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

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

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

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

where ℬ₀ = ℬ(0, R₀) ⊂ E_μ is a closed ball centered at 0 with radius R₀.

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 $$ 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

where $$.

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],

(II) There exist two positive constants T^* > t₀ and γ ∈ (0,1/2), and a 4(2N^* + 1)-dimensional orthogonal projection $$ for some N^* ∈ ℕ, such that for every τ ∈ ℝ and $$,

Proof. (I) For any τ ∈ ℝ, let

be two solutions of (21)-(22) with initial conditions $$, respectively. Set

From (21)-(22), we get

Taking the real part of the inner product of (35) with ψ_d in E_μ, we obtain

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

Combining (36) and (37), we get

where K₂ = 2(C₁ + K₁). Applying Gronwall inequality to (38) on [τ, τ + t] with t ∈ [t₀, T], we obtain Thus, where $$ does not depend on τ.

(II) Define a smooth function χ(x) ∈ 𝒞¹(ℝ₊, [0,1]), such that

Set

where M is a positive integer that will be specified later. From (16), we see that

Taking the imaginary part of the inner product of (43) with w_d in L², we get

Now, we have

where $$ 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 which implies that when t ≥ t₁ and M > M₁(ε, τ, ℬ₀)

Then, taking (44)–(47) into account, we obtain for every t ≥ t₁ and M > M₁(ε, τ, ℬ₀) that

From (17) and (19), we obtain

Taking the inner product of (49) with q_d in ℓ², we obtain

It is clear that $$. Then, (50) can be rewritten as

Also, we have

By some computations, we get

Here $$ locates between |m + 1| and |m|. Inserting (53) into (52), we get

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),

It follows from (51) and (54)-(55) that when M > M₂(ε, τ, ℬ₀) and t ≥ t₂, we have

Combining (48) and (56), when t ≥ t₂, we obtain

Since δ²ν² = μδ(ν/2 − δ), we get for any m ∈ ℤ that

Thus,

We then conclude from (57) and (59) that when M > M₂(ε, τ, ℬ₀) and t ≥ t₂,

Choosing ϑ = min {δ/2, ν/2 − δ, α/2}, we obtain

Applying Gronwall inequality to (61) from τ + t₂ to τ + t with t > t₂, we get for every $$ and t ≥ t₂ that

By (39), when M > M₂(ε, τ, ℬ₀) and t ≥ t₂ we have

Thus, it follows from (62)-(63) that for any t ≥ t₂ and M > M₂(ε, τ, ℬ₀),

Pick two sufficient large numbers T^* ≥ t₂ and M^* > M₂(ε, τ, ℬ₀) to satisfy

Then, from (64), we have for N^* > 2M^* that

that is, where γ < 1/2. The proof is complete.

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,

  $$ ()

• (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,

  $$ ()

where $$ 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.