Random attractors for non-autonomous fractional stochastic parabolic equations on unbounded domains

We study the asymptotic behavior of a class of non-autonomous non-local fractional stochastic parabolic equation driven by multiplicative white noise on the entire space \begin{document}$\mathbb{R}^n$\end{document} . We first prove the pathwise well-posedness of the equation and define a continuous non-autonomous cocycle in \begin{document}$L^2({\mathbb{R}} ^n)$\end{document} . We then prove the existence and uniqueness of tempered pullback attractors for the cocycle under certain dissipative conditions. The periodicity of the tempered attractors is also proved when the deterministic non-autonomous external terms are periodic in time. The pullback asymptotic compactness of the cocycle in \begin{document}$L^2({\mathbb{R}} ^n)$\end{document} is established by the uniform estimates on the tails of solutions for sufficiently large space and time variables.


1.
Introduction. This paper is concerned with the asymptotic behavior of solutions of the following non-autonomous, non-local, fractional stochastic equations on R n : ∂u ∂t + (−∆) s u + λu = f (t, x, u) + g(t, x) + αu • dW dt , x ∈ R n , t > τ, (1.1) with initial condition u(τ, x) = u τ (x), x ∈ R n , (1.2) where λ and α are positive constants, g ∈ L 2 loc (R, L 2 (R n )), W is a two-sided realvalued Wiener process on a probability space, and f : R × R n × R → R is a smooth nonlinearity. Note that the stochastic equation (1.1) is understood in the sense of Stratonovich's integration.
The main difficulty of this paper lies in the non-compactness of Sobolev embedding H s (R n ) → L 2 (R n ) with s > 0 due to the unboundedness of R n , which introduces a major obstacle for establishing the pullback asymptotic compactness of the solution operator. We overcome this difficulty by using the method of uniform estimates on the tails of solutions [49]. More precisely, for every ε > 0, we show that there exists a large open ball O K in R n with center at origin and radius K > 0 such that the solutions are uniformly less than 1 4 ε in L 2 (R n \ O K ) when time is sufficiently large. Since O K is bounded and the embedding H s (O K ) → L 2 (O K ) is compact with s > 0, by the uniform estimates, we can prove that the solutions are compact in L 2 (O K ). Consequently, the solutions are covered by a finite number of open balls in L 2 (O K ) with radii less than 1 4 ε. This along with the uniform tailestimates implies that the solutions are covered by a finite number of open balls in L 2 (R n ) with radii less than ε, and hence the solutions are asymptotically compact in L 2 (R n ), see Lemma 5.4 for more details. Compared with the equations with standard Laplace operator, the uniform estimates on the tails of solutions are much more involved because of the non-local nature of the fractional Laplace operator (−∆) s , see Lemma 4.4 in Section 4.
The rest of the paper is organized as follows. In Section 2, we review some basic results on the existence of random attractors for non-autonomous random dynamical systems. In Section 3, we prove the pathwise well-posedness of problem (1.1)-(1.2) in L 2 (R n ) and define a continuous non-autonomous cocycle over a metric dynamical system. The uniform estimates of solutions are contained in Section 4, and the proof of existence of tempered random attractors is given in Section 5.

2.
Preliminaries. In this section, we briefly review some notations and results for non-autonomous random dynamical systems for the sake of readers' convenience. We assume that (Ω, F, P) is a probability space, and (X, d) is a separable metric space. We use d(A, B) to denote the Hausdorff semi-distance for nonempty subsets A and B of X.
Definition 2.1. Let (Ω, F, P, (θ t ) t∈R ) be a metric dynamical systems. A mapping Φ : R + × R × Ω × X → X is called a continuous cocycle on X over (Ω, F, P, (θ t ) t∈R ) if for all τ ∈ R, ω ∈ Ω and t, s ∈ R + , the following conditions are satisfied: In addition, if there exists a positive number T such that for every t ∈ R + , τ ∈ R and ω ∈ Ω, then Φ is called a continuous periodic cocycle on X with periodic T .
Definition 2.2. Let D be a collection of some families of nonempty subsets of X. Then Φ is said to be D-pullback asymptotically compact in X if for all τ ∈ R, ω ∈ Ω and any sequences t n → +∞, x n ∈ D(τ − t n , θ −tn ω), the sequence has a convergent subsequence in X. Definition 2.3. Let D be a collection of some families of nonempty subsets of X and A = {A(τ, ω) : τ ∈ R, ω ∈ Ω} ∈ D. Then A is called a D-pullback attractor of Φ if the following conditions are satisfied: (i) A is measurable and A(τ, ω) is compact for all τ ∈ R and ω ∈ Ω; (ii) A is invariant, that is, for every τ ∈ R and ω ∈ Ω, (iii) A attracts every member of D, that is, given B ∈ D, τ ∈ R and ω ∈ Ω, In addition, if there exists T > 0 such that then we say A is periodic with period T .
The following results can be found in [52,53] (see also [16,17,45] for related results). Proposition 2.4. Let D be an inclusion-closed collection of some families of nonempty subsets of X, and Φ be a continuous cocycle on X over (Ω, F, P, (θ t ) t∈R ). If Φ is D-pullback asymptotically compact in X and has a closed measurable Dpullback absorbing set K in D, then Φ has a D-pullback attractor A in D. The D-pullback attractor A is unique and is given by, for each τ ∈ R and ω ∈ Ω, For the periodicity of D-pullback attractors, we have the following proposition from [52].
Proposition 2.5. Let D be an inclusion-closed collection of some families of nonempty subsets of X. Suppose Φ is a continuous periodic cocycle with period T > 0 on X over (Ω, F, P, (θ t ) t∈R ). If Φ is D-pullback asymptotically compact in X and has a closed measurable T -periodic D-pullback absorbing set K in D, then Φ has a unique T -periodic D-pullback attractor A in D.
Next, we recall some notations related to the fractional derivatives and fractional Sobolev spaces. Given 0 < s < 1, the fractional Laplace operator (−∆) s is defined by provided the integral exists, where C(n, s) is a positive constant depending on n and s as given by It follows from [18] that where F is the Fourier transform. Let H s (R n ) be the fractional Sobolev space defined by Throughout this paper, we denote the norm and the inner product of L 2 (R n ) by · and (·, ·), respectively. For convenience, the Gagliardo semi-norm of H s (R n ) is denoted · Ḣs (R n ) , i.e., We also use the notation Then for all u ∈ H s (R n ) we have u 2 H s (R n ) = u 2 + u 2Ḣ s (R n ) . Note that H s (R n ) is a Hilbert space with inner product given by By [18], we have and hence This implies that u 2 + (−∆) 3. Cocycles. In this section, we establish the existence of a continuous cocycle for the following non-autonomous fractional stochastic equation with s ∈ (0, 1): 2) where λ and α are positive constants, g ∈ L 2 loc (R, R n ), W is a two-sided real-valued Wiener process on a probability space. The nonlinearity f : R × R n × R → R is a continuous function which satisfies, for all t, u ∈ R and x ∈ R n , where β > 0 and p ≥ 2 are constants, : ω(0) = 0}, F is the Borel σ-algebra induced by the compact-open topology of Ω, and P is the Wiener measure on (Ω, F). Denote by θ t : Ω → Ω the transformation Then (Ω, F, P, {θ t } t∈R ) is a metric dynamical system. Consider the one-dimensional stochastic equation: dy + ydt = dW. It follows from [3] that this equation has a unique stationary solution y(t) = z(θ t ω) where z : Ω → R is a random variable given by z(ω) = − 0 −∞ e τ ω(τ )dτ for ω ∈ Ω. Moreover, there exists a θ t -invariant set of full measure Ω 0 such that z(θ t ω) is pathwise continuous for every ω ∈ Ω 0 and lim t→±∞ |z(θ t ω)| |t| = 0 and lim For convenience, in the sequel, we will not distinguish Ω 0 and Ω and use the same notation Ω for both Ω 0 and Ω. For our purpose, we need to convert the stochastic equation (3.1) into a deterministic one parametrized by ω ∈ Ω. To that end, we introduce where τ ∈ R is a deterministic initial time, t ≥ τ , ω ∈ Ω, u τ ∈ L 2 (R n ), and u = u(t, τ, ω, u τ ) is a solution of (3.1)-(3.2). Then we find that for t > τ , dv dt By a solution v of (3.8)-(3.9), we mean v satisfies the equation in the following sense.
To prove the existence of solutions of (3.8)-(3.9) in the sense of Definition 3.1, we will approximate the entire space R n by a bounded domain O k = {x ∈ R n : |x| < k} and then take the limit as k → ∞. Let ρ : [0, ∞) → R be a smooth function such that 0 ≤ ρ(s) ≤ 1 for all 0 ≤ s < ∞ and ρ(s) = 1 for 0 ≤ s ≤ 1 2 and ρ(s) = 0 for s ≥ 1. (3.11) Consider the following non-autonomous fractional equation on O k : where v τ ∈ L 2 (R n ). Note that in the boundary condition (3.13), we require v k = 0 on the complement of O k (i.e., on R n \ O k ), not just on the boundary of O k . This boundary condition is consistent with the definition of the non-local fractional operator (−∆) s . To present the existence of solutions of problem (3.12)-(3.14), for (3.15) By using the bilinear form a, we define A: where (·, ·) (H −s ,H s ) is the duality pairing of H −s (R n ) and H s (R n ). Since H k and V k are subspaces of L 2 (R n ) and H s (R n ), respectively, we find that a : in the sense of distribution on (τ, ∞). Next, we derive uniform estimates of the solution v k with respect to k ∈ N and prove the existence of solutions of (3.8)-(3.9) by taking the limit of v k when k → ∞.
Then for every τ ∈ R, ω ∈ Ω and v τ ∈ L 2 (R n ), problem (3.8)-(3.9) has a unique solution v(t, τ, ω, v τ ) in the sense of Definition 3.1. This solution is (F, B(L 2 (R n )))-measurable in ω and continuous in initial data v τ in L 2 (R n ). Moreover, the solution v satisfies the energy equation: Proof. The proof is similar to the case of bounded domains as in [54]. Of course, for problem (3.8)-(3.9) defined on the unbounded domain R n , we must show that all estimates on the solutions of (3.12)-(3.14) are uniform with respect to all k ∈ N.
Step (i). Uniform estimates of solutions of (3.12)-(3.14). By (3.12) we obtain By the boundary condition (3.13), all above integrals over the bounded domain O k can be replaced by that over the entire space R n , and hence we get 1 2 By (3.21) and (3.16) we see that for every fixed ω ∈ Ω and and (3.23) By (3.4) and (3.22) one can verify that As a consequence of (3.12) and (3.23)-(3.24) we find that for each fixed K ∈ N , Note that 1 < q ≤ 2 since p ≥ 2 and p and q are conjugate exponents.
Step (ii). Existence of solutions of problem (3.8)-(3.9). By a diagonal process, from (3.22)-(3.24), we find that there exists and and ) * is continuous. Then by (3.22), (3.25) and the compactness result in [37], after an appropriate diagonal process we find that, up to a subsequence, (3.31) By (3.31) and a diagonal process again, there exists a further subsequence (which Since f is continuous, by (3.32) we get (3.33) By (3.24) and (3.33) we infer from Mazur's lemma that (3.34) It follows from (3.28) and (3.34) that By simple computations, one can verify that for each K ∈ N, Taking the limit of (3) as k → ∞, by (3.26)-(3.28) and (3.35) we get Taking the limit of (3.38) as K → ∞, by (3.36) we find for all ξ ∈ H s (R n ) L p (R n ), in the sense of distribution on (τ, τ + T ).
The collection of all tempered families of bounded nonempty subsets of L 2 (R n ) is denoted D, that is, In this case, a D-pullback attractor is also called a tempered attractor since D given by (3.51) contains all tempered families of bounded nonempty subsets of L 2 (R n ). From now on, we assume that for every τ ∈ R, 0 −∞ e λs g(s + τ, ·) 2 + ψ 1 (s + τ, ·) L 1 (R n ) ds < ∞. When deriving the existence of tempered pullback absorbing sets, we will further assume that g and ψ 1 are tempered in the sense that for every c > 0, It is clear that (3.52) and (3.53) do not imply that g is bounded in L 2 (R n ) when t → ∞.

Uniform estimates of solutions.
In this section, we derive uniform estimates on the solutions of the non-local fractional stochastic equations in H s (R n ) as well as the uniform estimates on the tails of solutions for large space and time variables. The estimates in L 2 (R n ) are given below.
and M 1 is a positive number independent of τ , ω and D.
Proof. The proof is similar to the case of bounded domains as in [54]. For the reader's convenience, we here sketch the main idea. First, by (3.3) and (3.18) Note that the Young inequality implies which along with (4.1) yields After changing of variables, the desired estimates follows from (4.7) immediately.
As a consequence of Lemma 4.1, we see that problem (3.8)-(3.9) has a tempered pullback absorbing set in L 2 (R n ).   Proof. (4.8) follows from Lemma 4.1 when σ = τ , and the convergence of (4.10) can be proved in the same way as in the case of bounded domains which can be found in [54]. The details are omitted here.
Next, we derive uniform estimates of solutions in H s (R n ) for which we further assume that the function ψ 4 in (3.5) belongs to L ∞ (R, L ∞ (R n )) and the nonlinearity f satisfies, for all t, u ∈ R and x, y ∈ R n , where ψ 5 ∈ H s (R n ).
where M 2 is a positive number independent of τ , ω and D.
To prove the pullback asymptotic compactness of the cocycle associated with the problem (3.8)-(3.9) on the unbounded domain R n , we need to derive the uniform estimates on the tail parts of the solutions for large space variables when time is large enough.
Proof. Let χ(s) = 1 − ρ(s) for all 0 ≤ s < ∞ where ρ is the smooth function given by (3.11). Then we find that 0 ≤ χ(s) ≤ 1 for all s ≥ 0 and Note that there exists a positive constant c such that |χ (s)| ≤ c for all s ≥ 0. (4.20) For the third term on the left-hand side of (4.20), we have Note that the right-hand side of (4.21) is controlled by (4.22) We now estimate the first integral in (4.22). Given y ∈ R, for z = x k we have Since s ∈ (0, 1), we see that the above integrals are convergent. Thus we get where c 2 is a positive constant independent of k ∈ N and y ∈ R n By (4.22)-(4.23) we get For the first term on the right-hand side of (4.20), by (3.3) one has (4.25) For the second term on the right-hand side of (4.20), one has e −αz(θtω) )dr e −2αz(θsω) (g 2 (s + τ, x) + |ψ 1 (s + τ, x)|)dxds.
(4.33) We now estimate the right-hand side of (4). For the first term, since e αz(θ−tω By (3.6) and the fact that D ∈ D we find that the right-hand side of the above inequality converges to zero as t → ∞, and hence there exists By Lemma 4.1 with σ = τ , we see that there exists where R(τ, ω) is the number as in (4.9). For the last term in (4), by (3.6) and (3.52) we find that It follows from (4)-(4.36) that for all k ≥ K 2 and t ≥ T 2 , This completes the proof.

5.
Existence of random attractors. In this section, we prove the existence and uniqueness of tempered pullback attractors for the non-local fractional stochastic equation (3.1)-(3.2). To that end, we need to establish the existence of tempered random absorbing sets and the pullback asymptotic compactness of the cocycle Φ. , the cocycle Φ has a closed measurable pullback absorbing set K = {K(τ, ω) : τ ∈ R, ω ∈ Ω} ∈ D where for every τ ∈ R and ω ∈ Ω, the set K(τ, ω) is defined by where R(τ, ω) is the same number as in (4.9).
This shows that K is a D-pullback absorbing set of Φ. It is clear that R(τ, ω) is measurable in ω ∈ Ω, which implies the measurability of K(τ, ω) in ω ∈ Ω.
The uniform estimates of the solutions of problem (3.1)-(3.2) in H s (R n ) is given below. Proof. This estimate follows from (5.1) and Lemma 4.3 directly.
Proof. This is an immediate consequence of Lemma 4.4 together with the arguments of the proof of Lemma 5.1. The details are omitted here.
We now present our main result of this paper as follows. Proof. This is an immediate consequence of Lemmas 5.1,5.4 and Proposition 2.4.
Regarding the periodicity of D-pullback attractors, we have the following result.
Proof. Since f (t, x, s) and g(t, x) are T -periodic in t ∈ R, we find that the cocycle Φ is also T -periodic. Since g(t, x) and ψ 1 (t, x) are T -periodic in t ∈ R, by Lemma 5.1 we see that the D-pullback absorbing set K is also T -periodic. Then the Tperiodicity of A follows from Proposition 2.5 immediately.