑢²_𝑚+𝑣²_𝑚+|𝑧_𝑚|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 𝐸_𝜇^(𝑁)=𝜓=𝜓_{𝑚𝑚∈ℤ}∈𝐸_𝜇∣𝜓_𝑚=(0,0,0)^𝑇if|𝑚|>𝑁, (29) then 𝐸_𝜇^(𝑁) is a 4(2𝑁+1)-dimensional subspace of 𝐸_𝜇. Define a bounded projection 𝑃_𝑁∶𝐸_𝜇↦𝐸_𝜇^(𝑁)⊂𝐸_𝜇 by 𝑃_𝑁𝜓_𝑚=𝜓_𝑚,|𝑚||𝑚|𝜓≤𝑁;0,>𝑁,𝜓=_{𝑚𝑚∈ℤ}∈𝐸_𝜇. (30)

Lemma 5. (I) For any 𝑇>𝑡₀, there exists some 𝐿_𝑇>0 (independent of 𝜏), such that for every 𝜏∈ℝ and any 𝑡∈[𝑡₀,𝑇], 𝑈(𝑡+𝜏,𝜏)𝜓_𝜏⁽¹⁾−𝑈(𝑡+𝜏,𝜏)𝜓_𝜏⁽²⁾_𝐸𝜇≤𝐿_𝑇𝜓_𝜏⁽¹⁾−𝜓_𝜏⁽²⁾_𝐸𝜇,∀𝜓_𝜏⁽¹⁾,𝜓_𝜏⁽²⁾∈ℬ₀. (31)
(II) There exist two positive constants 𝑇^∗>𝑡₀ and 𝛾∈(0,1/2), and a 4(2𝑁^∗+1)-dimensional orthogonal projection 𝑃_𝑁^∗∶𝐸_𝜇↦𝐸^(𝑁∗)_𝜇 for some 𝑁^∗∈ℕ, such that for every 𝜏∈ℝ and 𝜓_𝜏⁽¹⁾,𝜓_𝜏⁽²⁾∈ℬ₀, 𝐼−𝑃_𝑁^∗𝑈𝑇^∗+𝜏,𝜏𝜓_𝜏⁽¹⁾−𝑈𝑇^∗+𝜏,𝜏𝜓_𝜏⁽²⁾_𝐸𝜇≤𝛾𝜓_𝜏⁽¹⁾−𝜓_𝜏⁽²⁾_𝐸𝜇. (32)

Proof. (I) For any 𝜏∈ℝ, let 𝜓⁽¹⁾(𝑡)=𝑈(𝑡,𝜏)𝜓_𝜏⁽¹⁾,𝜓⁽²⁾(𝑡)=𝑈(𝑡,𝜏)𝜓_𝜏⁽²⁾,∀𝑡≥𝜏 (33) be two solutions of (21)-(22) with initial conditions 𝜓_𝜏⁽¹⁾,𝜓_𝜏⁽²⁾∈ℬ₀, respectively. Set 𝑢_𝑑(𝑡)=𝑢⁽¹⁾(𝑡)−𝑢⁽²⁾(𝑡),𝑣_𝑑(𝑡)=𝑣⁽¹⁾(𝑡)−𝑣⁽²⁾(𝑡),𝑧_𝑑(𝑡)=𝑧⁽¹⁾(𝑡)−𝑧⁽²⁾(𝑡),𝜓_𝑑(𝑡)=𝜓⁽¹⁾(𝑡)−𝜓⁽²⁾(𝑡). (34)
From (21)-(22), we get 𝜓̇_𝑑+Θ𝜓_𝑑=𝐹𝜓⁽¹⁾,𝑡−𝐹𝜓⁽²⁾𝜓,𝑡,∀𝑡>𝜏._𝑑(𝜏)=𝜓_𝜏⁽¹⁾−𝜓_𝜏⁽²⁾. (35)
Taking the real part of the inner product of (35) with 𝜓_𝑑 in 𝐸_𝜇, we obtain 12d𝜓d𝑡^{𝑑2𝐸𝜇}+ReΘ𝜓_𝑑+𝐹𝜓⁽²⁾𝜓,𝑡−𝐹⁽¹⁾,𝑡,𝜓_𝑑𝐸𝜇=0,∀𝑡≥𝜏. (36)
Since Θ∶𝐸_𝜇↦𝐸_𝜇 is a bounded linear operator, 𝐹∶𝐸_𝜇×ℝ↦𝐸_𝜇 is a locally Lipschitz continuous operator (see Lemma 2.2 in [28]), and ℬ₀ is a bounded set in 𝐸_𝜇, we see that there exist two positive constants 𝐶₁ and 𝐾₁=𝐾₁(ℬ₀), such that ReΘ𝜓_𝑑+𝐹𝜓⁽²⁾,𝑡−𝐹𝜓⁽¹⁾,𝑡,𝜓_𝑑𝐸𝜇≤𝐶₁𝜓_𝑑𝐸𝜇+𝐹𝜓⁽²⁾,𝑡−𝐹𝜓⁽¹⁾,𝑡_𝐸𝜇𝜓_𝑑𝐸𝜇≤𝐶₁+𝐾₁𝜓_𝑑²_𝐸𝜇,∀𝑡≥𝜏. (37)
Combining (36) and (37), we get d𝜓d𝑡^{𝑑2𝐸𝜇}−𝐾₂𝜓_𝑑²_𝐸𝜇≤0,∀𝑡≥𝜏, (38) where 𝐾₂=2(𝐶₁+𝐾₁). Applying Gronwall inequality to (38) on [𝜏,𝜏+𝑡] with 𝑡∈[𝑡₀,𝑇], we obtain 𝜓_𝑑(𝑡+𝜏)²_𝐸𝜇≤e^𝐾2𝑡𝜓_𝑑(𝜏)²_𝐸𝜇,∀𝑡∈𝑡₀,𝑇. (39) Thus, 𝜓_𝑑(𝑡+𝜏)_𝐸𝜇=𝜓⁽¹⁾(𝑡+𝜏)−𝜓⁽²⁾(𝑡+𝜏)_𝐸𝜇=𝑈(𝑡+𝜏,𝜏)𝜓_𝜏⁽¹⁾−𝑈(𝑡+𝜏,𝜏)𝜓_𝜏⁽²⁾_𝐸𝜇≤𝐿_𝑇𝜓_𝜏⁽¹⁾−𝜓_𝜏⁽²⁾_𝐸𝜇,∀𝑡∈𝑡₀,𝑇, (40) where 𝐿_𝑇=e^𝐾2𝑇 does not depend on 𝜏.
(II) Define a smooth function 𝜒(𝑥)∈𝒞¹(ℝ₊,[0,1]), such that 𝜒(𝑥)=𝜒0,0≤𝑥≤1;1,𝑥≥2,_′(𝑥)≤𝜒₀positiveconstant,∀𝑥∈ℝ₊. (41)
Set 𝑝_𝑑=𝑝_{𝑑𝑚𝑚∈ℤ}with𝑝_𝑑𝑚|𝑚|=𝜒𝑀𝑢_𝑑𝑚,𝑞_𝑑=𝑞_{𝑑𝑚𝑚∈ℤ}with𝑞_𝑑𝑚|𝑚|=𝜒𝑀𝑣_𝑑𝑚,𝑤_𝑑=𝑤_{𝑑𝑚𝑚∈ℤ}with𝑤_𝑑𝑚|𝑚|=𝜒𝑀𝑧_𝑑𝑚,𝜙_𝑑=𝜙_{𝑑𝑚𝑚∈ℤ}with𝜙_𝑑𝑚=𝑝_𝑑𝑚,𝑞_𝑑𝑚,𝑤_𝑑𝑚, (42) where 𝑀 is a positive integer that will be specified later. From (16), we see that 𝑖𝑧̇_𝑑−𝐴𝑧_𝑑+𝑖𝛼𝑧_𝑑+𝑧⁽¹⁾𝑢⁽¹⁾−𝑧⁽²⁾𝑢⁽²⁾=0. (43)
Taking the imaginary part of the inner product of (43) with 𝑤_𝑑 in 𝐿², we get 12dd𝑡_𝑚∈ℤ𝜒|𝑚|𝑀𝑧_𝑑𝑚²−Im𝐴𝑧_𝑑,𝑤_𝑑+𝛼_𝑚∈ℤ𝜒|𝑚|𝑀𝑧_𝑑𝑚²+Im𝑧⁽¹⁾𝑢⁽¹⁾−𝑧⁽²⁾𝑢⁽²⁾,𝑤_𝑑=0. (44)
Now, we have −Im𝐴𝑧_𝑑,𝑤_𝑑=−Im𝐵𝑧_𝑑,𝐵𝑤_𝑑=−Im_𝑚∈ℤ𝑧_𝑑𝑚+1−𝑧_𝑑𝑚×𝜒|𝑚+1|𝑀𝑧_𝑑𝑚+1|𝑚|−𝜒𝑀𝑧_𝑑𝑚=Im_𝑚∈ℤ𝜒|𝑚|𝑀𝑧_𝑑𝑚+1𝑧_𝑑𝑚||+𝜒𝑚+1𝑀𝑧_𝑑𝑚+1𝑧_𝑑𝑚=Im_𝑚∈ℤ𝜒|𝑚+1|𝑀|𝑚|−𝜒𝑀𝑧_𝑑𝑚+1𝑧_𝑑𝑚≥−_𝑚∈ℤ𝜒^′𝑚𝑀1𝑀𝑧_𝑑𝑚+1𝑧_𝑑𝑚≥−2𝜒₀𝑀𝜓_𝑑²_𝐸𝜇,∀𝑡≥𝜏, (45) 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 Im_𝑚∈ℤ𝑧_𝑚⁽¹⁾𝑢_𝑚⁽¹⁾−𝑧_𝑚⁽²⁾𝑢_𝑚⁽²⁾|𝑚|,𝜒𝑀𝑧_𝑚⁽¹⁾−𝑧_𝑚⁽²⁾≤_𝑚∈ℤ𝜒|𝑚|𝑀𝑧_𝑚⁽¹⁾𝑢_𝑚⁽¹⁾−𝑧_𝑚⁽²⁾𝑢_𝑚⁽²⁾𝑧_𝑚⁽¹⁾−𝑧_𝑚⁽²⁾=_𝑚∈ℤ𝜒|𝑚|𝑀𝑧_𝑚⁽¹⁾𝑢_𝑚⁽¹⁾−𝑢_𝑚⁽²⁾+𝑢_𝑚⁽²⁾𝑧_𝑚⁽¹⁾−𝑧_𝑚⁽²⁾×𝑧_𝑚⁽¹⁾−𝑧_𝑚⁽²⁾≤𝛼4_𝑚∈ℤ𝜒|𝑚|𝑀𝑧_𝑚⁽¹⁾−𝑧_𝑚⁽²⁾²+𝛼𝜇𝛿2×_𝑚∈ℤ𝜒|𝑚|𝑀𝑢_𝑚⁽¹⁾−𝑢_𝑚⁽²⁾𝑧_𝑚⁽¹⁾−𝑧_𝑚⁽²⁾≤𝛼2_𝑚∈ℤ𝜒|𝑚|𝑀𝑧_𝑚⁽¹⁾−𝑧_𝑚⁽²⁾²+𝜇𝛿4_𝑚∈ℤ𝜒|𝑚|𝑀𝑢_𝑚⁽¹⁾−𝑢_𝑚⁽²⁾², (46) which implies that when 𝑡≥𝑡₁ and 𝑀>𝑀₁(𝜀,𝜏,ℬ₀)Im𝑧⁽¹⁾𝑢⁽¹⁾−𝑧⁽²⁾𝑢⁽²⁾,𝑤_𝑑=Im_𝑚∈ℤ𝜒|𝑚|𝑀𝑧_𝑚⁽¹⁾𝑢_𝑚⁽¹⁾−𝑧_𝑚⁽²⁾𝑢_𝑚⁽²⁾,𝑧_𝑚⁽¹⁾−𝑧_𝑚⁽²⁾𝛼≥−2_𝑚∈ℤ𝜒|𝑚|𝑀𝑧_𝑚⁽¹⁾−𝑧_𝑚⁽²⁾²−𝜇𝛿4_𝑚∈ℤ𝜒|𝑚|𝑀𝑢_𝑚⁽¹⁾−𝑢_𝑚⁽²⁾². (47)
Then, taking (44)–(47) into account, we obtain for every 𝑡≥𝑡₁ and 𝑀>𝑀₁(𝜀,𝜏,ℬ₀) that dd𝑡_𝑚∈ℤ𝜒|𝑚|𝑀𝑧_𝑑𝑚²+𝛼_𝑚∈ℤ𝜒|𝑚|𝑀𝑧_𝑑𝑚²≤𝜇𝛿2_𝑚∈ℤ𝜒|𝑚|𝑀𝑢²_𝑑𝑚+4𝜒₀𝑀𝜓_𝑑²_𝐸𝜇. (48)
From (17) and (19), we obtain 𝑣̇_𝑑+(𝜈−𝛿)𝑣_𝑑+𝛿(𝛿−𝜈)+𝜇𝑢_𝑑+𝐴𝑢_𝑑−𝛽𝑧⁽¹⁾²−𝑧⁽²⁾²=0. (49)
Taking the inner product of (49) with 𝑞_𝑑 in ℓ², we obtain 12dd𝑡_𝑚∈ℤ𝜒|𝑚|𝑀𝑣²_𝑑𝑚+(𝜈−𝛿)×_𝑚∈ℤ𝜒|𝑚|𝑀𝑣²_𝑑𝑚+𝛿(𝛿−𝜈)+𝜇𝑢_𝑑,𝑞_𝑑+𝐴𝑢_𝑑,𝑞_𝑑−𝛽𝑧⁽¹⁾²−𝑧⁽²⁾²,𝑞_𝑑=0. (50)
It is clear that 𝑞_𝑑=𝑝̇_𝑑+𝛿𝑝_𝑑. Then, (50) can be rewritten as dd𝑡_𝑚∈ℤ𝜒|𝑚|𝑀𝜇𝑢²_𝑑𝑚+𝑣²_𝑑𝑚+2(𝜈−𝛿)_𝑚∈ℤ𝜒|𝑚|𝑀𝑣²_𝑑𝑚+2𝜇𝛿_𝑚∈ℤ𝜒|𝑚|𝑀𝑢²_𝑑𝑚+2𝛿(𝛿−𝜈)𝑢_𝑑,𝑞_𝑑+2𝐴𝑢_𝑑,𝑞_𝑑−2𝛽𝑧⁽¹⁾²−𝑧⁽²⁾²,𝑞_𝑑=0. (51)
Also, we have 𝐴𝑢_𝑑,𝑞_𝑑=𝐴𝑢_𝑑,𝑝̇_𝑑+𝛿𝐴𝑢_𝑑,𝑝_𝑑=𝐵𝑢_𝑑,𝐵𝑝̇_𝑑+𝛿𝐵𝑢_𝑑,𝐵𝑝_𝑑. (52)
By some computations, we get 𝐵𝑢_𝑑,𝐵𝑝̇_𝑑=_𝑚∈ℤ𝐵𝑢_𝑑𝑚𝐵𝑝̇_𝑑𝑚=_𝑚∈ℤ𝐵𝑢_𝑑𝑚𝜒|𝑚+1|𝑀𝑢̇_𝑑𝑚+1|𝑚|−𝜒𝑀𝑢̇_𝑑𝑚=_𝑚∈ℤ𝐵𝑢_𝑑𝑚𝐵𝑢̇_𝑑𝑚𝜒|𝑚|𝑀+𝜒|𝑚+1|𝑀|𝑚|−𝜒𝑀𝑢̇_𝑑𝑚+1=_𝑚∈ℤ𝜒|𝑚|𝑀𝐵𝑢_𝑑𝑚𝐵𝑢̇_𝑑𝑚+_𝑚∈ℤ𝜒^′𝑚𝑀1𝑀𝑢_𝑑𝑚+1−𝑢_𝑑𝑚×𝑣_𝑑𝑚+1−𝛿𝑢_𝑑𝑚+1≥12dd𝑡_𝑚∈ℤ𝜒|𝑚|𝑀𝐵𝑢_𝑑𝑚²−2𝜒₀𝜇+𝛿+1𝜓𝑀𝜇^{𝑑2𝐸𝜇}𝛿,∀𝑡≥𝜏,𝐵𝑢_𝑑,𝐵𝑝_𝑑=𝛿_𝑚∈ℤ𝐵𝑢_𝑑𝑚𝐵𝑝_𝑑𝑚=𝛿_𝑚∈ℤ𝐵𝑢_𝑑𝑚𝜒|𝑚+1|𝑀𝑢_𝑑𝑚+1|𝑚|−𝜒𝑀𝑢_𝑑𝑚=𝛿_𝑚∈ℤ𝐵𝑢_𝑑𝑚𝐵𝑢_𝑑𝑚𝜒|𝑚|𝑀+𝜒|𝑚+1|𝑀|𝑚|−𝜒𝑀𝑢_𝑑𝑚+1=𝛿_𝑚∈ℤ𝜒|𝑚|𝑀𝐵𝑢_𝑑𝑚²+𝛿_𝑚∈ℤ𝜒^′𝑚𝑀1𝑀𝑢_𝑑𝑚+1−𝑢_𝑑𝑚𝑢_𝑑𝑚+1≥𝛿_𝑚∈ℤ𝜒|𝑚|𝑀𝐵𝑢_𝑑𝑚²−2𝜒₀𝛿𝜓𝑀𝜇^{𝑑2𝐸𝜇},∀𝑡≥𝜏. (53) Here 𝑚 locates between |𝑚+1| and |𝑚|. Inserting (53) into (52), we get 𝐴𝑢_𝑑,𝑞_𝑑≥12dd𝑡_𝑚∈ℤ𝜒|𝑚|𝑀𝐵𝑢_𝑑𝑚²+𝛿_𝑚∈ℤ𝜒|𝑚|𝑀𝐵𝑢_𝑑𝑚²−2𝜒₀𝜇+2𝛿+1𝜓𝑀𝜇^{𝑑2𝐸𝜇},∀𝑡≥𝜏. (54)
According to Lemma 4, 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), 𝛽𝑧⁽¹⁾²−𝑧⁽²⁾²,𝑞_𝑑=𝛽_𝑚∈ℤ𝜒|𝑚|𝑀𝑧_𝑚⁽¹⁾²−𝑧_𝑚⁽²⁾²𝑣_𝑑𝑚≤𝛽_𝑚∈ℤ𝜒|𝑚|𝑀𝑧_𝑚⁽¹⁾+𝑧_𝑚⁽²⁾𝑧_𝑑𝑚𝑣_𝑑𝑚√≤𝛽2𝛼𝜈2𝛽_𝑚∈ℤ𝜒|𝑚|𝑀𝑧_𝑑𝑚𝑣_𝑑𝑚≤𝜈2_𝑚∈ℤ𝜒|𝑚|𝑀𝑣²_𝑑𝑚+𝛼4_𝑚∈ℤ𝜒|𝑚|𝑀𝑧_𝑑𝑚². (55)
It follows from (51) and (54)-(55) that when 𝑀>𝑀₂(𝜀,𝜏,ℬ₀) and 𝑡≥𝑡₂, we have dd𝑡_𝑚∈ℤ𝜒|𝑚|𝑀𝐵𝑢_𝑑𝑚²+𝜇𝑢²_𝑑𝑚+𝑣²_𝑑𝑚+2𝛿_𝑚∈ℤ𝜒|𝑚|𝑀𝐵𝑢_𝑑𝑚²+2𝜇𝛿_𝑚∈ℤ𝜒|𝑚|𝑀𝑢²_𝑑𝑚+(𝜈−2𝛿)×_𝑚∈ℤ𝜒|𝑚|𝑀𝑣²_𝑑𝑚−2𝛿(𝜈−𝛿)𝑢_𝑑,𝑞_𝑑≤𝛼2_𝑚∈ℤ𝜒|𝑚|𝑀𝑧_𝑑𝑚²+4𝜒₀𝜇+2𝛿+1𝜓𝑀𝜇^{𝑑2𝐸𝜇}. (56)
Combining (48) and (56), when 𝑡≥𝑡₂, we obtain dd𝑡_𝑚∈ℤ𝜒|𝑚|𝑀𝐵𝑢_𝑑𝑚²+𝜇𝑢²_𝑑𝑚+𝑣²_𝑑𝑚+𝑧_𝑑𝑚²+2𝛿_𝑚∈ℤ𝜒|𝑚|𝑀𝐵𝑢_𝑑𝑚²+3𝜇𝛿2_𝑚∈ℤ𝜒|𝑚|𝑀𝑢²_𝑑𝑚+(𝜈−2𝛿)×_𝑚∈ℤ𝜒|𝑚|𝑀𝑣²_𝑑𝑚+𝛼2_𝑚∈ℤ𝜒|𝑚|𝑀𝑧_𝑑𝑚²−2𝛿(𝜈−𝛿)𝑢_𝑑,𝑞_𝑑≤4𝜒₀2𝜇+2𝛿+1×𝜓𝑀𝜇_{𝑑2𝐸𝜇},∀𝑀>𝑀₂𝜀,𝜏,ℬ₀. (57)
Since 𝛿²𝜈²=𝜇𝛿(𝜈/2−𝛿), we get for any 𝑚∈ℤ that 𝜇𝛿𝑢²_𝑑𝑚+𝜈2−𝛿𝑣²_𝑑𝑚−2𝛿(𝜈−𝛿)𝑢_𝑑𝑚𝑣_𝑑𝑚≥𝜇𝛿𝑢²_𝑑𝑚+𝜈2−𝛿𝑣²_𝑑𝑚−2𝛿𝜈𝑢_𝑑𝑚𝑣_𝑑𝑚≥0. (58) Thus, 𝜇𝛿_𝑚∈ℤ𝜒|𝑚|𝑀𝑢²_𝑑𝑚+𝜈2−𝛿_𝑚∈ℤ𝜒|𝑚|𝑀𝑣²_𝑑𝑚−2𝛿(𝜈−𝛿)𝑢_𝑑,𝑞_𝑑≥0. (59)
We then conclude from (57) and (59) that when 𝑀>𝑀₂(𝜀,𝜏,ℬ₀) and 𝑡≥𝑡₂, dd𝑡_𝑚∈ℤ𝜒|𝑚|𝑀𝐵𝑢_𝑑𝑚²+𝜇𝑢²_𝑑𝑚+𝑣²_𝑑𝑚+𝑧_𝑑𝑚²+2𝛿_𝑚∈ℤ𝜒|𝑚|𝑀𝐵𝑢_𝑑𝑚²+𝜇𝛿2_𝑚∈ℤ𝜒|𝑚|𝑀𝑢²_𝑑𝑚+𝜈2−𝛿_𝑚∈ℤ𝜒|𝑚|𝑀𝑣²_𝑑𝑚+𝛼2_𝑚∈ℤ𝜒|𝑚|𝑀𝑧_𝑑𝑚²≤4𝜒₀2𝜇+2𝛿+1𝜓𝑀𝜇^{𝑑2𝐸𝜇}. (60)
Choosing 𝜗=min{𝛿/2,𝜈/2−𝛿,𝛼/2}, we obtain dd𝑡_𝑚∈ℤ𝜒|𝑚|𝑀𝜓_𝑑𝑚²_𝐸𝜇+𝜗_𝑚∈ℤ𝜒|𝑚|𝑀𝜓_𝑑𝑚²_𝐸𝜇≤4𝜒₀2𝜇+2𝛿+1𝜓𝑀𝜇^{𝑑2𝐸𝜇},∀𝑡≥𝑡₂,∀𝑀>𝑀₂𝜀,𝜏,ℬ₀. (61)
Applying Gronwall inequality to (61) from 𝜏+𝑡₂ to 𝜏+𝑡 with 𝑡>𝑡₂, we get for every 𝜓_𝜏⁽¹⁾,𝜓_𝜏⁽²⁾∈ℬ₀,𝑀>𝑀₂(𝜀,𝜏,ℬ₀) and 𝑡≥𝑡₂ that _𝑚∈ℤ𝜒|𝑚|𝑀𝜓_𝑑𝑚(𝑡+𝜏)²_𝐸𝜇≤e^{−𝜗(𝑡−𝑡2)}_𝑚∈ℤ𝜒|𝑚|𝑀𝜓_𝑑𝑚𝜏+𝑡₂²_𝐸𝜇+4𝜒₀2𝜇+2𝛿+1𝑀𝜇^𝜏+𝑡_𝜏+𝑡2𝜓_𝑑(𝑠)²_𝐸𝜇e^{−𝜗(𝑡+𝜏−𝑠)}d𝑠. (62)
By (39), when 𝑀>𝑀₂(𝜀,𝜏,ℬ₀) and 𝑡≥𝑡₂ we have e^{−𝜗(𝑡−𝑡2)}_𝑚∈ℤ𝜒|𝑚|𝑀𝜓_𝑑𝑚𝜏+𝑡₂²_𝐸𝜇≤e^{−𝜗(𝑡−𝑡2)}𝜓_𝑑𝜏+𝑡₂²_𝐸𝜇≤e^{−𝜗(𝑡−𝑡2)+𝐾2𝑡2}𝜓_𝑑(𝜏)²_𝐸𝜇,4𝜒₀2𝜇+2𝛿+1𝑀𝜇^𝜏+𝑡_𝜏+𝑡2𝜓_𝑑(𝑠)²_𝐸𝜇e^{−𝜗(𝑡+𝜏−𝑠)}≤d𝑠4𝜒₀2𝜇+2𝛿+1𝜓𝑀𝜇^𝑑(𝜏)²_𝐸𝜇×^𝜏+𝑡_𝜏+𝑡2e^{−𝜗(𝑡+𝜏)+(𝜗+𝐾2)𝑠−𝐾2𝜏}=d𝑠4𝜒₀2𝜇+2𝛿+1𝜓𝑀𝜇^𝑑(𝜏)²_𝐸𝜇e^{−𝜗(𝑡+𝜏)−𝐾2𝜏}×^𝜏+𝑡_𝜏+𝑡2e^{(𝜗+𝐾2)𝑠}≤d𝑠4𝜒₀2𝜇+2𝛿+1𝑀𝜇𝜗+𝐾₂𝜓_𝑑(𝜏)²_𝐸𝜇e^𝐾2𝑡. (63)
Thus, it follows from (62)-(63) that for any 𝑡≥𝑡₂ and 𝑀>𝑀₂(𝜀,𝜏,ℬ₀), _𝑚∈ℤ𝜒|𝑚|𝑀𝜓_𝑑𝑚(𝑡+𝜏)²_𝐸𝜇≤e^{−𝜗(𝑡−𝑡2)+𝐾2𝑡2}𝜓_𝑑(𝜏)²_𝐸𝜇+4𝜒₀2𝜇+2𝛿+1𝑀𝜇𝜗+𝐾₂𝜓_𝑑(𝜏)²_𝐸𝜇e^𝐾2𝑡. (64)
Pick two sufficient large numbers 𝑇^∗≥𝑡₂ and 𝑀^∗>𝑀₂(𝜀,𝜏,ℬ₀) to satisfy e^{−𝜗(𝑇∗−𝑡2)+𝐾2𝑡2}+4𝜒₀2𝜇+2𝛿+1𝑀^∗𝜇𝜗+𝐾₂e^𝐾2𝑇∗=̇𝛾²<14. (65)
Then, from (64), we have for 𝑁^∗>2𝑀^∗ that _{|𝑚|>𝑁∗}𝜓_𝑑𝑚𝑇^∗+𝜏²_𝐸𝜇≤_𝑚∈ℤ𝜒|𝑚|𝑀^∗𝜓_𝑑𝑚𝑇^∗+𝜏²_𝐸𝜇≤𝛾²𝜓_𝑑(𝜏)²_𝐸𝜇; (66) that is, 𝐼−𝑃_𝑁^∗𝑈𝑇^∗+𝜏,𝜏𝜓_𝜏⁽¹⁾−𝑈𝑇^∗+𝜏,𝜏𝜓_𝜏⁽²⁾_𝐸𝜇≤𝛾𝜓_𝜏⁽¹⁾−𝜓_𝜏⁽²⁾_𝐸𝜇, (67) where 𝛾<1/2. The proof is complete.

Now, we can state the main result of this paper.

Theorem 6. Let the conditions of Lemma 4 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 each 𝑡∈ℝ, {𝒜(𝑡)} is a compact set of 𝐸_𝜇, and the fractal dimension dim_𝐹𝒜(𝑡) is finite and uniformly bounded in 𝑡; that is, sup_𝑡∈ℝdim_𝐹𝒜(𝑡)<∞; (68)(2) (positive invariant property) 𝑈(𝑡,𝜏)𝒜(𝜏)⊂𝒜(𝑡) for all 𝑡≥𝜏;(3)(pullback exponential attractivity) there exist an exponent 𝜂>0 and two positive-valued functions 𝑄,ℱ∶ℝ₊↦ℝ₊, such that for any bounded set ℬ⊂𝐸_𝜇, Dist_𝐸𝜇(𝑈(𝑡,𝜏)ℬ,𝒜(𝑡))≤𝑄‖ℬ‖_𝐸𝜇e^{−𝜂(𝑡−𝜏)},𝜏∈ℝ,𝜏+ℱ‖ℬ‖_𝐸𝜇≤𝑡<∞, (69)where Dist_𝐸𝜇(⋅,⋅) is the Hausdorff semidistance between two subsets of 𝐸_𝜇.

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.

Acknowledgments

The authors warmly thank the anonymous referee for useful comments and bringing them the issue on Lyapunov exponents for the trajectories in the pullback attractor into their attention. This paper is supported by National NSFC (no. 11271290), the National Key Basic Research Program of China (973 Program with Grant no. 2012CB426510), and NSF of Wenzhou University (2008YYLQ01).