Non-autonomous and random attractors for delay random semilinear equations without uniqueness

Manuscript submitted to Website: http://AIMsciences.org AIMS’ Journals Volume 00, Number 0, Xxxx XXXX pp. 000–000 NON–AUTONOMOUS AND RANDOM ATTRACTORS FOR DELAY RANDOM SEMILINEAR EQUATIONS WITHOUT UNIQUENESS T. Caraballo1, M.J. Garrido-Atienza1, B. Schmalfuß2, & J. Valero3 1Departamento de Ecuaciones Diferenciales y An´alisis Num´erico, Universidad de Sevilla, Apdo. de Correos 1160, 41080–Sevilla, Spain 2Institut f¨ur Mathematik, Fakult¨at EIM, Universit¨at Paderborn, Warburger Strasse 100, 33098 Paderborn, Germany 3Departamento de Estad´ıstica y Matem´atica Aplicada, Universidad Miguel Hern´andez, Avda. del Ferrocarril s/n, 03202–Elche, Spain (Communicated by Aim Sciences) Abstract. We first prove the existence and uniqueness of pullback and random attractors for abstract multi-valued non-autonomous and random dynamical systems. The standard assumption of compactness of these systems can be replaced by the assumption of asymptotic compactness. Then, we apply the abstract theory to handle a random reaction-diffusion equation with memory or delay terms which can be considered on the complete past defined by R−. In particular, we do not assume the uniqueness of solutions of these equations. 1. Introduction. The intention of this article is to study the asymptotic behaviour of multi-valued non–autonomous and random dynamical systems. The long-time behaviour of these systems can be expressed by terms like pullback attractor and random attractor. The theories of these attractors are now well established as have been extensively developed over the last one and a half decades (see, e.g. Caraballo et al. [13], Cheban [17], Chueshov [19], Crauel and Flandoli [20], Flandoli and Schmalfuß [21], Kloeden [24], Kloeden and Schmalfuß [25], Robinson [28], Schmalfuß[30], amongst many others). Pullback and/or random attractors have 2000 Mathematics Subject Classification. 60H15, 35K40. Key words and phrases. Cocycles, multi-valued non-autonomous and random dynamical systems, pullback and random attractors, delay differential equations. Partially supported by Ministerio de Educaci´on y Ciencia (Spain), FEDER (European Community) under grants MTM2005-01412, MTM2005-03868 and HA2005-0082, by Deutschen akademischen Austauschdienst ppp Austauchprogramm Az: 314/Al-e-dr and Consejer´ıa de Cultura y Educaci´on (Comunidad Aut´onoma de Murcia) grant 00684/PI/04. M. J. G.-A. thanks her collaborators for their great hospitality during her stays in Paderborn and Elche. 1 2 T. CARABALLO, M.J. GARRIDO-ATIENZA, B. SCHMALFUSS, & J. VALERO proven to be appropriate concepts to describe the long–time behaviour of many dynamical systems arising in science, especially those exhibiting non-autonomous (see also Chepyzhov and Vishik [18]) and/or random features. In this article we assume very weak assumptions on the dynamical systems under study. We deal with multi–valued systems connected with the fact that the standard assumption of compactness on this system is replaced by an asymptotic compactness hypothesis. We would also like to stress that, within a random set-up, it is more complicated to show that a multi–valued non–autonomous dynamical system is a random dynamical system. In this sense our article is a generalization of the results in Bates et al. [4] where asymptotically compact single valued systems are studied. Therefore, we are interested in a general model which can cover several of the previously mentioned situations at the same time. In other words, our non-autonomous or random partial differential equations will cover non–linearities with very weak assumptions where non-uniqueness of solutions may happen as well as some hereditary (memory terms) properties, being the delay eventually infinite. Then one of the main difficulties in the random case is due to the fact that the natural phase space to be considered is not separable in general. On the other hand, solution operators are only asymptotically compact. We also include in our theory several variants as those containing some hereditary characteristics as finite or bounded delays (see e.g. [14], [7], [6], [15], [26]) or others with non-uniqueness of solutions or modelled by differential inclusions (Caraballo et al. [10], [11], [12]). Our first aim is to develop a joint theory for both multi-valued non-autonomous and random dynamical systems, pointing out the main differences between both frameworks. Needless to say that a partial differential equation coming from a stochastic partial differential equation with additive white noise after having performed a suitable transformation or change of variable, is non-autonomous. Thus, the theory of pullback attractors for non-autonomous dynamical systems can be applied to analyse the long–time behaviour. However, random or stochastic models usually need additional measurability properties in order to be well-posed. This introduces an additional and important difference into the analysis. However, dealing with differential equations with white noise terms would go beyond the content of this article. We refer to the forthcoming article [8] for more details. Consequently, we have structured the content of the paper as follows. In Section 2 we include some preliminaries concerning the definitions of multi-valued non-autonomous dynamical systems (MNDS) and multi-valued random dynamical systems (MRDS) which turns to be an MNDS with an additional measurability property. We also prove a sufficient condition guaranteeing that an MNDS becomes an MRDS. Section 3 is devoted to prove a general result for the existence and uniqueness of pullback and random attractors for abstract MNDS. The crucial property ensuring this is the pullback asymptotic compactness. In Section 4, a rather general semilinear non-autonomous/random partial differential equation containing (eventually) infinite delays, and which may be related to a semilinear reaction-diffusion equation with memory and with random coefficients, is considered. We prove the existence of globally defined solutions and that these generate an MNDS which is to be studied in the next section. Indeed, in Section 5 we first prove the existence of a pullback attractor for the MNDS, and when the parameter space is a Polish space and has a probability structure, we are able to prove that the ATTRACTORS FOR DELAY RANDOM PDES WITHOUT UNIQUENESS 3 MNDS is in fact an MRDS which possesses a random attractor. Some illustrative examples are finally included in Section 7. 2. Preliminaries. In what follows we give some basic definitions for set–valued non–autonomous and random dynamical systems and formulate sufficient conditions for the existence of a pullback attractor for these systems which is a random set if the non–autonomous perturbation is a noise. Non–autonomous dynamical systems are systems under influence of a non–autonomous perturbation. If this non–autonomous perturbation is a noise term then we have a random dynamical system. We are going to describe these systems in the following. A pair (Ω, θ) where θ= (θt)t∈Ris a flow on Ω: θ:R×Ω→Ω θ0= idΩ, θt+τ=θt◦θτ=: θtθτfor t, τ ∈R is called a non–autonomous perturbation. As an example which describes typical non–autonomous perturbations we consider Ω = Rand θtτ=t+τfor τ=ω∈ Ω, t ∈R. Let P:= (Ω,F,P) be a probability space. On this probability space we consider ameasurable non–autonomous flow θ: θ: (R×Ω,B(R)⊗ F)→(Ω,F). In addition, Pis supposed to be ergodic with respect to θ, which means that every θt-invariant set has measure zero or one, t∈R. Hence Pis invariant with respect to θt. The quadruple (Ω,F,P, θ) which is the model for a noise is called a metric dynamical system. If we replace in the definition of a metric dynamical system the probability space Pby its completion Pc:= (Ω,¯ F,¯ P) the above measurability property is not true in general, see Arnold [1] Appendix A. But for fixed t∈Rwe have that the mapping θt: (Ω,¯ F)→(Ω,¯ F) is measurable. We also mention the following well known ergodic theorem. Theorem 2.1. Let Ybe a real random variable in L1. Then lim t→±∞ 1 tZt 0 Y(θτω)dτ =EY on a (θt)t∈R–invariant set of measure one. Outside this set of measure one we will replace the values of Yby EYso that this version of Yhas the above limit for all ω∈Ω. From now on, let X= (X, dX) be a Polish space. Let D:ω→D(ω)∈2Xbe a multi–valued mapping. The set of multi–functions D:ω→D(ω)∈2Xwith closed and non–empty images is denoted by C(X).Let also denote by Pf(X) the set of all non-empty closed subsets of the space X. Thus, it is equivalent to write that Dis in C(X), or D: Ω →Pf(X). 4 T. CARABALLO, M.J. GARRIDO-ATIENZA, B. SCHMALFUSS, & J. VALERO Let D:ω→D(ω) be a multi–valued mapping in Xover P. Such a mapping is called a random set if ω→inf y∈D(ω)dX(x, y) is a random variable for every x∈X. It is well known that a mapping is a random set if and only if for every open set Oin Xthe inverse image {ω:D(ω)∩O6=∅} is measurable, i.e., it belongs to F(see Hu and Papageorgiou [23, Proposition 2.1.4]). Clearly, all this is also valid if we replace Pby Pcand Fby F. Further, along the paper, if we do not specify which probability space we are using (Por Pc), it will mean that the result is valid for both cases. It is also evident that if Dis a random set with respect to P, then it is also random with respect to Pc. We now formulate properties for random sets that will be needed in the following (see Castaing and Valadier [16] Chapter III and Hu and Papageorgiou [23] Chapter 2.2). Lemma 2.2. (i) Let (Dn)n∈Nbe a family of random sets in C(X). Then [ n∈N Dn(ω) is a random set in C(X). If in addition (Dn)n∈Nis decreasing and every sequence (xn)with xn∈Dn(ω)is pre–compact, then C(ω) := \ n∈N Dn(ω) is non–empty and measurable. (ii) Let Dbe a random set in C(X). Then, there exists a countable number of random variables Yn, n ∈N, such that Yn(ω)∈D(ω)for all ω∈Ωand D(ω) = [ n∈N Yn(ω). (iii) Dis a random set with respect to Pcif and only if the graph of D Gr(D) := {(ω, x)∈Ω×X:x∈D(ω)} belongs to F ⊗ B(X). We now introduce non–autonomous and random dynamical systems. Definition 2.3. A multi–valued map U:R+×Ω×X→Pf(X) is called a multi– valued non–autonomous dynamical system (MNDS) if i) U(0, ω, ·) = idX, ii) U(t+τ, ω, x)⊂U(t, θτω, U(τ, ω, x)) (cocycle property) for all t, τ ∈R+, x ∈ X, ω ∈Ω. It is called a strict MNDS if iii) U(t+τ, ω, x) = U(t, θτω, U(τ, ω, x)) for all t, τ ∈R+, x ∈X, ω ∈Ω. An MNDS is called a multi–valued random dynamical system (MRDS) if the multi–valued mapping (t, ω, x)→U(t, ω, x) ATTRACTORS FOR DELAY RANDOM PDES WITHOUT UNIQUENESS 5 is B(R+)⊗ F ⊗ B(X) measurable, i.e. {(t, ω, x) : U(t, 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 non– empty set V⊂X,U(t, ω, V ) is defined by U(t, ω, V ) = [ x0∈V U(t, ω, x0). We also note that the above measurability hypothesis is not standard at least for single–valued random dynamical system. However, for MRDS it is more difficult to derive measurability than for single valued systems. We now introduce some topological properties of the MNDS U, but we first recall the definition of Hausdorff semi-distance of two non-empty sets A, B: distX(A, B) = sup x∈A inf y∈BdX(x, y). Definition 2.4. U(t, ω, ·) is called upper–semicontinuous at x0if for every neighborhood Uof the set U(t, ω, x0) there exists δ > 0 such that if dX(x0, y)< δ then U(t, ω, y)∈ U. U(t, ω, ·) is called upper–semicontinuous if it is upper–semicontinuous at every x0in X. Remark 1. (i) Note that if a mapping U(t, ω, ·) is upper–semicontinuous at x0, then for all ε > 0 there exists δ(ε)>0 such that distX(U(t, ω, y), U (t, ω, x0)) ≤ε, for any ysatisfying dX(y, x0)≤δ(ε). (ii) The converse is true when U(t, ω, x0) is compact, see Aubin and Cellina [2]. It is not difficult to extend Definition 2.4 if we consider the upper–semicontinuity with respect to all variables assuming that Ω is a Polish space. We now formulate a general condition ensuring that an MNDS defines an MRDS. We need the particular assumption that Ω is a Polish space and Fthe associated Borel–σ–algebra. Lemma 2.5. Let Ωbe a Polish space and let Fbe the Borel–σ–algebra. Suppose that (t, ω, x)7→ U(t, ω, x)is upper–semicontinuous. Then this mapping is measurable in the sense of Definition 2.3. Proof. Thanks to Proposition 1.2.5 in Hu and Papageorgiou [23], we have that for each closed subset C⊂X, the set U−1(C) = {(t, ω, x) : U(t, ω, x)∩C6=∅} is closed, and thus is a Borel set in B(R+)⊗ F ⊗ B(X) = B(R+×Ω×X). This implies that the inverse of a closed set is measurable and then, by Theorem 2.2.4 in [23], the map is measurable. 6 T. CARABALLO, M.J. GARRIDO-ATIENZA, B. SCHMALFUSS, & J. VALERO 3. Non–autonomous and random attractors for MNDS. In this section we generalize the concept of pullback and random attractors to the case of an MNDS and prove a general result for the existence and uniqueness of attractors. As a preparation we need the following definitions. A multi–valued mapping Dis said to be negatively, strictly, or positively invariant for the MNDS Uif D(θtω) ⊂ = ⊃ U(t, ω, D(ω)) for ω∈Ω, t ∈R+. Let Dbe the family of multi–valued mappings with values in C(X). We say that a family K∈ D is pullback D-attracting if for every D∈ D lim t→+∞distX(U(t, θ−tω, D (θ−tω)), K(ω)) = 0,for all ω∈Ω. B∈ D is said to be pullback D-absorbing if for every D∈ D there exists T= T(ω, D)>0 such that U(t, θ−tω, D (θ−tω)) ⊂B(ω),for all t≥T. (1) The following definition provides the main objective of this article. We have to introduce a particular set system (see Schmalfuß [31]): let Dbe a set of multi– valued mappings in C(X) satisfying the inclusion closed property: suppose that D∈ D and let D0be a multi–valued mapping in C(X) such that D0(ω)⊂D(ω) for ω∈Ω, then D0∈ D. Definition 3.1. A family A∈ D is said to be a global pullback D-attractor for the MNDS Uif it satisfies: 1. A(ω) is compact for any ω∈Ω; 2. Ais pullback D-attracting; 3. Ais negatively invariant. Ais said to be a strict global pullback D-attractor if the invariance property in the third item is strict. A natural modification of this definition for MRDS is Definition 3.2. Suppose Uis an MRDS and suppose that the properties of Definition 3.1 are satisfied. In particular, we consider Dto be a system of random sets. In addition, we suppose that Ais a random set, with respect to Pc.Then A is called a random global pullback D-attractor. Remark 2. (i) In contrast to the theory of random attractors for single valued random dynamical systems we have weaker assumptions on the measurability of A. Of course, it is desirable to obtain that Ais a random set with respect to P, but usually we need stronger assumptions in the applications to obtain this property. (ii) For the last Definition 3.2 we assume that the system Dfrom Definition 3.1 consists of random sets. So the inclusion property has to be checked only for random sets D0. A consequence of the pullback convergence and invariance of Pis that it reflexes the forward convergence to the attractor P−lim t→+∞distX(U(t, ω, D(ω)), A(θtω)) = 0 ATTRACTORS FOR DELAY RANDOM PDES WITHOUT UNIQUENESS 7 for all sets Dsuch that ω→U(t, ω, D(ω)) is measurable for t≥0. Indeed, we have to replace in the formula for the pullback convergence ωby θtω. However, this is only true in the weaker convergence in probability. There exist counterexamples which show that in general the forward convergence does not hold almost surely (see [1] page 488). The main tool to prove the existence of an attractor is the pullback-omega-limit set for the MNDS U. For some multi–valued mappings Dwe define a pullbackomega-limit set as an ω–dependent set Λ(D, ω) given by Λ (D, ω) = \ s≥0[ t≥s U(t, θ−tω, D (θ−tω)). This set is obviously closed, but in general it can be empty. It is not difficult to prove that y∈Λ (D, ω) if and only if there exist tn→+∞and yn∈ U(tn, θ−tnω, D (θ−tnω)) such that lim n→+∞yn=y. We then have the following lemma, which is a generalization of Theorem 6 and Lemma 8 in Caraballo et al. [10] to the case in which we consider the family D instead of the bounded sets of X. Lemma 3.3. Suppose that the MNDS U(t, ω, ·)is upper–semicontinuous for t≥0 and ω∈Ω. Let Bbe a multi–valued mapping such that the MNDS is asymptotically compact with respect to Bi.e. for every sequence tn→+∞, ω ∈Ωevery sequence yn∈U(tn, θ−tnω, B(θ−tnω)) is pre–compact. Then for ω∈Ωthe pullback–omega–limit set Λ (B, ω)is non–empty, compact, and lim t→+∞distX(U(t, θ−tω, B (θ−tω)),Λ (B, ω)) = 0,(2) Λ (B, θtω)⊂U(t, ω, Λ (B, ω)) , for all t≥0.(3) Proof. Take an arbitrary sequence yn∈U(tn, θ−tnω, B(θ−tnω)) with tn→+∞. Then, since Uis pullback–asymptotically compact with respect to B, there exists a converging subsequence and its limit ybelongs to Λ (B, ω), so that Λ (B, ω) is non–empty. Let us prove the compactness of Λ (B, ω). For any sequence {yn} ⊂ Λ (B, ω) there exist tn→+∞and zn∈U(tn, θ−tnω, B(θ−tnω)) , such that dX(yn, zn)<1 n. Using again the pullback asymptotic compactness of Uwe obtain the existence of a converging subsequence znk→z∈Λ (B, ω). It follows that ynk→z, so that Λ (B, ω) is compact. The attracting property (2) is proved by contradiction. If this is not the case, then there exist ε > 0 and yn∈U(tn, θ−tnω, B (θ−tnω)), tn→+∞, for which distX(yn,Λ (B, ω)) > ε. Again, since Uis pullback–asymptotically compact with respect to B, it follows that (up to a subsequence) yn→y∈Λ (B, ω), which is not possible. We prove now that (3) holds. If y∈Λ (B, θtω), then there exist sequences yn∈U(tn, θ−tnθtω, xn), xn∈B(θ−tnθtω), tn→+∞, such that yn→y. For tn≥t, the composition property implies U(tn, θ−tnθtω, xn)⊂U(t, ω, U (tn−t, θt−tnω, xn)) , 8 T. CARABALLO, M.J. GARRIDO-ATIENZA, B. SCHMALFUSS, & J. VALERO and then yn∈U(t, ω, zn), where zn∈U(tn−t, θt−tnω, xn). As before, accurate to a subsequence, zn→z∈Λ (B, ω). Since x7→ U(t, ω, x) is upper–semicontinuous with closed values, we have y∈U(t, ω, z)⊂U(t, ω, Λ (B, ω)). This is the main theorem of this section: Theorem 3.4. Assume the hypotheses in Lemma 3.3. In addition, suppose that B∈ D is pullback D–absorbing. Then, the set Agiven by A(ω) := Λ (B, ω) is a pullback D-attractor. Furthermore, Ais the unique element from Dwith these properties. In addition, if Uis a strict MNDS then Ais strictly invariant. Proof. We have to prove that lim t→+∞distX(U(t, θ−tω, D (θ−tω)), A(ω)) = 0 for every D∈ D.(4) Indeed by (2) for every ε > 0, ω ∈Ω there exists a T(ω, ε) such that for t≥T(ω, ε) distX(U(t, θ−tω, B (θ−tω)), A(ω)) < ε. But for every D∈ D we have that U(T(θ−tω, D), θ−t−T(θ−tω,D)ω, D(θ−t−T(θ−tω,D)ω)) ⊂B(θ−tω) so that, by Definition 2.3, distX(U(t, θ−tω, D (θ−tω)), A(ω)) < ε for tlarge. The third property from Definition 3.1 follows from (3). Since U(t, θ−tω, B(θ−tω)) ⊂B(ω) for t≥T(ω, B), we have the relation A(ω)⊂B(ω) for each ω∈Ω,so that A∈ D. But this shows that Ais unique. Indeed suppose we have another pullback D–attractor A0, then as A0(ω)⊂U(t, θ−tω, A0(θ−tω)) and lim t→+∞distX(U(t, θ−tω, A0(θ−tω)), A(ω)) = 0, we have that A0(ω)⊂A(ω). Exchanging Aand A0it follows that A=A0. Finally, assume that Uis a strict MNDS. Then Apullback attracts itself, so that U(t, ω, A (ω)) ⊂U(t, ω, U (τ, θ−τω, A (θ−τω))) =U(t+τ, θ−τω, A (θ−τω)) =U(t+τ, θ−t−τθtω, A (θ−t−τθtω)) , for any τ≥0, and then for each ε > 0 there is T(ε, ω, τ) such that distX(U(t, ω, A (ω)) , A (θtω)) < ε, if t+τ≥T. As ε > 0 is arbitrary we obtain that U(t, ω, A (ω)) ⊂A(θtω), as required. ATTRACTORS FOR DELAY RANDOM PDES WITHOUT UNIQUENESS 9 With respect to the measurability of Definition 3.2 and the applications in Section 6 we suppose for the next lemma a complete probability space Pc. However, the result is also valid for the space P. Lemma 3.5. Under the assumptions in Theorem 3.4, let ω→U(t, ω, B(ω)) be a random set for t≥0. Assume also that U(t, ω, B(ω)) is closed for all t≥0and ω∈Ω. Then the set Aintroduced in Theorem 3.4 is measurable. Proof. We introduce C(ω) = \ τ∈Z+[ t≥τ,t∈Z+ U(t, θ−tω, B(θ−tω)) which is a random set by Lemma 2.2 (i). We show that A(ω) = C(ω). We just know from the construction in Theorem 3.4 that A(ω)⊂B(ω). On account of the properties of Awe observe that C(ω)⊃\ τ∈Z+[ t≥τ,t∈Z+ U(t, θ−tω, A(θ−tω)) | {z } ⊃A(ω) ⊃A(ω). On the other hand we have that C(ω)⊂A(ω) from [ t≥τ U(t, θ−tω, B(θ−tω)) ⊃[ t≥τ,t∈Z+ U(t, θ−tω, B(θ−tω)) and the fact that the family τ∈R→[ t≥τ U(t, θ−tω, B(θ−tω)) is decreasing. 4. Mild solutions for abstract random evolution equations with nonuniqueness. In this section we consider the following evolution equation dy dt =Ay +f(θtω, yt).(5) Here we suppose that Ais the generator of a C0contraction semigroup (S(t))t≥0 on a separable Banach space (H, k·k): kS(t)xk ≤ kxke−αt,for some α > 0 and every t≥0. We need that the operators S(t) for t > 0 are compact. The non–linear term f depends on ωand on a delay term yt(s) = y(t+s) for s∈[−t, 0] x0(s+t)s < −t where t≥0. Here x0is a given continuous function on R−with values in H. According to the function x0we can equip (5) with an initial condition y(t) = x0(t), t ≤0. Before describing the assumptions on f, we first introduce the function space Cγ={u∈C((−∞,0]; H) : lim τ→−∞ u(τ)eγτ exists}, where γ > α, and set kukγ:= supτ∈(−∞,0] eγτ ku(τ)k<∞.This is a separable Banach space [22, p.15]. 16 T. CARABALLO, M.J. GARRIDO-ATIENZA, B. SCHMALFUSS, & J. VALERO Proof. In the case that U(t, ω, ·) is not upper–semicontinuous then there exist a neighborhood Mt,ω of U(t, ω, x0), a sequence {xn 0:n∈N}, xn 0→x0with convergence in Cγand elements yn t∈U(t, ω, xn)6∈ Mt,ω. We show that limn0→+∞yn0 t=: y0for some subsequence (n0) in N, which is an element in U(t, ω, x0). This is a contradiction. To see that yn tis relatively compact we apply the Arzel`a–Ascoli theorem. By the properties of the sequence xn 0(which is pre–compact in Cγ) it is sufficiently to show that yn t(s), s ∈[−t, 0] is pre–compact. We note that by Theorem 4.3 the set {yn t:n∈N} is bounded in Cγbecause {xn 0:n∈N}is bounded in Cγ. Hence sup n∈N,s∈[−t,0] kyn t(s)k<∞(12) The same argument as in the proof of Theorem 4.2 yields the relative compactness of Z(s) := {yn t(s) : n∈N}. In particular, {S(t+s)xn 0(0) : n∈N}is pre–compact. Similarly, we can apply the equicontinuity argument of Theorem 4.2. Indeed, to see the equicontinuity of {yn t:n∈N}at s∈(−t, 0] we still use the fact that r→S(r) is continuous in norm for r > 0. Following the idea of the proof of Theorem 4.2 to see equicontinuity at s=−twe have to study the following equation with r:= s+t > 0 yn t(s) = yn(r) = S(r)xn 0(0) + Zr 0 S(r−τ)f(θτω, yn τ)dτ. Note that the norm of the integral on the right hand side is small uniformly with respect to nif ris small applying (6) and (12). To see the equicontinuity of the functions formed by the first expression on the right hand side we have to show that for every ε > 0 there exists a δ > 0 such that for n∈Nand r≤δwe have that kS(r)xn 0(0)−xn 0(0)k ≤ ε. If not, there would exist ε > 0, sequences n→+∞, rn→0 such that kS(rn)xn 0(0) −xn 0(0)k ≥ ε. Choosing nsufficiently large such that for rin [0, t] the estimate kS(r)(xn 0(0) −x0(0))k ≤ ε 4 holds, we then have that kS(rn)xn 0(0) −xn 0(0)k ≤kS(rn)(xn 0(0) −x0(0))k+kS(rn)x0(0) −x0(0)k +kxn 0(0) −x0(0)k ≤ ε 4+ε 4+ε 4< ε for large n. This is a contradiction. Now we prove the following compactness conclusion for the MNDS U. Lemma 5.4. Assume that (10) holds. The multi–valued dynamical system Uis pullback asymptotically compact with respect to Bdefined in Lemma 5.2. Proof. Let ztbe the unique (mild) solution of dz dt =Az, z0=x0∈Cγ(13) given by zt(s) = S(t+s)x0(0) : s∈[−t, 0] x0(s+t) : s≤ −t ATTRACTORS FOR DELAY RANDOM PDES WITHOUT UNIQUENESS 17 so that kztkγ≤max(e−αtkx0(0)k, e−γtkx0kγ)≤e−αtkx0kγ.(14) Let yτbe a solution of (5) with initial function x0so that yτ∈U(τ, θ−tω, x0). Then there exists uτ∈Cγsuch that yτ=zτ+uτwhere uτis a mild solution of du dτ =Au +f(θτ−tω, yτ), u0(s) = 0 for s≤0.(15) Let tn→ ∞ and xn 0∈B(θ−tnω). The solution to this initial function of (5), with θ−tnωinstead of ω, is denoted by yn τ. According to Lemma 5.2, yn τ∈B(θ−tn+τω), hence kyn τkγ≤R(θ−tn+τω). Let unbe the solution of (15) with t=tnwhich can be written as un(tn+s) = Ztn+s 0 S(tn+s−τ)f(θ−tn+τω, yn τ)dτ, s ∈[−tn,0]. Similar to the proof of Theorem 4.2 and the above calculations we can find an estimate of kun(tn+s1)−un(tn+s2)k,−T≤s1< s2≤0 for an arbitrary T > 0 which gives us the equicontinuity of {un(tn+·) : n∈N}on [−T, 0]. In addition, we are also able to prove the pre–compactness of {un(tn+s) : n∈N} for s∈[−T, 0]. By the Arzel`a–Ascoli theorem there exist a subsequence {n0}and a function ψ:R−→Hwhich is the uniform limit of un0(tn0+·) on every interval [−T, 0]. The following a priori estimate holds kun(tn+s)k ≤ Z0 −∞ e−α(s−τ)(2c1(θτω) + c2(θτω)R(θτω))dτ =e−αsR(ω), s ∈R−.(16) The last inequality follows from Remark 4 including the existence of the integral where the properties of c1, c2and the continuity of t→R(θtω) are needed. From this inequality we can derive kun(tn+s)keγs ≤Z0 −∞ eατ (2c1(θτω) + c2(θτω)R(θτω))dτ, s ∈R−.(17) From (17), sup s∈[−T,0] kun0 tn0(s)keγs ≤R(ω) and then lim n0→∞ sup s∈[−T,0] kun0 tn0(s)keγs = sup s∈[−T,0] kψ(s)keγs ≤R(ω) for every T > 0.Hence sup T >0 sup s∈[−T,0] kψ(s)keγs ≤R(ω) that is, not only does ψbelong to Cγbut also kψkγ≤R(ω). In addition un0 tn0(·) converges to ψin Cγ.To prove that we have to check that for every ε > 0 there exists N(ε) such that sup s≤0 kun0 tn0(s)−ψ(s)keγs ≤εfor all n0≥N(ε).(18) Let us now consider γ > α. For every ε > 0 there exists Tε>0 such that e−(γ−α)TεR(ω)≤ε 2. 18 T. CARABALLO, M.J. GARRIDO-ATIENZA, B. SCHMALFUSS, & J. VALERO Since the convergence of un0 tn0(·) to ψholds in compact intervals, in order to prove (18) we only need to check that sup s≤−Tε kun0 tn0(s)−ψ(s)keγs ≤εfor all n0≥N(ε). But thanks to (16) kun0 tn0(s)keγs ≤e(γ−α)sR(ω) for s≤0, which together with the choice of Tεimplies sup s≤−Tε kun0 tn0(s)keγs ≤ε 2. Moreover, sup s∈(−T,−Tε] kψ(s)keγs ≤lim n0→∞ sup s∈[−T,−Tε] kun0 tn0(s)keγs ≤ε 2 for every T > Tε.Hence the convergence of {un0 tn0(·)}to ψis in Cγ. We then have yn t=un t+zn t where zn tis the solution of (13) with initial function xn 0. Since xn 0∈B(θ−tnω) it follows by (14) lim tn→+∞kzn tnkγ= 0 so that we have found the convergence of yn0 tn0to ψin Cγwhich is the conclusion of the lemma. According to Theorem 3.4 summarizing Lemmas 5.3, 5.2, 5.4 we have Theorem 5.5. Suppose (10). Then the MNDS generated by (5) has a pullback D–attractor Ain C(Cγ). Corollary 1. Suppose that γ0> α such that the assumptions on fgiven at the beginning of Section 4 are satisfied with respect to γ0. Then there exists a pullback Dγ0–attractor Aγ0where Dγ0consists of the subexponentially growing multi– functions in C(Cγ0). By the embedding kukγ0≤ kukγ,for u∈Cγ, for γ0> γ > α there exists a Dγ–pullback attractor Aγsuch that Aγ(ω)⊂Aγ0(ω). Remark 5. It is really interesting to stress out the relationship that there exists between the uniqueness of pullback attractors Aγand the systems of attracted sets Dγ.Observe that from the Definition 3.1 every pullback Dγ-attractor is an invariant set.According to the Corollary 1, the Dγ0-attractor Aγ0attracts the infinite number of Dγ-attractors Aγ, for γ0> γ > α, since Aγ∈ Dγ⊂ Dγ0.However, for γ0there exists a unique attractor Aγ0.Indeed, Aγdoes not have to attract the elements from Dγ0. ATTRACTORS FOR DELAY RANDOM PDES WITHOUT UNIQUENESS 19 6. Random attractors for equations with infinite delay. As before, along this section we assume the same conditions on Sand fgiven at the beginning of Section 4. We now apply the results proved in the previous sections to show the existence of a random attractor for (5). From now on in this section we suppose that Ω can be equipped with a metric providing a Polish space. Fis defined to be the Borel–σ–algebra of Ω. Finally θt is supposed to be continuous on Ω for t∈R. We assume also that the map Ω×Cγ3(ω, y)7→ f(ω, y)∈H(19) is continuous. We have to prove that the MNDS generated by (5) is an MRDS. Theorem 6.1. Assume condition (19) and also that for every ω0∈Ωand t0∈R there exists a neighborhood V=V(ω0, t0)such that for some µ > 1 Zt 0 (c1(θτω)µ+c2(θτω)µ)dτ ≤C(ω0, t0)<∞,for all (ω, t)∈V. (20) Then the mapping (t, ω, x0)→U(t, ω, x0) is B(R+)⊗ F ⊗ B(Cγ)measurable. Proof. According to Lemma 2.5 we show that the above mapping is upper–semicontinuous. We could follow exactly the proof of Lemma 5.3 except that we do not have fixed ω, t. However, condition (20) gives us some uniformity with respect to ω. Let tn→t0, xn→x0in Cγand xn, x0∈Cγ,ωn→ω0and ξn∈U(tn, ωn, xn). Choose Twith tn≤T. We consider ynto be yn(t) = S(t)xn(0) + Zt 0 S(t−τ)f(θτωn, yn∨τxn)dτ,t≤T, (21) for t≥0 and yn(t) = xn(t) for t < 0. This is a mild solution of (5) which satisfies yn tn=yn∨tnxn=ξn. Using condition (20) and the Arzel`a–Ascoli theorem we obtain, in a similar way as in Lemma 5.3, that ynis pre–compact in C([0, T ], H) for every T > 0. Then there exist a subsequence (n0) and a limit point y0∈C([0, T]; H) with yn0→y0uniformly on [0, T]. We extend y0by x0for t≤0. This new function is continuous at zero which follows from (21) for t= 0. Hence, on account of the uniform convergence of xnand yn,we have that for a given ε > 0 sup t∈[−T,0] kyn0∨tn0xn0(t)−y0∨t0x0(t)keγt <ε 4. On the other hand we have sup t≤−T kxn0(t+tn0)−x0(t+t0)keγt ≤sup t≤−T kxn0(t+tn0)−x0(t+tn0)keγt + sup t≤−T kx0(t+tn0)−x0(t+t0)keγt. Since the first term on the right hand side is bounded by kxn0−x0kCγ,we can make this term be less than ε/4 for large n0. Since x0∈Cγ,we know that there 20 T. CARABALLO, M.J. GARRIDO-ATIENZA, B. SCHMALFUSS, & J. VALERO exists limτ→−∞ x0(τ)eγτ =x∈H. Then we can choose a T0> T such that sup t≤−T0 kx0(t+tn0)−x0(t+t0)keγt ≤sup t≤−T0kx0(t+tn0)−xkeγt +kx−x0(t+t0)keγt <ε 4. But on the compact interval [−T0,−T] we have that sup t∈[−T0,−T] kx0(t+tn0)−x0(t+t0)keγt <ε 4 for n0large. This gives the convergence of ξn0in Cγto ξ0. Then ξ0=y0∨t0x0∈Cγ. Condition (19) gives for every τ f(θτωn0, yn0∨τxn0)→f(θτω0, y0∨τx0). To see that y0satisfies y0(t) = S(t)x0(0) + Zt 0 S(t−τ)f(θτω0, y0∨τx0)dτ, 0 ≤t≤t0, we mention that the integrands of (21) are bounded by the function eατ (c1(θτωn0) + c2(θτωn0)M), M := sup n0 sup τ∈[0,T ] kyn0 τkγ<∞. The uniform integrability condition (20) together with Vitali’s convergence theorem for finite measures give us the convergence of the above integrals. Hence ξ0∈ U(t, ω0, x0) so that Uis upper–semicontinuous. The following lemma is needed to prove the measurability of the pullback attractor. Lemma 6.2. In addition to (19), (10) and (20), assume that the mapping ω7→ R(ω)defined in (11) is the radius of a ball in Cγsuch that lim sup ω→ω0 R(ω)≤R(ω0)for ω0∈Ω.(22) Then the multi–function ω→U(t, ω, B(ω)) ⊂Cγis ¯ Fmeasurable for t≥0. In addition U(t, ω, B(ω)) ∈C(Cγ). Proof. According to Lemma 2.2(iii) we show that for fixed tthe graph Gr(U(t, ω, B(ω))) is closed. Suppose that lim n→∞(ωn, yn) = (ω0, y0) in Ω ×Cγwhere yn∈U(t, ωn, B(ωn)). We have yn(s) = xn 0(s+t) for s≤ −t, xn 0∈B(ωn). Then xn 0→x0 0in Cγ. On account of the properties of Rfor every ε > 0 there exists n0(ε) such that for n≥n0we have kxn 0kγ≤R(ω0) + εand then x0 0∈ BCγ(0, R(ω0) + ε). This holds for every ε > 0 so that x0 0∈B(ω0). Now we have lim n→∞ kxn 0(0) −x0 0(0)k= 0, lim n→∞ kZt 0 S(t−τ)(f(θτωn, yn τ)−f(θτω0, y0 τ))dτk= 0,for t≥0, ATTRACTORS FOR DELAY RANDOM PDES WITHOUT UNIQUENESS 21 so that y0is a mild solution of (5) in Cγ. For the convergence we use a similar argument as in the proof of Theorem 6.1. Since x0 0∈B(ω0) we conclude that y0∈U(t, ω0, B(ω0)). Thus the graph of U(t, ω, B(ω)) is closed in Ω ×Cγ. Then Gr(U(t, ·, B(·))) ∈¯ F ⊗B(Cγ),so that U(t, ω, B(ω)) is ¯ Fmeasurable. We need here that Cγis separable. The second statement follows similarly setting ωn=ω0. This is the main theorem of the section. Theorem 6.3. Suppose that the assumptions of Lemma 6.2 hold. Then the pullback attractor Aintroduced in Theorem 5.5 is ¯ Fmeasurable with respect to images in the space Cγ, that is, Ais a random attractor. Proof. It is a consequence of Lemmas 6.2, 3.5 and 5.1. In the following we formulate conditions ensuring the assumptions in Lemma 6.2 on R(ω). Lemma 6.4. Let (10) hold. Suppose that there exists µ > 1such that for every ω0∈ΩZ0 τ c2(θsω)µds ≤c(τ, ω0)for every τ < 0,(23) Z0 −∞ eµ(α−ρ)τ+µR0 τc2(θsω)dsc1(θτω)µdτ ≤c(ω0),(24) for ωin some neighborhood Vof ω0and for some ρ > 0such that α−ρ > ¯c2. The mappings ω→c1(ω), ω →c2(ω)are assumed to be continuous. Then ω→R(ω)is continuous. Proof. We rewrite (11) as 2Z0 −∞ e(α−ρ)τ+R0 τc2(θsω)dsc1(θτω)eρτ dτ so that eρτ dτ is a finite measure on B(R−). Then the second inequality in the assumptions ensures that the integrand is uniformly integrable for ω∈V. The first inequality ensures lim n→∞ Z0 τ c2(θsωn)ds =Z0 τ c2(θsω0)ds for τ < 0. Vitali’s convergence theorem about uniformly integrable functions for finite measures gives the conclusion. For Lemma 6.2 and Theorem 6.3 we need that the underlying probability space is complete. In the following we would like to avoid this assumption. The price we have to pay are stronger conditions on c1, c2. Theorem 6.5. Suppose that conditions (19), (10) hold, so that there exists a D– pullback attractor Ain Cγ. Assume that for some µ > 1 Zt 0 (c1(θτω)µ+c2(θτω)µ)dτ ≤C(t)<∞, for all ω∈Ω,(25) and also that the radius of the absorbing set R(ω)is uniformly bounded on ω∈Ω. Then: 22 T. CARABALLO, M.J. GARRIDO-ATIENZA, B. SCHMALFUSS, & J. VALERO 1. ∪ω∈ΩA(ω)is pre–compact in Cγ. 2. The map ω→A(ω)is upper–semicontinuous. 3. The map ω→A(ω)is Fmeasurable in Cγ. Proof. Let {ξn:n∈N}be a sequence in ∪ω∈ΩA(ω). We know from the negative invariance of Aand A⊂Bthat there exists xn 0∈A(θ−tnωn)⊂B(θ−tnωn) and ξn= yn tn∈U(tn, θ−tnωn, xn 0), where yn τis the corresponding mild solution. According to Lemma 5.2 and the assumption on R(ω) we have that kyn τkγ≤R(θ−tn+τωn)≤R. Now put yn τ=zn τ+un τ, where un(tn+s) = Ztn+s 0 S(tn+s−τ)f(θ−tn+τωn, yn τ)dτ, s ∈[−tn,0]. With this ansatz, repeating the same arguments of Theorem 6.1 (but using (25) instead of (20)) and Lemma 5.4 we obtain that {ξn:n∈N}contains a convergent subsequence. Denote K=∪ω∈ΩA(ω)Cγ. Further, for ωn→ωwe have distCγ(A(ωn), A (ω)) ≤distCγ(U(t, θ−tωn, A (ωn)) , A (ω)) ≤distCγ(U(t, θ−tωn, K), U (t, θ−tω, K)) + distCγ(U(t, θ−tω, K), A (ω)) . Let ε > 0. By the pullback attraction property there exists T(ε, K, ω) such that distCγ(U(t, θ−tω, K), A (ω)) < ε, if t≥T. Fix such a t. Then distCγ(U(t, θ−tωn, K), U (t, θ−tω, K)) →0 as n→ ∞. In other case there would exist δ > 0 and xn∈K, such that distCγ(U(t, θ−tωn, xn), U (t, θ−tω, K)) > δ for all n. Noting that xn→x0∈K(up to a subsequence), the upper semicontinuity of (ω, x)→U(t, ω, x) (see the proof of Theorem 6.1 and (25)) implies that distCγ(U(t, θ−tωn, xn), U (t, θ−tω, K)) ≤distCγ(U(t, θ−tωn, xn), U (t, θ−tω, x0)) →0 as n→ ∞, which is a contradiction. Hence, distCγ(A(ωn), A (ω)) →0 as n→ ∞. Since ω→A(ω) has compact values, the upper semicontinuity follows by Remark 1. The measurability then follows from Lemma 2.5. 7. Examples. To illustrate our theory we consider the following situation. Assume O ⊂ Rnis a bounded open set with smooth boundary. Let us set H=L2(O),denote by || · || the norm in H , and let −Abe the Laplace operator ∆ with homogeneous Dirichlet conditions. It is worth remembering that D(A) = {u∈H1 0(O) : Au ∈L2(O)}, α= inf{||∇u||L2(O):u∈H1 0(O),||u|| = 1}. Here H1 0(O) is the Sobolev space of functions in Hwith generalized derivatives in Hwhich are zero on the boundary ∂O. Then Ais the generator of a C0contraction semigroup (S(t))t≥0on Hthat actually satisfies ||S(t)ψ|| ≤ e−αt||ψ||,for every t≥0 and ψ∈H, being α > 0 the first eigenvalue of Ain H1 0(O).Moreover, S(t), t > 0,are compact operators on H. ATTRACTORS FOR DELAY RANDOM PDES WITHOUT UNIQUENESS 23 Assume that Ω is a Polish space on which the mappings θtare continuous and let Fbe the Borel–σ–algebra of Ω. 7.1. Example 1. Let l: Ω×R→Rbe such that for a fixed real number a∈Rthe mapping l(·, a) is measurable, and for fixed ω∈Ω,the mapping a∈R7−→l(ω, a)∈ Ris continuous. Suppose also that there exist two non-negative tempered random variables ec1,ec2(w.r.t. F) and that ec1is such that t7−→ ec1(θtω) is integrable with respect to every finite interval (t1, t2) and ω∈Ω. In addition suppose that |l(ω, a)| ≤ ec1(ω) + ec2(ω)|a|, ω ∈Ω, a ∈R, where |·|denotes the absolute value in R. Let %: Ω →R−be a random variable (w.r.t. F) in general unbounded, γ > α and consider f: Ω ×Cγ→Hgiven by f(ω, ξ)(x) = l(ω, ξ(%(ω))(x)),for ω∈Ω, ξ ∈Cγand x∈ O. It is easy to see that the function fis well-defined. Moreover, ||f(ω, ξ)|| =ZO |f(ω, ξ)(x)|2dx1/2 ≤ZO(1 + δ)ec1(ω)2+ (1 + δ−1)ec2(ω)2|ξ(%(ω))(x)|2dx1/2 ≤(1 + δ)1/2ec1(ω)|O|1/2+ (1 + δ−1)1/2ec2(ω)e−γ%(ω)eγ%(ω)||ξ(%(ω))|| ≤(1 + δ)1/2ec1(ω)|O|1/2+ (1 + δ−1)1/2ec2(ω)e−γ%(ω)||ξ||γ, where δ > 0 and |O| denotes the Lebesgue measure of O. Let us define the Nemitskii operator J: Ω×H→Hby J(ω, y) (x) = l(ω, y (x)), for x∈ O. Lemma 7.1. ω→J(ω, y)is measurable for all y∈Hand y→J(ω, y)is continuous for all ω∈Ω. Proof. Take first a constant function y(x)≡u∈R. The map ω7→ l(ω, u) is measurable by assumption. Define the map G: Ω →Hby G(ω) (x) = l(ω, u), for all x∈ O. We claim that G(ω) is measurable. Indeed, for any v∈Hwe have (G(ω), v) = ZO l(ω, u)v(x)dx =l(ω, u)ZO v(x)dx =l(ω, u)v0. Since the last map is measurable and the space His separable, G(ω) is a measurable map by Pettis’ theorem (see [32, p.131]). The equality G(ω) = J(ω, y) is obvious. Further, let ybe a step function, that is, y(x) =      u1,if x∈ O1, . . . um,if x∈ Om, where O=∪m i=1Oiand Oi∩ Oj=∅for i6=j. For each uiwe can take the measurable map ω→l(ω, ui). Define the map G: Ω →Hby G(ω) (x) =      l(ω, u1),if x∈ O1, . . . l(ω, um),if x∈ Om. 24 T. CARABALLO, M.J. GARRIDO-ATIENZA, B. SCHMALFUSS, & J. VALERO We claim that G(ω) is measurable. Indeed, for any v∈Hwe have (G(ω), v) = m X i=1 ZOi l(ω, ui)v(x)dx = m X i=1 l(ω, ui)vi, so that G(ω) is measurable and again the equality G(ω) = J(ω, y) is obvious. Finally, take a sequence of step functions ynconverging to yin H. Passing to a subsequence it holds: yn(x)→y(x) for a.e. x∈ O,(26) there exists h∈Hsuch that |yn(x)| ≤ h(x) for a.e. x∈ O,(27) (see Brezis [5], Th. IV.9). We know that the maps Gn(ω) = J(ω, yn) are measurable. It is clear from (26)-(27) that (passing to a subsequence) for all ω∈Ω J(ω, yn) (x) = l(ω, yn(x)) →l(ω, y (x)) = J(ω, y) (x) , for a.e. x∈ O, and |l(ω, yn(x))| ≤ ec1(ω) + ec2(ω)|h(x)|,for a.e. x∈ O. Thus, it follows by Lebesgue’s theorem that J(ω, yn)→J(ω, y) in H, for every ω∈Ω. Then ω→J(ω, y) is measurable. In a similar way as in the previous lines one can prove that if yn→yin H, then J(ω, yn)→J(ω, y) in H. From this property the continuity of y→J(ω, y) follows. Further, the map ξ→f(ω, ξ) is continuous. Indeed, suppose that ξn→ξin Cγ. Note then that ξn(%(ω)) →ξ(%(ω)) in H, so that arguing as in the proof of Lemma 7.1 we obtain f(ω, ξn)→f(ω, ξ), and then the continuity follows. Since Jis a Caratheodory map and f(ω, ξ) = J(ω, ξ (%(ω))), it follows from [3, Lemma 8.2.3] that ω→f(ω, ξ) is measurable. Let us define c1(ω) := (1 + δ)1/2|O|1/2ec1(ω), c2(ω) := (1 + δ−1)1/2e−γ%(ω)ec2(ω), and assume that Ee−γ%(ω)ec2(ω)<α (1 + δ−1)1/2. Then, it is straightforward to check that all the assumptions on fin Sections 4 and 5 are fulfilled. The only condition we need to check is the temperedness of c2: lim t→±∞ log+e−γ%(θtω)ec2(θtω) |t|≤γlim t→±∞ −%(θtω) |t|+ lim t→±∞ log+(ec2(θtω)) |t| =γlim t→±∞ −%(θtω) |t|. The last term is equal to zero if −%has a sublinear growth θ–almost surely, for which, as a consequence of the Birkhoff ergodic theorem, a sufficient condition is that Esup t∈[0,1] (−%(θtω)) <∞ (see Arnold [1], Page 165). ATTRACTORS FOR DELAY RANDOM PDES WITHOUT UNIQUENESS 25 As a particular case, we can choose for %the negative absolute value of a one– dimensional stationary Ornstein-Uhlenbeck process, that is, the unique stationary solution z∗of the stochastic differential equation dz =−zdt +dW, since z∗has continuous trajectories and satisfies lim t→±∞ |z∗(θtω)| |t|= 0, which follows by the Burkholder inequality (see Caraballo et al. [9]). Notice that the analysis we have just done ensures the existence of a pullback attractor Afor the corresponding equation (5). We are now interested in checking that Ais also a random attractor. Let then the maps (ω, a)→l(ω, a), ω→%(ω) be continuous and let the maps ec1,ec2be locally bounded. We can prove that the map Ω ×Cγ3(ω, ξ)7→ f(ω, ξ)∈His continuous. Suppose ωn→ω,ξn→ξin Cγ. Note then that ξn(%(ωn)) →ξ(%(ω)) in H, so that arguing as in the proof of Lemma 7.1 we obtain that (up to a subsequence) f(ωn, ξn)(x) = l(ωn, ξn(%(ωn))(x)) →l(ω, ξ(%(ω))(x)) = f(ω, ξ)(x),for a.e. x∈ O, and |f(ωn, ξn)(x)| ≤ ec1(ωn) + ec2(ωn)|h(x)| ≤ e C1(ω) + e C2(ω)|h(x)|, for some h∈H, which is a majorant for f(ωn, ξn). Thanks to Lebesgue’s majorant theorem we obtain f(ωn, ξn)→f(ω, ξ), and then we obtain the required continuity, so that condition (19). Next we will prove that under suitable assumptions we can apply Theorem 6.1 to have an MRDS. Assume that eci(ω) are uniformly bounded in the following sense: ec1(ω)≤C1,ec2(ω)e−γ%(ω)≤C2,for all ω∈Ω, with C2<α (1 + δ−1)1/2. Then, it follows immediately that ec1(ω),ec2(ω) satisfy (20). Thus for any t0∈R and ω0∈Ω there exists a neighborhood V(t0, ω0) such that Zt 0 (c1(θτω))µ+ (c2(θτω))µ)dτ =Zt 0 ((1 + δ)1/2|O|1/2ec1(θτω))µ+ ((1 + δ−1)1/2e−γ%(θτω)ec2(θτω))µ)dτ ≤C(t0, ω0), for any (t, ω)∈V. Therefore, we have an MRDS. To prove the measurability of the pullback attractor we apply Theorem 6.5. Indeed, thanks to our assumptions on eci(ω), condition (25) holds and R(ω) = 2(1 + δ)1/2|O|1/2Z0 −∞ eατ+(1+δ−1)1/2R0 τe−γ%(θrω)˜c2(θrω)drec1(θτω)dτ ≤2(1 + δ)1/2|O|1/2C1Z0 −∞ e(α−(1+δ−1)1/2C2)τdτ =2(1 + δ)1/2|O|1/2C1 α−(1 + δ−1)1/2C2 ,(28) so that the radius R(ω) is uniformly bounded.