Multi-valued random dynamics of stochastic wave equations with infinite delays

DISCRETE AND CONTINUOUS doi: DYNAMICAL SYSTEMS SERIES B MULTI-VALUED RANDOM DYNAMICS OF STOCHASTIC WAVE EQUATIONS WITH INFINITE DELAYS Jingyu Wang College of Applied Mathematics Shanxi University of Finance and Economics Taiyuan 030006, PR China Yejuan Wang∗ School of Mathematics and Statistics Gansu Key Laboratory of Applied Mathematics and Complex Systems Lanzhou University, Lanzhou 730000, PR China 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 Abstract. This paper is devoted to the asymptotic behavior of solutions to a non-autonomous stochastic wave equation with infinite delays. The nonlinear terms of the equation are not expected to be Lipschitz continuous, but only satisfy continuity assumptions along with growth conditions, under which the uniqueness of the solutions may not hold. Using the theory of multi-valued non-autonomous random dynamical systems, we prove the existence and measurability of a compact global pullback attractor. 1. Introduction. In this paper, we study the existence and measurability of pullback attractors for the following non-autonomous stochastic wave equation with infinite delays and additive white noise defined on a bounded domain D⊂Rnwith smooth boundary: 2010 Mathematics Subject Classification. Primary: 35L05, 37L30, 37L55. Key words and phrases. Random attractor, multi-valued non-autonomous random dynamical system, stochastic delay wave equation, nonlinear damping, infinite delay. This work was supported by NSF of China (Grants No. 11801335, 41875084). The research of T. Caraballo has been partially supported by Ministerio de Ciencia Innovaci´on y Universidades (Spain), FEDER (European Community) under grant PGC2018-096540-B-I00, and by FEDER and Junta de Andaluc´ıa (Consejer´ıa de Econom´ıa y Conocimiento) under projects US-1254251 and P18-FR-4509. ∗Corresponding author. 1 2 JINGYU WANG, YEJUAN WANG AND TOM´ AS CARABALLO                        ∂2u ∂t2+J∂u ∂t −∆u+λu =f(x, u(t−ρ(t))) +Z0 −∞ F(x, s, u(t+s))ds +g(x, t) + m X j=1 hj(x) ˙wj, t > τ, x ∈D, u(t, x) = 0, t > τ, x ∈∂D, u(t, x) = φ(t−τ, x), t ⩽τ, x ∈D, ∂u ∂t (t, x) = ∂φ ∂t (t−τ, x), t ⩽τ, x ∈D, (1.1) where λ > 0, τ∈Ris an initial time, φis an initial datum on (−∞,0], for each j= 1, . . . , m,hj(x)∈H1 0(D), {wj}m j=1 are independent two-sided real-valued Wiener processes on a probability space which will be specified below, and the other symbols satisfy the following conditions: (H1) There exist two constants β1,β2such that J(0) = 0,0< β1⩽J0(v)⩽β2<∞,∀v∈R. (H2) There exist a function k1∈L2(D) and a positive constant k2such that f∈C(D×R;R) and ρ∈C1(R; [0, h]) satisfy |f(x, ν)|2⩽|k1(x)|2+k2 2|ν|2,∀x∈D, ν ∈R, |ρ0(t)|⩽ρ∗<1,∀t∈R, where h > 0 is a given positive number. (H3) There exist a positive scalar function e−2γ·m1(·)∈L1((−∞,0]; R) and a function m0∈L1((−∞,0]; L2(D)) such that the function F∈C(D×R×R;R) satisfies |F(x, s, ν)|⩽m1(s)|ν|+|m0(s, x)|,∀x∈D, s, ν ∈R,(1.2) and we will denote m0=Z0 −∞ km0(s, ·)kL2(D)ds and m1=Z0 −∞ e−2γsm1(s)ds. (H4) The external force g∈C(R;L2(D)) is such that Zt −∞ ZD eα−8m2 1 β1δ2r|g(r, x)|2dxdr < ∞,∀t∈R, where α > 0 will be given in Lemma 11. Wave equations with some delay terms are considered suitable models in analysis of oscillatory phenomena including aftereffects, times lags or hereditary characteristics [19,22,38,44], as the deformation of viscoelastic materials [13,14] or the retarded control of the dynamics of flexible structures [23,26,28]. Global attractors, uniform attractors or pullback attractors for deterministic autonomous or nonautonomous damped wave equations with deterministic have been studied by many authors, see [2,3,4,5,9,11,16,18,21,27,29,31,32,41,46,47] and the references therein. The existence of random attractors for autonomous or non-autonomous stochastic wave equations has been considered in [12,15,20,24,30,33,35,43,45,48]. The existence of pullback attractors for deterministic damped wave equations with bounded delays has been initially established in [6]. Very recently, the asymptotic behavior of non-autonomous damped wave equations with bounded delays and without the uniqueness of solutions has been investigated in [37,42]. The goal STOCHASTIC WAVE EQUATIONS WITH INFINITE DELAYS 3 of this paper is to study the multi-valued random dynamics of stochastic damped wave equations with inifinite delays. Since the non-Lipschitz continuity of the nonlinearities leads to the non-uniqueness of solutions of (1.1), comparing with the case of uniqueness the new difficulty appears in the proof of the measurability of solutions as well as the measurability of pullback attractors. In additions, the presence of both variable and distributed delays also make the analysis of the asymptotic compactness of solutions more complicated. Therefore, in order to obtain the random attractor, we shall use the general theory of attractors for multi-valued random dynamical systems developed in [7], and then more generally for multi-valued non-autonomous random dynamical systems investigated in [36,40]. For multi-valued random dynamical systems, the reader is referred to [8,10,39] for the existence and measurability of pullback attractors for parabolic equations with delays or first-order lattice systems. This paper is organized as follows. In Section 2, we recall some basic concepts and results related to multi-valued non-autonomous random dynamical systems and global pullback attractors. In Section 3, we define a multi-valued non-autonomous random dynamical system for (1.1). Section 4 is devoted to the existence and uniqueness of the pullback attractor. The measurability of the pullback attractor is given in Section 5. 2. Preliminaries. We recall some basic definitions for multi-valued non-autonomous random dynamical systems and some results ensuring the existence and measurability of pullback attractors for these systems. The reader is referred to [7,34,36,40] for more details. Let Qbe a nonempty set, (Ω,F,P) be a probability space, and (X, d) be a Polish space with Borel σ-algebra B(X). The Hausdorff semi-distance between two nonempty subsets Aand Bof Xis defined by d(A, B) = sup{d(a, B) : a∈A}, where d(a, B) = inf{d(a, b) : b∈B}. Denote by Nr(A) the open r-neighborhood {y∈X:d(y, A)< r}of radius r > 0 of a subset Aof X. Let 2Xbe the collection of all subsets of X. Assume that there are two groups {σt}t∈Rand {θt}t∈Racting on Qand Ω, respectively. Specifically, σ:R×Q→Q is a mapping such that σ0is the identity on Q,σt+τ=σt◦στfor all t, τ ∈R. Similarly, θ:R×Ω→Ω is a (B(R)× F,F)-measurable mapping such that θ0is the identity on Ω, θt+τ=θt◦θτfor all t, τ ∈Rand θtP=Pfor all t∈R. In the sequel, we will call both (Q, {σt}t∈R) and (Ω,F,P,{θt}t∈R) parametric dynamical systems. Definition 1. Let (Q, {σt}t∈R) and (Ω,F,P,{θt}t∈R) be parametric dynamical systems. A multi-valued mapping Φ : R+×Q×Ω×X→2Xwith nonempty closed images is called a multi-valued cocycle on Xover (Q, {σt}t∈R) and (Ω,F,P,{θt}t∈R) if for all q∈Q,ω∈Ω and t, τ ∈R+, the following conditions are satisfied: (1) Φ(0, q, ω, ·) is the identity on X; (2) Φ(t+τ, q, ω, ·) = Φ(t, στq, θτω, Φ(τ, q, ω, ·)). It is well known that the usual definition of multi-valued cocycle requires Φ(t+ τ, q, ω, ·)⊂Φ(t, στq, θτω, Φ(τ, q, ω, ·)), and the one in Definition 1is referred to as strict multi-valued cocycle. As we will deal with strict multi-valued cocycles in our analysis, we will omit the word strict. 4 JINGYU WANG, YEJUAN WANG AND TOM´ AS CARABALLO A multi-valued cocycle is called a random cocycle if the multi-valued mapping (t, ω, x)→Φ(t, q, ω, x) is B(R+)×F×B(X) measurable, i.e., {(t, ω, x) : Φ(t, q, ω, x)∩ O6=∅} ∈ B(R+)× F × B(X) for every open set Oof the topological space X. For the above composition of multi-valued mappings, we use that for any nonempty set V⊂X, Φ(t, q, ω, V ) is defined by Φ(t, q, ω, V ) = [ x0∈V Φ(t, q, ω, x0). Definition 2. (See [34].) A collection Dof some families of nonempty subsets of Xis said to be neighborhood closed if for each D={D(q, ω) : q∈Q, ω ∈Ω} ∈ D, there exists a positive number εdepending on Dsuch that the family {B(q, ω) : B(q, ω) is a nonempty subset of Nε(D(q, ω)),∀q∈Q, ∀ω∈Ω}(2.1) also belongs to D. Note that the neighborhood closedness of Dimplies for each D∈ D, {˜ D(q, ω) : ˜ D(q, ω) is a nonempty subset of D(q, ω),∀q∈Q, ∀ω∈Ω} ∈ D.(2.2) A collection Dsatisfying (2.2) is said to be inclusion-closed in the literature, see, e.g., [17]. Definition 3. (See [7,34,36,40].) (1) A set-valued mapping K:Q×Ω→2Xis called measurable with respect to Fin Ω if the value K(q, ω) is a closed nonempty subset of Xfor all q∈Q and ω∈Ω, and the mapping ω∈Ω→d(x, K(q, ω)) is (F,B(R))-measurable for every fixed x∈Xand q∈Q. (2) Let Dbe a collection of some families of nonempty subsets of Xand K= {K(q, ω) : q∈Q, ω ∈Ω} ∈ D. Then Kis called a D-pullback absorbing set for Φ if for all q∈Q,ω∈Ω and for every B={B(q, ω) : q∈Q, ω ∈Ω}∈D, there exists T=T(B, q, ω)>0 such that Φ(t, σ−tq, θ−tω, B(σ−tq, θ−tω)) ⊆K(q, ω),∀t⩾T. If, in addition, for all q∈Qand ω∈Ω, K(q, ω) is a closed nonempty subset of Xand Kis measurable with respect to the P-completion of Fin Ω, then we say Kis a closed measurable D-pullback absorbing set for Φ. (3) Let Dbe a collection of some families of nonempty subsets of X. Then Φ is said to be D-pullback asymptotically upper-semicompact in Xif for all q∈Q and ω∈Ω, any sequence yn∈Φ(Tn, σ−Tnq, θ−Tnω, xn) has a convergent subsequence in Xwhenever Tn→+∞(n→ ∞), xn∈B(σ−Tnq, θ−Tnω) with B={B(q, ω) : q∈Q, ω ∈Ω}∈D. Definition 4. Let Dbe a collection of some families of nonempty subsets of Xand A={A(q, ω) : q∈Q, ω ∈Ω} ∈ D. Then Ais called a D-pullback attractor for Φ if it satisfies: (1) A(q, ω) is compact for all q∈Qand ω∈Ω. (2) Ais invariant, that is, for every q∈Qand ω∈Ω, Φ(t, q, ω, A(q, ω)) = A(σtq, θtω),∀t⩾0. (3) Aattracts every member of D, that is, for every B={B(q, ω) : q∈Q, ω ∈ Ω}∈Dand for all q∈Qand ω∈Ω, lim t→+∞d(Φ (t, σ−tq, θ−tω, B(σ−tq, θ−tω)),A(q, ω)) = 0. STOCHASTIC WAVE EQUATIONS WITH INFINITE DELAYS 5 Definition 5. Let B={B(q, ω) : q∈Q, ω ∈Ω}be a family of nonempty subsets of X. For every q∈Qand ω∈Ω, let Θ(B, q, ω) = \ τ⩾0[ t⩾τ Φ (t, σ−tq, θ−tω, B(σ−tq, θ−tω)). Then the family {Θ(B, q, ω) : q∈Q, ω ∈Ω}is called the Θ-limit set of Band is denoted by Θ(B). Theorem 6. (See [7,34,36,40].)Let Dbe a neighborhood closed collection of some families of nonempty subsets of X, and let Φbe a multi-valued cocycle on Xover (Q, {σt}t∈R)and (Ω,F,P,{θt}t∈R)satisfying the norm-to-weak upper-semicontinuity on X, i.e., if xn→xin X, then for any yn∈Φ(t, q, ω, xn), there exist a subsequence ynkand a y∈Φ(t, q, ω, x)such that ynk* y (weak convergence). Then Φhas a D-pullback attractor Ain Dif and only if Φis D-pullback asymptotically upper-semicompact in Xand Φhas a closed D-pullback absorbing set Kin D. The D-pullback attractor Ais unique and is given by, for each q∈Qand ω∈Ω, A(q, ω) = Θ(K, q, ω) = [ B∈D Θ(B, q, ω).(2.3) Remark 7. The concept of neighborhood closedness of Dis used to consider the necessary condition for the existence of a D-pullback attractor. For the sufficient condition for the existence of such attractor, we only need the concept of inclusionclosedness of D. Theorem 8. (See [7,34,36,40].)Let Φbe a multi-valued random cocycle. Under the assumptions of Theorem 6, let ω→Φ(t, q, ω, K(q, ω)) be measurable (w.r.t. the P-completion of F)for t⩾0and q∈Q, and let Φ(t, q, ω, K(q, ω)) be closed for all t⩾0,q∈Qand ω∈Ω. Then for every fixed q∈Q,A(q, ·)defined by (2.3)is measurable with respect to the P-completion of F. Let (X, k·kX) be a Banach space. The following result will be used to check the D-pullback asymptotically upper-semicompactness of multi-valued cocycles. Theorem 9. Let Dbe a collection of same families of nonempty subsets of X, and let Φbe a multi-valued cocycle on a Banach space X. Suppose that Φcan be written as Φ=Φ1+ Φ2 and for any fixed q∈Qand ω∈Ω, (1) lim t→+∞kΦ2(t, σ−tq, θ−tω, K(σ−tq, θ−tω))kX= 0; (2) for any fixed t > 0, every sequence un∈Φ1(t, σ−tq, θ−tω, K(σ−tq, θ−tω)) is a Cauchy sequence in X, where K={K(q, ω) : q∈Q, ω ∈Ω}is a D-pullback absorbing set for Φ. Then Φ is D-pullback asymptotically upper-semicompact in X. Proof. Let q∈Q,ω∈Ω, B={B(q, ω) : q∈Q, ω ∈Ω} ∈ D, sequences Tn→+∞ (n→+∞) and un∈Φ(Tn, σ−Tnq, θ−Tnω, B(σ−Tnq, θ−Tnω)) be given arbitrarily. In order to prove the precompactness of unin X, it suffices to show that unis a Cauchy sequence in X. By using the assumption (1), we can choose t1>0 such that kΦ2(t1, σ−t1q, θ−t1ω, K(σ−t1q, θ−t1ω))kX<ε 3.(2.4) 6 JINGYU WANG, YEJUAN WANG AND TOM´ AS CARABALLO On the other hand, we obtain that for sufficiently large n, Φ(Tn, σ−Tnq, θ−Tnω, B(σ−Tnq, θ−Tnω)) = Φ(t1, σ−t1q, θ−t1ω, Φ(Tn−t1, σ−Tnq, θ−Tnω, B(σ−Tnq, θ−Tnω))) ⊂Φ(t1, σ−t1q, θ−t1ω, K(σ−t1q, θ−t1ω)). This implies that un∈Φ(t1, σ−t1q, θ−t1ω, K(σ−t1q, θ−t1ω)) for sufficiently large n. We write unas un=yn+xn, where yn∈Φ1(t1, σ−t1q, θ−t1ω, K(σ−t1q, θ−t1ω)), xn∈Φ2(t1, σ−t1q, θ−t1ω, K(σ−t1q, θ−t1ω)). It follow from assumption (2) that there exists an N > 0 such that kyn−ymkX<ε 3,∀n, m > N. (2.5) By (2.4) and (2.5), we deduce that for all n, m > N, kun−umkX⩽kyn−ymkX+kxnkX+kxmkX< ε, (2.6) which completes the proof.  3. The multi-valued cocycle associated to the model. For the stochastic term in (1.1), we assume that for each j= 1, . . . , m,{wj}m j=1 are independent two-sided real-valued Wiener processes on a probability space (Ω,F,P), where Ω = {ω= (ω1, ω2, . . . , ωm)∈C(R,Rm) : ω(0) = 0}, Fis the Borel σ-algebra generated by the compact-open topology of Ω, and Pis the corresponding Wiener measure on (Ω,F). Then we will identify ωwith W(t), i.e., W(t, ω) = (w1(t), w2(t), . . . , wm(t)) = ω(t) for t∈R. Define a group {θt}t∈Racting on (Ω,F,P) by θtω(·) = ω(·+t)−ω(t), ω ∈Ω, t ∈R.(3.1) Then (Ω,F,P,{θt}t∈R) is a parametric dynamical system. Suppose Q=R. Define a family {σt}t∈Rof shift operators by σt(s) = s+tfor all t, s ∈R.(3.2) Given j= 1, . . . , m, consider the one-dimensional Ornstein-Uhlenbeck equation dzj+ ˜αzjdt =dwj(t).(3.3) One may easily verify that a solution to (3.3) is given by zj(t) = zj(θtωj)≡ −˜αZ0 −∞ e˜ατ (θtωj)(τ)dτ, t ∈R. It is known that there exists a θt-invariant set ˜ Ω⊆Ω of full Pmeasure such that zj(θtωj) is continuous in tfor every ω∈˜ Ω, and the random variable |zj(ωj)|is tempered. Hereafter, we will not distinguish ˜ Ω and Ω, and write ˜ Ω as Ω. It follows from Proposition 4.3.3 in [1] that there exists a tempered function r(ω)>0 such that m X j=1 |zj(ωj)|2⩽r(ω),(3.4) where r(ω) satisfies, for every ω∈Ω, r(θtω)⩽e˜α 2|t|r(ω), t ∈R.(3.5) STOCHASTIC WAVE EQUATIONS WITH INFINITE DELAYS 7 Then (3.4) and (3.5) imply that, for every ω∈Ω, m X j=1 |zj(θtωj)|2⩽e˜α 2|t|r(ω), t ∈R.(3.6) Putting z(θtω) = Pm j=1 hjzj(θtωj), by (3.3) we have dz + ˜αzdt = m X j=1 hjdwj. Let ξ=∂u ∂t +δu where δ > 0 will be fixed later. Then (1.1) can be written as                ∂u ∂t +δu =ξ, ∂ξ ∂t −δξ +J(ξ−δu)−∆u+ (λ+δ2)u=f(x, u(t−ρ(t))) +Z0 −∞ F(x, s, u(t+s))ds +g(x, t) + m X j=1 hj(x) ˙wj, (3.7) with bounded and initial conditions          u(t, x) = ξ(t, x) = 0, t > τ, x ∈∂D, u(t, x) = uτ(t−τ, x) = φ(t−τ, x), t ⩽τ, x ∈D, ξ(t, x) = ξτ(t−τ, x) = ∂φ(t−τ, x) ∂t +δφ(t−τ, x), t ⩽τ, x ∈D. (3.8) To convert the stochastic wave equation to a deterministic one with random parameters, let us consider a new variable given by v(t, x) = ξ(t, x)−z(θtω), where z(θtω) = Pm j=1 hjzj(θtωj). Then the system (3.7)-(3.8) becomes              ∂u ∂t +δu =v+z(θtω), ∂v ∂t −δv +J(v−δu +z(θtω)) −∆u+ (λ+δ2)u=f(x, u(t−ρ(t))) +Z0 −∞ F(x, s, u(t+s))ds +g(x, t) + (˜α+δ)z(θtω), (3.9) with bounded and initial conditions          u(t, x) = v(t, x) = 0, t > τ, x ∈∂D, u(t, x) = uτ(t−τ, x) = φ(t−τ, x), t ⩽τ, x ∈D, v(t, x) = vτ(t−τ, x) = ∂φ(t−τ, x) ∂t +δφ(t−τ, x)−z(θtω), t ⩽τ, x ∈D. (3.10) We denote by Cγ,X the space Cγ,X ={ψ∈C((−∞,0]; X)|lim s→−∞ ψ(s)eγs exists}, where the parameter γ > 0 will be determined later on, and set kψkCγ,X := sup s∈(−∞,0] eγskψ(s)kX<∞. This is a separable Banach space. Given T > τ and u: (−∞, T)→X, for each t∈[τ, T) we denote by utthe function defined on (−∞,0] by the relation ut(s) = u(t+s), s∈(−∞,0]. 8 JINGYU WANG, YEJUAN WANG AND TOM´ AS CARABALLO Let H=L2(D) with norm | · | and inner product (·,·), and let V=H1 0(D) with norm k·k. Set E=Cγ,V ×Cγ,H . In the sequel, Cdenotes an arbitrary positive constant, which may be different from line to line and even in the same line. Thanks to Lemma 11, by the standard Galerkin approximation and compactness method, the following existence result of solutions follows immediately from the similar arguments of Theorem 3.1 in [42]. Theorem 10. Suppose that (H1)-(H3)hold true and g∈L2 loc(R;H). Then for each τ∈R,ω∈Ωand for any (uτ, vτ)∈E, there exists a solution (u(t), v(t)) to problem (3.9)-(3.10), and u(·, τ, ω, uτ)∈C((−∞, T]; V), v(·, τ, ω, vτ)∈C((−∞, T ]; H),∀T > τ. Now we define a multi-valued mapping Φ : R+×R×Ω×E→Eby Φ(t, τ, ω, (uτ, vτ)) = ut+τ(·, τ, θ−τω, uτ), vt+τ(·, τ, θ−τω, vτ) u(·), v(·)is a solution of (3.9)-(3.10) with (uτ, vτ)∈E, where ut+τand vt+τare defined for θ∈[−∞,0] as ut+τ(θ) = u(t+τ+θ) and vt+τ(θ) = v(t+τ+θ) respectively, and vt+τ(·, τ, θ−τω, vτ) = ξt+τ(·, τ, θ−τω, ξτ)− z(θt+·ω) with vτ(·) = ξτ(·)−z(θ·ω). By a standard way as in [7] (Lemma 5.1), we see that Φ satisfies conditions (1)-(2) in Definition 1. It suffices to show that for any t∈R+,τ∈R,ω∈Ω and (uτ, vτ)∈E, the set Φ(t, τ, ω, (uτ, vτ)) is closed in E. Let (un t+τ(·), vn t+τ(·)) ∈Φ(t, τ, ω, (uτ, vτ)) and (ut+τ(·), vt+τ(·)) ∈Esuch that (un t+τ(·), vn t+τ(·)) →(ut+τ(·), vt+τ(·)) in E. (3.11) By slightly modifying the proof of the existence of weak solutions, in view of Lemma 11, we find that there exists a solution (eu, ev) of problem (3.9)-(3.10) with initial condition (uτ, vτ) such that, up to a subsequence un t+τ(·)*eut+τ(·) in L2(−t, 0; V), vn t+τ(·)*evt+τ(·) in L2(−t, 0; H). Combining this with (3.11), we conclude that (ut+τ(·), vt+τ(·)) = (eut+τ(·),evt+τ(·)) ∈ Φ(t, τ, ω, (uτ, vτ)). Hence, Φ is a multi-valued cocycle on Eover (R,{σt}t∈R) and (Ω,F,P,{θt}t∈R). 4. Uniform estimates of solutions. In this section, we derive uniform estimates on the solutions of (3.9)-(3.10) for the purpose of proving the existence of a pullback absorbing set of the multi-valued random dynamical system. Assume that D={D(τ, ω) : τ∈R, ω ∈Ω}is a family of bounded nonempty subsets of Esatisfying, for every τ∈Rand ω∈Ω, lim t→−∞ eα−8m2 1 β1δ2tsup (ϕ,ψ)∈D(τ+t,θtω)kϕk2 CV+kψk2 CH= 0,(4.1) where αwill be given in Lemma 11. Denote by Dthe collection of all families of bounded nonempty subsets of Ewhich fulfill condition (4.1), i.e., D={D={D(τ, ω) : τ∈R, ω ∈Ω}:Dsatisfies (4.1)}. Obviously Dis neighborhood closed. STOCHASTIC WAVE EQUATIONS WITH INFINITE DELAYS 9 Lemma 11. In addition to the assumptions (H1)-(H4), assume that there exist positive constants α,δand γsuch that ˜α 2+8m2 1 β1δ2< α < 2γ, (4.2) α < β1 4−2δ, (4.3) α < δ, (4.4) and α(λ+δ2)< δ(λ+δ2)−2β2 2δ2 β1 −4k2 2eαh β1(1 −ρ∗).(4.5) Then for every τ∈Rand ω∈Ω, the solution of (3.9)-(3.10)with ωreplaced by θ−τωsatisfies for all t⩾0, kvτk2 Cγ,H +kuτk2 Cγ,V + (λ+δ2)kuτk2 Cγ,H ⩽Ce−α−8m2 1 β1δ2tkvτ−tk2 Cγ,H +kuτ−tk2 Cγ,V +kuτ−tk2 Cγ,H  +Ce8m2 1 β1δ2−ατZτ −∞ eα−8m2 1 β1δ2r|g(r)|2dr +C+CZ0 −∞ eα−8m2 1 β1δ2rm X j=1 |zj(θrωj)|2dr. where Cis a positive constant independent of τand ω. Remark 12. We note that (4.2) holds for γlarge, whereas (4.3) holds if δis small enough. On the other hand, (4.2) and (4.5) are satisfied for λlarge and m1,k2 small enough. These conditions can be read as: a combination of strong damping (λlarge) and small effects of the delay (in terms of k2and m1small) ensure the existence of the attractor. Proof. Taking the inner product in Hof the second equation of (3.9) with v, we find that 1 2 d dt|v|2−δ|v|2+ (J(v−δu +z(θtω)), v)+(λ+δ2)(u, v)+(∇u, ∇v) (4.6) = (f(x, u(t−ρ(t))), v) + Z0 −∞ F(x, s, u(t+s)), v+g(t) + (˜α+δ)z(θtω), v. Recall that v=∂u ∂t +δu −z(θtω), hence by Young’s inequality we have (λ+δ2)(u, v)⩾(λ+δ2) 2 d dt|u|2+δ(λ+δ2) 2|u|2−C|z(θtω)|2,(4.7) (∇u, ∇v)⩾1 2 d dtkuk2+δ 2kuk2−Ckz(θtω)k2,(4.8) (g(t) + (˜α+δ)z(θtω), v)⩽β1 8|v|2+C|g(t)|2+C|z(θtω)|2,(4.9) and using (H2)-(H3), we find that (f(x, u(t−ρ(t))), v)⩽β1 8|v|2+2|k1|2 β1 +2k2 2 β1 |u(t−ρ(t))|2,(4.10) 16 JINGYU WANG, YEJUAN WANG AND TOM´ AS CARABALLO ⩽CZτ τ−tf(x, u1(r−ρ(r))) −f(x, u2(r−ρ(r))), vdr +CZτ τ−tZ0 −∞ F(x, s, u1(r+s))ds −Z0 −∞ F(x, s, u2(r+s))ds, vdr ⩽C f(x, u1(r−ρ(r))) −f(x, u2(r−ρ(r))) L2(D×[τ−t,τ]) (5.18) × kv(r, τ −t, θ−τω, vτ−t)kL2(D×[τ−t,τ]) +C   Z0 −∞ F(x, s, u1(r+s))ds −Z0 −∞ F(x, s, u2(r+s))ds   L2(D×[τ−t,τ]) × kv(r, τ −t, θ−τω, vτ−t)kL2(D×[τ−t,τ]) . Multiplying by (5.18) by e2γT 0e−2γT 0and replacing Tby T+T0, in view of the boundary condition (5.15), we deduce that, for all T∈[τ−t, t], kvτk2 Cγ,H +kuτk2 Cγ,V ⩽C f(x, u1(r−ρ(r))) −f(x, u2(r−ρ(r))) L2(D×[τ−t,τ]) (5.19) × kv(r, τ −t, θ−τω, vτ−t)kL2(D×[τ−t,τ]) +C   Z0 −∞ F(x, s, u1(r+s))ds −Z0 −∞ F(x, s, u2(r+s))ds   L2(D×[τ−t,τ]) × kv(r, τ −t, θ−τω, vτ−t)kL2(D×[τ−t,τ]) . Let un r(·, τ−t, θ−τω, un τ−t), vn r(·, τ−t, θ−τω, vn τ−t)∈Φ (r−τ+t, τ −t, θ−tω, K(τ−t, θ−tω)) with (un τ−t, vn τ−t)) ∈K(τ−t, θ−tω) be given arbitrarily. By (4.2) and (4.21)-(4.22) we obtain that {(un r, vn r)}is uniformly (w.r.t. r∈[τ−t, τ]) bounded in E. (5.20) We recall that e2γ·m1(·)∈L1((−∞,0]; R+) and m0∈L1((−∞,0]; L2(D)). Hence for any given ε > 0, there exists e T > h such that for any n, m ∈N, C    Z−e T −∞ F(x, s, un(r+s)) −F(x, s, um(r+s))ds     2 L2(D×[τ−t,τ]) ⩽CZτ τ−tZ−e T −∞ F(x, s, un(r+s)) −F(x, s, um(r+s))ds 2 dr ⩽CZτ τ−t Z−e T −∞ m1(s)|un(r+s)|+ 2|m0(s)|+m1(s)|um(r+s)|ds!2 dr ⩽CZτ τ−t kun rk2 Cγ,H +kun rk2 Cγ,H  Z−e T −∞ e2γsm1(s)ds!2 + Z−e T −∞ m1(s)ds!2 dr ⩽Ct sup r∈[τ−t,τ]kun rk2 Cγ,H +kun rk2 Cγ,H  Z−e T −∞ e2γsm1(s)ds!2 STOCHASTIC WAVE EQUATIONS WITH INFINITE DELAYS 17 +Ct Z−e T −∞ m1(s)ds!2 <ε 4,(5.21) thanks to (5.20). From (5.20), we find that {un(r, τ −t, θ−τω, un τ−t)}is bounded in L∞(τ−t−e T, τ;V),(5.22) {vn(r, τ −t, θ−τω, vn τ−t)}is bounded in L∞(τ−t−e T, τ;H).(5.23) Since ∂u ∂r =v−δu +z(θr−τω),in view of the continuity of z(θr−τω) in r, we obtain that for every ω∈Ω, ∂un(r, τ −t, θ−τω, un τ−t) ∂r is bounded in L∞(τ−t−e T, τ;H).(5.24) Hence, without loss of generality, we can assume that un(r)→u(r)∗-weakly in L∞(τ−t−e T, τ;V), and ∂un(r) ∂r →∂u(r) ∂r ∗-weakly in L∞(τ−t−e T, τ;H). Consequently, un→uin L2(τ−t−e T, τ;H), and un(r, x)→u(r, x) as n→ ∞ for almost every (T, x)∈[τ−t−e T, τ]×D. Since f∈C(D×R;R) and F∈ C(D×R×R;R), we have f(x, un(r)) →f(x, u(r)) as n→ ∞, and F(x, s, un(r)) →F(x, s, u(r)) as n→ ∞ for almost every (r, x, s)∈[τ−t−e T, τ]×D×R. Thanks to the Lebesgue convergence theorem, we deduce that lim n→∞ lim m→∞ kf(x, un(r−ρ(r)) −f(x, um(r−ρ(r))kL2(D×[τ−t,τ]) = 0,(5.25) and by (5.21) we obtain that there exists N1, such that for all n, m ⩾N1, C   Z0 −∞ F(x, s, un(r+s))ds −Z0 −∞ F(x, s, um(r+s))ds    2 L2(D×[τ−t,τ]) ⩽ε 2+CZτ τ−tZ0 −e T F(x, s, un(r+s)) −F(x, s, um(r+s))ds 2 dr (5.26) ⩽ε 2+Ce TZτ τ−tZDZ0 −e T |F(x, s, un(r+s)) −F(x, s, um(r+s))|2dsdxdr < ε. Inserting (5.25)-(5.26) into (5.19), in view of (5.23), we have lim n→∞ lim m→∞ kvn 1,τ −vm 1,τ k2 Cγ,H +kun 1,τ −um 1,τ k2 Cγ,V = 0.(5.27) This implies that condition (2) in Theorem 9holds true, and thus the D-pullback asymptotically upper-semicompactness of Φ in Efollows immediately. 18 JINGYU WANG, YEJUAN WANG AND TOM´ AS CARABALLO Finally, by slightly modifying the proof of Lemma 15, we can show that Φ is norm-to-weak upper-semicontinuous. Then by using Lemma 13, the assertion of this theorem follows immediately from Theorem 6. 6. The measurability of the pullback attractors. In order to prove the measurability of the pullback attractor A(τ, ω) for each fixed τ∈R, we need to show that Φ is a multi-valued random cocycle and the mapping ω→Φ(t, τ, ω, K(τ, ω)) is closed, where K={K(τ, ω) : τ∈R, ω ∈Ω} ∈ D is the closed measurable D-pullback absorbing set given in Lemma 13. For N∈N, we consider the sets ΩN=  ω∈Ω : m X j=1 |ωj(t)|⩽Neζ|t|,for t∈R  ,(6.1) where 0 < ζ < ˜α 4and ˜αis given in (3.2). Then, we can see that Ω = SNΩNand ΩN∈ F. It is shown in [10] that ΩNis a Polish space for any N. By the similar arguments of Lemmas 6.1-6.2 in [42], we have the following properties: (P1) For any n∈N, there exists β=β(N) such that kz(θt+·ω)kCγ,V ⩽βeζ|t|,(6.2) for all ω∈ΩN. (P2) The map R×ΩN3(t, ω)→z(θt+·ω) is continuous in the topology of Cγ,V . Let FΩNbe the trace σ-algebra of Fwith respect to ΩNand let BΩN(a, r), a∈ ΩN,r > 0 be a ball in ΩN. These balls can be generated by BΩ(a, r)TΩN, where BΩ(a, r) is a ball in Ω. The same is true for all open sets in ΩN. Therefore, FΩNis just the Borel σ-algebra of ΩN. Moreover, since ΩN∈ F we have FΩN⊂ F. Also, let ¯ FΩNbe the completion of FΩNwith respect to PΩN, where PΩN(A) := P(A), for A∈ FΩN, that is, PΩNis just the restriction of Pto FΩN. The following facts are given in [10]: (1) PΩNis a finite measure on (ΩN,FΩN). (2) If A∈¯ FΩN, then A∈¯ F. Lemma 15. Assume that the hypotheses in Lemma 11 hold. Then for each τ∈R, the mapping [0,+∞)×ΩN×E3(t, ω, (uτ, vτ)) →Φ(t, τ, ω, (uτ, vτ)) is uppersemicontinuous in E. Proof. Suppose not. Then there exist (uτ, vτ), t > 0, ω∈ΩN, a neighborhood Oof Φ(t, τ, ω, (uτ, vτ)) and sequences tn→t,ωn→ωin ΩN, (un τ, vn τ)→(uτ, vτ) in E,ξn∈Φ(tn, τ, ωn,(un τ, vn τ)) such that ξn/∈O. We shall prove that, up to a subsequence, ξn→ξ∈Φ(t, τ, ω, (uτ, vτ)), which is a contradiction. Without loss of generality, we can assume that 0⩽tn⩽1 + tand kvn τk2 Cγ,H +kun τk2 Cγ,V ⩽1+2kvτk2 Cγ,H + 2kuτk2 Cγ,V ,∀n∈N. (6.3) Let un s+τ(·, τ, θ−τωn, un τ), vn s+τ(·, τ, θ−τωn, vn τ)∈Φs, τ, ωn,(un τ, vn τ)be such that ξn=un tn+τ(·, τ, θ−τωn, un τ), vn tn+τ(·, τ, θ−τωn, vn τ). Arguing as in the proof of Lemma 11, we deduce that for all s∈[τ, 1 + t+τ] and n∈N, kvn sk2 Cγ,H +kun sk2 Cγ,V ⩽Ce8m2 1 β1δ2−α(s−τ)kvn τk2 Cγ,H +Ce8m2 1 β1δ2−α(s−τ)kun τk2 Cγ,V STOCHASTIC WAVE EQUATIONS WITH INFINITE DELAYS 19 +C+Ce8m2 1 β1δ2−αsZs τ eα−8m2 1 β1δ2r|g(r)|2dr (6.4) +Ce8m2 1 β1δ2−αsZs τ eα−8m2 1 β1δ2rkz(θr−τωn)k2dr. By using (6.2) we have Zs τ eα−8m2 1 β1δ2rkz(θr−τωn)k2dr ⩽CZs τ eα−8m2 1 β1δ2re2ζ(r−τ)dr ⩽Ceα−8m2 1 β1δ2se2ζ(s−τ).(6.5) Inserting (6.5) into (6.4) yields kvn sk2 Cγ,H +kun sk2 Cγ,V ⩽Ce8m2 1 β1δ2−α(s−τ)kvn τk2 Cγ,V +Ce8m2 1 β1δ2−α(s−τ)kun τk2 Cγ,V +C+Ce8m2 1 β1δ2−αsZs τ eα−8m2 1 β1δ2r|g(r)|2dr +Ce2ζ(s−τ)(6.6) for all s∈[τ, 1 + t+τ] and n∈N. By the property (P2), we have z(θtn+·ωn)→ z(θt+·ω) in Cγ,V . Analogous to the proof of Theorem 3.1 in [11] Section XV.3 and the argument in [32] Sections II.4 and IV.4.4, in view of the continuity of the mappings J,F,fand (t, ω)→z(θtω), by a standard argument we deduce that there exists a solution u(·, τ, θ−τω, uτ), v(·, τ, θ−τω, vτ)∈L∞(τ, 1+t+τ;V)×L∞(τ, 1+t+τ;H) of problem (3.9)-(3.10) and a subsequence of (un(·), vn(·)) (which we still denote (un, vn)) such that un(·)∗ * u(·) *-weakly in L∞(τ, 1 + t+τ;V),(6.7) vn(·)∗ * v(·) *-weakly in L∞(τ, 1 + t+τ;H) (6.8) as n→ ∞. Now we need to prove that there exists a subsequence (unk(·), vnk(·)) such that (unk(·), vnk(·)) converges to some function (u0(·), v0(·)) in C([τ, 1 + t+τ]; V)× C([τ, 1 + t+τ]; H). Indeed, if this holds true, then by (6.7) and (6.8) we have u0(·) = u(·), v0(·) = v(·) and unk(·, τ, θ−τωnk, unk τ)→u(·, τ, θ−τω, uτ) in C([τ, 1 + t+τ]; V),(6.9) vnk(·, τ, θ−τωnk, vnk τ)→v(·, τ, θ−τω, vτ) in C([τ, 1 + t+τ]; H),(6.10) where (u(·), v(·)) is a solution of problem (3.9). Let ε > 0 be given arbitrarily. Then we have sup s0∈[−1−t,0] eγs0|vnk(τ+tnk+s0, τ, θ−τωnk, vnk τ)−v(τ+t+s0, τ, θ−τω, vτ)| + sup s0∈[−1−t,0] eγs0kunk(τ+tnk+s0, τ, θ−τωnk, unk τ)−u(τ+t+s0, τ, θ−τω, uτ)k ⩽sup s0∈[−1−t,0] eγs0|vnk(τ+tnk+s0, τ, θ−τωnk, vnk τ)−v(τ+tnk+s0, τ, θ−τω, vτ)| + sup s0∈[−1−t,0] eγs0|v(τ+tnk+s0, τ, θ−τω, vτ)−v(τ+t+s0, τ, θ−τω, vτ)|(6.11) 20 JINGYU WANG, YEJUAN WANG AND TOM´ AS CARABALLO + sup s0∈[−1−t,0] eγs0kunk(τ+tnk+s0, τ, θ−τωnk, unk τ)−u(τ+tnk+s0, τ, θ−τω, uτ)k + sup s0∈[−1−t,0] eγs0ku(τ+tnk+s0, τ, θ−τω, uτ)−u(τ+t+s0, τ, θ−τω, uτ)k⩽ε 2 for ksufficiently large. On the other hand, since (un τ, vn τ)→(uτ, vτ) in E, in view of lim s0→−∞ uτ(s0)eγs0=u∈Vand lim s0→−∞ vτ(s0)eγs0=v∈H, we can choose T > 1 + t and ksufficiently large such that sup s0⩽−1−t eγs0|vnk(τ+tnk+s0)−v(τ+t+s0)| ⩽sup s0⩽−1−t eγs0|vnk τ(tnk+s0)−vτ(tnk+s0)| + sup s0∈[−T,−1−t] eγs0|vτ(tnk+s0)−vτ(t+s0)|(6.12) + sup s0⩽−Teγs0|vτ(tnk+s0)−v|+eγs0|v−vτ(t+s0)|⩽ε 4. and in a similar way, sup s0⩽−1−t eγs0kunk(τ+tnk+s0)−u(τ+t+s0)k⩽ε 4.(6.13) Combining (6.11)-(6.13) together, we have unk tnk+τ(·, τ, θ−τωnk, unk τ), vnk tnk+τ(·, τ, θ−τωnk, vnk τ) →ut+τ(·, τ, θ−τω, uτ), vt+τ(·, τ, θ−τω, vτ) in E, and consequently, ξnk=unk tnk+τ(·, τ, θ−τωnk, unk τ), vnk tnk+τ(·, τ, θ−τωnk, vnk τ) →ξ=ut+τ(·, τ, θ−τω, uτ), vt+τ(·, τ, θ−τω, vτ) ∈Φ(t, τ, ω, (uτ, vτ)) ⊂O as k→ ∞. This is in contradiction with ξnk/∈Ofor all k. Therefore, it only remains to show that there exists a subsequence (unk, vnk) of (un, vn) such that (unk(·), vnk(·)) →(u0(·), v0(·)) in C([τ, 1 + t+τ]; V)×C([τ, 1 + t+τ]; H) (6.14) for some function (u0(·), v0(·)) as k→ ∞. Let u=u1+u2,v=v1+v2, we decompose Eqs. (3.9)-(3.10) as follows:      ∂u2 ∂t +δu2=v2, ∂v2 ∂t −δv2−∆u2+ (λ+δ2)u2+J(v−δu +z(θtω)) −J(v1−δu1+z(θtω)) = 0, (6.15) with boundary and initial conditions              u2(t, x) = v2(t, x) = 0, t > τ, x ∈∂D, u2(t, x) = u2,τ (t−τ, x) = φ(t−τ, x), t ⩽τ, x ∈D, v2(t, x) = v2,τ (t−τ, x) = ∂φ(t−τ, x) ∂t +δφ(t−τ, x)−z(θtω), t⩽τ, x ∈D, (6.16) STOCHASTIC WAVE EQUATIONS WITH INFINITE DELAYS 21 and the non-homogeneous equations              ∂u1 ∂t +δu1=v1+z(θtω), ∂v1 ∂t −δv1+J(v1−δu1+z(θtω)) −∆u1+ (λ+δ2)u1=f(x, u(t−ρ(t))) +g(x, t) + Z0 −∞ F(x, s, u(t+s))ds + (˜α+δ)z(θtω), (6.17) with boundary and initial conditions      u1(t, x) = v1(t, x)=0, t > τ, x ∈∂D, u1(t, x) = u1,τ (t−τ, x)=0, t ⩽τ, x ∈D, v1(t, x) = v1,τ (t−τ, x)=0, t ⩽τ, x ∈D. (6.18) We divide the proof into two steps. Step 1. Since (−∆)−1is a continuous compact operator in H, by the classical spectral theory, there exist a sequence {λj}∞ j=1, 0< λ1⩽λ2⩽· · · ⩽λj⩽· · · , λj→+∞,as j→ ∞, and a family of elements {ej}∞ j=1 of D(−∆) which are orthogonal in Hand Vbut orthonormal in Hsuch that −∆ej=λjejfor all j∈N. Let Hm=span{e1, e2,· · · , em},Pmbe the canonical projector on Hmand Ibe the identity. Then for any u2∈Vand v2∈H,u2= ˆu2+ ˜u2,v2= ˆv2+ ˜v2, where ˆu2=Pmu2, ˜u2= (I−Pm)u2, ˆv2=Pmv2, ˜v2= (I−Pm)v2. We decompose (6.15)-(6.16) as follows:          ∂ˆu2 ∂t +δˆu2= ˆv2, ∂ˆv2 ∂t −δˆv2−∆ˆu2+ (λ+δ2)ˆu2+PmJ(v−δu +z(θtω)) −PmJ(v1−δu1+z(θtω)) = 0, (6.19) with boundary and initial conditions              ˆu2(t, x) = ˆv2(t, x)=0, t > τ, x ∈∂D, ˆu2(t, x) = ˆu2,τ (t−τ, x) = Pmφ(t−τ, x), t ⩽τ, x ∈D, ˆv2(t, x) = ˆv2,τ (t−τ, x) = Pm ∂φ(t−τ, x) ∂t +δPmφ(t−τ, x)−Pmz(θtω), t⩽τ, x ∈D, (6.20) and          ∂˜u2 ∂t +δ˜u2= ˜v2, ∂˜v2 ∂t −δ˜v2−∆˜u2+ (λ+δ2)˜u2+ (I−Pm)J(v−δu +z(θtω)) −(I−Pm)J(v1−δu1+z(θtω)) = 0, (6.21) 22 JINGYU WANG, YEJUAN WANG AND TOM´ AS CARABALLO with boundary and initial conditions              ˜u2(t, x) = ˜v2(t, x)=0, t > τ, x ∈∂D, ˜u2(t, x) = ˜u2,τ (t−τ, x) = (I−Pm)φ(t−τ, x), t ⩽τ, x ∈D, ˜v2(t, x) = ˜v2,τ (t−τ, x)=(I−Pm)∂φ(t−τ, x) ∂t +δ(I−Pm)φ(t−τ, x) −(I−Pm)z(θtω), t ⩽τ, x ∈D, (6.22) Taking the inner product in Hof the second equation of (6.21) with ˜v2, we have 1 2 d dt|˜v2|2−δ|˜v2|2+ (∇˜u2,∇˜v2) + (I−Pm)J(v−δu +z(θtω)) −(I−Pm)J(v1−δu1+z(θtω)),˜v2+ (λ+δ2)(˜u2,˜v2)=0.(6.23) Arguing as in (5.8)-(5.11) we deduce that d dteαt |˜v2|2+k˜u2k2+ (λ+δ2)|˜u2|2+ (β1−2δ−α)eαt|˜v2|2 + (2δ−α)eαtk˜u2k2+2δ(λ+δ2)−α(λ+δ2)−β2 2δ2 β1eαt|˜u2|2⩽0.(6.24) Integrating (6.24) over (τ, s) with s∈[τ, τ +t+ 1], and replacing ωby θ−τωn, in view of (4.3)-(4.5), we obtain that for all s∈[τ, τ +t+ 1] and n∈N, |˜vn 2(s, τ, θ−τωn,˜vn 2,τ )|2+k˜un 2(s, τ, θ−τωn,˜un 2,τ )k2 ⩽Ce−α(s−τ)|˜vn 2(τ, τ, θ−τωn,˜vn 2,τ )|2+k˜un 2(τ, τ, θ−τωn,˜un 2,τ )k2.(6.25) Recall that (un τ, vn τ)→(uτ, vτ) in E, hence for any ε > 0, there exist e N, such that for all n⩾e N, kvn τ−vτk2 Cγ,H +kun τ−uτk2 Cγ,V ⩽ε. (6.26) Since (uτ, vτ)∈E, we can choose msufficiently large such that |(I−Pm)v(τ)|2+k(I−Pm)u(τ)k2⩽ε, (6.27) By using (6.25)-(6.27), we can take msufficiently large such that for all s∈[τ, τ + t+ 1] and n⩾e N, |˜vn 2(s, τ, θ−τωn,˜vn 2,τ )|2 2+k˜un 2(s, τ, θ−τωn,˜un 2,τ )k2⩽C|vn(τ)−v(τ)|2 +Ckun(τ)−u(τ)k2+C|(I−Pm)v(τ)|2+Ck(I−Pm)u(τ)k2⩽Cε. (6.28) Now we consider the finite-dimensional system (6.19). Without loss of generality, we assume that s1,s2∈[τ, τ +t+ 1] with 0 < s2−s1<1. In view of | − ∆ˆun 2|2⩽ λmkˆun 2k2⩽λ2 m|ˆun 2|2, then by the second equation of (6.19) we have |ˆvn 2(s2, τ, θ−τωn,ˆvn 2,τ )−ˆvn 2(s1, τ, θ−τωn,ˆvn 2,τ )|⩽Zs2 s1 dˆvn 2(s0, τ, θ−τωn,ˆvn 2,τ ) ds0ds0 =Zs2 s1δˆvn 2(s0, τ, θ−τωn,ˆvn 2,τ )−PmJ(vn−δun+z(θtωn)) + PmJ(vn 1−δun 1+z(θtωn)) + ∆ˆun 2(s0, τ, θ−τωn,ˆun 2,τ )−(λ+δ2)ˆun 2(s0, τ, θ−τωn,ˆun 2,τ )ds0 STOCHASTIC WAVE EQUATIONS WITH INFINITE DELAYS 23 ⩽Zs2 s1δ|ˆvn 2(s0, τ, θ−τωn,ˆvn 2,τ )|+ (λ+δ2)|ˆun 2(s0, τ, θ−τωn,ˆun 2,τ )|ds0 +Zs2 s1 J0(ζ1)|ˆvn 2(s0, τ, θ−τωn,ˆvn 2,τ )|+δ|ˆun 2(s0, τ, θ−τωn,ˆun 2,τ )|ds0(6.29) +Zs2 s1 |∆ˆun 2(s0, τ, θ−τωn,ˆun 2,τ )|ds0 ⩽(δ+β2)Zs2 s1 |ˆvn 2(s0, τ, θ−τωn,ˆvn 2,τ )|ds0 + (λm+β2δ+ (λ+δ2)) Zs2 s1 |ˆun 2(s0, τ, θ−τωn,ˆun 2,τ )|ds0, where ζ1is between vn−δun+z(θtωn) and vn 1−δun 1+z(θtωn). By the similar arguments as in (6.24) and (6.25), in view of (6.3), we deduce that for all s∈ [τ, τ +t+ 1] and n∈N, |ˆvn 2(s, τ, θ−τωn,ˆvn 2,τ )|2 2+kˆun 2(s, τ, θ−τωn,ˆun 2,τ )k2 ⩽Ce−α(s−τ)|ˆvn 2(τ, τ, θ−τωn,ˆvn 2,τ )|2 2+kˆun 2(τ, τ, θ−τωn,ˆun 2,τ )k2 ⩽Ce−α(s−τ).(6.30) Inserting (6.30) into (6.29) gives |ˆvn 2(s2, τ, θ−τωn,ˆvn 2,τ )−ˆvn 2(s1, τ, θ−τωn,ˆvn 2,τ )|⩽Ce−α 2s1−e−α 2s2,(6.31) for all n∈Nand s1,s2∈[τ, τ +t+ 1] with 0 < s2−s1<1. By the first equation of (6.19) and (6.30), we obtain that for all n∈Nand s1,s2∈[τ, τ +t+ 1] with 0< s2−s1<1, kˆun 2(s2, τ, θ−τωn,ˆun 2,τ )−ˆun 2(s1, τ, θ−τωn,ˆun 2,τ )k ⩽pλm|ˆun 2(s2, τ, θ−τωn,ˆun 2,τ )−ˆun 2(s1, τ, θ−τωn,ˆun 2,τ )| ⩽pλmZs2 s1 dˆun 2(s0, τ, θ−τωn,ˆun 2,τ ) ds0ds0 ⩽pλmZs2 s1 (|ˆvn 2(s0, τ, θ−τωn,ˆvn 2,τ )|+δ|ˆun 2(s0, τ, θ−τωn,ˆun 2,τ )|)ds0 ⩽Ce−α 2s1−e−α 2s2.(6.32) Combining (6.28), (6.31) and (6.32) together, we find that {(un 2(·, τ, θ−τωn, un 2,τ ), vn 2(·, τ, θ−τωn, vn 2,τ ))}∞ n=1 is a Cauchy sequence in C([τ, 1 + t+τ]; V)×C([τ, 1 + t+τ]; H). Step 2. Arguing as in (5.18), (5.25) and (5.26), in view of (6.6) and the property (P2), we deduce that for all s∈[τ, 1 + t+τ], |vn 1(s, τ, θ−τωn, vn 1,τ )−vm 1(s, τ, θ−τωm, vm 2,τ )|2 +kun 1(s, τ, θ−τωn, un 1,τ )−um 1(s, τ, θ−τωm, um 1,τ )k2 ⩽Ckf(x, un(r−ρ(r))) −f(x, um(r−ρ(r)))kL2(D×[τ,τ+t+1]) × vn 1(r, τ, θ−τωn, vn 1,τ )−vm 1(r, τ, θ−τωm, vm 1,τ ) L2(D×[τ,τ+t+1]) (6.33) 24 JINGYU WANG, YEJUAN WANG AND TOM´ AS CARABALLO +C   Z0 −∞ F(x, s, un(r+s))ds −Z0 −∞ F(x, s, um(r+s))ds   L2(D×[τ,τ+t+1]) × vn 1(r, τ, θ−τωn, vn 1,τ )−vm 1(r, τ, θ−τωm, vm 1,τ ) L2(D×[τ,τ+t+1]) +Ckz(θr−τωn)−z(θr−τωm)kL2(D×[τ,τ+t+1]) × vn 1(r, τ, θ−τωn, vn 1,τ )−vm 1(r, τ, θ−τωm, vm 1,τ ) L2(D×[τ,τ+t+1]) →0 as n, m → ∞. Therefore, {(un(·, τ, θ−τωn, un τ), vn(·, τ, θ−τωn, vn τ))}∞ n=1 is a Cauchy sequence in C([τ, 1 + t+τ]; V)×C([τ, 1 + t+τ]; H). The proof of this lemma is complete.  Remark 16. It is worth mentioning that if we reduce the nonlinear term Jin (3.9) to the linear case, i.e., J=β1vfor all v∈R, the proof of Step 2 in Lemma 15 can be simplified. In fact, by a similar way as in (6.24)-(6.25), we obtain that for all s∈[τ, 1 + t+τ], |vn 2(s, τ, θ−τωn, vn 2,τ )−vm 2(s, τ, θ−τωm, vm 2,τ )|2 +kun 2(s, τ, θ−τωn, un 2,τ )−um 2(s, τ, θ−τωm, um 2,τ )k2 ⩽Ce−α(s−τ)kvn τ−vm τk2 Cγ,H +kun τ−um τk2 Cγ,V →0, as n, m → ∞, since (un τ, vn τ)→(uτ, vτ) in E. By slightly modifying the proofs of Lemma 6.5 and Theorem 6.1 in [39], we have Theorem 17. Assume that the hypotheses in Lemma 11 hold. Let Abe the D- pullback attractor given in Theorem 14. Then for every fixed τ∈R,A(τ, ·)is measurable with respect to the P-completion of F. REFERENCES [1] L. Arnold, Random Dynamical Systems, Springer-Verlag, New York, Berlin, 1998. [2] J. M. Arrieta, A. N. Carvalho, J. K. Hale, A damped hyperbolic equation with critical exponent, Comm. Partial Differential Equations, 17 (1992), 841-866. [3] A. 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