We first present some sufficient conditions for the existence of a pullback exponential attractor for continuous process on the product space of the weighted spaces of infinite sequences. Then we prove the existence and continuity of a pullback exponential attractor for second order lattice system with time-dependent coupled coefficients in the weighted space of infinite sequences. Moreover, we obtain the upper bound of fractal dimension and attracting rate for the attractor.

1. Introduction

Lattice dynamical systems (LDSs), which include coupled systems of infinite ordinary differential equations and coupled map lattices, have drawn more and more attention because these systems appear in various fields [1, 2]. In recent years, global attractors, uniform attractors, pullback attractors (or kernel sections), and random attractor for autonomous, nonautonomous, and stochastic LDSs have been studied; see [3–12]. However, these attractors sometimes attract orbits at a relatively slow speed, so that it might take an unexpected long time to be reached. Besides, it is usually difficult to estimate the attracting rate in terms of physical parameters of the model. Appropriate alternatives are exponential attractors and pullback exponential attractors which contain the global attractors and pullback attractors and attract all bounded sets exponentially [13–20]. For LDSs, [13, 16, 21] studied the existence of exponential attractors for first order, second order, and partly dissipative autonomous LDSs, respectively. Zhou and Han [20] presented some sufficient conditions for the existence of pullback exponential attractors for LDSs in

and provided applications to first order and partly dissipative nonautonomous LDSs. As we know, there are no results on pullback exponential attractors for second order nonautonomous LDSs which have been studied extensively [4–7, 9–13, 21–23]. In this paper, we study the following second order nonautonomous LDSs:

()

where for i ∈ ℤ, u_i ∈ ℝ, g_i ∈ C(ℝ, ℝ); f_i ∈ C¹(ℝ × ℝ, ℝ); η_i,j(t), j = −q, …, q (q ∈ ℕ) are locally integrable in t, and λ_i, α are positive constants. First we present some sufficient conditions for the existence of a pullback exponential attractor for a continuous process on the product space

of infinite sequences, which are direct results of [20]. Then we prove the existence of a pullback exponential attractor for system (1) in weighted space

; moreover, we obtain the upper bound of fractal dimension and attracting rate for the attractor.

2. Preliminaries

In this section, we present some sufficient conditions for the existence of a pullback exponential attractor for a continuous process on the product space of m weighted space of infinite sequences.

Let ρ_j : ℤ → ℝ₊ be positive-valued function such that 0 < ρ_j(i) = ρ_j,i ≤ M_j0, where M_j0 are positive constants, j = 1, …, m, and let

()

which is Banach space with norm

for u = (u_i) _i ∈ℤ,

. Let

be a product space of m spaces

, j = 1, …, m. Write

()

Then

is (2N + 1)-dimensional subspaces of

. Let

and define a bounded projection

as follows: for

()

Consider a two-parameter continuous process on E: W(t, τ) : E → E, t ≥ τ ∈ ℝ, satisfying the following: W(τ, τ) = I, for all τ ∈ ℝ; W(t, r)W(r, τ) = W(t, τ), −∞ < τ ≤ r ≤ t < ∞; (t, τ, φ_τ) → W(t, τ)φ_τ is continuous for −∞ < τ ≤ t < ∞.

Definition 1. A family {ℳ(t)} _t ∈ℝ of subsets of is called a pullback exponential attractor for the continuous process {W(t, τ)} _t≥τ, if

(i)
each ℳ(t) (t ∈ ℝ) is a compact set of and its fractal dimension is uniformly bounded in t; that is, sup⁡_t ∈ℝdim⁡_fℳ(t) < ∞;
(ii)
it is positively invariant; that is, W(t, τ)ℳ(τ) ⊂ ℳ(t) for all −∞ < τ ≤ t < ∞;
(iii)
there exist an exponent α > 0 and two positive-valued functions Q, 𝒯 : ℝ₊ → ℝ₊ such that for any bounded set
()

where “d_h(·, ·)” is the Hausdorff semidistance between two subsets of

As a direct consequence of Theorem 2 of [20], we have the following theorem.

Theorem 2. Let {W(t, τ)} _t≥τ be a continuous process in . Assume that there exists a uniformly bounded absorbing set for {W(t, τ)} _t≥τ: for any bounded set , there exists a constant T_B ≥ 0 such that W(t, τ)B⊆B₀ for all t ≥ τ + T_B, τ ∈ ℝ. For any t ∈ ℝ, set . If

(A1) there exist and such that, for every τ ∈ ℝ, any t ∈ [0, T^*], and φ_τ, ψ_τ ∈ X(τ),
()
(A2) there exist a positive constant γ ∈ [0, 1/2) and a (2N + 1)m-dimensional bounded projection such that, for every τ ∈ ℝ and φ_τ, ψ_τ ∈ X(τ),
()

Then {W(t, τ)} _t≥τ possesses a pullback exponential attractor {ℳ(t)} _{t ∈ R} satisfying

()

and, for any bounded set

()

where N_θ is the minimal number of closed balls of E with radius θ covering the closed unit ball B_N(0,1) of

centered at 0, and

with

, α = −ln⁡θ/T^*, and

3. Pullback Exponential Attractor for Second Order Nonautonomous Lattice System

In this section, we study the existence of a pullback exponential attractor for the continuous processes associated with the second order nonautonomous lattice system (1). Note that system (1) with initial conditions can be written as a vector form

()

where u = (u_i) _i ∈ℤ,

, λu = (λ_iu_i) _i ∈ℤ, f(u, t) = (f_i(u_i, t)) _i ∈ℤ, g(t) = (g_i(t)) _i ∈ℤ.

Throughout this section, let ρ be a positive weight function from ℤ to (0, M₀) ⊂ ℝ⁺ satisfying

(P0)
0 < ρ(i) ≤ M₀, ρ_i = ρ(i) ≤ c₀ρ (i ± 1), i ∈ ℤ, for some positive constants M₀ and c₀.

Consider the weighted Hilbert space of infinite sequences

()

endowed with inner product (u, v) _ρ = ∑_{i ∈ Z} ρ_iu_iv_i and norm

for u = (u_i) _{i ∈ Z},

. For any u,

, define an inner product on

by (u, v) _λ,ρ = ∑_{i ∈ Z} ρ_iλ_iu_iv_i, then the norm ∥·∥_λ,ρ include by (·, ·) _λ,ρ is equivalent to the norm ∥·∥_ρ include by (·, ·) _ρ. Let

()

We make the following assumptions on λ_i, η_i,j, f_i, and g_i, for i ∈ ℤ, j = −q, …, q.

(H1)
There exist two positive constants and such that
()
(H2)
Let η(t) = {sup⁡|η_i,j(t)| : i ∈ ℤ, j = −q ⋯ q} < ∞ (q ∈ ℕ) satisfy the following:
- (H2a)
  in t;
- (H2b)
  there exists a continuous positive valued function Q : ℝ₊ → ℝ₊ such that
  ()
  where is a positive constant;
- (H2c)
  there exist I₀ ∈ ℕ and η₀ > 0 such that
  ()
(H3)
For all i ∈ ℤ, let f_i satisfy the following:
- (H3a)
  f_i(x, t) is differentiable in x and continuous in t;
- (H3b)
  for any i ∈ ℤ, f_i(0, t) = 0, sup⁡_x,t ∈ℝ | f_i(x, t)| ≤ β_i(t);
- (H3c)
  there exist functions Γ ∈ C(ℝ₊ × ℝ, ℝ₊) and such that
  ()
(H4)
g(t) = (g_i(t)) _i ∈ℤ ∈ G, β(t) = (β_i(t)) _i ∈ℤ ∈ G, where
()

and

denotes the space of all continuous bounded functions from ℝ into

Letting

()

then the system (10) is equivalent to the following evolution equation:

()

where

()

Definition 3. The function φ : [τ, τ + T) → E (T > 0) is called a mild solution of the following lattice differential equations:

()

if φ ∈ C([τ, τ + T), E) and

()

Theorem 4. Assume that (P0) and (H1)–(H4) hold. Then for any fixed τ ∈ ℝ and any initial data φ(τ) = (u_τ, v_τ) ∈ E, problem (19) admits a unique mild solution φ(·, τ; φ_τ) ∈ C([τ, +∞), E) with φ(τ, τ; φ_τ) = φ_τ and φ(·, τ; φ_τ) being continuous in φ_τ ∈ E, and the mapping

()

generates a continuous process {U(t, τ)} _t≥τ on E.

Proof. Let

()

and then

is continuous in u and locally integrable in t from

into

. For any bounded set B ⊂ E with sup⁡_{φ ∈ B}∥φ∥ ≤ r, let

()

Then for φ = (u, v), φ^(j) = (u^(j), u^(j)) ∈ B, j = 1, 2, by (P0), (H1)–(H4), we have

()

By the approximating method [24], it follows that problem (19) possesses a unique local mild solution φ(·, τ; φ_τ) ∈ C([τ, T_max⁡), E). Similar to the proof of Theorem 5 below concerning the existence of a uniformly bounded absorbing set, we can prove that T_max⁡ = +∞.

In the following part of this section, we assume that (P0) and (H1)–(H4) and

()

hold.

Theorem 5. Assume that (P0) and (H1)–(H4) hold. There exists a uniform bounded closed absorbing ball B₀ = B₀(0, r₀) ⊂ E of {U(t, τ)} _t≥τ centered at 0 with radius r₀ (independent of τ, t) such that, for any bounded subset B ⊂ E, there exists T_B ≥ 0 yielding U(t + τ, τ)B⊆B₀ for all t ≥ T_B.

Proof. Let be a mild solution of (19), t ≥ τ. Since the set of continuous functions is dense in , by (H2a), there exist sequences of continuous functions , m ∈ ℕ, such that

()

Consider the following lattice differential equations:

()

where

()

Combining the continuity of F^m(φ^m, t) in t with the proof of Theorem 4, (36) has a unique strong solution φ^m(·, τ, φ_τ) = (u^m(·, τ, φ_τ), v^m(·, τ, φ_τ)) ∈ C([τ, +∞), E)∩C¹((τ, +∞), E) satisfying

()

Taking the inner product (·, ·) _E of (29) with φ^m(t), we obtain

()

By some computation, we have that

()

Then we have that, for t ≥ τ,

()

Applying Gronwall’s inequality to (35) on [τ, t] (t ≥ τ), we obtain

()

It then follows that, for T > 0,

()

for some κ(T, τ) > 0 independent of m and i, and then, for any t, s ∈ [τ, τ + T), m ∈ ℕ,

()

implies the equicontinuity of

. By the Arzela-Ascoli Theorem, there exists a convergent subsequence

, such that

()

and

is continuous in t ∈ [τ, τ + T]; moreover, by (36),

for t ∈ [τ, τ + T]. By the Lebesgue Convergence Theorem, we have

()

Thus, by replacing m with m_k in (31) and letting k → +∞, we have

()

By the uniqueness of the mild solutions of (19), we have

for t ∈ [τ, τ + T]. By replacing m with m_k and letting k → ∞ in (36), we have that for, t ≥ τ,

()

where

()

By (H2b), there exists T₁ > 0 such that for, s ≥ T₁ and t ∈ ℝ,

()

Let

()

and then

()

Thus, for any bounded subset B of E and φ_τ ∈ B,

()

where

()

Again, by (H2b),

()

It then follows that

()

is a uniformly bounded closed absorbing set for {U(t, τ)} _t≥τ.

Lemma 6. Assume that (P0) and (H1)–(H4) hold. For any ε > 0, there exist and I(ε, B₀) ∈ ℕ such that, for t ≥ T(ε, B₀), the mild solution U(τ + t, t)φ_τ = φ(τ + t, τ; φ_τ) = (u_i(τ + t, τ; u_τ), v_i(τ + t, τ; v_τ)) _i ∈ℤ ∈ E of system (19) with φ_τ ∈ B₀ ⊂ E satisfies

()

Proof. Let ξ ∈ C¹(R⁺, R) be a smooth increasing function which satisfies

()

Let

be a solution of (29) with φ^m(τ) = φ_τ. Let M ≥ I₀ + q (I₀ is in (H2c)) be a suitable large integer, and set

()

Taking the inner product (·, ·) _E of (29) with z^m, we have

()

And we have the following estimates:

()

where

()

Therefore,

()

Applying Gronwall’s inequality to (57) on

, we have that, for φ_τ ∈ B₀,

()

By (44)-(45), for

, we have

()

as t → +∞. This means that, for any ε > 0, there exists

such that when t ≥ T₂(ε, B₀),

()

By (t), β(t) ∈ G, and (H2b), there exists I₁(ε) ∈ ℕ such that, for M > I₁(ε),

()

From (47),

()

By (44), we have

()

And, for any ε > 0, there exists I₂(ε, B₀) > I₀ + q such that, for M > I₂(ε, B₀),

()

Let T(ε, B₀) = max⁡{T₁, T₂(ε, B₀)} and I(ε, B₀) = 2max⁡{I₁(ε), I₂(ε, B₀). It follows immediately that

()

In the following, we prove the existence of a pullback exponential attractor for {U(t, τ)} _t≥τ by applying Theorem 2. For any t ∈ ℝ, set

()

where B₀ is the uniform absorbing set defined in Theorem 5.

Theorem 7. Assume that (P0) and (H1)–(H4) hold.

(a)
There exists a continuous positive value function L_t in t such that, for every τ ∈ ℝ,
()
(b)
There exist positive constants T^* > 0, γ ∈ [0, 1/2), and a 2(2N + 1)-dimensional orthogonal projection P_N : E → E_N (N ∈ ℤ), such that, for every τ ∈ ℝ and φ_τ, ψ_τ ∈ Y(τ),
()
(c)
For each fixed t ∈ ℝ, .

Proof. For any τ ∈ ℝ, initial data φ_τ = (u_τ, v_τ), ψ_τ = (x_τ, y_τ). Let φ(t) = U(t, τ)φ_τ, ψ(t) = U(t, τ)ψ_τ, and ϕ(t) = φ(t) − ψ(t) = (ϕ_i(t)) _i ∈ℤ; then, φ(t), ψ(t), ϕ(t) ∈ C([τ, +∞), E).

(a) For any t ≥ τ, let φ^m(t), ψ^m(t) be two solutions of (29) with initial data φ_τ, ψ_τ and set ϕ^m(t) = φ^m(t) − ψ^m(t), and then ϕ^m(t) ∈ C([τ, +∞), E)∩C¹((τ, +∞), E) satisfies

()

Taking the inner product (·, ·) _E of (69) with ϕ^m, we have

()

Since φ_τ, ψ_τ ∈ Y(τ), by (36) and (66), for t ≥ τ, φ^m(t) = (u^m(t), v^m(t)), ψ^m(t) = (x^m(t), y^m(t)) ∈ Y(t)⊆B₀, thus

, and

for t ≥ τ. By (33), we have

()

and, by (H2)-(H3),

()

Thus,

()

Applying Gronwall’s inequality to (73) on [τ, τ + t], (t > 0), we have

()

that is,

()

where

()

By replacing m with m_k in (75) and letting m_k → ∞, we have

()

(b) For i ∈ ℤ, let and , where ξ is as in (52) and M > I₀ + q. Taking the inner product of (69) with ω^m in E, we have that, for t ≥ τ,

()

For the terms in the right hand of (78), we have

()

where

()

By the continuity of Γ(u, t) and Γ(0, t) = 0 (see (H3c)), there exists

such that

()

By Lemma 6 and for

, there exist

such that, for

()

implying that, for

()

and thus

()

For

and

()

Therefore, we obtain that, for all

()

Applying Gronwall’s inequality to (86) from

to τ + t (t ≥ T₂ + T₁), we have

()

Thus, for t ≥ T₂ + T₁, we have

()

where

()

where

()

Thus, for

, we have

()

Taking

()

then

()

Thus, by (91), we have

()

and the continuity of Q(·),

As a direct consequence of Theorem 2, Theorem 4, Theorem 5, Theorem 7, and Theorem 3.1 of [15], we have the following result.

Theorem 8. Assume that (P0) and (H1)–(H4) hold. The process {U(t, τ)} _t≥τ associated with (19) possesses a pullback exponential attractor {𝒜(t)} _t ∈ℝ with the following properties:

(i)
for any t ∈ ℝ, 𝒜(t)⊆Y(t)⊆B₀;
(ii)
;
(iii)
and for any bounded set B ⊂ E, d_h(U(t, τ)Y(τ), , −∞ < τ + T_B < t ≤ +∞;
(iv)
lim⁡_t→τd_h(𝒜(t), 𝒜(τ)) = 0;

where

()

and

is the minimal number of closed balls of E with radius θ covering the closed unit ball B_N(0,1) of E_N centered at 0.

Remark 9. It should be pointed out that, for some special cases, (H2) and (27) can be removed. For example, if the coupled linear operator A(t) ≡ A in (10) is a constant operator A satisfying −A = D^*D = DD^*, where , for all (constant), − m₀ ⩽ j ⩽ m₀, and D^* is the adjoint of D. A simple example of such an operator A is (Au) _i = u_i−1 + u_i+1 − 2u_i and (Du) _i = u_i+1 − u_i. Choose a positive weight function ρ : ℤ → ℝ₊ satisfying the properties (P0) and |ρ(i ± 1) − ρ(i)| ≤ bρ(i), for all i ∈ ℤ for some small b ≥ 0. Two examples of such weight functions are ρ(i) = 1/(1 + κ²i²) ^q, q > 1/2, and ρ(i) = e^−κ|i|, i ∈ ℤ, with κ > 0 being a parameter. If the number b satisfies

()

where

, then conditions (H2) and (27) can be removed.

Conflict of Interests

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

Acknowledgments

This work is supported by the National Natural Science Foundation of China under Grant no. 11071165, 11326114 and Zhejiang Normal University (ZC304011068).

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Pullback Exponential Attractor for Second Order Nonautonomous Lattice System

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1. Introduction

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Pullback Exponential Attractor for Second Order Nonautonomous Lattice System

Abstract

1. Introduction

2. Preliminaries

3. Pullback Exponential Attractor for Second Order Nonautonomous Lattice System

Conflict of Interests

Acknowledgments

References

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