Pullback random attractors of stochastic strongly damped wave equations with variable delays on unbounded domains

: In this paper, we consider the asymptotic behavior of solutions to stochastic strongly damped wave equations with variable delays on unbounded domains, which is driven by both additive noise and deterministic non-autonomous forcing. We ﬁrst establish a continuous cocycle for the equations. Then we prove asymptotic compactness of the cocycle by tail-estimates and a decomposition technique of solutions. Finally, we obtain the existence of a tempered pullback random attractor.


Introduction
The aim of this paper is to establish the existence of pullback random attractors of the following stochastic non-autonomous strongly damped wave equation with variable delays and with additive noise in R d : u tt − α∆u t − ∆u + u t + λu = f (x, u(t − ρ(t))) + g(t, x) + where x ∈ R d , t τ, τ ∈ R; α > 0 is the strong damping coefficient, λ is a positive constant; g ∈ L 2 loc (R, L 2 (R d )) and h j ∈ H 2 (R d ); f is a nonlinear function satisfying some conditions, ρ is a given delay function; {W j } m j=1 are independent real-valued two-sided Wiener process on a complete probability space (Ω, F , P), which will be specified later.
As we know, the concept of random attractor, as an extension of the global attractor for the deterministic systems, was first introduced in [8], which has been studied in many papers, see [2, 3, 9-12, 20, 21, 23, 28, 30, 31, 39] and references therein.
Time delay differential equations arise from some evolution phenomena in physics, biology and life science, which depend not only on the current states but also on their past history. There have been many works on the asymptotic behavior of delay differential equations, see [6,13,19] in the deterministic case and [4,7,16,17,29,35,36] in the stochastic case and references therein.
The asymptotic behavior of solutions of stochastic wave equation have been studied extensively in [15,22,25,37] in the autonomous case. For the non-autonomous stochastic wave equation, the existence of random attractors was obtained in [33] on bounded domains and in [5,18,27,32,34] on unbounded domains. Wave equations with delays are widely used in biology, physics, engineering and chemistry. Therefore, it is important for us to study the asymptotic behavior of stochastic delay wave equation. In [14,38], stochastic wave equations with delays on bounded domains are considered. However, the results for the stochastic delay wave equation on unbounded domains are very few.
In this work, we study the pullback random attractors of stochastic non-autonomous strongly damped wave equations with variable delays on unbounded domains. To prove the existence of pullback random attractors, we need to derive some kind compactness. The main difficulty in this paper is to establish the asymptotic compactness since the Sobolev embedding is no longer compact on unbounded domains. We here overcome the difficulty by showing that the uniform tail-estimates of solutions are sufficiently small (see Lemma 2.3). On bounded domains, we decompose the solutions into a sum of two parts. One part decays exponentially and the other part has higher regularity. For the higher regularity part, we first give some uniform estimates (see Lemma 2.4) and obtain the Hölder continuity of solutions in time (see Lemma 2.5). Then we apply Arzela-Ascoli theorem to prove the precompactness (see Lemma 3.2) and hence establish our main result (see Theorem 3.3). In addition, the strongly damped term α∆u t and the delay term f (x, u(t − ρ(t))) introduce an additional difficulty in deriving the uniform estimates, which needs some nontrivial arguments.
The paper is organized as follows. In the next section, we establish a continuous cocycle for problem (1.1) and (1.2) and some uniform estimates of solutions are derived. Then we prove the existence and uniqueness of the tempered pullback attractors for (1.1) and (1.2) in Section 3. In Section 4, we make conclusion as well as some comments on our results.
Throughout this paper, we use (·) and · to denote the inner product and norm of L 2 (R d ), respectively, and use · X to denote the norm of a general Banach space X. For h > 0, let C h be the Banach space C([−h, 0]; L 2 (R d )) endowed with the norm ϕ C h = sup s∈[−h,0] ϕ(s) and u t be the function defined by u t = u(t + s), s ∈ [−h, 0]. Let C be a positive constant whose value may be different from line to line or even in the same line.
where σ > 0 is a fixed constant such that

Preliminaries and cocycles
In this subsection, we first show that the system (1.1) and (1.2) generates a continuous cocycle. Then, we recall some results for the existence of pullback random attractors for non-autonomous random dynamical systems.
For our purpose, we transform the system (1.1) and (1.2) into a deterministic system with random parameters but without white noise, and then show that it generates a continuous cocycle on E over R and (Ω, F , P, {θ t } t∈R ).
For j = 1, 2, · · · , m, consider the one-dimensional Ornstein-Uhlenbeck equation: dz j + z j dt = dW j , (2.1) whose solution is given by It is known that the random variable |z j (ω j )| is tempered and there exists a θ t −invariant subset Ω ⊂ Ω of full measure such that z j (θ t ω j ) is continuous in t for each ω ∈ Ω. From now on, we will not distinguish Ω and Ω, and write the space Ω as Ω. Set |z j (ω j )| 2 r(ω), (2.4) where r(ω) > 0 satisfies for each ω ∈ Ω, here σ is a positive constant which will be fixed later. Then by (2.4) and (2.5), we obtain, for each In the rest of this subsection, we show that there is a continuous cocycle generated by the system (1.1) and (1.2). Firstly we give the following assumptions on f and g: (A1) There exist a function k 1 (x) ∈ L 2 (R d ) and a positive constant k 2 such that the functions f ∈ and |ρ (t)| ρ * < 1, ∀t ∈ R; (A2) There exists a constant L > 0, such that (A3) The deterministic forcing g(t, x) ∈ L 2 loc (R, L 2 (R d )), and τ −∞ e λr g(r, ·) 2 dr < ∞, ∀τ ∈ R, which implies lim k→∞ τ −∞ |x| k e λr |g(r, ·)| 2 dxdr = 0, ∀τ ∈ R.
Let v = u t + σu − z(θ t ω), where σ is given in (1.3), then (1.1) and (1.2) can be rewritten as the following equivalent form: with the initial conditions . By the classical theory in [24], we may show the following existence results of solutions of (2.7) and (2.8).
Next, we provide the following result for non-autonomous random dynamical systems from [26].
Proposition 2.1. Let Φ be a continuous cocycle on X over R and (Ω, F , P, {θ t } t∈R ). Suppose Φ is D(X)pullback asymptotically compact in X and has a closed measurable D(X)-pullback absorbing set K in D(X). Then Φ has an unique D(X) pullback attractor A in D(X). For each τ ∈ R and ω ∈ Ω, A is given by, In the rest of this paper, we will use Proposition 2.1 to prove the existence and uniqueness of a pullback random attractor for the continuous cocycle Φ in E .

Uniform tail-estimates
In this subsection, we derive some uniform tail-estimates of solutions of problem (2.7) and (2.8).
Hereafter we suppose that D is the collection of all tempered families of nonempty bounded subsets of X.
Proof. Taking the inner product of the second equation of (2.7) by v in L 2 (R d ), we have Then by (2.17), we derive that Now we estimate the terms on the right-hand of (2.20). By Young's inequality, Cauchy-Schwarz inequality and (A1), we have where ε 1 , ε 2 , ε 3 , ε 4 are fixed positive constants which will be chosen later. Combining these estimates with (2.20), we have Recalling the definition of norm · E , from (2.21), we get Set r = r − ρ(r). Combining ρ(r) ∈ [0, h] and the fact 1 It follows from (2.23) and (2.24) that . By (2.11)-(2.13), we can choose ε 1 small enough such that Replacing ω with θ −τ ω and by (2.25)-(2.28), we get Combining (2.32), (2.30) and (2.29), we obtain for all t T Thus, we complete the proof of Lemma 2.2.
Now we establish the following estimates on the exterior of a ball.
Then the proof of Lemma 2.3 is finished.

Estimates on bounded domain
We now decompose the solutions of (2.7) and (2.8) in bounded domains and derive some uniform estimates.
Proof. From the first equation in (2.61), we have Taking the inner product of the second equation in (2.61) and (2.67) with −∆v 1 and ∆u 1 respectively, we have Next, we give the estimates of the terms on the right-hand of (2.70). It follows from Cauchy-Schwarz inequality and Young's inequality that and Using the properties of the cutoff function ξ, we infer that and similarly, Thus we have In the same way we obtain Combining these estimates with (2.70), we get d dt Let σ > 0 be a constant satisfying (2.11)-(2.13). Multiplying e σ t on both sides of (2.71) and integrating over ( Now we replace ω by θ −τ ω and estimate the terms on the right-hand of (2.72). By Lemma 2.2, there exists T 1 = T 1 (τ, ω, D) > 0, such that for all t T 1 , Let T =max{T 1 , T 2 }. It follows from (2.72)-(2.76) that for all t T, Thus we complete the proof of Lemma 2.4.
Next, we establish the Hölder continuity of ϕ 1 in time, which will be useful to show the equicontinuity of solutions in C([−h, 0], E(Ω 2k )) based on the Arzela-Ascoli theorem.

Results and discussion
In this section, we aim to prove the existence of tempered pullback random attractors for the system (1.1) and (1.2) in E . Firstly we show the existence of the pullback absorbing set as follows.
We are now to give our main result. Proof. By Proposition 2.1, Lemma 3.1 and Lemma 3.2, we can obtain the existence and uniqueness of D-pullback random attractor of Φ in E immediately.

Conclusions
Since the Sobolev embedding is no longer compact on unbounded domains, we obtained the existence of random attractor for the problem (1.1) and (1.2) by using the uniform tail-estimates of solutions and the decomposition technique as well as the compactness argument. In addition, to derive the uniform estimates, we make some nontrivial arguments due to the presence of strongly damped term α∆u t and the delay term f (x, u(t − ρ(t))) in (1.1).