Asymptotically autonomous robustness of random attractors for a class of weakly dissipative stochastic wave equations on unbounded domains

Proceedings of the Royal Society of Edinburgh,151, 1700–1730, 2021 DOI:10.1017/prm.2020.77 Asymptotically autonomous robustness of random attractors for a class of weakly dissipative stochastic wave equations on unbounded domains Tom´as Caraballo Departamento de Ecuaciones Diferenciales y An´alisis Num´erico, Facultad de Matem´aticas, Universidad de Sevilla, C/ Tarfia s/n, 41012-Sevilla, Spain Boling Guo Institute of Applied Physics and Computational Mathematics, PO Box 8009, Beijing 100088, China Nguyen Huy Tuan Department of Mathematics and Computer Science, University of Science-VNUHCM, 227 Nguyen Van Cu Str., Dist. 5, Ho Chi Minh City, Viet Nam Renhai Wang∗ Institute of Applied Physics and Computational Mathematics, PO Box 8009, Beijing 100088, China ([email protected]) (received 4 April 2020; accepted 29 September 2020) This paper is concerned with the asymptotic behaviour of solutions to a class of non-autonomous stochastic nonlinear wave equations with dispersive and viscosity dissipative terms driven by operator-type noise defined on the entire space Rn.The existence, uniqueness, time-semi-uniform compactness and asymptotically autonomous robustness of pullback random attractors are proved in H1(Rn)×H1(Rn) when the growth rate of the nonlinearity has a subcritical range, the density of the noise is suitably controllable, and the time-dependent force converges to a time-independent function in some sense. The main difficulty to establish the time-semi-uniform pullback asymptotic compactness of the solutions in H1(Rn)×H1(Rn) is caused by the lack of compact Sobolev embeddings on Rn,as well as the weak dissipativeness of the equations is surmounted at light of the idea of uniform tail-estimates and a spectral decomposition approach. The measurability of random attractors is proved by using an argument which considers two attracting universes developed by Wang and Li (Phys. D 382: 46–57, 2018). Keywords: Weakly dissipative wave equation; pullback random attractors; asymptotically autonomous robustness; time-semi-uniform compactness; operator-type noise 2020 Mathematics subject classification: Primary 37L55 Secondary 37B55; 35B41; 35B40 ∗Corresponding author. c The Author(s), 2020. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh 1700 https://www.cambridge.org/core/terms. https://doi.org/10.1017/prm.2020.77 Downloaded from https://www.cambridge.org/core. Universidad de Sevilla, on 02 Mar 2022 at 10:13:19, subject to the Cambridge Core terms of use, available at Asymptotically autonomous robustness of random attractors 1701 1. Introduction In this article we investigate existence, uniqueness, time-semi-uniform compactness as well as asymptotically autonomous robustness of pullback random attractors of the following non-autonomous stochastic dispersive-dissipative wave equations perturbed by operate-type noise define on Rn: utt +αut−Δut−βΔutt +λu −Δu+f(x, u)=g(t, x)+εSu◦dW dt, u(τ,x)=uτ(x),u t(τ,x)=uτ,1(x),x∈Rn,t>τ,τ∈R,(1.1) where n∈Nis arbitrary, α,βand λare positive constants, ε>0 is the density of noise, S=I−βΔ, Wis a two-sided real-valued Wiener process defined on the probability space (Ω,F,P), g∈L2 loc(R,L 2(Rn)), and the nonlinear function f:Rn×R→Rhas a subcritical growth rate in its second argument. The symbol ◦means that the stochastic equation is interpreted in the sense of Stratonovich integration. The two terms (−Δ)sutt and (−Δ)sutare referred to as the dispersive and viscosity dissipative terms respectively due to their own physical background. For deterministic version of (1.1) defined on bounded domains, the well-posedness and existence of global attractors have been investigated by Carvalho and Cholewa [11] as well as Sun et al. [38]. For additive white noise driven version of (1.1) defined on unbounded domain Rnfor n=1,2,3, the existence of random attractors was recently examined by Jones and Wang [22] when the force gis time-independent, and the stochastic term εSu◦(dW/dt) is replaced by h(dW/dt) with h∈L2(Rn) being a known function. As far as the authors are concerned, up to now, the existence of random attractors remains open for the non-autonomous stochastic version of problem (1.1)even for n=1,2,3 and the bounded domain case. The main reason here is that we can only transform the additive noise driven version of (1.1) into a pathwise random equation, but cannot transform the multiplicative noise driven version of (1.1) with S=Iinto a pathwise random one due to the dispersive and dissipative (−Δ)sutt and (−Δ)sut. This essentially distinguishes from the damped (or strong damped) wave equations as considered by many authors, see e.g., [21,37,40,42,48,49, 54,56–58]. Nevertheless, in this paper we are able to convert problem (1.1) with S=I−βΔ into a pathwise deterministic one, and hence study the random attractors of stochastic (1.1) with S=I−βΔ. As is well known, the concept of attractors investigated by many authors, see e.g., Robinson et al. [5,6,12,13,27,28,34–36], plays an important role in the study of asymptotic behaviour of solutions to differential equations. Autonomous random attractors proposed by Brze´zniak et al. [2], Crauel and Flandoli [14], Crauel et al. [15], Flandoli and Schmalfuß [18] and Caraballo and Langa [8] can be viewed as a generation of global attractors form deterministic to random. Non-autonomous random attractors developed by Caraballo et al. [7], Caraballo and Langa [3]and Wang [41,42] can be regarded as an extension of autonomous random attractors form autonomous to non-autonomous. In light of those theoretical frameworks, random attractors have reached a flourishing development in recent years, see https://www.cambridge.org/core/terms. https://doi.org/10.1017/prm.2020.77 Downloaded from https://www.cambridge.org/core. Universidad de Sevilla, on 02 Mar 2022 at 10:13:19, subject to the Cambridge Core terms of use, available at 1702 T. Caraballo, B. Guo, N.H. Tuan and R. Wang e.g., [1,4,9,10,19,23,29,40,49,58] and [16,20,33,46,48,50,52,53,57,58]for autonomous and non-autonomous PDEs, respectively. In general, a non-autonomous random attractor typically takes the form Aα= {Aα(τ,ω):τ∈R,ω∈Ω}, where αis an external parameter that comes from various perturbations. Notice that, in the literature aforementioned, many properties of Aαsuch as compactness, attraction, regularity as well as finite fractal dimension were usually discussed for each fixed time-section Aα(τ,ω), and the robustness of Aα(τ,ω) was only investigated with respect to the external parameter αbut not the internal parameter τ. This kind of researches are just analogous to the autonomous case, and thereby the time-dependence character related to non-autonomous random attractors are not well-understood. In the present paper we will not only establish the existence and uniqueness but also the time-semi-uniform compactness as well as asymptotically autonomous robustness of non-autonomous random attractors of problem (1.1). Our first aim is to prove that problem (1.1) admits a unique pullback random attractor A={A(τ,ω):τ∈R,ω∈Ω}in H1(Rn)×H1(Rn) under the theoretical framework established in [41]. In order to achieve the goal, as usual, we must prove the usual pullback asymptotic compactness of solutions to (1.1)inH1(Rn)×H1(Rn). There are three difficulties we need to surmount. 1. The compact Sobolev embeddings on unbounded domain Rnare not available. 2. Equation (1.1) is a weakly dissipative one due to the dispersive and dissipative terms (−Δ)sutt and (−Δ)sut, which is essentially different from the damped wave equations as widely considered in the literature aforementioned. 3. The uniform estimates of solutions to (1.1) cannot be derived by differentiating the equation with respect to time since the Wiener process is almost surely nowhere differentiable with respect to time variable. We combine the ideal of uniform tail-estimates developed by Wang [39] and a spectral decomposition approach to overcome the three difficulties, and hence establish the desired pullback asymptotic compactness in H1(Rn)×H1(Rn) for our purpose. Another significant goal of this article is to prove the following time-semi-uniform compactness of A:  s⩽τA(s, ω) is precompact in H1(Rn)×H1(Rn) (1.2) for each (τ,ω)∈R×Ω as well as the following asymptotically autonomous robustness of the time-section A(τ,ω)ofAas time τgoes to negative infinity: lim τ→−∞distH1(Rn)×H1(Rn)(A(τ,ω),A∞(ω)) = 0,P-a.s. ω∈Ω,(1.3) where A∞={A∞(ω):ω∈Ω}is a random attractor of the autonomous version of problem (1.1). Furthermore, we prove that such a robustness is basically uniform in the probability space Ω for discrete time sequence, and we also prove that for https://www.cambridge.org/core/terms. https://doi.org/10.1017/prm.2020.77 Downloaded from https://www.cambridge.org/core. Universidad de Sevilla, on 02 Mar 2022 at 10:13:19, subject to the Cambridge Core terms of use, available at Asymptotically autonomous robustness of random attractors 1703 any discrete time sequence τn→−∞, there exist {τnk}∞ k=1 of {τn}∞ n=1 such that lim k→∞distH1(Rn)×H1(Rn)(A(τnk,θ τnkω),A∞(θτnkω)) = 0,P-a.s. ω∈Ω. Notice that the usual pullback asymptotic compactness of solutions to (1.1)is no longer useful (or say not enough) to establish (1.2) and (1.3). To solve this problem, at present, we first introduce a time-semi-uniform attracting universe (see (2.20)) that is indeed small than the usual attracting universe (see (2.19)), and then derive the time-semi-uniform pullback asymptotic compactness of solutions to (1.1) in H1(Rn)×H1(Rn). Based upon this we are able to prove (1.2) and (1.3). It is worth mentioning that the radii sups⩽τR(s, ω) of the absorbing set in the case is taken the supremum over an uncountable set (−∞,τ]. This introduces difficulties to prove the measurability of attractors. Our idea to solve this issue is to prove that the two attractors with respect to the two different universes are equal, see theorem 4.3. We remark that the time-semi-uniform compactness of non-autonomous attractors and kernel sections has been recently investigated in [30,32,55,56]and [43,44] for deterministic and stochastic PDEs, respectively. The asymptotically autonomous robustness of non-autonomous attractors was studied for deterministic equations [24–26,31,45]. In this paper we study both time-semi-uniform compactness and asymptotically autonomous robustness of non-autonomous random attractors of stochastic equation (1.1). The structure of the paper is as follows. In the next section we define a nonautonomous cocycle for (1.1). In §3we derive two types of long time uniform estimates. In §4we establish the existence, uniqueness and time-semi-uniform compactness of random attractors. In the last section we discuss the asymptotically autonomous robustness of random attractors. In Appendix we provide the proof of measurability of the solution operators. 2. Non-autonomous cocycle generated by stochastic wave equations In this section we consider the following wave equation perturbed by operate-type noise on unbounded domain Rn: utt +αut−Δut−βΔutt +λu −Δu+f(x, u)=g(t, x)+εSu◦dW dt, u(τ,x)=uτ(x),u t(τ,x)=uτ,1(x),x∈Rn,t>τ,τ∈R, (2.1) where α, β, λ, ε > 0, S=I−βΔ, g∈L2 loc(R,L 2(Rn)), the nonlinearity fwill be specified later, and the two-sided real-valued Wiener process Wis defined on the probability space (Ω,F,P), where Ω = {ω∈C(R,R): ω(0) = 0}equipped with the compact-open topology, F=B(Ω) is the Borel sigma-algebra of Ω, and Pis the Wiener measure. Define a family of shift operators {θt}t∈Racting on Ω defined by θtω(·)=ω(·+t)−ω(t)for(ω,t)∈Ω×R. Then (Ω,F,P,{θt}t∈R) forms a metric dynamical system. https://www.cambridge.org/core/terms. https://doi.org/10.1017/prm.2020.77 Downloaded from https://www.cambridge.org/core. Universidad de Sevilla, on 02 Mar 2022 at 10:13:19, subject to the Cambridge Core terms of use, available at 1704 T. Caraballo, B. Guo, N.H. Tuan and R. Wang 2.1. First-order random wave equations Denote by z:= ut+δu with δ>0 to be determined latter, then we have the following equivalent system: ⎧ ⎪ ⎨ ⎪ ⎩ ut=−δu +z, zt−βΔzt+δ1z−δ2Δz+δ3u−δ4Δu+f(x, u)=g(t, x)+εSu◦dW dt, u(τ,x)=uτ(x),z(τ,x)=uτ,1(x)+δuτ(x), (2.2) where δ1:= α−δ,δ2:= 1 −βδ,δ3:= λ−αδ +δ2and δ4:= 1 −δ+βδ2.Let v(t, τ, ω, vτ):=z(t, τ, ω, zτ)−εy(θtω)u(t, τ, ω, uτ),(2.3) where vτ=zτ−εy(θτω)uτand y(θtω)=−δ0 −∞ eδτ (θtω)(τ)dτis the stationary solution of the one-dimensional Ornstein–Uhlenbeck equation dy+δydt=dW(t). By Fan [17], Caraballo and Langa [8], and Wang and Zhou [47], there exists {θt}t∈R-invariant subset which we still denoted by Ω of full measure such that lim t→±∞ y(θtω) t= lim t→±∞ 1 t0 −t y(θsω)ds=0,for every ω∈Ω,(2.4) lim t→±∞ 1 t0 −t|y(θsω)|mds=Γ(1+m 2) √πδm,for every ω∈Ωandm>0.(2.5) Then one can find that ϕ=(u, v) solves the following random system: ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ ut=(εy(θtω)−δ)u+v, vt−βΔvt+δ1v−δ2Δv+δ3u−δ4Δu+f(x, u) =g(t, x)−εy(θtω)v+εβy(θtω)Δv−εδ5y(θtω)+ε2y2(θtω)u +εδ6y(θtω)+ε2βy2(θtω)Δu, u(τ,x)=uτ(x),v(τ,x)=vτ(x)=uτ,1(x)+δuτ(x)−εy(θτω)uτ(x), (2.6) where δ5:= α−3δ,δ6:= 1 −3βδ. 2.2. Assumptions Next, we list the hypotheses on the nonlinearity, on the density of noise and on the time-dependent force. Hypothesis F. The smooth function f:Rn×R→Rhas a subcritical growth range such that |f(x, s)|⩽γ1|s|p+φ1(x),φ 1∈L2(Rn),γ 1>0,(2.7a) f(x, s)s⩾γ2F(x, s)+φ2(x),φ 2∈L1(Rn),γ 2>0,(2.7b) F(x, s)⩾γ3|s|p+1 −φ3(x),φ 3∈L1(Rn),γ 3>0,(2.7c)  ∂f ∂s(x, s) ⩽γ4|s|p−1+φ4(x),φ 4∈L2(Rn),γ 4>0,(2.7d) https://www.cambridge.org/core/terms. https://doi.org/10.1017/prm.2020.77 Downloaded from https://www.cambridge.org/core. Universidad de Sevilla, on 02 Mar 2022 at 10:13:19, subject to the Cambridge Core terms of use, available at Asymptotically autonomous robustness of random attractors 1705 where F(x, s)=s 0f(x, σ)dσ,and 1⩽p<∞for n=1,2; 1 ⩽p<n+2 n−2for n⩾3.(2.8) By (2.7a)–(2.7d), for all (x, s)∈Rn×R, there is a constant c, independent of x and s, such that F(x, s)⩽c(1 + |s|p+1 +|φ1(x)|2+|φ2(x)|).(2.9) Let δ>0 be small enough such that the constants δi>0(i=1,2,3,4,5,6), and denote by κ1:= min δ1,δ2 β,δ,δγ 2,(2.10a) κ2:= max 2(δ5+1),2(δ5+1) δ3 ,2(δ6+β) β,2(δ6+β) δ4 ,γ1 γ3 ,4.(2.10b) Hypothesis S. The size of the noise is suitably controllable: ε∈(0,ε 0] with ε0:= min 1,κ1 2(p+1) 2(2 √πδ +1 δ)κ2.(2.11) The following lemma is useful when establishing the existence of pullback random attractors. Lemma 2.1. Let (2.11)hold. Then for each ω∈Ω, there are T0(ω)>0and C0(ω)> 0such that |y(θ−tω)|+sup ε∈(0,ε0] εκ20 −t Y(θσω)dσ ⩽κ1t (p+1) 2,∀t⩾T0(ω),(2.12a) |y(θ−tω)|+sup ε∈(0,ε0] εκ20 −t Y(θσω)dσ⩽κ1t (p+1) 2+C0(ω),∀t⩾0,(2.12b) where Y(θσω)=|y(θσω)|+|y(θσω)|2for σ∈R. Proof. By (2.11) and (2.4) and (2.5) with m=1,2, there exists T0(ω)>0 such that for all t⩾T0, |y(θ−tω)|+sup ε∈(0,ε0] εκ20 −t Y(θσω)dσ⩽1 2(p+1) 2κ1t+ε0κ22Γ(1) √πδ +2Γ(3/2) √πδ t ⩽1 2(p+1) 2κ1t+κ1κ2 2(p+1) 2(2 √πδ +1 δ)κ22 √πδ +1 δt=1 (p+1) 2κ1t, (2.13) which implies (2.12a). Take C0(ω)=sup t∈[0,T0]|y(θ−tω)|+ε0κ20 −T0Y(θσω)dσ. Then we have (2.12b).  https://www.cambridge.org/core/terms. https://doi.org/10.1017/prm.2020.77 Downloaded from https://www.cambridge.org/core. Universidad de Sevilla, on 02 Mar 2022 at 10:13:19, subject to the Cambridge Core terms of use, available at 1706 T. Caraballo, B. Guo, N.H. Tuan and R. Wang Throughout this paper, the inner product and norm of L2(Rn) are written as (·,·)and·respectively. The norm of Lp(Rn)forp⩾1 is written as ·p.The letter c>0 denotes a generic constant which may change its values from line to line or even in the same line. Hypothesis G. The time-dependent function g∈L2 loc(R,L 2(Rn)) converges to a time-independent function g∞∈L2(Rn) in the sense that lim τ→−∞τ −∞ g(r)−g∞2dr=0.(2.14) A typical and simple example for the functions gand g∞which satisfy condition (2.14), and illustrates that the new condition used here is reasonable is the following. Choose a function g0∈L2(Rn), we set g(t, x)=(et+1)g0(x)andg∞(x)=g0(x) for (t, x)∈R×Rn. Then we have lim τ→−∞τ −∞ g(r)−g∞2dr=g02lim τ→−∞τ −∞ e2rdr=0. We remark that, under hypothesis Gonly, we can not only show the convergence of solutions to (2.2) from nonautonomous to autonomous, but also can show the following properties on the time-dependent force, which are important to discuss the existence and time-semi-uniform compactness of the random attractors. Proposition 2.2. Let hypothesis Ghold. Then we have the following assertions. (i) gis κ-integrable: τ −∞ eκ(r−τ)g(r)2dr < +∞,for all κ>0and τ∈R.(2.15) (ii) gis κ-tail-small: lim k→∞τ −∞ eκ(r−τ)|x|⩾k|g(r, x)|2dxdr < +∞,for all κ>0and τ∈R. (2.16) (iii) gis time-semi-uniformly κ-integrable: sup s⩽τs −∞ eκ(r−s)g(r)2dr < +∞,for all κ>0and τ∈R.(2.17) (iv) gis time-semi-uniformly κ-tail-small: lim k→∞sup s⩽τs −∞ eκ(r−s)|x|⩾k|g(r, x)|2dxdr < +∞,for all κ>0and τ∈R. (2.18) Proof. The proof is similar to that as considered in [45], we do not repeat it again.  https://www.cambridge.org/core/terms. https://doi.org/10.1017/prm.2020.77 Downloaded from https://www.cambridge.org/core. Universidad de Sevilla, on 02 Mar 2022 at 10:13:19, subject to the Cambridge Core terms of use, available at Asymptotically autonomous robustness of random attractors 1707 Remark 2.3. Conditions (2.15) and (2.16) are used to ensure the existence of pullback attractors of PDEs defined on unbounded domains, see e.g.,[41,42]. Condition (2.17) is used to ensure the existence of time-semi-uniformly compact pullback attractors of PDEs defined on bounded domains, see [32,55]. Both (2.17)and (2.18) are used in [30,56] to ensure the existence of time-semi-uniformly compact pullback attractors of PDEs defined on unbounded domains. 2.3. Non-autonomous cocycle In this article we consider the energy space E=H1(Rn)×H1(Rn) equipped with the equivalent norm: ϕE=(v2+β∇v2+δ3u2+δ4∇u2)1/2,∀ϕ=(u, v)∈E. By Carvalho and Cholewa [11], we are able to show that for each (τ,ω)∈ R×Ωandϕτ=(uτ,v τ)∈E, problem (2.6) has a unique solution ϕ(·,τ,ω,ϕ τ)= (u(·,τ,ω,u τ),v(·,τ,ω,v τ)) ∈C([τ,∞),E) such that the solution continuously depends on ϕτ∈E. In addition, we can also prove the (F,B(Hs(Rn)))- measurability of the solutions. Then we find that Φ : R+×R×Ω×E→Egiven by Φ(t, τ, ω, (uτ,z τ)) = (u(t+τ,τ,θ−τω,uτ),z(t+τ,τ,θ−τω,zτ)) is a continuous cocycle over Rand (Ω,F,P,{θt}t∈R) in the sense of [41, Def. 1.1]. Let D={D(τ,ω):τ∈R,ω ∈Ω}be a family of bounded nonempty subsets in H1(Rn)×H1(Rn) satisfying lim t→+∞e−κ1t/(p+1)2D(τ−t, θ−tω)H1(Rn)×H1(Rn)=0.(2.19) Let D={D ={∅ =D(τ,ω)⊆E:τ∈R,ω∈Ω}:Dsatisfies (2.19)}.Wealsointroduce B={B(τ,ω):τ∈R,ω ∈Ω}, which is a family of bounded nonempty subsets of H1(Rn)×H1(Rn) satisfying lim t→+∞e−κ1t/(p+1)2sup s⩽τB(s−t, θ−tω)H1(Rn)×H1(Rn)=0.(2.20) Denote by B={B ={∅ =B(τ,ω)⊆E:τ∈R,ω∈Ω}:Bsatisfies (2.20)}. 3. Long time (B,D)-uniform estimates This section is devoted to several kinds of long time (B,D)-uniform estimates of solutions to problem (2.6). 3.1. (B,D)-uniform estimates in the entire space Let us start with the following long time (B,D)-uniform estimates of solutions of problem (2.6)inH1(Rn)×H1(Rn). https://www.cambridge.org/core/terms. https://doi.org/10.1017/prm.2020.77 Downloaded from https://www.cambridge.org/core. Universidad de Sevilla, on 02 Mar 2022 at 10:13:19, subject to the Cambridge Core terms of use, available at 1708 T. Caraballo, B. Guo, N.H. Tuan and R. Wang Lemma 3.1. Let hypotheses F,Sand Gbe satisfied. Then for each (τ,ω,B,D)∈ R×Ω×B×D, there are TB=T(τ,ω)>0and TD=T(τ,ω)>0such that for all t⩾TBand t⩾TD, sup s⩽τ(u(s, s −t, θ−sω,us−t),z(s, s −t, θ−sω,zs−t))2 H1(Rn)×H1(Rn) ⩽Me|y(ω)|(1 + sup s⩽τ R(s, ω)),(3.1) and (u(τ,τ −t, θ−τω,uτ−t),z(τ,τ −t, θ−τω,zτ−t))2 H1(Rn)×H1(Rn) ⩽Me|y(ω)|(1 + R(τ,ω)),(3.2) where (us−t,z s−t)∈B(s−t, θ−tω)for s⩽τ,(uτ−t,z τ−t)∈D(τ−t, θ−tω),M>0 is a constant independent of τ,ω,Band D,andR(s, ω)is given by R(s, ω):=0 −∞ eκ1r+|y(θrω)|+εκ20 rY(θσω)dσ(1 + g(r+s)2)dr. (3.3) Proof. Taking the inner product of the second equation of (2.6) with vin L2(Rn), we have d dt(v2+β∇v2)+2δ1v2+2δ2∇v2+2δ3(u, v)−2δ4(Δu, v)+2(f(x, u),v) =2(g(t),v)−2εyv2−2εβ∇v2−2(εδ5y+ε2y2)(u, v) +2(εδ6y+ε2βy2)(Δu, v),(3.4) where y:= y(θtω). Thanks to (2.10a) we see from (3.4) that d dtϕ2 E+2Rn F(x, u)dx+2κ1ϕ2 E+2δ(f(x, u),u) ⩽2εy(f(x, u),u)+2(g(t),v)−2εyv2−2εβy∇v2+2εδ3yu2 +2εδ4y∇u2−2(εδ5y+ε2y2)(u, v)+2(εδ6y+ε2βy2)(Δu, v).(3.5) By Young’s inequality, we see 2(g(t),v)⩽2vg(t)⩽cϕEg(t)⩽1 2κ1ϕ2 E+cg(t)2.(3.6) By (2.11) and (2.10b), we have −2(εδ5y+ε2y2)(u, v)+2(εδ6y+ε2βy2)(Δu, v) ⩽ε(δ5+ 1)(|y|+|y|2)(v2+u2)+ε(δ6+β)(|y|+|y|2)(∇v2+∇u2) ⩽1 2εκ2(|y|+|y|2)(v2+δ3u2+β∇v2+δ4∇u2)=1 2εκ2Y(θtω)ϕ2 E. (3.7) https://www.cambridge.org/core/terms. https://doi.org/10.1017/prm.2020.77 Downloaded from https://www.cambridge.org/core. Universidad de Sevilla, on 02 Mar 2022 at 10:13:19, subject to the Cambridge Core terms of use, available at Asymptotically autonomous robustness of random attractors 1715 for all p⩾1. Then by (us−t,z s−t)∈B(s−t, θ−tω)fors⩽τand B∈Bwe see from (2.9), (3.13) and (2.12a) that as t→+∞, I1⩽cr1(ω)e−p p+1 p+3 κ1t+pεκ20 −tY(θσω)dσ ×1+ϕs−t2 E+Rn (F(x, us−t)+φ3(x)) dxp ⩽cr1(ω)e−p p+1 p+3 κ1t+pεκ20 −tY(θσω)dσus−tp(p+1) H1(Rn)+vs−tp(p+1) H1(Rn)+1  ⩽cr1(ω)e−p p+1 p+3 κ1t+p|y(θ−tω)|+pεκ20 −tY(θσω)dσ ×(us−t,z s−t)p(p+1) H1(Rn)×H1(Rn)+1  ⩽cr1(ω)e−p p+1 p+3 κ1t+p (p+1)2κ1tsup s⩽τB(s−t, θ−tω)p(p+1) H1(Rn)×H1(Rn)+1  ⩽cr1(ω)e−1 (p+1)2κ1tsup s⩽τB(s−t, θ−tω)H1(Rn)×H1(Rn)p(p+1) +ce−p p+1 κ1t→0.(3.34) Note that 1−p p+1 p+1 −1 (p+1) 2>0 for all p⩾1. Then by (2.12b)wesee I2⩽c1+sup s⩽τ ˆ Rp(s, ω)0 −∞ e1−p p+1 p+1 κ1r+|y(θrω)|+εκ20 rY(θσω)dσdr<∞. (3.35) By (3.33)–(3.35), the remaining term on the right-hand side satisfies, as t, k →+∞, rksup s⩽τs s−t eκ1(r−s)+|y(θr−sω)|+εκ20 r−sY(θσω)dσϕ(r, s −t, θ−sω,ϕs−t2p Edr→0. (3.36) Finally, we take the supremum over s∈(−∞,τ]in(3.28), then the desired result (3.17) follows from (3.29)–(3.31) and (3.36). By the same argument, we can show (3.18), the details are omitted.  3.3. (B,D)-uniform estimates inside a large ball In this subsection we derive (B,D)-uniform estimates of solutions to problem (2.2) on bounded domains. Denote by ξk(x):=1−ρk(x)fork∈Nwith ρgiven in (3.16). Let ¯ϕ=(¯u, ¯v):=ξkϕ=(ξku, ξkv), where ϕ=(u, v) is the solution of https://www.cambridge.org/core/terms. https://doi.org/10.1017/prm.2020.77 Downloaded from https://www.cambridge.org/core. Universidad de Sevilla, on 02 Mar 2022 at 10:13:19, subject to the Cambridge Core terms of use, available at 1716 T. Caraballo, B. Guo, N.H. Tuan and R. Wang problem (2.6). Multiplying (2.6)byξkwe find ⎧ ⎨ ⎩ ¯ut=(εy −δ)¯u+¯v, ¯vt−βΔ¯vt+δ1¯v−δ2Δ¯v+δ3¯u−δ4Δ¯u+ξkf(x, u) =−εy¯v+εβyΔ¯v−(εδ5y+ε2y2)¯u+(εδ6y+ε2βy2)Δ¯u+J, (3.37) where y=y(θtω) and the remaining terms are given by J:= ξkg(t)−βvtΔξk−2β∇ξk·∇vt−δ2vΔξk−2δ2∇ξk·∇v −δ2uΔξk−2δ2∇ξk·∇u−εβyvΔξk−2εβy∇ξk·∇v −(εδ4y+ε2βy2)uΔξk−2(εδ4y+ε2βy2)∇ξk·∇u. (3.38) Note that the eigenvalue problem: −Δu=λu in O2kwith u|∂O2k= 0 has a family of eigenvalues {λi}∞ i=1 such that 0 <λ 1⩽λ2⩽···λi→∞as i→∞and the corresponding eigenfunctions {ei}∞ i=1 in H1(O2k) form an orthonormal basis of L2(O2k). Let Pi:L2(O2k)→span{e1,e 2...,e i}be the canonical projection. Lemma 3.4. Let hypotheses F,Sand Gbe satisfied. Then for each (k,τ,ω,B,D)∈ N×R×Ω×B×D, lim t,i→+∞sup s⩽τ sup (us−t,zs−t)∈B(s−t,θ−tω)(I−Pi)ξku(s, s −t, θ−sω,us−t), (I−Pi)ξkz(s, s −t, θ−sω,zs−t)H1(Rn)×H1(Rn)=0,(3.39) and lim t,i→+∞sup (uτ−t,zτ−t)∈D(τ−t,θ−tω)(I−Pi)ξku(τ,τ −t, θ−τω,uτ−t), (I−Pi)ξkz(τ,τ −t, θ−τω,zτ−t)H1(Rn)×H1(Rn)=0.(3.40) Proof. Let ¯ui=(I−Pi)ξku,¯zi=(I−Pi)ξkzand ¯vi=(I−Pi)ξkv. Applying I− Pito the second equation of (3.37) and taking the inner product of the resulting equation with ¯viin L2(O2k), we find d dt(¯vi2+β∇¯vi2)+2δ1¯vi2+2δ2∇¯vi2+2δ3(¯ui,¯vi) −2δ4(Δ¯ui,¯vi)+2(ξkf(x, u),¯vi)=−2εy¯vi2−2εβy∇¯vi2 −2(εδ5y+ε2y2)(¯u, ¯vi)+2(εδ6y+ε2βy2)(Δ¯u, ¯vi)+2 J, ¯vi. Let ¯ϕi=(¯ui,¯vi), and then we find that d dt¯ϕi2 E+2κ1¯ϕi2 E⩽−2(ξkf(x, u),¯vi)−2εy¯vi2 −2εβy∇¯vi2+2εδ3y¯ui2+2εδ4y∇¯ui2 −2(εδ5y+ε2y2)(¯u, ¯vi)+2(εδ6y+ε2βy2)(Δ¯u, ¯vi)+2(J, ¯vi).(3.41) https://www.cambridge.org/core/terms. https://doi.org/10.1017/prm.2020.77 Downloaded from https://www.cambridge.org/core. Universidad de Sevilla, on 02 Mar 2022 at 10:13:19, subject to the Cambridge Core terms of use, available at Asymptotically autonomous robustness of random attractors 1717 Note that μ:= ((np −n)/(2p+ 2)) ∈[0,1) for all n⩾1 due to (2.8). Then we see from (2.7a) and the Gagliardo–Nirenberg inequality that −2(ξkf(x, u),¯vi)⩽cup p+1¯vip+1 +cφ1¯vi ⩽cup p+1∇¯viμ¯vi1−μ+c¯vi ⩽cλ μ−1 2 i+1 up H1(Rn)∇¯vi+cλ−1 2 i+1|∇¯vi ⩽cλ μ−1 2 i+1 ϕp E¯ϕiE+cλ−1 2 i+1¯ϕiE ⩽1 2κ1¯ϕi2 E+cλμ−1 i+1 ϕ2p E+cλ−1 i+1.(3.42) Note that −2εy¯vi2−2εβy∇¯vi2+2εδ3y¯ui2+2εδ4y∇¯ui2 ⩽2ε|y|¯ϕi2 E⩽1 2εκ2Y(θtω)¯ϕi2 E.(3.43) By Young’s inequality, (2.11) and (2.10b), we have −2(εδ5y+ε2y2)(¯u, ¯vi)+2(εδ6y+ε2βy2)(Δ¯u, ¯vi) ⩽ε(δ5|y|+|y|2)(¯ui2+¯vi2)+ε(δ6|y|+β|y|2)(∇¯ui2+∇¯vi2) ⩽1 2εκ2Y(θtω)¯ϕi2 E.(3.44) Finally, by lemma 3.2,wesee (J, ¯vi)⩽J¯vi⩽cλ−1 2 i+1J¯ϕiE⩽1 4κ1¯ϕi2 E+cλ−1 i+1e|y|(1 + g(t)2+ϕ2p E). (3.45) Given τ∈Rand ω∈Ω, we substitute (3.42)–(3.45)into(3.41) to find that ϕi(ς):=ϕi(ς,s−t, θ−sω) with ς⩾s−t,s⩽τand t⩾0 satisfies the following energy inequality: d dςϕi2 E+(κ1−εκ2Y(θς−sω))ϕi2 E⩽cθie|y(θς−sω)|(1 + g(ς)2+ϕ2p E). (3.46) where θi:= λμ−1 i+1 +λ−1 i+1 →0asi→+∞. Multiplying (3.46)by eς s−t(κ1−εκ2Y(θσ−sω)) dσand integrating over (s−t, s), then we take the supremum https://www.cambridge.org/core/terms. https://doi.org/10.1017/prm.2020.77 Downloaded from https://www.cambridge.org/core. Universidad de Sevilla, on 02 Mar 2022 at 10:13:19, subject to the Cambridge Core terms of use, available at 1718 T. Caraballo, B. Guo, N.H. Tuan and R. Wang over s∈(−∞,τ], finally, we obtain (3.13) from that sup s⩽τui(s−t, θ−sω,(I−Pi)(ξkus−t)), ×zi(s−t, θ−sω,(I−Pi)(ξkzs−t))2 H1(Rn)×H1(Rn) ⩽ce|y(ω)|e−κ1t+|y(θ−tω)|+εκ20 −tY(θσω)dσ ×sup s⩽τ((I−Pi)ξkus−t,(I−Pi)ξkzs−t)2 H1(Rn)×H1(Rn) +cθie|y(ω)|sup s⩽τ0 −∞ eκ1r+|y(θrω)|+εκ20 rY(θσω)dσ(1 + g(r+s)2)dr +cθie|y(ω)|sup s⩽τs s−t eκ1(r−s)+|y(θr−sω)|+εκ20 r−sY(θσω)dσ ×ϕ(r, s −t, θ−sω,ϕs−t2p Edr. (3.47) By I−Pi⩽1, ξk∞⩽1 and (us−t,z s−t)∈B(s−t, θ−tω) for all s⩽τ, we find from (2.12a) that as t→+∞, e−κ1t+|y(θ−tω)|+εκ20 −tY(θσω)dσsup s⩽τ(I−Pi)(ξkus−t,ξ kzs−t)2 H1(Rn)×H1(Rn) ⩽ce−1 (p+1)2κ1tsup s⩽τB(s−t, θ−tω)H1(Rn)×H1(Rn)2→0. By (3.30) and (3.36) we find that the remaining terms on the right-hand side of (3.47) go to zero as i, t →+∞, and hence we have (3.39). By the same argument, we can show (3.40), the details are omitted here.  4. Existence, uniqueness and semi-uniform compactness of pullback random attractors In this section we establish the semi-uniform compactness of pullback random attractors of the cocycle Φ generated by the stochastic wave equation (2.2). 4.1. Existence of (B,D)-pullback absorbing set First, we establish the existence of (B,D)-pullback absorbing set in H1(Rn)× H1(Rn). Lemma 4.1. Assume hypotheses F,Sand Ghold. Then we have the following two conclusions for the cocycle Φgenerated by problem (2.2): (i) Φ has a B-pullback absorbing set KB={KB(τ,ω):τ∈R,ω∈Ω}∈B, which is given by, for (τ,ω)∈R×Ω, KB(τ,ω)=(u, z)∈H1(Rn)×H1(Rn):(u, z)2 H1(Rn)×H1(Rn) ⩽Me|y(ω)|1+sup s⩽τ R(s, ω).(4.1) https://www.cambridge.org/core/terms. https://doi.org/10.1017/prm.2020.77 Downloaded from https://www.cambridge.org/core. Universidad de Sevilla, on 02 Mar 2022 at 10:13:19, subject to the Cambridge Core terms of use, available at Asymptotically autonomous robustness of random attractors 1719 (ii) Φ has a D-pullback random absorbing set KD={KD(τ,ω):τ∈R,ω∈Ω}∈ D, which is given by, for every (τ,ω)∈R×Ω, KD(τ,ω)=(u, z)∈H1(Rn)×H1(Rn):(u, z)2 H1(Rn)×H1(Rn) ⩽Me|y(ω)|(1 + R(τ,ω)),(4.2) where Mand R(s, ω)are the same as given in lemma 3.1. Proof. (i) By (i) lemma 3.1, for each (τ,ω,B)∈R×Ω×B, there exists TB:= TB(τ,ω)>0 such that t⩾TBs⩽τΦ(t, s −t, θ−tω)B(s−t, θ−tω)⊂ KB(τ,ω). Next, we show KB∈B. For every (τ,ω)∈R×Ω, we deduce from (2.12b) as well as (iii) of proposition 2.2 that as t→+∞, e−2 (p+1)2κ1tsup s⩽τ R(s, θ−tω) =e−2 (p+1)2κ1tsup s⩽τ−t −∞ eκ1(r+t)+|y(θrω)|+εκ2−t rY(θσω)dσ(1 + g(r+s)2)dr ⩽e −κ1t 2(p+1)2sup s⩽τ0 −∞ e κ1r 2(p+1)2(1 + g(r+s)2)dr→0. Therefore, we have, as t→+∞, e−1 (p+1)2κ1tsup s⩽τKB(s−t, θ−tω)H1(Rn)×H1(Rn) ⩽ce|y(θ−tω)|e−2 (p+1)2κ1t1+sup s⩽τ R(s, ω)1/2 →0, which shows that Bis a B-pullback absorbing set of Φ. (ii) Since the mapping ω→R(τ,ω)isF-measurable and KD⊆K B∈B⊆D. Then by (ii) of lemma 3.1, we find that KDis a D-pullback random absorbing set for Φ.  4.2. (B,D)-pullback asymptotic compactness Then, we establish the (B,D)-pullback asymptotic compactness of Φ in H1(Rn)×H1(Rn). Lemma 4.2. Assume hypotheses F,Sand Ghold. Then we have the following two conclusions for the cocycle Φgenerated by (2.2): (i) Φ is B-pullback time-semi-uniformly asymptotically compact in H1(Rn)× H1(Rn), that is, for every τ∈R,ω∈Ωand B={B(τ,ω):τ∈R,ω∈Ω)}∈ B, the sequence {Φ(tn,s n−tn,θ −tnω,(u0,n,z 0,n)}∞ n=1 is pre-compact in H1(Rn)×H1(Rn)whenever tn→+∞,(u0,n,z 0,n)∈B(sn−tn,θ −tnω)and sn⩽τ. https://www.cambridge.org/core/terms. https://doi.org/10.1017/prm.2020.77 Downloaded from https://www.cambridge.org/core. Universidad de Sevilla, on 02 Mar 2022 at 10:13:19, subject to the Cambridge Core terms of use, available at 1720 T. Caraballo, B. Guo, N.H. Tuan and R. Wang (ii) Φ is D-pullback asymptotically compact in H1(Rn)×H1(Rn), that is, for every τ∈R,ω∈Ωand D={D(τ,ω):τ∈R,ω ∈Ω)}∈D, the sequence {Φ(tn,τ −tn,θ −tnω,(u0,n,z 0,n))}∞ n=1 is pre-compact in H1(Rn)×H1(Rn) whenever tn→+∞and (u0,n,z 0,n)∈D(τ−tn,θ −tnω). Proof. (i) Let ε>0 be an arbitrary number, we want to show that the sequence {(Un,Z n)}∞ n=1 := {Φ(tn,s n−tn,θ −tnω,(u0,n,z 0,n))}∞ n=1 has a finite open cover with radiis less then εin H1(Rn)×H1(Rn) provided tn→+∞,(u0,n,z 0,n)∈B(sn−tn,θ −tnω)andsn⩽τ.By(3.1), there are N1=N1(τ,ω,B)⩾1andc1=c1(τ,ω)>0 such that for all n⩾N1, (Un,Z n)H1(Rn)×H1(Rn)⩽c1.(4.3) By (3.17), there are N2=N2(τ,ω,B,ε)⩾N1and k=k(τ,ω,B,ε)⩾1 such that for all n⩾N2, (Un,Z n)H1(Rn\Ok)×H1(Rn\Ok)<ε 2.(4.4) Recall that ξk(x)=1−ρ(|x|/k) with the function ρas given in (3.16). By (3.39), there are N3=N3(τ,ω,B,ε)⩾N2and i=i(τ,ω,B,ε)⩾1 such that for all n⩾N3, ((I−Pi)ξkUn,(I−Pi)ξkZn)H1(Rn)×H1(Rn)<ε 4,(4.5) Then by (4.3), ξk∞⩽1 and the finite-dimensional range of Pi, we know that {(PiξkUn,PiξkZn)}∞ n=1 is pre-compact in Pm0(H1(O2k)×H1(O2k)), which along with (4.5) implies that {(ξkUn,ξ kZn)}∞ n=1 has a finite open cover with radiis less then 1 2εin H1(O2k)×H1(O2k). This along with the fact that (Un(x),Z n(x)) = (ξk(x)Un(x),ξ k(x)Zn(x)) for all x∈O k0implies that the sequence {(Un,Z n)}∞ n=1 has a finite open cover with radiis less then 1 2εin H1(Ok0)×H1(Ok0). This together with (4.4) further implies that the sequence {(Un,Z n)}∞ n=1 has a finite open cover with radiis less then εin H1(Rn)×H1(Rn). (ii) By (3.2), (3.18) and (3.40), we can similarly prove (ii), the details are omitted here.  4.3. Existence and time-semi-uniform compactness of pullback random attractors We are in the position to establish the existence, uniqueness and time-semiuniform compactness of pullback random attractors of Φ in H1(Rn)×H1(Rn). This will be used to discuss the asymptotically autonomous robustness of the pullback random attractors in the next section. Theorem 4.3. Let hypotheses F,Sand Gbe satisfied. Then the following two conclusions hold for the cocycle Φgenerated by problem (2.2): https://www.cambridge.org/core/terms. https://doi.org/10.1017/prm.2020.77 Downloaded from https://www.cambridge.org/core. Universidad de Sevilla, on 02 Mar 2022 at 10:13:19, subject to the Cambridge Core terms of use, available at Asymptotically autonomous robustness of random attractors 1721 (i) Φ has a B-pullback attractor AB={AB(τ,ω):τ∈R,ω ∈Ω}∈B,whichis given by AB(τ,ω)=  t0>0 t⩾t0 Φ(t, τ −t, θ−tω)KB(τ−t, θ−tω) H1(Rn)×H1(Rn) .(4.6) (ii) ABis time-semi-uniformly compact in H1(Rn)×H1(Rn)in the sense that the union  s⩽τAB(s, ω)is pre-compact in H1(Rn)×H1(Rn)for every (τ,ω)∈R×Ω. (iii) Φ has a D-pullback random attractor AD={AD(τ,ω):τ∈R,ω ∈Ω}∈D, which is given by AD(τ,ω)=  t0>0 t⩾t0 Φ(t, τ −t, θ−tω)KD(τ−t, θ−tω) H1(Rn)×H1(Rn) .(4.7) (iv) AB=AD, and thus Φhas a unique pullback random attractor which is timesemi-uniformly compact in H1(Rn)×H1(Rn). Proof. (i) By (i) of lemma 4.1 we find that KB={KB(τ,ω):τ∈R,ω ∈Ω}∈B is a B-pullback absorbing set for Φ. By (i) of lemma 4.2 we know that Φ is B-pullback asymptotically compact in H1(Rn)×H1(Rn). Hence by the abstract result as given by Wang [41, proposition 3.8] we know that Φ has a unique B-pullback attractor AB={AB(τ,ω):τ∈R,ω ∈Ω}∈Bgiven by (4.6) in the sense of [41, definition 2.15]. However, we remark that only the Fmeasurability of ABis unknown, that is why we here say ABis a B-pullback attractor but not a B-pullback random attractor. (ii) It suffices to show that ∪s⩽τAB(s, ω) is pre-compact in H1(Rn)×H1(Rn) for each τ∈Rand ω∈Ω. Let {(Un,Z n)}∞ n=1 be an arbitrary sequence taken from ∪s⩽τAB(s, ω). Then there exists sn⩽τsuch that (Un,Z n)∈ AB(sn,ω)foreachn∈N.Now,welettn→∞, and by the invariance of ABwe have (Un,Z n)=Φ(tn,s n−tn,θ −tnω)AB(sn−tn,θ −tnω), which implies that there exists (u0,n,z 0,n)∈A B(sn−tn,θ −tnω) such that (Un,Z n)=Φ(tn,s n−tn,θ −tnω,(u0,n,z 0,n)). Note that (u0,n,z 0,n)∈ AB(sn−tn,θ −tnω)⊆K B(sn−tn,θ −tnω) with sn⩽τand KB∈B, then by the B-pullback time-semi-uniform asymptotic compactness of Φ as proved in lemma 4.2 we know that the sequence {(Un,Z n)}∞ n=1 is pre-compact in H1(Rn)×H1(Rn), which means that ∪s⩽τAB(s, ω) is pre-compact in H1(Rn)×H1(Rn). (iii) By (i) of lemma 4.1 we know that KD={KD(τ,ω):τ∈R,ω ∈Ω}∈Dis a D-pullback random absorbing set for Φ. By (ii) of lemma 4.2 we find that Φ is D-pullback asymptotically compact in H1(Rn)×H1(Rn). And therefore, by the abstract result established by Wang [41, definition 2.15] and [42], we know that Φ has a D-pullback random attractor AD∈Din H1(Rn)×H1(Rn), which is given by (4.7). https://www.cambridge.org/core/terms. https://doi.org/10.1017/prm.2020.77 Downloaded from https://www.cambridge.org/core. Universidad de Sevilla, on 02 Mar 2022 at 10:13:19, subject to the Cambridge Core terms of use, available at 1722 T. Caraballo, B. Guo, N.H. Tuan and R. Wang (iv) Given (τ,ω)∈R×Ω. Note that by the construction of KBand KDwe find KB(τ,ω)⊇K D(τ,ω). Hence, by (4.6) and (4.7), we have AB(τ,ω)⊆ AD(τ,ω). On the other hand, since AB∈B⊆D, then by the invariance of ABas well as the attraction of ADwe have, as t→+∞, distE(AB(τ,ω),AD(τ,ω)) =dist E(Φ(t, τ −t, θ−tω)AB(τ−t, θ−tω),AD(τ,ω)) →0, which implies AB(τ,ω)⊂ AD(τ,ω)E=AD(τ,ω).Hence we have AB=AD, which along with the F-measurability of ADimplies the F-measurability of AB.  5. Asymptotically autonomous robustness of pullback random attractors In this section we discuss the asymptotically autonomous robustness of the timesection AB(τ,ω) of pullback random attractor AB={AB(τ,ω):τ∈R,ω∈Ω}as time τgoes to negative infinity. 5.1. Random attractors of autonomous stochastic wave equations To be more specific, we also consider an autonomous version of problem (2.1): ˆutt +αˆut−Δˆut−βΔˆutt +λˆu−Δˆu+f(x, ˆu)=g∞(x)+εSˆu◦dW dt, ˆu(0,x)=ˆu0(x),ˆut(0,x)=ˆu1(x),x∈Rn,t>0, (5.1) where g∞∈L2(Rn) is the same function as in (2.14). Denote by ˆz:= ˆut+δˆuwith same δ>0 as given in above sections. Then we have the following equivalent system: ⎧ ⎪ ⎨ ⎪ ⎩ ˆut=−δˆu+ˆz, ˆzt−βΔˆzt+δ1ˆz−δ2Δˆz+δ3ˆu−δ4Δˆu+f(x, ˆu)=g∞(x)+εSˆu◦dW dt, ˆu(0,x)=ˆu0(x),ˆz(0,x)=ˆu1(x)+δˆu0(x). (5.2) Let ˆv:= ˆz−εy(θtω)ˆuto find that ˆϕ=(ˆu, ˆv) satisfies the following random system ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ˆut=(εy(θtω)−δ)ˆu+ˆv, ˆvt−βΔˆvt+δ1ˆv−δ2Δˆv+δ3ˆu−δ4Δˆu+f(x, ˆu)=g∞(x)−εy(θtω)ˆv +εβy(θtω)Δˆv−εδ5y(θtω)+ε2y2(θtω)ˆu+εδ6y(θtω)+ε2βy2(θtω)Δˆu, ˆu(0,x)=ˆu0(x),ˆv(0,x)=ˆv0(x)=ˆu1(x)+δˆu0(x)−εy(θτω)ˆu0(x). (5.3) In fact, the well-posedness of problem (5.3) permits us to define an autonomous cocycle Φ∞:R+×Ω×H1(Rn)×H1(Rn)→H1(Rn)×H1(Rn) given by, for every https://www.cambridge.org/core/terms. https://doi.org/10.1017/prm.2020.77 Downloaded from https://www.cambridge.org/core. Universidad de Sevilla, on 02 Mar 2022 at 10:13:19, subject to the Cambridge Core terms of use, available at Asymptotically autonomous robustness of random attractors 1723 t⩾0andω∈Ω, Φ∞(t, ω, (ˆz0,ˆv0)) = ˆu(t, ω, ˆu0),ˆz(t, ω, ˆz0) =ˆu(t, ω, ˆu0),εy(θtω)ˆu(t, ω, ˆu0)+ˆv(t, ω, ˆv0).(5.4) Denote by D∞={D∞(ω):ω∈Ω}a family of bounded nonempty subsets of H1(Rn)×H1(Rn) satisfying lim t→+∞e(−κ1t)/((p+1)2)D∞(θ−tω)H1(Rn)×H1(Rn)=0.(5.5) Let D∞be the universe of all families of bounded nonempty random subsets of H1(Rn)×H1(Rn) satisfying (5.5). By the standard method as in [51], it is not difficult to prove that Φ∞has a unique D∞-pullback random attractor A∞={A∞(ω):ω∈Ω}∈D∞. The main goal of this section is to prove that the time-section AB(τ,ω)ofABis upper semi-continuous to A∞(ω)asτ→−∞in the sense of the Hausdorff semi-distance of H1(Rn)×H1(Rn). 5.2. Asymptotically autonomous convergence of stochastic wave equations In this subsection we establish the asymptotically autonomous convergence of solutions to problem (2.6)inH1(Rn)×H1(Rn). Lemma 5.1. Let hypotheses Fand Gbe satisfied. Then the solutions to the nonautonomous equations of (2.6)converge to the solutions of autonomous equations (5.3)in the sense that for every (t, ω)∈R+×Ω, lim τ→−∞Φ(t, τ, ω, (uτ,z τ)) −Φ∞(t, ω, (ˆu0,ˆz0))H1(Rn)×H1(Rn)=0, whenever (uτ−ˆu0,z τ−ˆz0)H1(Rn)×H1(Rn)→0as τ→−∞. Proof. Given T>0, for t∈(0,T), we let u(t):=u(t+τ,τ,θ−τω,uτ)−ˆu(t, ω, ˆu0), z(t):=z(t+τ,τ,θ−τω,zτ)−ˆz(t, ω, ˆz0)andv(t):=v(t+τ,τ,θ−τω,vτ)− ˆv(t, ω, ˆv0)By(2.6)–(2.10a) and (2.7a) and (2.7b)wehaveut=v−δu+εyuand vt−βΔvt+δ1v−δ2Δv+δ3u−δ4Δu=f(x, ˆu(t)) −f(x, u(t+τ)) +g(t+τ)−g∞−εyv+εβyΔv−(εδ5y+ε2y2)u+(εδ6y+ε2βy2)Δu,(5.6) where y:= y(θtω). Taking the inner product of (5.6) with vin L2(Rn)weget 1 2 d dt(v2+β∇v2)+δ1v2+δ2∇v2+δ3(u,v)−δ4(Δu,v) =(f(x, ˆu(t)) −f(x, u(t+τ)),v)+(g(t+τ)−g∞,v)−εyv2−εβy∇v2 −(εδ5y+ε2y2)(u,v)+(εδ6y+ε2βy2)(Δu,v).(5.7) https://www.cambridge.org/core/terms. https://doi.org/10.1017/prm.2020.77 Downloaded from https://www.cambridge.org/core. Universidad de Sevilla, on 02 Mar 2022 at 10:13:19, subject to the Cambridge Core terms of use, available at 1724 T. Caraballo, B. Guo, N.H. Tuan and R. Wang Let Ψ(t)=(u(t),v(t)), then we find from (5.7) that 1 2 d dtΨ2 E⩽(f(x, ˆu(t)) −f(x, u(t+τ)),v)+(g(t+τ)−g∞,v) +εδ3yu2+εδ4y∇u2−εyv2−εβy∇v2 −(εδ5y+ε2y2)(u,v)+(εδ6y+ε2βy2)(Δu,v).(5.8) Let ˆ f(x, s)=(∂/∂s)f(x, s), then by hypothesis F,H1(Rn)→Lp+1(Rn) and the H¨older inequality, we have (f(x, ˆu(t)) −f(x, u(t+τ)),v)⩽c1+u(t+τ)p−1 H1(Rn)+ˆu(t)p−1 H1(Rn)Ψ2 E. (5.9) Note that the remaining terms on the right-hand side of (5.8) are bounded by (g(t+τ)−g∞,v)+εδ3yu2+εδ4y∇u2−εyv2 −εβy∇v2−(εδ5y+ε2y2)(u,v)+(εδ6y+ε2βy2)(Δu,v) ⩽c(1 + |y|+|y|2)Ψ2 E+cg(t+τ)−g∞2.(5.10) Substituting (5.9) and (5.10)into(5.8), we obtain d dtΨ2 E⩽ce|y(θtω)|+u(t+τ)p−1 H1(Rn) +ˆu(t)p−1 H1(Rn)Ψ2 E+cg(t+τ)−g∞2.(5.11) Applying the Gronwall inequality to (5.11)over(0,t), we have Ψ(t)2 E⩽ceJ(τ,ω)Ψ(0)2 E+T 0g(r+τ)−g∞2dr, where J(τ,ω):=cT 0ec|y(θrω)|+ˆu(r)H1(Rn)+u(r+τ)H1(Rn)dr. Then we have u(t)H1(Rn)+z(t)H1(Rn)⩽ce|y(θtω)|+J(τ,ω)((uτ−ˆu0,z τ−ˆz0)H1×H1 +T 0g(r+τ)−g∞2dr). By hypothesis G, we find T 0g(r+τ)−g∞2dr⩽τ+T −∞ g(r)−g∞2dr→0asτ→−∞.(5.12) Note that uτ−ˆu0H1(Rn)+zτ−ˆz0H1(Rn)→0asτ→−∞. Then it suffices to show that J(τ,ω) is bounded as τ→−∞. Note that ϕ=(u, v) satisfies d dtϕ(t+τ)2 E+2Rn F(x, u(t+τ)) dx ⩽c1ϕ2 E+2Rn F(x, u)dx+c2g(t+τ)2+c2. https://www.cambridge.org/core/terms. https://doi.org/10.1017/prm.2020.77 Downloaded from https://www.cambridge.org/core. Universidad de Sevilla, on 02 Mar 2022 at 10:13:19, subject to the Cambridge Core terms of use, available at