Pullback Attractors for Bi-Spatial Continuous Random Dynamical Systems and Application to Stochastic Fractional Power Dissipative Equation on an Unbounded Domain

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DISCRETE AND CONTINUOUS doi:10.3934/dcdsb.2018326 DYNAMICAL SYSTEMS SERIES B Volume 24, Number 7, July 2019 pp. 3395–3438

PULLBACK ATTRACTORS FOR BI-SPATIAL CONTINUOUS RANDOM DYNAMICAL SYSTEMS AND APPLICATION TO STOCHASTIC FRACTIONAL POWER DISSIPATIVE EQUATION ON AN UNBOUNDED DOMAIN

Wenqiang Zhao School of Mathematics and Statistics Chongqing Technology and Business University Chongqing 400067, China (Communicated by Bj¨ornSchmalfuß)

Abstract. In this article, a notion of bi-spatial continuous random dynamical system is introduced between two completely separable metric spaces. It is show that roughly speaking, if such a random dynamical system is asymptotically compact and random absorbing in the initial space, then it admits a bi- spatial pullback attractor which is measurable in two spaces. The measurability of pullback attractor in the regular spaces is completely solved theoretically. As applications, we study the dynamical behaviour of solutions of the non- N autonomous stochastic fractional power dissipative equation on R with additive white noise and a polynomial-like growth nonlinearity of order p, p ≥ 2. We 2 N s N p N prove that this equation generates a bi-spatial (L (R ),H (R ) ∩ L (R ))- continuous random dynamical system, and the random dynamics for this system is captured by a bi-spatial pullback attractor which is compact and at- s N p N s N tracting in H (R ) ∩ L (R ), where H (R ) is a fractional Sobolev space with s ∈ (0, 1). Especially, the measurability of pullback attractor is individu- s N p N ally derived by proving the the continuity of solutions in H (R ) and L (R ) with respect to the sample. A diﬀerence estimates approach, rather than the usual truncation estimate and spectral decomposition technique, is employed s N p N to overcome the loss of Sobolev compact embedding in H (R ) ∩ L (R ), s ∈ (0, 1),N ≥ 1.

1. Introduction. This article is concerned with the random dynamics of solutions of the non-autonomous stochastic dissipative equation involving the fractional power Laplacian on the whole space RN du + λu + (−∆)su + f(t, x, u) = g(t, x) + φ(x)W˙ (t), t > τ, (1) dt 2 2 N where s ∈ (0, 1), λ > 0, τ ∈ R, g ∈ Lloc(R,L (R )), W (t) is a two-sided real-valued Brownian motion deﬁned on a probability space (Ω, F,P ) speciﬁed later, and φ is a given function on RN . The equation is understood in the sense of Stratonovich integration.

2010 Mathematics Subject Classiﬁcation. Primary: 35R60, 35B40, 35B41; Secondary: 35B65. Key words and phrases. Bi-spatial continuous random dynamical system, fractional power dissipative equation, fractional Sobolev space, pullback attractor, measurability. The author is supported by NSF grant 11601046, CTBU Grant 1751041 and Chongqing key laboratory of social economy and applied statistics.

3395 3396 WENQIANG ZHAO

The operator (−∆)s with s ∈ (0, 1) is called a fractional power Laplacian, whose limit as s ↑ 1 is the classic Laplacian ∆, see [17]. It is a nonlocal generalization of the classic Laplacian that is often used to model the diffusive processes, see [17, 27, 40] and the references therein. As for the long-time dynamics, Hu et al [32] discussed the existence of a random attractor in L2(RN ) for the fractional power dissipative stochastic equation with additive noises. Wang [46] discussed the existence and uniqueness of random attractor for the same equation with multiplicative noise on bounded domain. The dynamics of 3D Ginzburg-Landau equation involving fractional power Laplacian ware studied in [33, 34]. Most recently, Gu et al [21] obtained the regularity of the random attractor for problem (1) in Hs(RN ) in the case of multiplicative noise, where the authors employed the tail estimate and spectral decomposition technique to overcome the loss of compactness on unbounded domains. But little is known about the continuity and measurability of its solutions in Hs(RN ) and Lp(RN ) for s ∈ (0, 1) and p ≥ 2. In this paper, we prove the existence, regularity and measurability of pullback attractors for the stochastic fractional power dissipative equation (1) defined on the whole space RN . Most of the stochastic partial differential equations (SPDEs) present a regular solution, namely, if the initial data belongs to a separable Banach space, called an initial space, the corresponding solution may enter into another space, called a target space or regular space in the terminology as [31]. Thus the random dynamical system always possesses a bi-spatial characteristic property. In general, the continuity and measurability of a solution of an SPDE in the initial space are easy to check, and therefore to derive a (measurable) random attractor in such a space we only need to look for the compact condition and the absorption property, which has been extensively studied ever since [14, 15, 37] first introduced the concept of random attractor. For example, we may refer to [8,9, 10, 19] for the autonomous case and [6, 43, 44, 45, 48] for the non-autonomous case. However, it is challenging to prove the continuity and measurability of solution of an SPDE in the corresponding regular spaces, see the statements in the introductions of the literature [7,8, 31]. Therefore, the measurability of a random attractor in the regular space is not explicitly studied in the literature, although the existence of bi-spatial random attractor has been obtained for a large volume of stochastic partial equations (cf. [29, 30, 31, 41, 49, 50, 53, 54, 57]). Firstly, we introduce a new notion of a bi-spatial continuous random dynamical system (briefly, bi-spatial continuous RDS) between two completely separable metric spaces X and Y , and establish the existence criterion of its bi-spatial (random) pullback attractor. By definition, a bi-spatial continuous RDS is continuous from X to X, and in addition is continuous from X to Y with respect to the initial data. It is showed that roughly speaking, if such a random dynamical system is asymptotically compact and absorbing in the initial space X, then it admits automatically a bi- spatial “true random” pullback attractor. The expression “true random” means that this bi-spatial pullback attractor (as a set-valued mapping) is not only compact and attracting but also measurable in these spaces, which is different from the parallel concept of bi-spatial random attractor developed in [31, 56], where the measurability of attractors in the regular space Y is unknown. In a sense, the au- tomatical presentation of a bi-spatial “true random” pullback attractor for such a RDS is attractive and interesting. In other word, the measurability of pullback attractor in the regular space is completely solved without requesting any embedding relations between X and Y . PULLBACK ATTRACTORS FOR BI-SPATIAL SYSTEMS 3397

Secondly, we derive the continuity of solutions to problem (1) in the regular spaces Lp(RN ) and Hs(RN ), respectively, when the initial datum is given in L2(RN ), where Hs(RN ) is a fractional Sobolev space with s ∈ (0, 1). By means of the equation satisfied by the difference of two solutions at the common initial time, and using some technical iteration, we obtain that the difference (t − τ)¯u(t) in Lp(RN ) is not larger than the difference of the initial data which belongs to L2(RN ), for any 2 2 N t > τ and the non-autonomous force g ∈ Lloc(R,L (R )). On the other hand, we overcome the difficulty caused by the fractional power Laplacian to obtain a semi- 2p−2 2p−2 N intervals estimate of the bound of the solution in Lloc (R; L (R )), and then 2p t+τ p−1 an appropriate multiplier (r − 2 ) is chosen to prove the continuity of solutions in Hs(RN ). This approach was used to prove the asymptotical compactness of solutions of reaction-diffusion equation, p-Laplacian equation and Fitzhugh-Nagumo system on bounded or unbounded domain, see [7, 53, 54, 55, 59]. Thirdly, we consider the measurability of solutions. It is well known that the existence theorem of pullback attractor for the deterministic dynamical system can be applied to analyse the long-time dynamics of random dynamical system with some modifications. However, in the case of random (stochastic) differential equations, we need additional measurability properties to assure that the solutions are well-posed [8]. This leads to an additional and great difference in comparison with the non-random case. We here prove the measurability of solution as a single valued mapping from Ω to Lp(RN ) ∩ Hs(RN ). This is achieved by proving the continuity of solution with respect to the sample ω ∈ Ωk, where Ωk, k ∈ N is a countable decomposition of the sample sapce Ω. Note that Ωk is a Polish space, possessing some nice analytical properties to serve us. In particular, the properties of the Ornstein- Uhlenbeck process over Ωk are substantially employed. The reader is referred to [8,9, 10, 47] for the idea to restrict Ω to Ω k. Finally, as direct results of our theoretic criterion of section 2, we derive the existence of a measurable pullback attractor (as a set-valued mapping) in Lp(RN ) and Hs(RN ), p ≥ 2 and s ∈ (0, 1), see Theorem 7.1 and Theorem 7.2. The asymptotical compactness of solution in the initial space L2(RN ) is proved by estimate of the far-field values of solutions and some locally compact embedding on bounded domain. In comparison with the results recently published in the literature, the novelty and difficulties of this work are in three aspects: (i) We study the dynamics by proving of the bi-spatial continuity of solution with respect to the initial data to obtain the asymptotical compactness, which is substantially different from [21, 57] by tails estimate and spectral approaches. (ii) We give a detailed analysis of the measurability of solution and the measurability of pullback attractor in the regular spaces without embedding relation for the considered spaces, e.g.Lp(RN ) and L2(RN ) for p > 2, with a special interest in the fractional Sobolev space Hs(RN ), s ∈ (0, 1). (iii) In the case of additive noise, the structure of the equation is varied and some new term emerges. With these problems, some new estimates involving the fractional power Laplacian and the nonlinear function are developed here. It is also pointed out that the approaches utilized in this paper are applicable for a variety of stochastic differential equations, such as stochastic fractional Burgers equation, stochastic fractional Ginzburg-Landau equation and stochastic fractional p-Laplacian equation and etc. 3398 WENQIANG ZHAO

This article is arranged as follows. In section 2, we introduce the concepts of bi-spatial continuous RDS and bi-spatial pullback attractor. A criterion on the existence of bi-spatial pullback attractor for the bi-spatial continuous RDS is demon- strated. In section 3, we first introduce the fractional power Laplacian and its related fractional power Sobolev space. Then we transform the stochastic equation into a deterministic one by the well-known O-U transformation. A random cocycle through the solution of problem (1) is defined, especially the domain of attraction is given. In section 4, we prove the existence of random absorbing set and the asymptotical compactness in L2(RN ). In section 5, we prove the the bi-spatial continuity of solutions in the p-times integrable functions spaces Lp(RN ) and the fractional Sobolev spaces Hs(RN ) with respect to the initial data belonging to L2(RN ), for any s ∈ (0, 1) and p ≥ 2. In section 6, we prove the continuity and measurability of solutions in the spaces Lp(RN ) and Hs(RN ) with respect to the sample ω. The final section is concerning the existence of bi-spatial (measurable) pullback attractor in the spaces Lp(RN ) and Hs(RN ).

2. Abstract results on the bi-spatial continuous RDS. In this section, we present some concepts related to the bi-spatial RDS, and prove a theoretic results on the existence of a bi-spatial pullback attractor for a bi-spatial continuous RDS. For a comprehensive survey of the theory for the RDS, the readers may refer to [1, 13] and the deterministic dynamical system to [11, 36, 42].

2.1. Preliminaries. We introduce two Polish spaces (completely separable metric space) (X, dX ) and (Y, dY ) with the σ-algebras of Borel subsets B(X) and B(Y ), respectively, where X serves as the initial space, and Y as the target space or regular space, whose intersection is nontrivial. We assume that each of them is continuously imbedded in a Hausdorff topological vector space X . This implies that for every sequence xn ∈ X ∩ Y, n = 1, 2, ..., such that xn → x in X and xn → y in Y respectively, then we have x = y ∈ X ∩ Y . For example, the usual Lebesgue spaces Lp(RN )(p > 2) and L2(RN ) are continuously embedded into the separately topological space ℘0(RN ) (the space of all continuous linear functionals on the space ∞ N C0 (R ) equipped with a particular topology as in [51, p28]). Thus it makes sense to consider the common topology in these targeted spaces, see [56]. Let (Ω, F,P ) be a probability space on which there is a (B(R)×F, F)-measurable mapping ϑt : R × Ω → Ω such that ϑ0 = I, ϑt+τ = ϑt ◦ ϑτ and ϑtP = P for all t, τ ∈ R. In particular, the mapping ϑt, t ∈ R is called a driving system on Ω, whereas the sample space Ω of the probability space (Ω, F,P ) is a called a base space of the driving system ϑt, see [16]. The quadruple (Ω, F,P, {ϑt}t∈R) is called a metric (or measurable) dynamical system (briefly, MDS ϑ), which is of importance to deal with the white noises. Let R+ = {t ∈ R : t ≥ 0} and 2X be the collection of all subsets of X. Definition 2.1. A family of single-valued mappings ϕ : R+ × R × Ω × X → X, (t, τ, ω, x) 7→ ϕ(t, q, ω, x) is called a random cocycle on X over an MDS ϑ if for all s, t ∈ R+, τ ∈ R and ω ∈ Ω, the following statements are satisfied: • ϕ(., τ, ., .): R+ × Ω × X → X is (B(R+) × F × B(X), B(X))-measurable; • ϕ(0, τ, ω, .) is the identity on X; • ϕ(t + s, τ, ω, .) = ϕ(t, τ + s, ϑsω, ϕ(s, τ, ω, .)). A random cocycle ϕ on X is said to take its values into the target space Y if ϕ(t, τ, ω, .) maps X into Y for every t > 0, τ ∈ R and ω ∈ Ω. PULLBACK ATTRACTORS FOR BI-SPATIAL SYSTEMS 3399

Definition 2.2. Let ϕ be a random cocycle on X over an MDS ϑ, taking its values into Y . A random cocycle ϕ is said to be (X,X)-continuous (briefly, continuous in X) if the mapping ϕ(t, τ, ω, .): X → X is continuous for each t ∈ R+, τ ∈ R and ω ∈ Ω. A random cocycle ϕ is said to be (X,Y )-continuous if ϕ is (X,X)- continuous, and in addition the mapping ϕ(t, τ, ω, .): X → Y is continuous for each t > 0, τ ∈ R and ω ∈ Ω. The universe of sets in the initial space X is denoted by D, which is a collection of some families of nonempty subsets of the initial space X, parameterized by the initial time τ and the sample ω such that D = {D = {∅= 6 D(τ, ω) ∈ 2X : τ ∈ R, ω ∈ Ω}}. The universe D will serve as the domain of attraction (in the terminology in [38]). In practice, the element of D has particular growth property t 7→ D(τ − t, ϑ−tω) for each fixed τ ∈ R and ω ∈ Ω. The universe D is called inclusion closed if for every D = {D(τ, ω): τ ∈ R, ω ∈ Ω} ∈ D with every D0(τ, ω) ⊂ D(τ, ω) for each τ ∈ R and ω ∈ Ω, then the family = D0 = {D0(τ, ω): τ ∈ R, ω ∈ Ω} ∈ D. Note that the inclusion closed universe D is also called for short an IC system which was first introduced in [2, 19, 38]. In this paper, it suffices to define the universe D in the initial space X, not necessary in the target space Y

Definition 2.3. Let D : R × Ω → 2X \∅; D :(τ, ω) 7→ D(τ, ω) ∈ 2X be a set-valued mapping. We say D :(τ, ω) 7→ D(τ, ω) is measurable with respect to F (briefly, measurable) in X if for every fixed x ∈ X and τ ∈ R, the mapping

ω → inf dX (x, z) z∈D(τ,ω) is (F, B(R))-measurable. If the set-valued mapping D is measurable, then the family of its images D = {D(τ, ω): τ ∈ R, ω ∈ Ω} is also called a random set. If for every fixed τ ∈ R and ω ∈ Ω, the image D(τ, ω) is closed (resp. compact) in X, then the family D = {D(τ, ω): τ ∈ R, ω ∈ Ω} is called a closed (resp. compact) random set in X. It is well known that a set-valued mapping D is measurable X if and only if for every open subset O ⊂ X, the inverse image {ω : D(τ, ω) ∩ O 6= ∅} ∈ F for every fixed τ ∈ R, see [3, 12, 22]. Recall that a single-valued mapping f :Ω → X is called measurable (precisely, (F, B(X))-measurable) if for every open subset O ⊂ X, the inverse image f −1(O) ∈ F (or equivalently for every closed set C ⊂ X, f −1(C) ∈ F) (cf [3, p307]). Definition 2.4. Let D be a collection of some families of nonempty subsets of the initial space X. K = {K(τ, ω): τ ∈ R, ω ∈ Ω} ∈ D is called a D-pullback absorbing set in X if for every τ ∈ R, ω ∈ Ω and D ∈ D , there exists an absorbing time T = T (τ, ω, D) > 0 such that

ϕ(t, τ − t, ϑ−tω, D(τ − t, ϑ−tω)) ⊆ K(τ, ω) for all t ≥ T. Furthermore, if K is a random set, the family K is called a D-pullback random absorbing set in X. Deﬁnition 2.5. Let D be a collection of some families of nonempty subsets of the initial space X. Let ϕ be a random cocycle on X over an MDS ϑ. (i) A family of sets A = {A(τ, ω): τ ∈ R, ω ∈ Ω} ∈ D is called a (X,X)-pullback attractor for ϕ if the next three statements hold: •A is a random set in X and A(τ, ω) is compact in X for every τ ∈ R and ω ∈ Ω; 3400 WENQIANG ZHAO

•A is invariant, that is, for every τ ∈ R and ω ∈ Ω, ϕ(t, τ, ω, A(τ, ω)) = A(τ + t, ϑtω), ∀ t ≥ 0; •A is attracting in X, namely, for every τ ∈ R and ω ∈ Ω and D ∈ D,

lim distX (ϕ(t, τ − t, ϑ−tω, D(τ − t, ϑ−tω)), A(τ, ω)) = 0. t→∞ (ii) Suppose further that ϕ takes its values into Y . Then a family of sets A = {A(τ, ω): τ ∈ R, ω ∈ Ω} ∈ D is called a (X,Y )-pullback attractor for ϕ if A is a (X,X)-pullback attractor, and in addition there hold •A is a random set in Y , A(τ, ω) is compact in Y for every τ ∈ R and ω ∈ Ω; •A is attracting in Y , namely, for every τ ∈ R and ω ∈ Ω and D ∈ D,

lim distY (ϕ(t, τ − t, ϑ−tω, D(τ − t, ϑ−tω)), A(τ, ω)) = 0, t→∞ Y where distY is the Hausdorff semi-metric in 2 with distY (A, B) = supx∈A infy∈B dY (x, y). The concept of attractors in different spaces was first introduced by Babin and Bishik [4, p119]. However, here we emphasize that the bi-spatial (X,Y )-attractor is a pullback attractor in both X and Y , which is stronger than the one in [4]. Moreover in the random setting, just as the above definition, the measurability in the regular space Y is included, which is also largely different from the bi-spatial notion in [31, 49, 50, 52]. We also note that the attracting in Lemma 2.5 is called pullback attraction, which implies the forward attraction:

lim distY (ϕ(t, τ, ω, D(τ, ω)), A(τ + t, ϑtω)) = 0 in probability. (2) t→∞ As for the attractor with forward attraction of point-wise convergence is considered by Kloeden [16, 23, 24, 25]. However, the considered convergence in this paper is in the sense of pullback attraction, which is stronger than the forward attraction in probability, defined by (2). Definition 2.6. Let D be a collection of some families of nonempty subsets of the initial space X. Let ϕ be a random cocycle ϕ on X over an MDS ϑ. (i) The random cocycle ϕ is said to be (X,X)-pullback asymptotically compact if for every τ ∈ R, ω ∈ Ω,D ∈ D and whenever tn → ∞, xn ∈ D(τ − tn, ϑ−tn ω), the sequence ∞ {ϕ(tn, τ − tn, ϑ−tn ω, xn)}n=1 is precompact in X. (ii) Suppose further that ϕ takes its values into Y . Then ϕ is called (X,Y )- pullback asymptotically compact if ϕ is (X,X)-pullback asymptotically compact and in addition the sequence ∞ {ϕ(tn, τ − tn, ϑ−tn ω, xn)}n=1 is precompact in Y, for every τ ∈ R, ω ∈ Ω, D ∈ D and whenever tn → ∞ and xn ∈ D(τ − tn, ϑ−tn ω). 2.2. Pullback attractors for the random dynamical systems with the bi- spatial continuity. The existence theorem of bi-spatial random attractor seems to be established in [31] for the autonomous case and [52] for the non-autonomous case. However, the measurability of a bi-spatial random attractor is assumed to hold on the initial space X, not considering the measurability in the regular space Y . In fact, the measurability of a bi-spatial random attractor can be proved in the regular space Y if we impose some measurability of the cocycle on ω and the continuity of cocycle in Y with respect to the initial values belonging to X. We will PULLBACK ATTRACTORS FOR BI-SPATIAL SYSTEMS 3401 prove this in this subsection. We fist give an interesting lemma which claims that the bi-spatial continuity implies the bi-spatial asymptotical compactness. Lemma 2.7. Let ϕ be a random cocycle on X over an MDS ϑ, taking its values into Y . Suppose that ϕ is (X,X)-pullback asymptotically compact, and in addition ϕ is (X,Y )-continuous. Then ϕ is (X,Y )-pullback asymptotically compact.

Proof. Given τ ∈ R and ω ∈ Ω, let {ϕ(tn, τ − tn, ϑ−tn ω, un)}n∈N be a sequence and D ∈ D. Since ϕ is (X,X)-pullback asymptotically compact, then there exists a point u0 ∈ X such that, up to a subsequence,

ϕ(tn − 1, τ − tn, ϑ−tn ω, un) → u0 in X, (3) whenever tn → ∞ and un ∈ D(τ − tn, ϑ−tn ω). By the (X,Y )-continuity of ϕ, we know that ϕ(t, τ, ω, .): X → Y is continuous for every t > 0, τ ∈ R and ω ∈ Ω. Therefore by (3), we deduce that

ϕ(1, τ − 1, ϑ−1ω, ϕ(tn − 1, τ − tn, ϑ−tn ω, un)) → ϕ(1, τ − 1, ϑ−τ ω, u0) in Y. Then the ﬁnal proof is as a result of the cocycle property of ϕ. The following lemma is concerning the properties for a random set, which will be needed in the following arguments. See [8, 12, 22]. Lemma 2.8. (i) For any random set E = {E(ω), ω ∈ Ω} ⊂ Y with nonempty ∞ closed images, there exist a sequence {fn(ω)}n=1 of (F, B(Y ))-measurable variables such that fn(ω) ∈ E(ω) for every ω ∈ Ω, and moreover, Y E(ω) = ∪n∈N{fn(ω)} . Y (ii) Let En :Ω → 2 be a random set with nonempty closed images with n ∈ N. Then Y Ω 3 ω → ∪n∈NEn(ω) is measurable in Y with nonempty closed images. Y (iii) Let En :Ω → 2 be a random set with nonempty closed image with n ∈ N. ∞ Suppose that for each ﬁxed ω ∈ Ω, every sequence {yn}n=1 with yn ∈ En(ω) has a ∞ convergence subsequence in Y , and in addition {En}n=1 is a decreasing sequence. Then ∞ Ω 3 ω → ∩n=1En(ω) is measurable in Y with nonempty closed images. The following lemma is adapted from Lemma 8.2.3 in [3]. Lemma 2.9. ([3]) Let the mapping Ψ:Ω × X → Y . Suppose that for every x, Ψ(., x) is (F, B(Y ))-measurable and for every ω ∈ Ω, Ψ(ω, .) is continuous. If χ : Ω → X is (F, B(X))-measurable, then the mapping ω → Φ(ω, χ(ω)) is (F, B(Y ))- measurable. We are now at the point to present a new result on the existence of a bi-spatial pullback attractor for a random dynamical system with the bi-spatial continuity. Note that we do not impose any inclusion relations between X and Y . Theorem 2.10. Let ϕ be a random cocycle on X (over an MDS ϑ) which is (X,Y )- continuous, and D be inclusion closed universe in X. Suppose that (i) ϕ has a closed D-pullback random absorbing set K = {K(τ, ω): τ ∈ R, ω ∈ Ω} ∈ D in X; 3402 WENQIANG ZHAO

(ii) ϕ is (X,X)-pullback asymptotically compact in X; (iii) For every ﬁxed t > 0, τ ∈ R, x ∈ X, the mapping ϕ(t, τ, ., x):Ω → Y is (F, B(Y ))-measurable. Then the random cocycle ϕ possesses a unique (X,Y )-pullback attractor A = {A(τ, ω): τ ∈ R, ω ∈ Ω} ∈ D, where X A(τ, ω) = ∩s>0∪t≥sϕ(t, τ − t, ϑ−tω, K(τ − t, ϑ−tω)) . Furthermore, it can also be structured by the Y -metric, namely, Y A(τ, ω) = ∩s>0∪t≥sϕ(t, τ − t, ϑ−tω, K(τ − t, ϑ−tω)) . Proof. By (i) and (ii), it follows that ϕ admits a (X,X)-pullback attractor A = {A(τ, ω): τ ∈ R, ω ∈ Ω} ∈ D in X, where

X A(τ, ω) = ∩s>0∪t≥sϕ(t, τ − t, ϑ−tω, K(τ − t, ϑ−tω)) . For fixed τ ∈ R, the measurability of A(τ, .) in X is proved by Wang [44]. We here mainly consider the measurability, compactness and attracting properties of A in the target space Y . By Lemma 2.7, ϕ is (X,Y )-asymptotical compact. Therefore by [52], for every τ ∈ R and ω ∈ Ω, the omega-limits A(τ, ω) of K, X A(τ, ω) = ∩s>0∪t≥sϕ(t, τ − t, ϑ−tω, K(τ − t, ϑ−tω)) ⊂ Y, (4) is an invariant subset which is compact, attracting in Y . Furthermore, by the (X,Y )-asymptotic compactness, A(τ, ω) can also be reformulated in Y -metric, namely, Y A(τ, ω) = ∩s>0∪t≥sϕ(t, τ − t, ϑ−tω, K(τ − t, ϑ−tω)) , (5) under the condition that X and Y are continuous embedding into a Hausdorff topological vector space X . It remains to prove the A is a random set. To this end, we first prove for every τ ∈ R and ω ∈ Ω, the right hand side of (5) can be reformulated as a discrete form, namely, Y A(τ, ω) = ∩n∈N∪m=nϕ(m, τ − m, ϑ−mω, K(τ − m, ϑ−mω)) . (6) Obviously, by (5), we have Y A(τ, ω) ⊇ ∩n∈N∪m=nϕ(m, τ − m, ϑ−mω, K(τ − m, ϑ−mω)) . (7) We prove the inverse inclusive relation. First, by the definition A(τ, ω) in (4) and the absorption of K, it is easy to see that A(τ, ω) ⊂ K(τ, ω) for every τ ∈ R and ω ∈ Ω. Therefore by the invariance of A, we have

A(τ, ω) = ϕ(m, τ − m, ϑ−mω, A(τ − m, ϑ−mω))

⊂ ϕ(m, τ − m, ϑ−mω, K(τ − m, ϑ−mω)), (8) for every ﬁxed m ∈ N. Note that A(τ, ω) ⊂ Y and ϕ(t, τ, ω, .) maps X into Y for all t > 0. Then we have Y A(τ, ω) = A(τ, ω) ⊂ ∩n∈N∪m=nϕ(m, τ − m, ϑ−mω, K(τ − m, ϑ−mω)) . (9) Thus we have proved (6). For a ﬁxed τ ∈ R and each n ∈ N, denote by Y En(ω) = ∪m=nϕ(m, τ − m, ϑ−mω, K(τ − m, ϑ−mω)) . (10) PULLBACK ATTRACTORS FOR BI-SPATIAL SYSTEMS 3403

By Lemma 2.8( iii), we need to prove that for each n ∈ N and τ, En = {En(ω): ω ∈ Ω} is a random set in Y with nonempty closed images. Since K is a random set with nonempty closed images, then by Lemma 2.8( i), there exists a countable ∞ ∞ family {fi(ω)}i=1 of (F, B(X))-measurable variables, such that {fi(ω)}i=1 is dense in K(τ − m, ϑ−mω) for each fixed m ∈ N and τ. On the other hand, by our assumption (iii) and the continuity of ϕ(t, τ, ω, .): X → Y , using Lemma 2.9, it follows that for every fixed m ∈ N and τ ∈ R, the mapping ω → ϕ(m, τ − m, ϑ−mω, fi(ω)) is (F, B(Y ))-measurable. Therefore by Lemma 2.8( ii), it follows that Y Ω 3 ω 7→ ∪i∈N{ϕ(m, τ − m, ϑ−mω, fi(ω))} , (11) is measurable in Y for fixed m ∈ N and τ ∈ R. We now claim that, for every fixed m ∈ N and τ ∈ R, Y ϕ(m, τ − m, ϑ−mω, K(τ − m, ϑ−mω)) = ∪i∈N{ϕ(m, τ − m, ϑ−mω, fi(ω))} . (12)

In fact, since K(τ − m, ϑ−mω) is closed and ϕ(t, τ, ω, .) continuous from X to Y , the left side hand is closed in Y . It is easy to say, for every ﬁxed m ∈ N, Y ϕ(m, τ − m, ϑ−mω, K(τ − m, ϑ−mω)) ⊇ ∪i∈N{ϕ(m, τ − m, ϑ−mω, fi(ω))} . (13)

For the inverse inclusion, let y ∈ ϕ(m, τ − m, ϑ−mω, K(τ − m, ϑ−mω)). Then there exists a point u ∈ K(τ − m, ϑ−mω) such that y = ϕ(m, τ − m, ϑ−mω, u). ∞ Note that {fi(ω)}i=1 is dense in K(τ − m, ϑ−mω). Therefore there is a sequence ∞ ∞ {fik (ω)}k=1 ⊂ {fi(ω)}i=1 with fik (ω) → u in X as k → ∞. By the the continuity of ϕ(t, τ, ω, .): X → Y , it follows that

ϕ(m, τ − m, ϑ−mω, fik (ω)) → ϕ(m, τ − m, ϑ−mω, u) in Y. That is to say, we ﬁnd a sequence

yk = ϕ(m, τ − m, ϑ−mω, fik (ω)) ∈ ∪i∈N{ϕ(m, τ − m, ϑ−mω, fi(ω))} Y with yk → y in Y . Therefore y ∈ ∪i∈N{ϕ(m, τ − m, ϑ−mω, fi(ω))} . Then the inverse inclusion of (13) is proved. Thus (12) holds true. By (12), we rewrite (10) as Y En(ω) = ∪m=nϕ(m, τ − m, ϑ−mω, K(τ − m, ϑ−mω)) Y Y = ∪m=n∪i∈N{ϕ(m, τ − m, ϑ−mω, fi(ω))} . (14) Thanks to (11) and using Lemma 2.8( ii) again, we know that Y Y Ω 3 ω 7→ ∪m=n∪i∈N{ϕ(m, τ − m, ϑ−mω, fi(ω))} (15) is measurable in Y for fixed τ ∈ R. Then by (14) and (15) we get Ω 3 ω 7→ En(ω) is measurable in Y for fixed τ ∈ R. On the other hand, for each n ∈ N, taking a yn ∈ En(ω), by Lemma 2.8( iii), ∞ we need to prove that {yn}n=1 has a convergence subsequence in Y . It seems that this can not be derived directly from the asymptotical compactness of ϕ in Y . We now give the detained proof. Indeed, for every fixed τ ∈ R, ω ∈ Ω, by (10), for each n n n ∈ , there exist 3 m ≥ n and z n ∈ K(τ − m , ϑ n ω) such that N N k mk k −mk n n d (ϕ(m , τ − m , ϑ n ω, z n ), y ) → 0, (16) Y k k −mk mk n 3404 WENQIANG ZHAO

∞ as k → ∞. We now choose a sequence {jn}n=1 ⊂ N by the following rules.

1 1 1 1 j = m ≥ 1 if d (ϕ(m , τ − m , ϑ 1 ω, z 1 ), y ) ≤ ; 1 k1 Y k1 k1 −m m 1 k1 k1 2 2 2 2 1 j = m ≥ max{j , 2} if d (ϕ(m , τ − m , ϑ 2 ω, z 2 ), y ) ≤ ; 2 k2 1 Y k2 k2 −m m 2 2 k2 k2 2 ......

n n n 1 jn = mk ≥ max{jn−1, n} if dY (ϕ(mk , τ − mk , ϑ−mn ω, zmn ), yn) ≤ ; n n n kn kn 2n ...... ,

∞ ∞ where {ki}i=1 is the subsequence of {k}k=1. Therefore we get a sequence {ϕ(jn, τ − j , ϑ ω, z )}∞ satisfying that j is increasing in n, z ∈ K(τ − z , ϑ ω) n −jn jn n=1 n jn jn −zjn and 1 d (ϕ(j , τ − j , ϑ ω, z ), y ) ≤ . (17) Y n n −jn jn n 2n since as n → ∞, we have jn → ∞, then by the (X,Y )-asymptotical compactness of ϕ, there exists a point y ∈ Y such that, up to a subsequence (denoted by itself for simplicity),

dY (ϕ(jn, τ − jn, ϑ−jn ω, zjn ), y) → 0 (18) as n → ∞. By the triangle inequality, along with (17) and (18) we have dY (yn, y) ≤ dY (ϕ(jn, τ − jn, ϑ−jn ω, zjn ), yn) + dY (ϕ(jn, τ − jn, ϑ−jn ω, zjn ), y) → 0 ∞ as n → ∞, and consequently the sequence {yn}n=1 converges in Y (thus En(ω) is compact in Y ). So by Lemma 2.8( iii) and (10) it follows that

Ω 3 ω 7→ A(τ, ω) = ∩n∈NEn(ω). (19) is measurable in Y for ﬁxed τ ∈ R, and therefore by Deﬁnition 2.3, the family A = {A(τ, ω): τ ∈ R, ω ∈ Ω} is a random set in Y . Since A ∈ D, then the uniqueness is proved by the fact that A attracts itself. The proof is completed.

3. Fractional power dissipative equation on RN with additive noise. In this section, we ﬁrst present the concepts of fractional power Laplacian and fractional Sobolev space. Then we give the condition on the nonlinearity and make a parameter transformation of the fractional power dissipative equation. Finally the random dynamical system generated by the solution is deﬁned.

3.1. Fractional power Laplacian and fractional Sobolev space. The fractional power Laplacian, as a positive power of the classical Laplacian, has been introduced for a long time from several areas of mathematics and real-world applications, such as potential theory [27], fractional calculus [39], probability [20, 35] and etc. However, until the past decade this operator became a very popular ob- ject in the field of partial differential equations. There are two ways to define the fractional power Laplacian in the literature. One is the spectral fractional operator (cf.[28, 58] and the references therein), the other is the integral fractional operator (cf.[17]), which is one of main topics in this paper. Let J be the Schwartz space of rapidly decaying C∞ functions on RN . For s ∈ (0, 1), the integral fractional power Laplacian (−∆)s is defined as, for u ∈ J , PULLBACK ATTRACTORS FOR BI-SPATIAL SYSTEMS 3405

Z u(x) − u(y) (−∆)su(x) = C(N, s)P.V. dy |x − y|N+2s N R 1 Z u(x + y) − 2u(x) + u(x − y) = − C(N, s) dy, x ∈ N , (20) 2 |y|N+2s R N R where P.V. means the principal value of the integral, and C(N, s) is an explicit positive constant depend only on N and s, see [17]. Deﬁne Z |u(x) − u(y)|2 Hs( N ) = {u ∈ L2( N ): dxdy < ∞}. (21) R R |x − y|N+2s 2N R Then Hs(RN ) is a Hilbert space equipped with the inner product, for u, v ∈ Hs(RN ), Z (u(x) − u(y))(v(x) − v(y)) (u, v) s N = (u, v) + dxdy H (R ) |x − y|N+2s 2N R and the norm Z 2 2 2 |u(x) − u(y)| kuk s N = kuk + dxdy, (22) H (R ) |x − y|N+2s 2N R where and later the notations (., .) and k.k denote the inner product and norm in L2(RN ), respectively. From Proposition 3.6 in [17], we have Z 2 s 2 1 |u(x) − u(y)| k(−∆) 2 uk = C(N, s) dxdy, 2 |x − y|N+2s 2N R namely, the norm in (22) is rewritten as

2 s 2 2 2 2 s N kuk s N = kuk + k(−∆) uk , u ∈ H ( ), H (R ) C(N, s) R s 2 2 2 2 s N and consequently kuk s N = kuk +k(−∆) uk is an equivalent norm of H ( ). H (R ) R 3.2. Fractional power dissipative stochastic equation. In this subsection, we present some settings about the the non-autonomous fractional power dissipative stochastic equation on the whole space RN , driven by an additive noise: du + λu + (−∆)su + f(t, x, u) = g(t, x) + φ(x)W˙ (t), t > τ, τ ∈ , (23) dt R with the initial condition 2 N u(τ, x) = uτ ∈ L (R ), (24) where s ∈ (0, 1) and λ > 0. The nonlinear function f is continuous on R × RN × R N satisfying the following conditions, for all t, u1, u2 ∈ R and x ∈ R , f(t, x, 0) = 0 and p 2 (f(t, x, u1) − f(t, x, u2))(u1 − u2) ≥ γ|u1 − u2| − ψ1(t, x)|u1 − u2| , (25)

p−2 p−2 |f(t, x, u1) − f(t, x, u2)| ≤ ψ2(t, x)|u1 − u2|(1 + |u1| + |u2| ), (26) ∞ N+1 p−2 where γ > 0, p ≥ 2 and ψ1, ψ2 ∈ L (R ). For example, if f(t, x, u) = |u| u − u, u ∈ R, then f satisﬁes (25) and (26). 3406 WENQIANG ZHAO

We further assume that

λ > µ = 2(kψ1k∞ + kψ2k∞), (27) where ψ and ψ are given in (25) and (26). Here k.k = k.k ∞ N+1 . The 1 2 ∞ L (R ) 2 2 N non-autonomous term g ∈ Lloc(R,L (R )) satisﬁes, for every τ ∈ R, τ Z eα0rkg(r, .)k2dr < ∞, (28)

−∞ where α0 = λ − µ > 0. For the noise coefficient, we assume that φ satisfies that, for any s ∈ (0, 1) and p ≥ 2, s N 2p−2 N s 2 N φ ∈ H (R ) ∩ L (R ), (−∆) φ ∈ L (R ). (29) Then by the Sobolev interpolation, we have φ ∈ Lp(RN ). To model the random noise in (23), we introduce the classic Wiener probability space (Ω, F,P ), where Ω = {ω ∈ C(R, R): ω(0) = 0} with the compact-open topology, F is its Borel σ-algebra, P is the Wiener measure on (Ω, F). Then the Brown motion is identified as W (t) = W (t, ω) = ω(t). The driving system is a shift operator ϑt on Ω by

ϑtω(.) = ω(t + .) − ω(t), ∀ t ∈ R.

It is known that the Wiener measure P is ergodic and invariant under the drift ϑt, and the quadruple form (Ω, F,P, {ϑt}t∈R) is a metric dynamical system, see Arnold [1]. By the law of the iterated logarithm (cf.[18]), there exists a ϑt-invariant set Ω1 ⊂ Ω with P (Ω)˜ = 1, and in addition, |ω(t)| → 0, as |t| → ∞, ω ∈ Ω . (30) t 1 Thanks to the Brownian motion W (t) = ω(t) is not differentiable in t, the stochastic equation need to be transformed into an equation with random coefficient, usually by an Ornstein-Uhlenbeck process over (Ω,P, F, {ϑt}t∈R), which is a sta- tionary process Z 0 λr y(ϑtω) = −λ e ϑtω(r)dr, t ∈ R, (31) −∞ satisfying the stochastic differential equation:

dy(ϑtω) + λy(ϑtω)dt = dω(t).

In addition, it follows that from [1], that there exits a ϑt-invariant set Ω2 ⊂ Ω of full measure such that y(ϑtω) is pathwise continuous in t for every ﬁxed ω ∈ Ω2, and t Z |y(ϑtω)| 1 lim = 0, and lim y(ϑrω)dr = 0, ω ∈ Ω2. (32) t→±∞ |t| t→±∞ |t| 0

Let Ω˜ = Ω1 ∩ Ω2. Then Ω˜ ⊂ Ω is a ϑt-invariant set of full measure, and in addition, for every ω ∈ Ω,˜ (30) and (32) hold. In the sequel, all arguments are understood to hold on this Ω,˜ but we keep the notation Ω for Ω.˜ PULLBACK ATTRACTORS FOR BI-SPATIAL SYSTEMS 3407

Put 2 p 2p−2 J(ϑtω) = |y(ϑtω)| + |y(ϑtω)| + |y(ϑtω)| . (33) Then by Proposition 4.3.3 in [1] there exists a random variable %(ω) such that α0 |t| J(ϑtω) ≤ %(ϑtω) ≤ %(ω)e 2 , ∀ω ∈ Ω, (34) where α0 = λ − µ > 0 with µ as in (27). Set z(ϑtω) = φ(x)y(ϑtω) and let v(t) = u(t) − z(ϑtω), where u(t) satisﬁes the equations (23) and (24). Then the stochastic equations (23) and (24) are transformed into a pathwise deterministic one, namely, v(t) satisﬁes dv + λv + (−∆)sv + f(t, x, v + z(ϑ ω)) = g(t, x) − (−∆)sz(ϑ ω), (35) dt t t with the initial condition

v(τ, x) = vτ = uτ − z(ϑτ ω). (36) + 2 N Given t ∈ R , τ ∈ R, ω ∈ Ω and uτ ∈ L (R ), deﬁne

ϕ(t, τ, ω, uτ ) = u(t + τ, τ, ϑ−τ ω, uτ ) = v(t + τ, τ, ϑ−τ ω, vτ ) + z(ϑtω), (37) where uτ = vτ + z(ω). In the later sections, we will show that the random cocycle ϕ(t, τ, ., uτ ) is (F, B(Z))-measurable, and in addition the mapping ϕ(t, τ, ω, .): L2(RN ) → Z is continuous with respect to the initial data, where Z = L2(RN ),Lp (RN ) and Hs(RN ). Then ϕ is a measurable bi-spatially continuous RDS. For the universe of sets, we consider a family D = {D(τ, ω) ⊂ L2(RN ): τ ∈ R, ω ∈ Ω} such that for each τ ∈ R and ω ∈ Ω,

−α0t 2 lim e kD(τ − t, ϑ−tω)k = 0, (38) t→∞ where α0 = λ − µ > 0 and kDk = sup{kuk : u ∈ D}. Such a set D(τ, ω) satisfying (38) is called tempered set. Denote by D the collection of all tempered families of nonempty bounded subsets of L2(RN ). Then it is obvious that D is inclusion closed.

4. Absorbing set and asymptotical compactness in L2(RN ). In this section, we prove the existence of a random pullback absorbing set and the pullback asymptotical compactness of ϕ deﬁned by (37) in L2(RN ). 4.1. Existence of random absorbing set.

Lemma 4.1. Let (25)-(29) hold. Let τ ∈ R, ω ∈ Ω and D = {D(τ, ω); τ ∈ R, ω ∈ Ω} ∈ D. Then there exists a constant T = T (τ, ω, D) ≥ 1 such that for all t ≥ T and uτ−t ∈ D(τ − t, ϑ−tω), the solution u of equations (23) and (24) satisﬁes 2 ku(σ, τ − t, ϑ−τ ω, uτ−t)k ≤ c(1 + C(τ, ω)), σ ∈ [τ − 1, τ]; (39)

τ Z α0(r−τ) 2 p e (kv(r)k s N + kv(r)k )dr ≤ c(1 + C(τ, ω)), (40) H (R ) p τ−t 0 where c > 0 is a deterministic generic constant and C(τ, ω) = R eµrkg(r + −∞ τ, .)k2dr + %(ω). Deﬁne 2 N 2 K(τ, ω) = {u ∈ L (R ): kuk ≤ c(1 + C(τ, ω))}. (41) 3408 WENQIANG ZHAO

Then K = {K(τ, ω): τ ∈ R, ω ∈ Ω} ∈ D is a closed D-pullback random absorbing set for the random cocycle ϕ in L2(RN ). Proof. Taking the inner product of (35) in L2(RN ) with v, we obtain Z 1 d 2 2 s 2 kvk + λkvk + k(−∆) 2 vk + f(t, x, u)vdx 2 dt N R Z s = g(t, x)vdx − ((−∆) z(ϑtω), v). (42)

N R For the nonlinearity, from (25) and (26), by using Young inequality repeatedly, we have p 2 f(t, x, u)v ≥ γ|u| − |ψ1(t, x)| |u| − |f(t, x, u)| |φ(x)z(ϑtω)| γ ≥ |u|p − (|ψ (t, x)| + |ψ (t, x)|) |u|2 − |ψ (t, x)| |φ(x)z(ϑ ω)|2 2 1 2 2 t p p − c|ψ2(t, x)| |φ(x)z(ϑtω)| γ ≥ |v|p − 2(|ψ (t, x)| + |ψ (t, x)|) |v|2 2p 1 2 2 − (2|ψ1(t, x)| + 3|ψ2(t, x)| |φ(x)z(ϑtω)| γ − (c|ψ (t, x)|p + )|φ(x)z(ϑ ω)|p, (43) 2 2 t where we used the inequality |a+b|r ≥ 21−r|a|r −|b|r for any r ≥ 1 in the ﬁrst term of the last inequality on the right hand side of (43). By φ ∈ L2(RN ) ∩ Lp(RN ) and ∞ N+1 ψ1, ψ2 ∈ L (R ), along with (27), we deduce from (43) that Z γ f(t, x, u)vdx ≥ kvkp − µkvk2 − c(|z(ϑ ω)|2 + |z(ϑ ω)|p) 2p p t t N R γ ≥ kvkp − µkvk2 − cJ(ϑ ω), (44) 2p p t where J(.) is given as in (33). On the other hand, by integration by parts and Young inequality, along with φ ∈ Hs(RN ), we have

d 2 2 α0 2 s 2 γ p kvk + α kvk + kvk + k(−∆) 2 vk + kvk dt 0 2 2p p 2 ≤ ckg(t, .)k + cJ(ϑtω), (47) where α0 = λ − µ as in (28). Apply Gronwall lemma to (47) over the interval [τ − t, σ] for σ ∈ [τ − 1, τ] and t ≥ 1, meanwhile, using ϑ−τ ω to replace ω, to ﬁnd that σ Z 2 α (r−σ) α0 2 s 2 α p kv(σ)k + e 0 ( kv(r)k + k(−∆) 2 v(r)k + kv(r)k )dr 2 2p p τ−t τ Z α0 −α0t 2 α0 α0(r−τ) 2 ≤ e e kvτ−tk + ce e kg(r, .)k dr τ−t τ Z α α (r−τ) α0 |r−τ| + ce 0 %(ω) e 0 e 2 dr

τ−t

α0 −α0t 2 2 2 ≤ 2e e (kuτ−tk + kφk |y(ϑ−tω)| ) 0 Z + c eα0rkg(r + τ, .)k2dr + c%(ω), (48)

−∞ where we used (34) in the last term of the ﬁrst inequality. By noting that uτ−t ∈ D(τ − t, ϑ−tω) and |y(ω)| is tempered, we have

−α0t 2 2 2 2e (kuτ−tk + kφk |y(ϑ−tω)| ) → 0, (49)

0 as t → ∞. By (28), we have R eα0rkg(r + τ, .)k2dr < ∞. Therefore, it follows −∞ from (48) and (49) that, for every ﬁxed τ ∈ R, ω ∈ Ω and all uτ−t ∈ D(τ − t, ϑ−tω), there exists a T = T (τ, ω, D) ≥ 1 such that for all t ≥ T , σ Z s 2 α0(r−τ) 2 2 2 p kv(σ)k + e (kv(r)k + k(−∆) v(r)k + kv(r)kp)dr τ−t 0 Z ≤ c(1 + eα0rkg(r + τ, .)k2dr + %(ω)). (50)

−∞

Note that u(t) = v(t) + z(ϑtω). Then by (34) it follows that, for σ ∈ [σ − 1, σ], 2 2 2 2 ku(σ)k ≤ 2kv(σ)k + 2kφk |y(ϑσ−τ ω)| 2 2 α0 |σ−τ| ≤ 2kv(σ)k + 2kφk %(ω)e 2 2 2 α0 ≤ 2kv(σ)k + 2kφk %(ω)e 2 . (51) Therefore, by (50) and (51), we get (39) and (40). On the other hand, for any x ∈ L2(RN ) and a ﬁxed τ ∈ R, we have 1 inf dL2( N )(x, K(τ, ω)) = kxk + c(1 + C(τ, ω)) 2 z∈K(τ,ω) R 3410 WENQIANG ZHAO

0 R α r 2 is (F, B(R))-measurable since C(τ, ω) = e 0 kg(r + τ, .)k dr + %(ω) as a random −∞ variable is (F, B(R))-measurable for fixed τ ∈ R. Then the set-valued mapping K defined by (41) is a closed random set in L2(RN ), which completes the proof. 4.2. Estimate of the tail of solution. To prove the asymptotical compactness of ϕ in L2(RN ) , we need to prove that the far-field value of solution to (35) vanishes when the time and radius of ball are large enough. Let ξ(.): R+ → [0, 1] be a smooth function satisfying ( 0, if r ∈ [0, 1]; ξ(r) = (52) 1, if r ∈ [2, ∞). Then there exists a positive constant c such that |ξ0(.)| ≤ c. We first give a lemma about the cutoff function.

Lemma 4.2. Let ξ be deﬁned by (52). Then for any y ∈ RN and k ∈ N, we have 2 ξ( |x| ) − ξ( |y| ) Z k k c dx ≤ , (53) |x − y|N+2s k2s N R where the constant c > 0 is independent of k and y. Proof. Make a variable change x − y = kz. Then dx = kN dz, and hence we have 2 2 ξ( |x| ) − ξ( |y| ) ξ(|z + y |) − ξ( |y| ) Z k k 1 Z k k dx = dz |x − y|N+2s k2s |z|N+2s N N R R 2 ξ(|z + y |) − ξ( |y| ) 1 Z k k = dz k2s |z|N+2s |z|≤1 2 ξ(|z + y |) − ξ( |y| ) 1 Z k k + dz. (54) k2s |z|N+2s |z|>1 By the Lagrange mean theorem and |ξ0(.)| ≤ c, we get 2 2 ξ(|z + y |) − ξ( |y| ) |ξ0(.)| |z + y | − |y| ) 1 Z k k 1 Z k k dz ≤ dz k2s |z|N+2s k2s |z|N+2s |z|≤1 |z|≤1 c Z 1 ≤ dz < ∞, (55) k2s |z|N+2s−2 |z|≤1 1 where we used the fact that |z|N+2s−2 is integrable for N + 2s − 2 < N. On the other hand, since |ξ(.)| ≤ 1, we have 2 ξ(|z + y |) − ξ( |y| ) 1 Z k k 4 Z 1 dz ≤ dz < ∞, (56) k2s |z|N+2s k2s |z|N+2s |z|>1 |z|>1 PULLBACK ATTRACTORS FOR BI-SPATIAL SYSTEMS 3411

1 where we used the fact that |z|N+2s is integrable for N + 2s > N. Then combine (54)-(56) to get the desired. We now present the tail estimate of solutions, which demonstrates that the far- ﬁeld value of solution vanishes in L2(RN ) as the time and the radius larger enough. Lemma 4.3. Let (25)-(29) hold. Let τ ∈ R, ω ∈ Ω and D = {D(τ, ω); τ ∈ R, ω ∈ Ω} ∈ D. Then for any η > 0, there exist constants C0 = C0(ω),R = R(τ, ω, D, η) ≥ 1 and T = T (τ, ω, D, η) ≥ 1 such that the solution u of equations (23) and (24) satisﬁes, for all σ ∈ [τ − 1, τ] and t ≥ T , Z 2 |u(σ, τ − t, ϑ−τ ω, uτ−t)| dx ≤ C0(ω)η,

c QR c N N where QR = R − QR and QR is the ball of R centred at zero with radius R. |x| N Proof. Let ξk = ξ( k ) for x ∈ R and k ∈ N, where ξ(.) is given by (52). Take the 2 N inner product of (35) in L (R ) with ξkv to ﬁnd 1 d Z Z Z ξ |v|2dx + λ ξ |v|2dx = −((−∆)sv, ξ v) − f(t, x, u)ξ vdx 2 dt k k k k N N N R R R Z s + g(t, x)ξkvdx − ((−∆) z(ϑtω), ξkv). (57)

N R We now estimate every terms on the right hand side of (57). By some calculations, the ﬁrst term on the right hand side of (57) is rewritten as 1 Z (v(x) − v(y))(ξ( |x| )v(x) − ξ( |y| )v(y)) −((−∆)sv, ξ v) = − C(N, s) k k dxdy k 2 |x − y|N+2s 2N R 1 Z |v(x) − v(y)|2ξ( |x| ) = − C(N, s) k dxdy 2 |x − y|N+2s 2N R 1 Z (v(x) − v(y))(ξ( |x| ) − ξ( |y| ))v(y) − C(N, s) k k dxdy. (58) 2 |x − y|N+2s 2N R By H¨olderinequality and Lemma 4.2, the second term on the right hand side of (58) is estimated as 1 Z (v(x) − v(y))(ξ( |x| ) − ξ( |y| ))v(y) − C(N, s) k k dxdy 2 |x − y|N+2s 2N R |x| |y| 1 1 n Z Z (v(x) − v(y))(ξ( ) − ξ( )) 2 o 2 ≤ C(N, s)kvk k k dx dy 2 |x − y|N+2s N N R R 2 |x| |y| 2 1 1 n Z Z |v(x) − v(y)| Z |ξ( ) − ξ( )| o 2 ≤ C(N, s)kvk dx k k dx dy 2 |x − y|N+2s |x − y|N+2s N N N √ R R R √ Z 2 1 c |v(x) − v(y)| 2 c 2 ≤ C(N, s)kvk dxdy ≤ C(N, s)kvk s N , 2ks |x − y|N+2s 2ks H (R ) 2N R 3412 WENQIANG ZHAO which, along with (58), gives

1 Z |v(x) − v(y)|2ξ( |x| ) −((−∆)sv, ξ v) ≤ − C(N, s) k dxdy k 2 |x − y|N+2s 2N √ R c2 2 + C(N, s)kvk s N . (59) 2ks H (R ) For the last term on the right hand side of (57), we rewrite as

s − ((−∆) z(ϑtω), ξkv) 1 Z y(ϑ ω)(φ(x) − φ(y))(v(x)ξ( |x| ) − v(y)ξ( |y| )) = − C(N, s) t k k dxdy 2 |x − y|N+2s 2N R 1 Z y(ϑ ω)(φ(x) − φ(y))(v(x) − v(y))ξ( |x| ) = − C(N, s) t k dxdy 2 |x − y|N+2s 2N R 1 Z y(ϑ ω)(φ(x) − φ(y))(ξ( |x| ) − ξ( |y| ))v(y) − C(N, s) t k k dxdy. (60) 2 |x − y|N+2s 2N R The ﬁrst term on the right hand side of (60) is bounded by

1 Z y(ϑ ω)(φ(x) − φ(y))(v(x) − v(y))ξ( |x| ) − C(N, s) t k dxdy 2 |x − y|N+2s 2N R 1 Z |v(x) − v(y)|2ξ( |x| ) ≤ C(N, s) k dxdy 2 |x − y|N+2s 2N R 1 Z |φ(x) − φ(y)|2ξ( |x| ) + C(N, s)|y(ϑ ω)|2 k dxdy 2 t |x − y|N+2s 2N R 1 Z |v(x) − v(y)|2ξ( |x| ) ≤ C(N, s) k dxdy 2 |x − y|N+2s 2N R 1 Z Z |φ(x) − φ(y)|2 + C(N, s)|y(ϑ ω)|2 dy dx. (61) 2 t |x − y|N+2s N |x|≥k R By Lemma 4.2, along with (29), the second term on the right hand side of (60) is bounded by

1 Z y(ϑ ω)(φ(x) − φ(y))(ξ( |x| ) − ξ( |y| ))v(y) − C(N, s) t k k dxdy 2 |x − y|N+2s 2N R |x| |y| 1 1 n Z Z y(ϑ ω)(φ(x) − φ(y))(ξ( ) − ξ( )) 2 o 2 ≤ C(N, s)kvk t k k dx dy 2 |x − y|N+2s N N R R 1 n Z Z |y(ϑ ω)|2|φ(x) − φ(y)|2 ≤ C(N, s)kvk t dx 2 |x − y|N+2s N N R R PULLBACK ATTRACTORS FOR BI-SPATIAL SYSTEMS 3413

|x| |y| 2 1 Z |ξ( ) − ξ( )| o 2 × k k dx dy |x − y|N+2s N R 2 1 c Z |(φ(x) − φ(y))| 2 ≤ C(N, s)kvk|y(ϑ ω)| dxdy 2ks t |x − y|N+2s 2N R c ≤ (kvk2 + |y(ϑ ω)|2). (62) 2ks t Incorporate (61) and (62) into (60) to yield c −((−∆)sz(ϑ ω), ξ v) ≤ (kvk2 + |y(ϑ ω)|2)+ t k 2ks t 1 Z |v(x) − v(y)|2ξ( |x| ) + C(N, s) k dxdy 2 |x − y|N+2s 2N R 1 Z Z |φ(x) − φ(y)|2 + C(N, s)|y(ϑ ω)|2 dy dx. (63) 2 t |x − y|N+2s N |x|≥k R For the nonlinearity term in (57), by (25),(26) and (27), we deduce that Z 1 Z Z f(t, x, u)ξ vdx ≥ γ ξ |u|pdx − µ ξ |v|2dx k 2 k k N N N R R R Z Z p 2 − c ξk|φz(ϑtω)| dx − c ξk|φz(ϑtω)| dx. (64)

N N R R On the other hand, we have Z λ − µ Z 1 Z ξ g(t, x)vdx ≤ ξ |v|2dx + ξ |g(t, x)|2dx. (65) k 2 k 2(λ − µ) k N N N R R R Consequently, combine (57) with (59),(63)-(65) to yield Z Z d 2 2 c 2 2 ξk|v| dx + α0 ξk|v| dx ≤ (kvk s N + |y(ϑtω)| ) dt ks H (R ) N N R R Z Z p 2 2 + c ξk(|φ(x)z(ϑtω)| + |φ(x)z(ϑtω)| )dx + c ξk|g(t, x)| dx

N N R R Z Z |φ(x) − φ(y)|2 + c|y(ϑ ω)|2 dy dx, (66) t |x − y|N+2s N |x|≥k R

p N s N where α0 = λ − µ as in (28). Since φ ∈ L (R ) ∩ H (R ), then for every η > 0 there exists a R1 = R1(η) ≥ 1 such that for all k ≥ R1, Z Z p 2 p 2 c ξk(|φ(x)z(ϑtω)| + |φ(x)z(ϑtω)| )dx ≤ c (|φ(x)z(ϑtω)| + |φ(x)z(ϑtω)| )dx

N R |x|≥k η ≤ J(ϑ ω), (67) 2 t 3414 WENQIANG ZHAO and Z Z |φ(x) − φ(y)|2 η c|y(ϑ ω)|2 dy dx ≤ |y(ϑ ω)|2. (68) t |x − y|N+2s 2 t N |x|≥k R

|x|≥k Apply Gronwall lemma to (69) over the interval [τ − t, σ] with σ ∈ [τ − 1, τ] and t ≥ 1, replacing ω by ϑ−τ ω, to ﬁnd that, for all k ≥ R1, Z Z 2 α0 −α0t 2 ξk|v(σ)| dx ≤ e e ξk|vτ−t| dx

N N R R τ ceα0 Z α0(r−τ) 2 2 + e (kv(r)k s N + |y(ϑr−τ ω)| )dr ks H (R ) τ−t τ Z Z + ceα0 eα0(r−τ) |g(r, x)|2dxdr

τ−t |x|≥k τ Z α0 α0(r−τ) + e η e J(ϑr−τ ω)dr. (70) τ−t

By (38), for every η > 0 and uτ−t ∈ D(τ−t, ϑ−tω), there exists a T1 = T1(τ, ω, D, η) ≥ 1 such that for all t ≥ T1, Z η eα0 e−α0t ξ |v |2dx ≤ 2eα0 e−α0t(ku )k2 + kφz(ϑ ω)k2) ≤ . (71) k τ−t τ−t −t 4 N R

By (40), there is a T2 = T2(τ, ω, D) ≥ T1 such that for all t ≥ T2, τ Z α0(r−τ) 2 e kv(r)k s N dr ≤ c(1 + C(τ, ω)). H (R ) τ−t

Then there exists a R2 = R2(τ, ω, D, η) ≥ R1 such that for all t ≥ T2 and k ≥ R2, τ ceα0 Z c(1 + C(τ, ω)) η α0(r−τ) 2 e kv(r)k s N dr ≤ ≤ . (72) ks H (R ) ks 4 τ−t By (34), we have

τ 0 Z Z α0(r−τ) 2 α0r 2 e |y(ϑr−τ ω)| dr = e |y(ϑrω)| dr τ−t −t PULLBACK ATTRACTORS FOR BI-SPATIAL SYSTEMS 3415

0 Z α r − α0r 2%(ω) ≤ %(ω) e 0 e 2 dr ≤ < ∞, α0 −t and therefore there exists a R3 = R3(ω) such that for all k ≥ R3,

τ ceα0 Z ceα0 2%(ω) η α0(r−τ) 2 s e |y(ϑr−τ ω)| dr ≤ s ≤ . (73) k k α0 4 τ−t

0 R α0r R 2 By (28), e |g(r + τ, x)| dxdr < ∞, then there exists a R4 = R4(τ, η) ≥ R3 N −∞ R such that for all k ≥ R4,

τ 0 Z Z Z Z ceα0 eα0(r−τ) |g(r, x)|2dxdr = ceα0 eα0r |g(r + τ, x)|2dxdr

τ−t |x|≥k −∞ |x|≥k η ≤ . (74) 4 As for the last term on the right hand side of (70), we get from )33 that

τ 0 Z Z α0 α0(r−τ) α0 α0r e η e J(ϑr−τ ω)dr = e η e J(ϑrω)dr τ−t −t 0 Z α 1 α r 2 α ≤ e 0 η%(ω) e 2 0 dr ≤ %(ω)e 0 η. (75) α0 −∞

Thus, combine (70) to (75) to ﬁnd that for all t ≥ T2 and k ≥ R4 and σ ∈ [τ − 1, τ], Z 2 2 α0 ξk|v(σ)| dx ≤ (1 + %(ω)e )η, α0 N R which, along with u(t) = v(t) + z(ϑtω), implies the desired.

4.3. Asymptotical compactness. In order to prove the asymptotical compactness, we ﬁrst prove a convergence result of the solution to equations (23) and (24) if the initial value sequence is bounded in L2(RN ), which the idea is from [26].

N ∞ Lemma 4.4. Given R > 0, let QR = {x ∈ R : |x| ≤ R, R > 0} and {xn}n=1 ⊂ 2 N L (R ) be an initial values sequence with sup kxnk ≤ M for some ﬁxed M > 0. n Let τ ∈ R and ω ∈ Ω. Then for almost every σ ∈ [τ − 1, τ], there exits a ﬁxed 2 N ∞ point uσ ∈ L (R ) such that the sequence {un(σ, τ − 1, ω, xn)}n=1 has a convergent 2 subsequence with un(σ, τ − 1, ω, xn) → uσ in L (QR), where un(σ) is a solution to (23) and (24) corresponding the initial value xn.

Proof. Integrate (47) from τ − 1 to σ for σ ∈ [τ − 1, τ] to ﬁnd

2 kvn(σ, τ − 1, ω, xn − φz(ϑτ−1ω))k 3416 WENQIANG ZHAO

τ Z 2 + kvn(r, τ − 1, ω, xn − φz(ϑτ−1ω))k s N dr H (R ) τ−1 τ Z 2 2 2 2 ≤ 2(kxnk + kφk |z(ϑτ−1ω)| ) + c (J(ϑrω) + kg(r, .)k )dr. (76) τ−1 2 2 N 2 N s N Consider that g ∈ Lloc(R,L (R )), φ ∈ L (R ) ∩ H (R ), J(ϑrω) is continuous in r ∈ R and un(t) = vn(t) + φz(ϑtω). Then by (76), there exists a positive random constant C0(τ, ω) (independent of n) such that, for every σ ∈ [τ − 1, τ], 2 kun(σ, τ − 1, ω, xn)k ≤ C0(τ, ω), (77) and τ Z 2 kun(r, τ − 1, ω, xn)k s N dr ≤ C0(τ, ω). (78) H (R ) τ−1 2 N By (77), for every σ ∈ [τ − 1, τ], there exist uσ ∈ L (R ) and a subsequence (for simplicity denoted by itself) by a diagonal process such that

2 N un(σ, τ − 1, ω, xn) → uσ weakly in L (R ). (79) On the other hand, by (78) we deduce that

un(., τ − 1, ω, xn) is unformly bounded with respect to n in 2 s N L (τ − 1, τ; H (R )). (80) We now need to deﬁne a new space, for s ∈ (0, 1),

s N c V = {u ∈ H (R ): u = 0 in QR}. Then V is a Hilbert space, and by Theorem 7.1 in [17], the embedding of V,→ 2 L (QR) is compact. From (80), we have 2 un(., τ − 1, ω, xn) is unformly bounded with respect to n in L (τ − 1, τ; V ). Thus by the compact embedding, there is a subsequence (for simplicity labeled by itself) by a diagonal process such that ∞ 2 2 {un(., τ − 1, ω, xn)}n=1 strongly converges in L (τ − 1, τ; L (QR)). Thanks to (79), it follows that, for all most every σ ∈ [τ − 1, τ], 2 un(σ, τ − 1, ω, xn) → uσ strongly in L (QR). (81)

In addition, it is obvious that uσ is independent of the choice of R, and consequently (81) holds for any R > 0.

The following lemma is concerning the asymptotical compactness of random cocycle ϕ in L2(RN ). Lemma 4.5. Let (25)-(29) hold and D be deﬁned by (38). Then the random cocycle ϕ deﬁned by (37) is (L2(RN ),L2(RN ))-pullback asymptotically compact, namely, for ∞ every τ ∈ R and ω ∈ Ω, the sequence {ϕ(tn, τ−tn, ϑ−tn ω, u0,n)}n=1 has a convergent 2 N subsequence in L (R ) whenever tn → ∞ and u0,n ∈ D(τ −tn, ϑ−tn ω) with D ∈ D. PULLBACK ATTRACTORS FOR BI-SPATIAL SYSTEMS 3417

Proof. Let τ ∈ R, ω ∈ Ω and u0,n ∈ D(τ − tn, ϑ−tn ω). By Lemma 4.1, there exists a N 3 N1 = N1(τ, ω, D) ≥ 1 such that, for all n ≥ N1, 2 ku(τ − 1, τ − tn, ϑ−τ ω, u0,n)k ≤ c(1 + C(τ, ω)). (82)

Now replacing xn by u(τ − 1, τ − tn, ϑ−τ ω, u0,n) in Lemma 4.4, we find that for any 2 N fixed R > 0, there is a certain real number σ ∈ [τ − 1, τ], uσ ∈ L (R ) and up to a subsequence (for simplicity labeled by itself) such that 2 u(σ, τ − 1, ϑ−τ ω, u(τ − 1, τ − tn, ϑ−τ ω, u0,n)) → uσ strongly in L (QR), (83) where uσ is independent of R > 0. By the cocycle property of ϕ and (83), it follows that 2 u(σ, τ − tn, ϑ−τ ω, u0,n) → uσ in L (QR), (84) for every R > 0. Let η > 0 be arbitrary. Then by Lemma 4.3, there exist N 3 N2 = N2(τ, ω, D, η) ≥ N1 and R1 = R1(τ, ω, D, η) such that for all n ≥ N2 and R ≥ R1, Z η eµ0 |u(σ, τ − t , ϑ ω, u )|2dx ≤ , (85) n −τ 0,n 6 |x|≥R 2 N for this fixed σ ∈ [τ − 1, τ], where N1 is as in (82). Since uσ ∈ L (R ), there is a number R2 = R2(η) ≥ R1 such that for all R ≥ R2 Z η eµ0 |u |2dx ≤ . (86) σ 6 |x|≥R

By (84), there exists a N 3 N3 = N3(τ, ω, D, η) ≥ N2 such that for all n ≥ N3 and R ≥ R2, Z η eµ0 |u(σ, τ − t , ϑ ω, u ) − u |2dx ≤ . (87) n −τ 0,n σ 3 |x|≤R Consider that

u(τ, τ − tn, ϑ−τ ω, u0,n) = u(τ, σ, ϑ−τ ω, u(σ, τ − tn, ϑ−τ ω, u0,n)). Then by (94) in the following section, we deduce that, for this ﬁxed σ ∈ [τ − 1, τ], 2 ku(τ, τ − tn, ϑ−τ ω, u0,n) − u(τ, σ, ϑ−τ ω, uσ)k

(τ−σ)µ0 2 ≤ e ku(σ, τ − tn, ϑ−τ ω, u0,n) − uσk Z µ0 2 ≤ e |u(σ, τ − tn, ϑ−τ ω, u0,n) − uσ| dx

|x|≤R Z µ0 2 + e |u(σ, τ − tn, ϑ−τ ω, u0,n) − uσ| dx, (88)

|x|≥R where R ≥ R2 and n ≥ N3. Consequently, by a combination of (85)-(88), it follows that, for all n ≥ N3, 2 ku(τ, τ − tn,ϑ−τ ω, u0,n) − u(τ, σ, ϑ−τ ω, uσ)k ≤ η, which and (37) end the proof. 3418 WENQIANG ZHAO

5. Continuity of solutions.

5.1. Continuity w.r.t. the initial data in Lp(RN ). To consider the continuity of solutions, we turn to the diﬀerence equation. Let ui be the solution of problem (23) and (24) corresponding to the initial value uτ,i, i = 1, 2. Givenu ¯ = u1 − u2,u ¯ is the solution to the following diﬀerence equation: du¯ + λu¯ + (−∆)su¯ + f(t, x, u ) − f(t, x, u ) = 0 (89) dt 1 2 with the initial valueu ¯τ = uτ,1 − uτ,2 and 0 < s < 1.

Lemma 5.1. (See Bartsch [5], Lemma 3.6.) Assume that x, y ∈ RN . Then for any d ≥ 2, there exists a constant cN,d > 0 such that d−2 d−2 d (|x| x − |y| y).(x − y) ≥ cN,d|x − y| , where . is the inner product in RN . For the fractional power Laplacian (−∆)su¯ and the nonlinear term f in (89), we have the following characteristic properties, which is of importance for us to cope with the subsequential proofs. Lemma 5.2. For any d ≥ 2, ((−∆)su,¯ |u¯|d−2u¯) ≥ 0. Proof. Note that, by Lemma 5.1 for the case N = 1, we have Z (¯u(t)(x) − u¯(t)(y))(|u¯(t)(x)|d−2u¯(t)(x) − |u¯(t)(y)|d−2u¯(t)(y)) dxdy |x − y|N+2s 2N R Z |u¯(t)(x) − u¯(t)(y)|d ≥ c dxdy. 1,d |x − y|N+2s 2N R Therefore we have ((−∆)su,¯ |u¯|d−2u¯) ≥ 0, which is the desired.

Lemma 5.3. If f satisﬁes (25), then for any d ≥ 2 Z d−2 d+p−2 d (f(t, x, u1) − f(t, x, u2))|u¯| udx¯ ≥ γku¯kd+p−2 − kψ1k∞ku¯kd. (90)

N R Proof. It is clear by (25).

We are now at the point to prove the continuity of solution to problem (23) and (24) in Lp(RN ) with respect to the initial value belonging to L2(RN ). By Lemma 2.7, this implies the asymptotical compactness of solution in Lp(RN ).

Lemma 5.4. Let (25) hold. Let ui(t) = ui(t, τ, ω, uτ,i) be the solution of equations (23) and(24) with respect to the initial value uτ,i, i = 1, 2, respectively. Given τ ∈ R, ω ∈ Ω and T > τ, then there exists a positive deterministic constant CT −τ depending only on γ, p and µ0 such that p 2 (t − τ)k(t − τ)u(t)kp ≤ CT −τ ku¯τ k , t ∈ (τ, T ), PULLBACK ATTRACTORS FOR BI-SPATIAL SYSTEMS 3419 and T Z 2p 2p−2 2 (r − τ) ku(r)k2p−2dr ≤ CT −τ ku¯τ k , τ where u(t) = u1(t) − u2(t).

Proof. Take the inner product of (89) in L2(RN ) withu ¯ to get Z 1 d 2 s 2 ku¯k + k(−∆) 2 u¯k + (f(t, x, u ) − f(t, x, u ))¯udx ≤ 0. (91) 2 dt 1 2 N R By Lemma 5.3 for the case d = 2, we have Z p 2 (f(t, x, u1) − f(t, x, u2))¯udx ≥ γku¯k − kψ1k∞ku¯k . (92)

N R By a combination of (91)-(92), it gives

d 2 s 2 p 2 ku¯k + k(−∆) 2 u¯k + γku¯k ≤ µ ku¯k , (93) dt 0 where µ0 = 2kψ1k∞. By using Gronwall lemma in (93) over the interval (τ, t), we see that ku¯(t)k2 ≤ eµ0(t−τ)ku¯(τ)k2, t ∈ (τ, T ). (94) Integrate (93) over the interval (τ, T ) and then use (94) to ﬁnd

T Z 1 ku¯(r)kpdr ≤ (1 + eµ0(T −τ))ku k2. (95) p γ τ τ Then, we get

T Z 1 (r − τ)pku¯(r)kpdr ≤ (T − τ)p(1 + eµ0(T −τ))ku k2 =: C ku k2, (96) p γ τ T −τ τ τ where CT −τ > 0 is a generic constant, which is increasing with respect to T − τ. Now, taking the inner product of (89) in L2(RN ) with |u¯|p−2u¯, using Lemma 5.2, we obtain 1 d Z ku¯kp + (f(t, x, u ) − f(t, x, u )|u¯|p−2udx¯ ≤ 0. (97) p dt p 1 2 N R Here we let d = p in Lemma 5.3. Combining (97) and (90) , we get d ku¯kp + γku¯k2p−2 ≤ µ pku¯kp. (98) dt p 2p−2 0 p Multiply (98) by (t − τ)p to yield

p d p p 2p−2 p p (t − τ) ku¯k + γk(t − τ) 2p−2 u¯k ≤ µ p(t − τ) ku¯k . (99) dt p 2p−2 0 p Since d d (t − τ)p ku¯kp = k(t − τ)¯ukp − p(t − τ)p−1ku¯kp, (100) dt p dt p p 3420 WENQIANG ZHAO then by (99) and (100) we get, for t > τ,

d p p 2p−2 k(t − τ)¯uk + γk(t − τ) 2p−2 u¯k dt p 2p−2 p p p−1 p ≤ µ0p(t − τ) ku¯kp + p(t − τ) ku¯kp 1 ≤ p(µ + )k(t − τ)¯ukp. (101) 0 t − τ p Obviously, it follows that d (t − τ) k(t − τ)¯ukp ≤ C k(t − τ)¯ukp, t ∈ [τ, T ]. (102) dt p T −τ p Through integrating (102) over the interval (τ, t) for every t ∈ (τ, T ), by integration by parts for the left term, we get t t Z d Z (r − τ) k(r − τ)¯u(r)kpdr = (t − τ)k(t − τ)¯ukp − k(r − τ)¯u(r)kpdr dr p p p τ τ t Z p ≤ CT −τ k(r − τ)¯u(r)kpdr. (103) τ Therefore by (96) and (103), it follows that, for any t ∈ (τ, T ), t Z p p 2 (t − τ)k(t − τ)¯ukp ≤ (CT −τ + 1) k(r − τ)¯u(r)kpdr ≤ CT −τ ku¯τ k , (104) τ which shows that (A) holds true. Multiply (101) by (t − τ)p to yield, in conjunction with (104),

p d p 2p 2p−2 (t − τ) k(t − τ)¯u(t)k + γk(t − τ) 2p−2 u¯(t)k dt p 2p−2 p p−1 p ≤ p µ0(t − τ) + (t − τ) k(t − τ)¯u(t)kp

p−1 p−2 p ≤ p µ0(T − τ) + (T − τ) (t − τ)k(t − τ)¯u(t)kp 2 ≤ CT −τ ku¯τ k , (105) for all t ∈ (τ, T ) and p ≥ 2 (noting that p − 2 ≥ 0). Note that by integration by parts, we have T Z d (r − τ)p k(r − τ)¯u(r)kpdr = (T − τ)pk(T − τ)¯u(T )kp dr p p τ T Z p−1 p − p (r − τ) k(r − τ)¯u(r)kpdr. (106) τ Therefore, integrating the last inequality of (105) over the interval (τ, T ), along with (106) and (104), it follows that

T T Z Z b 2p−2 2 p−1 p γ k(r − τ) u¯(r)k2p−2dr ≤ CT −τ (T − τ)ku¯τ k + p (r − τ) k(r − τ)¯u(r)kpdr τ τ PULLBACK ATTRACTORS FOR BI-SPATIAL SYSTEMS 3421

T Z 2 2 p−2 ≤ (T − τ)ku¯τ k + pCT −τ ku¯τ k (r − τ) dr τ 2 ≤ CT −τ ku¯τ k , (107) 2p where b = 2p−2 . Then we conclude the proof. According to Lemma 5.4 and (94), we immediately get Theorem 5.5. Under the conditions (25)-(29), the solution u of problem (23) and (24) is bi-spatial (L2(RN ),Lp(RN ))-continuous with respect to the initial data. Namely, for each ﬁxed τ ∈ R, ω ∈ Ω, the solution u(t, τ, ω, .) is continuous from L2(RN ) to L2(RN ) for all t ≥ τ, and in addition u(t, τ, ω, .) is continuous from L2(RN ) to Lp(RN ) for all t > τ. 5.2. Continuity w.r.t. the initial data in Hs(RN ). We ﬁrst present a lemma in [55]. Lemma 5.6. Let y, g and h be tree nonnegative and locally integrable functions on dy R such that dt is also locally integrable and dy(t) + νy(t) + g(t) ≤ h(t), t ∈ , dt R for some constant ν ≥ 0. Then (i) for arbitrary r > 0 and τ ∈ R, τ τ 1 Z Z y(τ) ≤ eν(t−τ)y(t)dt + eν(t−τ)h(t)dt. r τ−r τ−r (ii) for arbitrary r, ε > 0 and σ ∈ [τ − r, τ], τ τ Z eνr + 1 Z y(σ) + eν(t−τ)g(t)dt ≤ eν(t−τ)y(t)dt ε τ−r τ−r−ε τ Z + (eνr + 2) eν(t−τ)h(t)dt.

τ−r−ε In particular, (i) and (ii) hold for ν = 0.

Lemma 5.7. Let (25) and (26) hold. Then for every τ ∈ R and ω ∈ Ω, the solution of problem (23) and (24) satisﬁes p kv(t, τ, ω, vτ )kp ≤ G(t, τ, ω, uτ ), (108) and t Z 2p−2 ku(r, τ, ω, uτ )k2p−2dr ≤ G(t, τ, ω, uτ ), (109)

t+τ 2 where t t c Z Z G(t, τ, ω, u ) ≤ kg(r, .)k2dr + J(ϑ ω)dr + ku k2 + c|y(ϑ ω)|2 τ t − τ r τ τ τ τ 3422 WENQIANG ZHAO

t t Z Z 2 + c kg(r, .)k dr + c J(ϑrω)dr. (110) τ τ Proof. By (47), we have d kvk2 + ckvkp ≤ ckg(t, .)k2 + cJ(ϑ ω), (111) dt p t where J(ϑtω) is as in (33). Integrating (111) from τ to t, we get t t Z Z p 2 2 kv(r)kpdr ≤ c (kg(r, .)k + J(ϑrω))dr + kvτ k τ τ t t Z Z 2 2 2 ≤ c( kg(r, .)k dr + J(ϑrω)dr + |y(ϑτ ω)| + kuτ k ). (112) τ τ Take the inner productof (36) in L2(RN ) with |v|p−2v, we obtain 1 d Z kvkp + λkvkp + ((−∆)sv, |v|p−2v) + f(t, x, u)|v|p−2vdx p dt p p N R Z p−2 s p−2 = g(t, x)|v| vdx − ((−∆) z(ϑtω), |v| v). (113)

N R By Lemma 5.1, it gives ((−∆)sv, |v|p−2v) C(N, s) Z (v(x) − v(y))(|v(x)|p−2v(x) − |v(y)|p−2v(y)) = dxdy 2 |x − y|N+2s 2N R C(N, s) Z |v(x) − v(y)|p ≥ c dxdy ≥ 0. (114) 1,p 2 |x − y|N+2s 2N R For the last term on the right hand side of (113), we have s p−2 ((−∆) z(ϑtω), |v| v) C(N, s) Z y(ϑ ω)(φ(x) − φ(y))(|v(x)|p−2v(x) − |v(y)|p−2v(y)) = t dxdy 2 |x − y|N+2s 2N R

C(N, s) Z y(ϑ ω)(φ(x) − φ(y))|v(x)|p−2v(x) = t dxdy 2 |x − y|N+2s 2N R C(N, s) Z y(ϑ ω)(φ(x) − φ(y))|v(y)|p−2v(y) − t dxdy. (115) 2 |x − y|N+2s 2N R Tanks to (−∆)sφ ∈ L2(RN ), by H¨olderinequality , we have C(N, s) Z y(ϑ ω)(φ(x) − φ(y))|v(t)(x)|p−2v(t)(x) t dxdy 2 |x − y|N+2s 2N R PULLBACK ATTRACTORS FOR BI-SPATIAL SYSTEMS 3423

1 C(N, s) n Z Z y(ϑ ω)(φ(x) − φ(y)) 2 o 2 ≤ kvkp−1 t dy dx 2 2p−2 |x − y|N+2s N N R R C(N, s) = kvkp−1 (y(ϑ ω)k(−∆)sφk) 2 2p−2 t γ ≤ kvk2p−2 + c|y(ϑ ω)|2. (116) 2p+4 2p−2 t Then it follows from (115) and (116) that γ ((−∆)sz(ϑ ω), |v|p−2v) ≤ kvk2p−2 + c|y(ϑ ω)|2. (117) t 2p+3 2p−2 t By (43), and using Young inequality repeatedly, we infer that, along with φ ∈ L2(RN ) ∩ L2p−2(RN ), Z γ Z f(t, x, u)|v|p−2v ≥ kvk2p−2 − 2 (|ψ (t, x)| + |ψ (t, x)|)|v|pdx 2p 1 2 N N R R Z 2 p−2 − (2|ψ1(t, x)| + 3|ψ2(t, x)| |φ(x)y(ϑtω)| |v| dx

N R Z γ − (c|ψ (t, x)|p + )|φ(x)y(ϑ ω)|p|v|p−2dx 2 2 t N R γ ≥ kvk2p−2 − ckvkp − c|y(ϑ ω)|p − c|y(ϑ ω)|2p−2. (118) 2p+1 t t On the other hand, we have Z γ g(t, x)|v|p−2vdx ≤ kvk2p−2 + ckg(t, .)k2. (119) 2p+3 2p−2 N R Incorporate (114) and (117)-(119) into (113) to yield d kvkp + ckvk2p−2 ≤ c(kvkp + kg(t, .)k2 + cJ(ϑ ω). (120) dt p 2p−2 p t t−τ By applying Lemma 5.6( ii) for the cases ν = 0 and ε = 2 to (120), we get t t Z c Z kv(t)kp + kv(r)k2p−2dr ≤ kv(r)kpdr p 2p−2 t − τ p t+τ τ 2 t Z 2 + c (kg(r, .)k + J(ϑrω))dr, (121) τ which by u(t) = v(t) + φ(x)y(ϑtω) obviously implies that t t t t Z c Z Z Z kv(t)kp + ku(r)k2p−2dr ≤ kv(r)kpdr + c kg(r, .)k2 + c J(ϑ ω)dr, p 2p−2 t − τ p r t+τ τ τ τ 2 which and (112) together give t Z p 2p−2 kv(t)kp + ku(r)k2p−2dr

t+τ 2 3424 WENQIANG ZHAO

t t c Z Z ≤ kg(r, .)k2dr + J(ϑ ω)dr + ku k2 + c|y(ϑ ω)|2 t − τ r τ τ τ τ t t Z Z 2 + c kg(r, .)k dr + c J(ϑrω)dr, τ τ which completes the proof.

Remark 1. Lemma 5.7 manifests that for every t > 0, τ ∈ R and ω ∈ Ω, the mapping ϕ(t, τ, ω, .) = u(t + τ, τ, ω, .) maps L2(RN ) into Lp(RN ), namely, it makes sense to investigate the dynamics of the random cocyle ϕ on Lp(RN ).

Lemma 5.8. Let (25) and (26) hold. Let ui(t) = ui(t, τ, ω, uτ,i) be the solution of problem (23) and (24) with respect to the initial value uτ,i, i = 1, 2. Given τ ∈ R 1 and ω ∈ Ω, there exist a positive deterministic constant Ct−τ depending on p, λ and 2 µ0; and a random constant Ct−τ depending on p, λ, µ0 and G(t, τ, ω, uτ,i) such that, for all t > τ

s 2 2 2 1 2 2 p−1 k(−∆) u¯(t, τ, ω, u¯τ )k ≤ Ct−τ ku¯τ k + Ct−τ ku¯τ k , where u(t) = u1(t) − u2(t) and G(t, τ, ω, uτ,i) is as in (110). 2 N Proof. Taking the inner product of (89) in L (R ) withu ¯t, we get Z Z 1 d s 2 2 k(−∆) 2 u¯k + ku¯ k = −λ u¯u¯ dx − (f(t, x, u ) − f(t, x, u ))¯u dx 2 dt t t 1 2 t N N R R 1 Z ≤ ku¯ k2 + cku¯k2 + (f(t, x, u ) − f(t, x, u ))¯u dx . (122) 4 t 1 2 t N R By employing first (26), then Young inequality and Hölder inequality, it follows that Z Z 1 2 p−2 p−2 (f(t, x, u ) − f(t, x, u ))¯utdx ≤ c (1 + |u1| + |u2| )|u¯||u¯t|dx N N R R 1 Z ≤ ku¯ k2 + cku¯k2 + c (|u |2p−4 + |u |2p−4)|u¯|2dx 4 t 1 2 N R 1 ≤ ku¯ k2 + cku¯k2 + c(ku k2p−4 + ku k2p−4)ku¯k2 . (123) 4 t 1 2p−2 2 2p−2 2p−2 Combine (122) and (123) to find that, replacing t by r,

d s 2 2 2p−4 2p−4 2 k(−∆) 2 u¯(r)k ≤ cku¯(r)k + c(ku (r)k + ku (r)k )ku¯(r)k . (124) dr 1 2p−2 2 2p−2 2p−2 2p t+τ p−1 t+τ Multiplying (124) by (r − 2 ) with r ∈ [ 2 , t], we get

t + τ 2p d s 2 t + τ 2p 2 (r − ) p−1 k(−∆) 2 u¯(r)k ≤ c(r − ) p−1 ku¯(r)k 2 dr 2 t + τ 2p 2p−4 2p−4 2 + c(r − ) p−1 (ku (r)k + ku (r)k )ku¯(r)k 2 1 2p−2 2 2p−2 2p−2 2p 2 ≤ c(t − τ) p−1 ku¯(r)k

t + τ 2p 2 2p−4 2p−4 + c(r − ) p−1 ku¯(r)k (ku (r)k + ku (r)k ). (125) 2 2p−2 1 2p−2 2 2p−2 PULLBACK ATTRACTORS FOR BI-SPATIAL SYSTEMS 3425

Note that by integration by parts, we have t Z t + τ 2p d s 2 t − τ 2p s 2 (r − ) p−1 k(−∆) 2 u¯(r)k dr = ( ) p−1 k(−∆) 2 u¯(r)k 2 dr 2 t+τ 2 t Z 2p t + τ p+1 s 2 − (r − ) p−1 k(−∆) 2 u¯(r)k dr. (126) p − 1 2 t+τ 2 t+τ Therefore integrating (125) over the interval [ 2 , t], we obtain, along with (126),

t − τ 2p s 2 ( ) p−1 k(−∆) 2 u¯(t)k 2 t t Z Z 2p t − τ p+1 s 2 2p 2 ≤ ( ) p−1 k(−∆) 2 u¯(r)k ds + c(t − τ) p−1 ku¯(r)k dr p − 1 2 t+τ t+τ 2 2 t Z 2p p−1 2 2p−4 2p−4 + c (r − τ) ku¯(r)k2p−2(ku1(r)k2p−2 + ku2(r)k2p−2)dr, (127)

t+τ 2 2p 2p p−1 t+τ p−1 t+τ where we use (r − τ) ≥ (r − 2 ) for any r ∈ [ 2 , t]. Integrating (93) from τ to t, and in conjunction with (94), we get t t Z s Z 2 2 2 µ0(t−τ) 2 k(−∆) 2 u¯(r)k dr ≤ ku¯(τ)k + µ0 ku¯(r)k dr ≤ (1 + e )ku¯τ k , τ τ which gives, for all t > τ, t Z 2p t − τ p+1 s 2 p+1 µ (t−τ) 2 ( ) p−1 k(−∆) 2 u¯(r)k dr ≤ c(t − τ) p−1 (1 + e 0 )ku¯ k . (128) p − 1 2 τ t+τ 2 In terms of (94), we also have t 2p Z 2p 2 µ0(t−τ) 2 c(t − τ) p−1 ku¯(r)k dr ≤ c(t − τ) p−1 e ku¯τ k , t > τ. (129)

t+τ 2 Note that by H¨olderinequality, it follows that t Z 2p p−1 2 2p−4 2p−4 (r − τ) ku¯(r)k2p−2(ku1(r)k2p−2 + ku2(r)k2p−2)dr

t+τ 2 t Z p−2 2p−2 2p−2 p−1 ≤ c (ku1(r)k2p−2 + ku2(r)k2p−2)dr

t+τ 2 t Z 1 2p 2p−2 p−1 × (r − τ) ku¯(r)k2p−2dr . (130) τ By Lemma 5.7, we have 3426 WENQIANG ZHAO

t Z 2p−2 2p−2 (ku1(r)k2p−2 + ku2(r)k2p−2)dr ≤ G(t, τ, ω, uτ,1) + G(t, τ, ω, uτ,2), (131)

t+τ 2 where G(t, τ, ω, uτ,i)(i = 1, 2) is given by (110). By Lemma 5.4, we ﬁnd that, there exist a deterministic constant Ct−τ such that T Z 2p 2p−2 2 (r − τ) ku(r)k2p−2dr ≤ Ct−τ ku¯τ k , (132) τ By a combination of (130) to (132), it follows that t Z 2p p−1 2 2p−4 2p−4 (r − τ) ku¯(r)k2p−2(ku1(r)k2p−2 + ku2(r)k2p−2)dr

t+τ 2 p−2 1 2 ≤ c(G(t, τ, ω, uτ,1) + G(t, τ, ω, uτ,2)) p−1 (Ct−τ ) p−1 ku¯τ k p−1 . (133) Incorporate (128), (129) and (133) into (127) to ﬁnd

t − τ 2p s 2 p+1 µ (t−τ) ( ) p−1 k(−∆) 2 u¯(t)k ≤ c((t − τ) p−1 (1 + µ e 0 ) 2 0 2p µ0(t−τ) 2 + c(t − τ) p−1 e )ku¯τ k + c(G(t, τ, ω, uτ,1)

p−2 1 2 + G(t, τ, ω, uτ,2)) p−1 (Ct−τ ) p−1 ku¯τ k p−1 , and therefore we have s 2 c µ (t−τ) 2 µ (t−τ) 2 k(−∆) 2 u¯(t)k ≤ (1 + e 0 )ku¯ k + ce 0 ku¯ k t − τ τ τ p−2 1 c(G(t, τ, ω, uτ,1) + G(t, τ, ω, uτ,2)) p−1 (Ct−τ ) p−1 2 + ku¯ k p−1 , (t − τ)2p/p−1 τ which is the desired result. In terms of Lemma 5.8 and (94), we have the continuity of solutions in the fractional Hs(RN ) for s ∈ (0, 1). Theorem 5.9. Under the conditions (25)-(29), the solution u of problem (23) and (24) is bi-spatial (L2(RN ),Hs(RN ))-continuous with respect to the initial data. 6. Measurability of solution in the sample ω. In this section, we study the measurability of solution with respect to the sample ω, which is a crucial condition to prove the measurability of pullback attractor in the regular space Lp(RN ) and Hs(RN ). To this end, the topological property and structure of the sample space Ω are analysed and furthermore the exponent growth property of the Ornstein- Uhlenbeck process deﬁned by (31) is substantially employed.

6.1. Properties of the sample space Ω. For k ∈ N, deﬁne the set ζ|t| Ωk = {ω ∈ Ω: |ω(t)| ≤ ke , t ∈ R}, ∀ k ∈ N, (134) where 0 < ζ(2p − 2) < λ. It is easy to check that Ω = ∪k∈NΩk and Ωk ∈ F.

Let FΩk be the trace σ-algebra of F with respect to Ωk, and let BΩk (ω0, r), ω0 ∈ Ωk, r > 0 be a ball in Ωk. This ball can be generated by BΩ(ω0, r) ∩ Ωk, where

BΩ(ω0, r) is a ball in Ω, namely, BΩk (ω0, r) = BΩ(ω0, r) ∩ Ωk. The same is true for PULLBACK ATTRACTORS FOR BI-SPATIAL SYSTEMS 3427

all open sets in Ωk. Therefore FΩk is just the Borel σ-algebra of Ωk. Furthermore, since Ωk ∈ F, then we have FΩk ⊂ F. The following is concerning some important properties about Ωk, k ∈ N, see [9, 10].

Lemma 6.1. (i) For every k ∈ N, the mapping Ωk 3 ω → y(ϑtω) is uniformly continuous on any t-bounded intervals, where y(ϑtω) is the Ornstein-Uhlenbeck process defined by (31). (ii) For every k ∈ N, the space (Ωk, ρ) is a Polish space, namely, a completely separable metric space, where ρ is the Frechétmetric defined by

X 1 kω1 − ω2kn ρ(ω1, ω2) = n , kω1 − ω2kn = sup kω1(t) − ω2(t)k. (135) 2 1 + kω1 − ω2kn −n≤t≤n n∈N To prove the continuity of solutions in ω, we also need the following lemma about the property of J(ϑtω), where J(ϑtω) is as in (33). The proof is similar to [10].

Lemma 6.3. Let ϕ be a mapping from Ω to Λ such that ϕ :Ωk → Λ is (FΩk , B(Λ))- measurable for every ﬁxed k ∈ N, then ϕ :Ω → Λ is (F, B(Λ))-measurable.

Proof. Note that Ω = ∪k∈NΩk. Then for any open set O ∈ B(Λ), we have −1 ϕ (O) = {ω ∈ Ω: ϕ(ω) ∈ O} = ∪k∈N{ω ∈ Ωk : ϕ(ω) ∈ O}.

Let Dk = {ω ∈ Ωk : ϕ(ω) ∈ O}. Since ϕ is (FΩk , B(Λ))-measurable, then Dk ∈ FΩk . −1 Consider that FΩk ⊂ F. Then ϕ (O) = ∪k∈NDk ∈ F. Thus ϕ :Ω → Λ is (F, B(Λ))-measurable.

6.2. Continuity w.r.t. the sample ω in L2(RN ). In order to obtain a measurable attractor in L2(RN ), it is necessary to prove the measurability of ϕ(t, τ, ., x) with respect to F in the target space, for every fixed t > 0, τ ∈ R, x ∈ L2(RN ). Nevertheless, this should be achieved by the continuous dependence of solutions on 2 N ω ∈ Ωk in the initial space L (R ), k ∈ N. 2 N Given ω1, ω2 ∈ Ωk, for any fixed t ≥ τ, τ ∈ R and vτ ∈ L (R ), denote by v1(t, τ, ω1, vτ ) and v2(t, τ, ω2, vτ ) the solutions of problem (23) and (24) at the sam- ples ω1 and ω2, respectively. Let V¯ = v1 − v2. Then V¯ is the solution to the following difference equation: dV¯ + λV¯ + (−∆)sV¯ + f(t, x, v + z(ϑ ω )) − f(t, x, v + z(ϑ ω )) dt 1 t 1 2 t 2 s = −(−∆) (z(ϑtω1) − z(ϑtω2)) (136) with the initial value V¯τ = 0. For convenience, we writey ¯ = y(ϑtω1) − y(ϑtω2) and U¯ = u1(t, τ, ω1, uτ ) − u2(t, τ, ω2, uτ ). Then we have 3428 WENQIANG ZHAO

2 N Lemma 6.4. Let t ≥ τ, τ ∈ R and uτ ∈ L (R ). Suppose that u(t, τ, ω, uτ ) is the solution to problems (23) and (24). Then the mapping ω → u(t, τ, ω, uτ ) is 2 N continuous from (Ωk, ρ) to L (R ) for each k ∈ N. Proof. By (136), we have

1 d 2 2 s 2 kV¯ k + λkV¯ k + k(−∆) 2 V¯ k 2 dt Z + (f(t, x, v1 + z(ϑtω1)) − f(t, x, v2 + z(ϑtω2)))V¯ dx

N R s = −((−∆) (z(ϑtω1) − z(ϑtω2)), V¯ ). (137) For the nonlinearity, by (25) and (26), we have

(f(t, x, u1) − f(t, x, u2))V¯

= (f(t, x, u1) − f(t, x, u2))U¯ − (f(t, x, u1) − f(t, x, u2))¯yφ(x) p 2 p−2 p−2 ≥ γ|U¯| − ψ1(t, x)|U¯| − ψ2(t, x)U¯(1 + |u1| + |u2| )|y¯||φ(x)| p 2 2 2 ≥ γ|U¯| − (ψ1(t, x) + ψ2(t, x))|U¯| − ψ2(t, x)|y¯| |φ(x)| p−2 p−2 − ψ2(t, x)U¯(|u1| + |u2| )|y¯||φ(x)|, (138) where ui = vi + z(ϑtωi), i = 1, 2 and U¯ = V¯ + φy.¯ By the generalized Young inequality: 1 1 1 |abc| ≤ |a|p + |b|q + |c|r if + + = 1, for any p, q, r > 1, (139) p q r we deduce that p−2 p−2 U¯(|u1| + |u2| )|y¯||φ(x)| p−1 p−1 p−2 p−2 ≤ (|u1| + |u2| + |u1||u2| + |u2||u1| )|φ(x)||y¯| p p p ≤ c(|u1| + |u2| + |φ| )|y¯|. (140) Therefore by (138) and (140), it follows that p 2 2 2 (f(t, x, u1) − f(t, x, u2))V¯ ≥ γ|U¯| − (ψ1(t, x) + ψ2(t, x))|U¯| − ψ2(t, x)|y¯| |φ(x)| p p p − cψ2(t, x)(|u1| + |u2| + |φ| )|y¯|, (141)

p N ∞ N+1 from which and φ ∈ L (R ), ψi ∈ L (R ), i = 1, 2, we derive Z ¯ ¯ p ¯ 2 2 (f(t, x, u1) − f(t, x, u2))V dx ≥ γkUkp − ckV k − c|y¯|

N R p p p − c(ku1kp + ku2kp + kφkp)|y¯|. (142) For the term on the right hand side of (137), we have

1 Z |φ(x) − φ(y)|2 + C(N, s)|y¯|2 dxdy 4 |x − y|N+2s 2N R 1 s 2 2 ≤ k(−∆) 2 V¯ k + c|y¯| . (143) 2 By a combination of (137), (142) and (143), it follows that

d 2 2 s 2 p kV¯ k + λkV¯ k + k(−∆) 2 V¯ k + γkU¯k dt p ¯ 2 2 p p ≤ ckV k + c|y¯| + c(ku1kp + ku2kp)|y¯|. (144) By using Gronwall lemma to (144) over the intervals [τ, t], we get

t t Z s Z ¯ 2 ct −cr 2 ¯ 2 ¯ 2 ct −cr ¯ p kV (t)k + e e (k(−∆) V (r)k + λkV (r)k )dr + γe e kU(r)kpdr τ τ t t Z Z ct −cr 2 ct −cr p p ≤ ce e |y¯(r)| dr + ce e (ku1(r)kp + ku2(r)kp)|y¯(r)|dr. (145) τ τ

Due to the deﬁnition of (Ωk, ρ), we know that if ρ(ω2, ω1) → 0, we have ω2 → ω1 on any t-bounded intervals. And hence by Lemma 6.1, the mapping y(ϑtω):Ωk 3 ω → R is uniformly continuous on the arbitrary bounded intervals [τ, t], for every k ∈ N, t ct R −cr 2 and therefore we deduce that ce e |y¯(r)| dr converges to zero as ω2 → ω1 in τ (Ωk, ρ). On the other hand, in terms of (112), in conjunction with Lemma 6.2, we ﬁnd that, for ωi ∈ Ωk, i = 1, 2,

t t Z Z −cr p −cτ p e kui(r)kpds ≤ e kui(r)kpds τ τ t t Z Z −cτ 2 2 2 ≤ ce ( kg(r, .)k dr + J(ϑrωi)dr + kuτ k + c|y(ϑτ ωi)| ) τ τ t t Z Z −cτ 2 λ|r| 2 λ|τ| ≤ ce ( kg(r, .)k dr + L e dr + kuτ k + cLe ) τ τ =: M(t, τ), (146) which is independent of ωi, i = 1, 2. In view of (146) and by Lebesgue dominated convergence, we have

t Z ct −cr p p ce e (ku1(r)kp + ku2(r)kp)|y¯(r)|dr → 0, τ as ω2 → ω1 in (Ωk, ρ). Consequently, from (145) we get

t Z 2 ct −cr s 2 2 kV¯ (t)k + e e (k(−∆) 2 V¯ (r)k + λkV¯ (r)k )dr

τ 3430 WENQIANG ZHAO

t Z ct −cr ¯ p + e e kU(r)kpdr → 0, (147) τ as ω2 → ω1 in (Ωk, ρ). Note that ui = vi + z(ϑtωi), i = 1, 2. Then we have

2 2 2 2 2 kU¯(t)k = ku1(t) − u2(t)k ≤ 2kV¯ (t)k + 2kφk |y¯| , which and (147) together give the desired result.

From the convergence in the above (147), we formulate a lemma which is useful in the sequel arguments. ¯ Lemma 6.5. For every ω1, ω2 ∈ Ωk for every ﬁxed k ∈ N. Let V (t) be the solution to (136). Then we have

t Z s 2 2 (k(−∆) 2 V¯ (r)k + kV¯ (r)k )dr → 0,

τ as ω2 → ω1.

6.3. Continuity w.r.t. the sample ω in Lp(RN ). In this subsection, we prove the continuous of solutions to problem (23) and (24) in Lp(RN ) with respect to ω ∈ Ωk for every k ∈ N, which is crucial for us to derive the measurability of pullback attractor in Lp(RN ). The proof is technical and involved. 2 N Lemma 6.6. Let t > τ, τ ∈ R and uτ ∈ L (R ). Suppose that u(t, τ, ω, uτ ) is the solution to problem (23) and (24). Then the mapping ω → u(t, τ, ω, vτ ) is p N continuous from (Ωk, ρ) to L (R ), p ≥ 2, for each k ∈ N. Proof. By (136), we have 1 d Z kV¯ kp + λkV¯ kp + (f(t, x, v + z(ϑ ω )) − f(t, x, v + z(ϑ ω )))|V¯ |p−2V¯ dx p dt p p 1 t 1 2 t 2 N R s p−2 s p−2 = −((−∆) V,¯ |V¯ | V¯ ) − ((−∆) (z(ϑtω1) − z(ϑtω2)), |V¯ | V¯ ). (148)

We need to estimate the terms in (148). First, from (141) and the relation |U¯|p ≥ 21−pV¯ |p − |y¯|p|φ|p it follows that

p 2 (f(t, x, u1) − f(t, x, u2))V¯ ≥ γ|U¯| − (ψ1(t, x) + ψ2(t, x))|U¯| 2 2 p p p − ψ2(t, x)|y¯| |φ(x)| − cψ2(t, x)(|u1| + |u2| + |φ| )|y¯| γ ≥ |V¯ |p − |y¯|p|φ|p − (ψ (t, x) + ψ (t, x))|U¯|2 2p−1 1 2 2 2 p p p − ψ2(t, x)|y¯| |φ(x)| − cψ2(t, x)(|u1| + |u2| + |φ| )|y¯|, (149) which gives γ (f(t, x, u ) − f(t, x, u ))|V¯ |p−2V¯ ≥ |V¯ |2p−2 − |y¯|p|φ|p|V¯ |p−2 1 2 2p−1 2 p−2 2 2 p−2 − (ψ1(t, x) + ψ2(t, x))|U¯| )|V¯ | − ψ2(t, x)|y¯| |φ(x)| |V¯ | p p p p−2 − cψ2(t, x)(|u1| + |u2| + |φ| ))|V¯ | |y¯|. (150) PULLBACK ATTRACTORS FOR BI-SPATIAL SYSTEMS 3431

The terms on the right hand side of (150) can be estimated by Young inequality, namely, we have

 p p ¯ p−2 γ ¯ 2p−2 2p−2 2p−2 |y¯| |φ| V | ≤ 2p+1 |V | + c|φ| |y¯| ;  p  ¯ 2 ¯ p−2 λ ¯ p 2 ¯ p ψi(t, x))|U| )|V | ≤ 4 |V | + |ψi| |U| , i = 1, 2;  p 2 2 ¯ p−2 λ ¯ p 2 p p ψ2(t, x)|y¯| |φ(x)| |V | ≤ 4 |V | + |ψ2| |φ| |y¯| ; cψ (t, x)(|u |p + |u |p + |φ|p))|V¯ |p−2|y¯| ≤ γ |V¯ |2p−2  2 1 2 2p+1  2p−2 2p−2 2p−2 2p−2 2p−2  +c(ψ2(t, x)) p (|u1| + |u2| + |φ| )|y¯| p , from which and (150), the nonlinearity in (148) is estimated as

Z γ λ (f(t, x, u ) − f(t, x, u ))|V¯ |p−2V¯ dx ≥ kV¯ k2p−2 − kV¯ kp 1 2 2p 2p−2 2 p N R 2p−2 ¯ p p 2p−2 p − c(kUkp + |y¯| + |y¯| + |y¯| ) 2p−2 2p−2 2p−2 p − c(ku1k2p−2 + ku2k2p−2)|y¯| . (151)

For the ﬁrst term on the right hand of (148), we have

((−∆)sV,¯ |V¯ |p−2V¯ ) 1 Z (V¯ (x) − V¯ (y))(|V¯ (x)|p−2V¯ (x) − |V¯ (y)|p−2V¯ (y)) = C(N, s) dxdy 2 |x − y|N+2s 2N R 1 Z |V¯ (x) − V¯ (y)|p ≥ C(N, s) dxdy ≥ 0. (152) 2 |x − y|N+2s 2N R On the other hand, for the second term on the right hand of (148), by a similar argument as in (117), we deduce

s p−2 − ((−∆) (z(ϑtω1) − z(ϑtω2)), |V¯ | V¯ ) 1 Z y¯(φ(x) − φ(y))(|V¯ (x)|p−2V¯ (x) − |V¯ (y)|p−2V¯ (y)) = − C(N, s) dxdy 2 |x − y|N+2s 2N R 1 Z y¯(φ(x) − φ(y))|V¯ (x)|p−2V¯ (x) ≤ C(N, s) dxdy 2 |x − y|N+2s 2N R 1 Z y¯(φ(x) − φ(y))|V¯ (y)|p−2V¯ (y) + C(N, s) dxdy 2 |x − y|N+2s 2N R γ ≤ C(N, s)|y¯|kV¯ kp−1 k(−∆)sφk ≤ kV¯ k2p−2 + c|y¯|2. (153) 2p−2 2p+1 2p−2 Therefore incorporation (151)- (153) into (148) to ﬁnd

d p γ 2p−2 p 2 p 2p−2 2p−2 kV¯ k + kV¯ k ≤ ckV¯ k + c(|y¯| + |y¯| + |y¯| + |y¯| p ) dt p 2p+1 2p−2 p 2p−2 2p−2 2p−2 p + c(ku1k2p−2 + ku2k2p−2)|y¯| . (154) 3432 WENQIANG ZHAO

t+τ Employing Gronwall lemma to (154) over the intervals (ς, t) with ς ∈ [ 2 , t], we get

t γ Z kV¯ (t)kp + ect e−crkV¯ (r)k2p−2dr p 2p+1 2p−2 ς t Z 2p−2 ct −cς ¯ p ct −cr 2 p 2p−2 p ≤ e e kV (ς)kp + ce e (|y¯(r)| + |y¯(r)| + |y¯| + |y¯(r)| )dr τ t Z 2p−2 ct −cr 2p−2 2p−2 p + ce e (ku1(r)k2p−2 + ku2(r)k2p−2)|y¯(r)| dr. (155)

t+τ 2

t+τ Integrate the last inequality of (155) with respect to ς over the intervals ( 2 , t) to yield

t t t γ 2ect Z Z 2 Z kV¯ (t)kp + e−crkV¯ (r)k2p−2drdς ≤ ect e−crkV¯ (r)kpdr p 2p+1 t − τ 2p−2 t − τ p t+τ ς τ 2 t Z ct −cr 2 p 2p−2 + ce e (|y¯(r)| + |y¯(r)| + |y¯(r)| p )dr

τ t Z 2p−2 ct −cr 2p−2 2p−2 p + ce e (ku1(r)k2p−2 + ku2(r)k2p−2)|y¯(r)| dr

t+τ 2

= I1 + I2 + I3. (156) By (147), we deduce that, for any t > τ,

t 2 Z I = ect e−crkV¯ (r)kpdr → 0, (157) 1 t − τ p τ as ω2 → ω1. By the Lebesgue dominated convergence theorem, it follows that, for any t > τ,

t Z ct −cr 2 p 2p−2 2p−2 I2 = ce e (|y¯(r)| + |y¯(r)| + |y¯(r)| + |y¯(r)| p )dr → 0, (158) τ as ω2 → ω1. On the other hand, in terms of (110), in conjunction with Lemma 6.2, we ﬁnd that, for ωi ∈ Ωk, i = 1, 2,

t t c Z Z G(t, τ, ω , u ) ≤ kg(r, .)k2dr + J(ϑ ω )dr + ku k2 + c|y(ϑ ω )|2 i τ t − τ r i τ τ i τ τ t t Z Z 2 + c kg(r, .)k dr + c J(ϑrωi))dr τ τ PULLBACK ATTRACTORS FOR BI-SPATIAL SYSTEMS 3433

t t c Z Z ≤ kg(r, .)k2dr + L eλ|r|dr + ku k2 + cLeλ|τ| t − τ τ τ τ t t Z Z + c kg(r, .)k2dr + cL eλ|r|dr = N(t, τ), (159)

τ τ which is independent of ωi, i = 1, 2. So by (109) and (159), we ﬁnd, for any ωi ∈ Ωk, i = 1, 2, t t Z Z −cr 2p−2 −cτ 2p−2 −cτ e kui(r)k2p−2dr ≤ e kui(r)k2p−2dr ≤ e N(t, τ). (160)

t+τ t+τ 2 2 Therefore, the Lebesgue dominated convergence theorem can be applied to yield t Z 2p−2 ct −cr 2p−2 2p−2 p I3 = ce e (ku1(r)k2p−2 + ku2(r)k2p−2)|y¯(r)| dr → 0, (161)

t+τ 2 as ω2 → ω1. This achieves the proof. From the above proof, we get the following lemma which is useful for our sequel discussions.

2 N Lemma 6.7. Let t > τ, τ ∈ R, ω ∈ Ω and uτ ∈ L (R ). Suppose that u(t, τ, ωi, uτ ) is the solution to problem (23) and (24) at the sample ωi ∈ Ωk for each ﬁxed k ∈ N, i = 1, 2. Then t Z t + τ (r − )kU¯(r)k2p−2dr → 0, 2 2p−2 t+τ 2 as ω2 → ω1, where U¯(r) = u(r, τ, ω1, uτ ) − u(r, τ, ω2, uτ ). Proof. From (156), it follow that t t Z Z −cr ¯ 2p−2 e kV (r)k2p−2drdς → 0, (162) t+τ ς 2 as ω2 → ω1. By exchanging the integral order of the integral in (162), we write t t t Z Z Z t + τ e−crkV¯ (r)k2p−2drdς = e−cr(r − )kV¯ (r)k2p−2dr, 2p−2 2 2p−2 t+τ ς t+τ 2 2 from which and (162) we derive t Z t + τ e−cr(r − )kV¯ (r)k2p−2dr → 0, (163) 2 2p−2 t+τ 2 ¯ ¯ 2p−2 N as ω2 → ω1. Note that U(t) = V (t) + φy¯(t) and φ ∈ L (R ). Then by (163) and Lebesgue dominate convergence theorem, we get 3434 WENQIANG ZHAO

t t Z t + τ Z t + τ e−cr(r − )kU¯(r)k2p−2dr ≤ c e−cr(r − )kV¯ (r)k2p−2dr 2 2p−2 2 2p−2 t+τ t+τ 2 2 t Z t + τ + c e−cr(r − )|y¯(r)|2p−2dr → 0, (164) 2 t+τ 2 as ω2 → ω1. This obviously implies that

t Z t + τ (r − )kU¯(r)k2p−2dr → 0, 2 2p−2 t+τ 2 as ω2 → ω1, which completes the proof.

6.4. Continuity w.r.t. the sample ω in Hs(RN ). To consider the measurability of solution in Hs(RN ), 0 < s < 1, we ﬁrst prove the continuity of solution with respect to sample ω for ω ∈ (Ωk, ρ).

2 N Lemma 6.8. Let t > τ, τ ∈ R, ω ∈ Ω and uτ ∈ L (R ). Suppose that u(t, τ, ω, uτ ) is the solution to problem (23) and (24). Then the mapping ω → u(t, τ, ω, uτ ) is s N continuous from (Ωk, ρ) to H (R ), 0 < s < 1, for each k ∈ N. Proof. We begin with by proving some priori estimates. First, by (26), we infer that Z Z ¯ p−2 p−2 ¯ ¯ (f1(t, x, u1) − f1(t, x, u2))Vtdx ≤ c (1 + |u1| + |u2| )|U||Vt|dx N N R R 1 Z ≤ kV¯ k2 + ckU¯k2 + c (|u |2p−4 + |u |2p−4|)U¯|2dx 4 t 1 2 N R 1 ≤ kV¯ k2 + ckU¯k2 + c(ku k2p−4 + ku k2p−4)kU¯k2 , (165) 4 t 1 2p−2 2 2p−2 2p−2 where V¯t is the derivative of V¯ with respect to the time t. By H¨olderinequality, Young inequality and (−∆)sφ ∈ L2(RN ), it follows that

s ¯ − (−∆) (z(ϑtω1) − z(ϑtω2)), Vt) 1 Z y¯(φ(x) − φ(y))(V¯ (x) − V¯ (y)) = C(N, s) t t dxdy 2 |x − y|N+2s 2N R 1 Z y¯(φ(x) − φ(y))V¯ (x) ≤ C(N, s) t dxdy 2 |x − y|N+2s 2N R 1 Z y¯(φ(x) − φ(y))V¯ (y) + C(N, s) t dxdy 2 |x − y|N+2s 2N R 1 ≤ C(N, s)|y¯|kV¯ kk(−∆)sφk ≤ kV¯ k2 + c|y¯|2. (166) t 4 t PULLBACK ATTRACTORS FOR BI-SPATIAL SYSTEMS 3435

2 N ¯ Taking the inner product of (136) in L (R ) with Vt, from (165) and (166) we deduce that, with t being replaced by r,

d s 2 2 2 k(−∆) 2 V¯ (r)k ≤ ckV¯ (r)k + c|y¯(r)| dr 2p−4 2p−4 ¯ 2 + c(ku1(r)k2p−2 + ku2(r)k2p−2)kU(r)k2p−2. (167) t+τ t+τ Multiply (167) by r − 2 with r ∈ ( 2 , t) to yield

t + τ d s 2 t + τ 2 t + τ 2 (r − ) k(−∆) 2 V¯ (r)k ≤ c(r − )kV¯ (r)k + c(r − )|y¯(r)| 2 dt 2 2 t + τ + c(r − )(ku (r)k2p−4 + ku (r)k2p−4)kU¯(r)k2 2 1 2p−2 2 2p−2 2p−2 ≤ c(t − τ)kV¯ (r)k2 + c(t − τ)|y¯(r)|2 t + τ + c(r − )(ku (r)k2p−4 + ku (r)k2p−4)kU¯(r)k2 . (168) 2 1 2p−2 2 2p−2 2p−2 t+τ By integrating (168) over the intervals ( 2 , t), we obtain t t Z Z t − τ s 2 s 2 2 2 k(−∆) 2 V¯ (t)k ≤ k(−∆) 2 V¯ (r)k dr + c(t − τ) (kV¯ (r)k + |y¯(r)| )dr 2 t+τ t+τ 2 2 t Z t + τ + c (r − )(ku (r)k2p−4 + ku (r)k2p−4)kU¯(r)k2 dr. (169) 2 1 2p−2 2 2p−2 2p−2 t+τ 2 We now estimate every terms on the right hand side of (169). By Lemma 6.5, we have t t Z Z s 2 2 k(−∆) 2 V¯ (r)k dr + c(t − τ) kV¯ (r)k dr → 0, (170)

t+τ t+τ 2 2 as ω2 → ω1. For the last term on the right hand side of (169), by using Hölder inequality, we deduce that t Z t + τ (r − )(ku (r)k2p−4 + ku (r)k2p−4)kU¯(r)k2 dr 2 1 2p−2 2 2p−2 2p−2 t+τ 2 t p−2 Z t + τ p−1 ≤ (r − )(ku (r)k2p−2 + ku (r)k2p−2)dr 2 1 2p−2 2 2p−2 t+τ 2 t 1 Z t + τ p−1 × (r − )kU¯(r)k2p−2dr , 2 2p−2 t+τ 2 so it follows from (160) and Lemma 6.7 that t Z t + τ (r − )(ku (r)k2p−4 + ku (r)k2p−4)kU¯(r)k2 dr → 0, (171) 2 1 2p−2 2 2p−2 2p−2 t+τ 2 3436 WENQIANG ZHAO as ω2 → ω1. Consequently in the end, incorporate (170) and (171) into (169) to produce s 2 k(−∆) 2 V¯ (t)k → 0, as ω2 → ω1. This and Lemma 6.4 together conclude the proof. 7. Existence of bi-spatial measurable pullback attractor. In this section, we are at the point to present the bi-spatial (L2(RN ),Hs(RN ))-pullback attractor and (L2(RN ),Lp(RN ))-pullback attractor, respectively. Especially, it is pointed out that this attractor is measurable (as a set-valued mapping) in the regular spaces Hs(RN ) and Lp(RN ), where s ∈ (0, 1) and p ≥ 2. Theorem 7.1. Suppose that (25)-(29) hold. Then the random cocycle ϕ defined 2 N s N by (37) admits a unique bi-spatial (L (R ),H (R ))-pullback attractor AHs = {AHs (τ, ω): τ ∈ R, ω ∈ Ω}, in the sense of Definition 2.5. Proof. By Theorem 5.9, the random cocycle ϕ is bi-spatial (L2(RN ),Hs(RN ))- continuous with respect to the initial data. By Lemma 4.1, ϕ has a closed and pullback random absorbing set in the initial space L2(RN ). By Lemma 4.5, ϕ is (L2(RN ),L2(RN ))-asymptotically compact. In view of Theorem 2.10, it remains to check that the mapping ϕ(t, τ, ., x):Ω 3 ω → Hs(RN ) is (F, B(Hs(RN )))- measurable for every fixed t > τ, τ ∈ R and x ∈ L2(RN ). Indeed, by Lemma s N 6.8, the mapping ϕ(t, τ, ., x):Ωk 3 ω → H (R ) is continuous and consequently s N s N measurable from (Ωk, FΩk ) to (H (R ), B(H (R ))) for each k ∈ N. Therefore by Lemma 6.3, the mapping ϕ(t, τ, ., x):Ω 3 ω → Hs(RN ) is (F, B(Hs(RN )))- measurable. This concludes the proof. Theorem 7.2. Suppose that (25)-(29) hold. Then the random cocycle ϕ defined 2 N p N by (37) admits a unique bi-spatial (L (R ),L (R ))-pullback attractor ALp = {ALp (τ, ω): τ ∈ R, ω ∈ Ω}, in the sense of Definition 2.5. Furthermore, ALp = AHs , 0 < s < 1 and p ≥ 2. Proof. By Theorem 5.5, Lemma 6.6, Lemma 4.1, Lemma 4.5 and Lemma 6.3. The result is followed from Theorem 2.10 by a similar argument as Theorem 7.1.

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