Pullback attractors of nonautonomous discrete p-Laplacian complex Ginzburg–Landau equations with fast-varying delays

In this paper, we consider a class of nonautonomous discrete p-Laplacian complex Ginzburg–Landau equations with time-varying delays. We prove the existence and uniqueness of pullback attractor for these equations. The existing results of studying attractors for time-varying delay equations require that the derivative of the delay term should be less than 1 (called slow-varying delay). By using differential inequality technique, our results remove the constraints on the delay derivative. So, we can deal with the equations with fast-varying delays (without any constraints on the delay derivative).


Introduction
Due to numerous applications in physics, biology, and engineering such as pattern formation, propagation of nerve pulses, electric circuits, and so on, see, e.g., [2,6,7,10,12], lattice differential equations have become a large and growing interdisciplinary area of research. For an understanding of the dynamical behavior of dissipative infinite lattice systems, attractors are especially important because they retain most of the dynamical information. The existence of global attractors for lattice systems was initialed by Bates et al. [1], followed by extensions in [3,8,13,16,19,24] and the references therein. Of those, the asymptotic behavior of an infinite-dimensional p-Laplacian lattice system was investigated in [8]. The dynamical behaviors for discrete complex Ginzburg-Landau equations were studied in [11,27].
Since time-delays are frequently encountered in many practical systems, which may induce instability, oscillation, and poor performance of systems, delay lattice systems then arise naturally while these delays are taken into account. Recently, attractors of delay lattice systems have been considered in [4,5,9,23,26].
The existence and uniqueness of solutions were proved for the complex Ginzburg-Landau equation with p-Laplacian in [17,18]. The dynamical behavior of p-Laplacian complex Ginzburg-Landau equations was considered in [25]. The existence and uniqueness of attractor for nonautonomous discrete p-Laplacian complex Ginzburg-Landau equations with fast-varying delays based on nonautonomous p-Laplacian complex Ginzburg-Landau equations with fast-varying delays are investigated in this paper. We prove the existence and uniqueness of pullback attractor for these equations. The existing results of studying attractors for time-varying delay equations require that the derivative of the delay term should be less than 1, see [14,15,[21][22][23]. By using the differential inequality technique, our results remove the constrains on the delay derivative. So, we can deal with the equations with fast-varying delays.
The plan of this paper is as follows. In the next section, we establish the existence of a continuous nonautonomous dynamical system in C([-ρ, 0], l 2 ) for nonautonomous equation (1.1) and (1.2). Section 3 contains all necessary uniform estimates of the solutions. We then prove the existence and uniqueness of a pullback attractor for the nonautonomous equations in Sect. 4.

Nonautonomous dynamical systems associated with nonautonomous lattice systems
In this section we show that there is a continuous nonautonomous dynamical system generated by the nonautonomous discrete p-Laplacian complex Ginzburg-Landau equations with time-varying delays. Denote and let l 2 be a Hilbert space with the inner produce and norm given by We define the linear operators A, B, B * : l 2 → l 2 as follows: (Au) n = |u nu n-1 | p-2 (u nu n-1 ) -|u n+1u n | p-2 (u n+1u n ), Then where u ⊗ v = (u i v i ) i∈Z and u p = ( n∈Z |u n | p ) 1 p . Denote by u t , t ∈ R, the function defined on [-ρ, 0] according to the relation Then problem (1.1)-(1.2) can be written as an equation in l 2 : for τ ∈ R and t > τ , where u = (u n ) n∈Z , |u| q u = (|u n | q u n ) n∈Z , f (u(tρ 0 (t))) = (f n (u n (tρ 0 (t)))) n∈Z , g(t) = (g n (t)) n∈Z , and ϕ = (ϕ n ) n∈Z . We make the following assumptions on f n . For each n ∈ Z, f n is a nonlinear function satisfying the following assumption: (H) f n (0) = 0 and f n (s) is Lipschitz continuous with respect to s, that is, there is a positive constant L such that, for all s 1 , s 2 ∈ C, In fact, by (H) we find that Lemma 2.1 For any p > 0 and a, b ∈ C, we have that there exists c = c(p) > 0 such that Proof Without loss of generality, we assume that |a| ≥ |b|. By mean value theorem, we have (2.4), and Young's inequality, we get This completes the proof.

Lemma 2.2
The operator A : l 2 → l 2 is locally Lipschitz continuous.
Proof Based on Lemma 2.1 we have, for any u, v ∈ l 2 , This completes the proof.
It follows from Lemma 2.2 that the right-hand side function in (2.1) is locally Lipschitz continuous from l 2 to l 2 . Therefore, by the standard theory of functional differential equations, one can show that, for every ϕ ∈ C ρ , there exists T > 0 such that system (2.1)-(2.2) has a unique solution u t (·, τ , ϕ) ∈ C([τ , T), C ρ ). As showed below, under some conditions this local solution is actually defined for all t > τ . Furthermore, one may show that u t (·, τ , ϕ) is continuous in ϕ with respect to the norm of C ρ .
In the sequence, we assume that

Lemma 2.3
Assume that (H) and (2.5) hold. Then, for every τ ∈ R, T > 0 and ϕ ∈ C ρ , there exists a positive constant c = c(τ , T, ϕ) such that the solution u of (2.1)-(2.2) satisfies Proof Taking the inner product of (2.1) with u in l 2 and keeping the real part, we find that for t > τ , For the last two terms on the right-hand side of (2.7), by (H) we have It follows from (2.10) and Gronwall's inequality that, for t ≥ τ , From condition (2.5), by using continuity, we obtain that there exist positive constants λ and N such that ϕ ρ ≤ N and hold. We will prove that, for t ≥ τ , where I(t) = max τ ≤ξ ≤t 1 γ ξ τ e -γ 2 (ξ -s) g(s) ds. To this end, we first prove, for any d > 1, (2.14) If inequality (2.14) is not true, from the fact that ϕ ρ ≤ N and u(t) are continuous, then there must be t * > τ such that and which contradicts inequality (2.15). So inequality (2.14) holds for all t ≥ τ . Letting d → 1 in inequality (2.14), we have inequality (2.13). The proof is complete.

Lemma 2.3 implies that the solution u is defined in any interval of [τ , τ + T] for any T > 0. It means that this local solution is, in fact, a global one.
Given t ∈ R, define a translation θ t on R by θ t (τ ) = τ + t for all τ ∈ R. (2.18) Then {θ t } t∈R is a group acting on R. We now define a mapping Φ : R + × R × C ρ → C ρ for problem (2.1)-(2.2). Given t ∈ R + , τ ∈ R, and u τ ∈ C ρ , let where u t+τ (s, τ , u τ ) = u(t + τ + s, τ , u τ ), s ∈ [-ρ, 0]. By the uniqueness of solutions, we find that for every t, s ∈ R + , τ ∈ R, and u τ ∈ C ρ , Then we see that Φ is a continuous nonautonomous dynamical system on C ρ . In the following two sections, we investigate the existence of pullback attractor for Φ. To this end, we need to define an appropriate collection of families of subsets of C ρ . Let B ρ = {B ρ (τ ) : τ ∈ R} be a family of nonempty subsets of C ρ . Then B ρ is called tempered (or subexponentially growing) if for every c > 0 the following holds: where B ρ ρ = sup x∈B ρ x ρ . In the sequel, we denote by D ρ the collection of all families of tempered nonempty subsets of C ρ , i.e., The following condition will be needed when deriving uniform estimates of solutions: and ϕ ∈ D ρ (τt), the solution u of (2.1)-(2.2) satisfies Proof From condition (2.5), by using continuity, we obtain that there exists a positive constant λ such that holds.
Replacing t and τ in (2.9) by and τt, respectively, we have, for > τt, For simplicity, we denote u( ) = u ( , τt, ϕ). Then, let us define functions V ( ) = e λ u( ) , ≥ τtρ, and If inequality (3.4) is not true from the fact that V (t) and U(t) are continuous, then there must be * > τt such that and there is a sufficiently small positive constant such that Calculating the upper right-hand Dini derivative of V (t) at * and considering (3.6) and (3.7), we obtain On the other hand, it follows from (3.3) that Notice that U( ) is monotone nondecreasing on [τtρ, +∞). This, together with (3.5) and (3.6), yields which implies It follows from (3.2), (3.9), and (3.11) that which contradicts (3.8). Until now, (3.4) has been proven to be true. Thus, we get, for t > ρ and -ρ ≤ ξ ≤ 0, Since ϕ ∈ D(τt) and D ∈ D, we find that for every τ ∈ R and D ∈ D there exists T = T(τ , D) > ρ such that, for all t ≥ T and -ρ ≤ ξ ≤ 0, This completes the proof. Taking the inner product of (2.1) with x = (ϑ( |n| k )u n ) n∈Z in l 2 and keeping the real part, where k is a fixed positive integer specified later, we get We now estimate the terms in (3.13) as follows. First, we have By the property of the function ϑ and Young's inequality, we have which implies that (3.14) For the last two terms on the right-hand side of (3.13), we get from (H)

Existence of pullback attractors
In this section, we establish the existence of D ρ -pullback attractor for the nonautonomous dynamical system Φ associated with problem (2.1)-(2.2).
As for the compactness in l 2 in [19], one can easily verify the following compactness criteria in C ρ by means of uniform tail estimates. Proof For τ ∈ R, denote by -∞ e λs g(s + τ ) ds .
First, we know from Lemma 3.1 that Φ has a D ρ -pullback absorbing set K(τ ). Second, since Lemmas 3.1, 3.2, and 4.1 coincide with all the conditions of Lemma 4.2, Φ is D ρpullback asymptotically compact in C ρ . Hence, the existence of a unique D ρ -pullback attractor for the nonautonomous dynamical system Φ follows from Proposition 2.7. in [20] immediately.