WONG-ZAKAI APPROXIMATIONS AND ATTRACTORS FOR FRACTIONAL STOCHASTIC REACTION-DIFFUSION EQUATIONS ON UNBOUNDED DOMAINS∗

In this paper, we investigate the Wong-Zakai approximations induced by a stationary process and the long term behavior of the fractional stochastic reaction-diffusion equation driven by a white noise. Precisely, one of the main ingredients in this paper is to establish the existence and uniqueness of tempered pullback attractors for the Wong-Zakai approximations of fractional stochastic reaction-diffusion equations. Thereafter the upper semicontinuity of attractors for the Wong-Zakai approximation of the equation as δ → 0 is proved.


Introduction
This paper considers the Wong-Zakai approximations and the long term behavior of the non-local, fractional stochastic reaction-diffusion equations on R n as following: ∂u ∂t + (−∆) s u + λu = f (t, x, u) + g(t, x) + h(t, x, u) • dW dt , t > τ, x ∈ R n , (1.1) with initial condition u(τ, x) = u τ (x), x ∈ R n . (1.2) Here, s ∈ (0, 1), λ > 0 is a fixed constant, f : R × R n × R → R is a smooth functon, g ∈ L 2 loc (R, L 2 (R n )), W is a one-dimensional two-sided Brownian motion, the symbol • means that the equation is understood in the sense of Stratonovich's integration.
Let (Ω, F, P) be the classical Wiener probability space, where with the open compact topology, F is its Borel σ-algebra, and P is the Wiener measure. The shift operator θ t is defined on (Ω, F, P) by θ t ω(·) = ω(t + ·) − ω(t).
As we all know, the probability measure P is an ergodic invariant measure for θ t . (Ω, F, P, {θ t } t∈R ) forms a metric dynamical system, see Arnold [1].
For given δ ∈ R, set G δ : Ω → R as the random variable: By checking, we know that G δ (θ t ω) is a stationary stochastic process. G δ (θ t ω) is an approximation of white noise in the sense for each T > 0, which was first introduced by K. Lu and Q. Wang in [13]. From then on, the same approximation is used in [14] for bounded domains and also in [15] and [17] for unbounded domains. More recently, this approximation was used by Shen, Zhao, Lu and Wang to investigate the invariant manifolds and stable foliations of the Wong-Zakai approximaitons, which turn out to converge to the invariant manifolds and stable foliations of the stochastic evolution equation, respectively in [18]. In the present paper, we also use the same approximation as in (1.4). (1.4) implies that equation (1.1) could be approximated by the following Wong-Zakai equation driven by a multiplicative noise of G δ (θ t ω) as δ → 0: ∂u ∂t + (−∆) s u + λu = f (t, x, u) + g(t, x) + h(t, x, u)G δ (θ t ω), t > τ, x ∈ R n . (1.5) To place our result in context, we review a few highlights from the random attractors of fractional stochastic equations. The authors of [9] and [10] obtained the existence of random attractors. Recently, the authors of [22] established the existence of the random attractors on bounded domains, and the authors of [12] solved the case of unbounded domains by applying diagonal processes for two times and tail-estimate. Moreover, by using the idea of spectral decomposition on bounded domains O in R n along with the uniform tail-estimates of solutions, Wang etc obtained the regularity of random attractors [6]. Note that the noise in the above five papers is either additive or linear multiplicative. On the other hand, for the nonlinear case, there are few results: the existence of random attractors for stochastic PDEs driven by a fractional Brownian motion was proved by the authors of [4], [5]. Also, very recently, Wang etc in [24] established the existence and uniqueness of pullback random attractors for the fractional nonclassical diffusion equations driven by colored noise via using the similar method in [6].
In this paper, strongly motivated by the work of Wang etc [15], we study the long term behavior of equations (1.1) and (1.2). In general, the stochastic equation (1.1) could generate a continuous cocycle only when h(t, x, u) either only depends on t and x or is linear in u. For general nonlinear function h(t, x, u), (1.1) may not generate continuous cocycle, hence the existence of attractor is unclear. Fortunately, we are able to show that (1.5) could generate a continuous cocycle and additionally it has a tempered attractor for a class of nonlinear functions h. Thus we could indirectly investigate the long term behavior of (1.1) via considering (1.5). This result is presented in Theorem 2.1.
To illustrate the advantage of random equation (1.5) over stochastic equation (1.1), in section 3, for linear multiplicative noise, we will prove the solutions of equation (1.5) converge to that of equation (1.1) as δ → 0 and furthermore we will obtain the upper semi-continuity of the attractors for equation (1.5) in L 2 (R n ). This property is contained in Theorem 3.1.
To obtain the uniform estimates of solutions of equation (1.5) in H s (R n ), a strong condition (2.9) for the noise term h(t, x, u) is necessary. Indeed, one needs (2.9) to ensure the regularity of h(t, x, u), which is a key step in the estimate of u. The uniform estimates and the uniform tail-estimates of solutions in L 2 (R n ) will yield the pullback asymptotic compactness of solutions in H s (R n ).
The pioneer work of approximating stochastic equations by pathwise deterministic equations could date back to Wong and Zakai [25,26]. So far, there has been a series of nice results about Wong-Zakai approxiamtions, for instance, the readers can consult [2,3,7,8,19].
The remaining part of this paper is organized as follows: In section 2, the existence and uniqueness of pullback random attractors for Wong-Zakai approximations are proved. In the last section, we obtain the upper semi-continuity of attractors of Wong-Zakai approximations (3.11) for multiplicative noise as δ → 0.
Notations. Before ending this introduction, let us recall some related notions about the integral fractional operator (−∆) s . Given s ∈ (0, 1), the fractional Laplace operator (−∆) s is defined by where C(n, s) is a positive number depending on n and s with For s ∈ (0, 1), the fractional Sobolev space H s (R n ) is defined by In this paper, the norm and the inner product of L 2 (R n ) are denoted by · and (·, ·), respectively. The Gagliardo semi-norm of H s (R n ) is denoted by Note that H s (R n ) is a Hilbert space, with inner product Moreover, by [16] we have is an equivalent norm of H s (R n ). Hereafter, the letters c, c i and C i may be different in different lines.
Now we comment on the proof of Theorem 2.1. We firstly prove the existence of a continuous cocycle for random fractional reaction-diffusion equations.

Continuous cocycles
As in the introduction, (Ω, F, P, {θ t } t∈R ) is a metric dynamical system, then there exists a {θ t } t∈R -invariant subset of full measure Ω 0 ⊆ Ω such that for all ω ∈ Ω 0 , For the concision of notation, in this paper, we don't distinguish Ω and Ω 0 . By Here is a list of properties of G δ (θ t ω), see [14].
Although some essential steps in the proof are similar to those in [12], we still give the details for reader's convenience. We will prove the result in the following three steps: (1) Uniform estimates By (2.19), we get that for t > τ , (2.24) For the simplicity of presentation, we define that Because of the boundary condition (2.20), the above (2.24) can be rewritten as By (2.3), (2.7) and Young's inequality, we obtain that (2.30) Multiplying (2.30) by e λt , then integrating over (τ, t) for t ≥ τ , we get Hence, (2.32), (2.34) and (2.35) imply (2) Limiting process Similar to the method in [12], by (2.32)-(2.38), we can get that there existsū ∈ L 2 (R n ), u ∈ L ∞ (τ, T ; L 2 (R n )) L 2 (τ, T ; H s (R n )) L p (τ, T ; L p (R n )) and w ∈ L q (τ, T ; L q (R n )) such that up to a subsequence, and * is continuous, and by (2.32), (2.38), after an diagonal process about k, we infer from [11] Again, by (2.44) and a diagonal process about K, there exists a subsequence of (2.46) From (2.36) and (2.46), we have (2.47) By (2.41) and (2.47), we have In (2.52), also letting K → ∞, by (2.50), we have To prove the continuity of u : Next, we prove u(τ ) = u τ and u(T ) =ū. For this aim, we let ϕ ∈ C 1 ([τ, T ]) and ξ ∈ H s (R n ) L p (R n ). Similar to (2.51), by (2.19)-(2.21), we get for every k > K, (2.56) In (2.56), letting K → ∞, by (2.50), we get for all ξ ∈ H s (R n ) L p (R n ),  Similar to (2.63), we can get that for all t ≥ τ , as k → ∞, which along with (2.5) and (2.8), we get that for every T > τ , there exists c > 0, such that for all t ∈ [τ, T ], Therefore, we get Thus, the uniqueness and continuity of solutions in initial data in L 2 (R n ) are proved.
By the three steps above, the proof of Lemma 2.2 is completed. Now we can define a mapping Φ : where u is a soluiton of equations (2.1)-(2.2). The uniqueness of the solution shows that Φ is a continuous cocycle in L 2 (R n ) for equations (2.1)-(2.2).

Pullback random attractors
In this subsection, we prove the existence and uniqueness of attractors of Φ in L 2 (R n ). By [20], our subsequent tasks are to prove the existence of a tempered pullback absorbing set for equations (2.1)-(2.2) in L 2 (R n ) as well as the asymptotic compactness of the solutions. Also, we assume ϕ 1 ∈ L ∞ (R, L p p−p 1 (R n )), ϕ 2 ∈ L ∞ (R, L q (R n )). In this process, a series of inequalities are derived via delicate computation and analysis.
We recall that a family of bounded nonempty subsets of
where u τ −t ∈ D(τ − t, θ −t ω) and C 2 is a positive constant independent of τ, ω and D.

Upper semi-continuity of attractors for multiplicative noise
In present section, for the case of linear multiplicative noise, we focus on the upper semi-continuity of attractors for the Wong-Zakai approximation as δ → 0.

Equation driven by white noise
This subsection concerns about the fractional reaction-diffusion equation (1.1) when h(t, x, u) = u: By the standard process, we introduce a new variable: Then we get where v τ (x) = e −ω(τ ) u τ (x). By the transform (3.3), we could do some computation which is similar to [12] but easier, to get the following result: First, equation (3.4) and (3.5) admit a unique Thus, a continuous cocycle Φ 1 could be defined as follows: for all t ∈ R + , τ ∈ R, ω ∈ Ω. Additionally the cocycle Φ 1 admits a unique random attractor A 1 .
For our later usage in proving the convergence of solutions, some necessary results are listed as follows: where u τ −t ∈ D(τ − t, θ −t ω) and C 1 is a positive constant independent of σ, τ , ω, and D.
From Lemma 3.1, one immediately has the following two results.