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Asymptotic Autonomy of Attractors for Stochastic Fractional Nonclassical Diffusion Equations Driven by a Wong–Zakai Approximation Process on Rn

In this paper, we consider the backward asymptotically autonomous dynamical behavior for fractional non-autonomous nonclassical diffusion equations driven by a Wong–Zakai approximations process in Hs(Rn) with s ∈ (0, 1). We first prove the existence and backward time-dependent uniform compactness of tempered pullback random attractors when the growth rate of nonlinearities have a subcritical range. We then show that, under the Wong–Zakai approximations process, the components of the random attractors of a non-autonomous dynamical system in time can converge to those of the random attractor of the limiting autonomous dynamical system in Hs(Rn).

A random pullback attractor is a bi-parametric random set in the form A = {A(τ, ω) : τ ∈ R, ω ∈ Ω}. The time-dependence character reflects the non-autonomous feature of the system, which should be the most significant characteristic distinguished from autonomous cases, so both the existence and some time-dependent properties of the pullback random attractor A(τ, ω) for the cocycle generated by system (3) in H s (R n ) with s ∈ (0, 1) are concerned. We are more specifically interested in the backward compactness of random attractor A(τ, ω), i.e., the compactness of ∪ ς≤τ A(ς, ω) for each τ ∈ R. It is worth noting that the absorbing set is a union of some random sets over an uncountable index set (−∞, τ], hence the measurability of attractor A is uncertain. To solve this difficulty, we shall prove a crucial fact that the attractor does not vary between the two attracted universes, see Theorem 1. The time-dependence of a pullback attractor, such as the backward compactness of attractor, has recently been studied in the literature, see Refs. [31][32][33][34][35]. In order to establish the existence result of a backward compact random attractor in H s (R n ), we shall prove the backward asymptotic compactness of solutions, which indicates that the usual asymptotic compactness is uniform in the past. The main difficulties for proving such compactness comes from the fact that the Sobolev embedding H r (R n ) → H s (R n ) with r > s is not compact, and the Wiener process is nowhere differentiable with respect to time. We cannot utilize the technique of differentiating the equation with respect to t. To get around these problems, we will use the methods of a cut-off technique [36] and flattening properties [37] of solutions to establish the desired backward asymptotic compactness in H s (R n ).
Finally, we will show that the random attractor A(τ, ω) is backward asymptotically autonomous to random attractor A ∞ (ω) in the sense of Hausdorff semi-distance. More precisely, for P-a.s. ω ∈ Ω, where A ∞ (ω) is a random attractor of the following limiting equation: whereg ∈ L 2 (R n ) andf ,h will be specified later, see Theorem 2.
Since system (3) contains time-dependent nonlinear terms whereas system (11) does not, it is more complicated to determine the limit process (10), see Lemma 7. As a result, we will make certain assumptions about nonlinear terms and forcing terms, as shown in (83)-(85), which differs from the earlier works on an asymptotically autonomous problem in Refs. [31,34,35,38,39].
The outline of the paper is as follows. In the next section, we give some necessary assumptions and define a non-autonomous co-cycle for system (3). In Section 3, we derive backward uniform estimates on the solutions in H s (R n ) including the backward uniform estimates on the tails and the bounded domains. In Section 4, we prove the existence, uniqueness, and backward compactness of random attractors for problem (3) in H s (R n ). In the Section 5, we establish the asymptotic upper semicontinuity of these attractors in H s (R n ) as τ → −∞. Finally, we illustrate with one example the results of the paper for n = 2 briefly.

Non-Autonomous Co-Cycles for Fractional Equations
In this section, we first briefly review the concepts of fractional derivatives and fractional Sobolev spaces. Let (−∆) s with s ∈ (0, 1) be the non-local, fractional Laplace operator defined by where P.V. means the principal value of the integral and C(n, s) is a constant given by Let H s (R n ) with s ∈ (0, 1) be the fractional Sobolev space given by which is a Hilbert space equipped with the inner product and the norm: By Ref. [40], we also have and thus The reader is referred to Ref. [40] for more details on fractional operators. It follows from Refs. [40,41] and (9) that where p is the number in (4) and µ = np−2n 2p ∈ (0, 1). Under conditions (4)-(8), for each τ ∈ R, ω ∈ Ω, then as in [42], the system (3) has a unique solution u(·, τ, ω, u τ ) ∈ C([τ, ∞), H s (R n )), where u τ ∈ H s (R n ). This allow us to define a continuous cocycle (see [43] We take a universe D on H s (R n ), which consists of all backward tempered set-valued where D denotes the supremum of norms of all elements. Meanwhile, let D be the usual universe of all tempered sets on H s (R n ), which satisfies Obviously, D is a subset of D.
In order to obtain the D-pullback attractor A D , we make the following assumption for some α 0 > 0, (18) where we use · r (resp. · ) to denote the norm of L r (R n ) (resp. L 2 (R n )) here and after. Indeed, by ( [35], Lemma 4.2), one can show that the growth rate α 0 can be arbitrary, which means for ∀α > 0, we have Based on this, (19) can imply the following tempered condition with arbitrary growth rates: Since the systems defined on the entire space R n , we also need the following assumptions

Backward Uniform Estimates of Solutions
In this section, we derive uniform estimates on the solutions of (3) when t → +∞, with the purpose of proving the existence of a bounded D-pullback absorbing set that is uniform in the past and the backward asymptotic compactness of the random dynamical system associated with the equation.
where M > 0 is a constant independent of τ, ω and D, R(τ, ω) is given by which is well-defined due to (19).
Proof. Let τ ∈ R be fixed, and ς ≤ τ. Multiplying Equation (3) by u and integrate over R n , we obtain We now consider the right hand side of (26) one by one. For the nonlinear term, we have For the last term, by Young's inequality, we have Substituting (27), (28) into (26), we see that where κ = min(λ, 1). Multiplying (29) by e κt , then integrating over (ς − t, ς) with t ≥ 0, and replace ω by θ −ς ω, we find that Taking the supremum with respect to the time over ς ≤ τ in (30) and using (2) and (19), we see that and Therefore, it follows from (30) to (32), and we have which implies (23).
Furthermore, taking (p − 1)-th power of (30) and multiplying by e κ(r−ς) , we integrate the result over ( Taking the supremum with respect to the time over ς ≤ τ in (34), and by which together with (34) implies the desired result (24). This completes the whole proof.
Proof. Multiplying (3) by u t and integrate over R n , we have Using the assumption (5) and the Sobolev embedding inequality given in (12), we obtain By (12) and (7), we obtain Applying Young's inequality, we have It follows from (36) to (39) that the desired result (35) follows immediately.
We now derive the uniform estimates on the tails of the solutions. To this end, we introduce a smooth function ρ(s) defined for 0 ≤ s < ∞ such that 0 ≤ ρ(s) ≤ 1 for s ≥ 0 and satisfies ρ(s) ≡ 0 for all 0 ≤ s ≤ 1 2 and ρ(s) ≡ 1 for all s ≥ 1. Letting ρ k (x) = ρ( |x| k ) for x ∈ R n and k ∈ N, by ( [44], Lemma 3.4), we have Lemma 3. Let (4)- (9) and (19) be satisfied. Then for every > 0, τ ∈ R, ω ∈ Ω and D = such that for all t ≥ T and k ≥ K, the solution of Proof. By Ref. [12], Lemma 2 and a slight modification, we have the following energy equation: We now multiply the above by e κt , then integrate the result over (ς − t, ς) with t ≥ 0 and replace ω by θ −ς ω in the resulting inequality to obtain For the first term on the right-hand side of (44), since u ς−t ∈ D(ς − t, θ −t ω) with D ∈ D and ς ≤ τ, we have, as t → +∞, For the second term on the right-hand side of (44), by Lemma 1, there exists T := T(ς, ω, D) > 0 such that for all t ≥ T, as k → ∞, By (2) and (19), we have tends to 0 as k → ∞. In addition, it follows from assumption (21), we get that For the last term on the right-hand side of (44), apply (2) and the assumption (22), we have (49) which tends to 0 as k → ∞.
In order to establish the backward pullback asymptotic compactness of solutions in H s (R n ), we need to approximate R n ba a family of bounded domains. For every x ∈ R n and k ∈ N, let Thenû(t, τ, ω,û τ )(x) = 0 for k ∈ N and x ∈ O c k , and û H s (R n ) ≤ c u H s (R n ) for some constant c > 0 independent of k ∈ N.

Existence of Backward Pullback Random Attractors
In this section, we show that the cocycle Φ, generated by the Wong-Zakai approximations for fractional nonclassical diffusion equations, has unique D-pullback random attractors in H s (R n ). We establish the existence of D-pullback absorbing sets in H s (R n ) firstly.
Proof. By Lemma 1, K D is an absorbing set and ω → R(τ, ω) is measurable since it is the integral of some random variables. Hence, K D is a random set. Next, we show K D ∈ D.
On the other hand, it is easy to see that K D is increasing, which means where we have used K D ∈ D. Then we have K D ∈ D as required.

Remark 1.
In this lemma, the measurability of K D is unknown since it is the union of some random sets K D (ς, ·) over an uncountable index set ς ≤ τ. However, we will prove the measurability of K D in Theorem 1.
We then show the D-pullback backward asymptotic compactness of Φ.
Proof. By Lemmas 1, 3 and 4, the proof is standard, so we omit it.
Finally, we will establish the existence, uniqueness, and backward compactness of pullback random attractors of Φ in H s (R n ). Theorem 1. Suppose that conditions (4)- (9) and (19) hold. Let Φ be the continuous co-cycle for problem (3) and D (resp. D) be the universe of backward tempered sets (resp. tempered sets). Then (i) Φ has a D-pullback bi-parametric attractor A D ∈ D such that A D is backward compact in H s (R n ); (ii) Φ has a D-pullback random attractor A D ∈ D; (iii) A D (τ, ω) = A D (τ, ω) for all (τ, ω) ∈ R × Ω, that is, A D ∈ D is a D-pullback random attractor with the backward compactness.
Proof. (i) By Lemma 5, we find that K D = {K D (τ, ω) : τ ∈ R, ω ∈ Ω} ∈ D is a Dpullback absorbing set for Φ in H s (R n ). On the other hand, by Lemma 6 we know that Φ is D-pullback backward asymptotically compact in H s (R n ). Then it follows from the abstract result as given by [43] we show that Φ has a D-pullback bi-parametric attractor A D ∈ D given by where the measurability of K D is unknown. However, we will prove that A D is a random attractor in (ii) and (iii). Next, we prove A D is backward compact. We define , ∀n ∈ N.
We choose three sequences by r n ≤ τ, t n ≥ n and x n ∈ K D (r n − t n , θ −t n ω) such that which implies that Let {y n } ⊂ W (τ, ω, K D ), then there are r n ≤ τ, t n ≥ max{t n−1 , n} and x n ∈ K D (r n − t n , θ −t n ω) such that d(Φ(t n , r n − t n , θ −t n ω)x n , y n ) ≤ 1 n , ∀n ∈ N.
By the D-backward asymptotic compactness of Φ given by Lemma 6, there is a x ∈ X and a index subsequence {n k }, such that Both (69) and (70) imply y n k → x ∈ W (τ, ω, K D ) as k → ∞ and thus W (τ, ω, K D ) is compact as desired.
On the other hand, we have Hence, A D is backward compact.
(ii) By Lemma 5, we know that co-cycle Φ has a D-pullback absorbing set K D ∈ D as given in Lemma 5 and K D is a random set. Using a similar process in Section 3, one can show that Φ is D-pullback asymptotically compact. Then, by the abstract result in [43], Φ has a D-pullback random attractor A D ∈ D given by (71) (iii) By the definition of K D (τ, ω) and K D (τ, ω) in Lemma 5, we have the inclusion K D (τ, ω) ⊂ K D (τ, ω). Therefore, by (67) and (71), we see that On the other hand, since D is a subset of D and A D ∈ D, we have A D ∈ D and thus the D-pullback attractor A D attracts A D . Since A D is invariant, we have By the compactness of A D , we have A D (τ, ω) ⊂ A D (τ, ω), which together with (72) implies that A D (τ, ω) = A D (τ, ω) for all τ ∈ R, ω ∈ Ω. These finish the proof.

Conclusions
In view of Theorem 1, we can see that fractional nonclassical diffusion equations driven by Wong-Zakai approximations in H s (R n ) with s ∈ (0, 1) exists in backward timedependent uniform compact random attractors when the growth rate of nonlinearities have a subcritical range. Theorem 2 implies that the components of the random attractors of a non-autonomous dynamical system in time can converge to those of the random attractor of the limiting autonomous dynamical system (time-independent) in H s (R n ). In this paper, system (3) contains time-dependent nonlinear terms, so it is more complicated to determine the limit process (10), which differs from the earlier works.
We give an example for n = 2 and the nonlinearity f and forcing term g, which satisfy all the assumptions in the paper.
In this case, Theorems 1 and 2 are clearly true.