RANDOM DYNAMICS OF FRACTIONAL NONCLASSICAL DIFFUSION EQUATIONS DRIVEN BY COLORED NOISE

. The random dynamics in H s ( R n ) with s ∈ (0 , 1) is investigated for the fractional nonclassical diﬀusion equations driven by colored noise. Both existence and uniqueness of pullback random attractors are established for the equations with a wide class of nonlinear diﬀusion terms. In the case of additive noise, the upper semi-continuity of these attractors is proved as the correlation time of the colored noise approaches zero. The methods of uniform tail-estimate and spectral decomposition are employed to obtain the pullback asymptotic compactness of the solutions in order to overcome the non-compactness of the Sobolev embedding on an unbounded domain.

To describe the colored noise, we introduce a probability space (Ω, F, P), where Ω = {ω ∈ C(R, R) : ω(0) = 0} equipped with the compact-open topology, F = B(Ω) is the Borel sigma-algebra of Ω, P is the Wiener measure. We define the measure-preserving transformation group by θ t ω(·) = ω(· + t) − ω(t) for all (ω, t) ∈ Ω × R. Let W be a two-sided real-valued Wiener process on (Ω, F, P), we define Then the process ζ δ (θ t ω) is called an Ornstein-Uhlenbeck process (or colored noise) which is the unique stationary solution of the one-dimensional stochastic differential equation dζ δ + 1 δ ζ δ dt = 1 δ dW . The colored noise was first introduced in [46] and then widely used in many applications to study the dynamics of physical systems, see e.g. [6,22] and the references therein. The nonclassical diffusion equation arises in physics and fluid mechanics, see, e.g., [2,3,32]. If s = 1, then the fractional operator (−∆) s becomes the standard Laplace operator. In this special case, the attractor of the nonclassical diffusion equation has been studied in [4,5,52,54,56,61] for the deterministic systems, and in [7,59,60] for the stochastic systems driven by linear white noise. However, even for s = 1, the attractors of the nonclassical diffusion equations driven by colored noise have not been discussed in the literature. The purpose of the present paper is to investigate this problem and prove the existence of attractors of (1) driven by colored noise with s ∈ (0, 1).
Fractional partial differential equations can be derived from a variety of applications in physics, biology, chemistry, finance and other fields of science, see e.g., [11,29,43,44,45] and the references therein. If the term (−∆) s u t is absent, then equation (1) reduces to a fractional parabolic equation. In this case, the existence of random attractors of the fractional equation driven by linear white noise has been studied in [27,39,40,41,50]. But, for the fractional equation like (1) involving (−∆) s u t , there is no result available on the existence of such attractors. We will deal with this problem in the present article.
In this paper, we will first prove the existence of attractors of the random equation (1) in H s (R n ) with s ∈ (0, 1) when the nonlinear diffusion term h satisfies some growth conditions. We then examine the limiting behavior of these attractors as the correlation time δ of the colored noise tends to zero. More precisely, we will also consider the following fractional nonclassical diffusion equation driven by additive white noise: du + d(−∆) s u + (−∆) s udt + λudt = f (t, x, u)dt + g(t, x)dt + h(x)dW, u(τ, x) = u τ (x), x ∈ R n , t > τ, τ ∈ R, where h ∈ L 2 (R n ). We prove the stochastic equation (2) has a random attractor A 0 = {A 0 (τ, ω) : τ ∈ R, ω ∈ Ω} in H s (R n ) which, in a sense, can be considered as the limit of the attractors of the following random equation driven by colored noise as δ → 0: Indeed, we will show the random attractors A δ = {A δ (τ, ω) : τ ∈ R, ω ∈ Ω} of (3) are upper semi-continuous at δ = 0 and the limit is given by A 0 = {A 0 (τ, ω) : τ ∈ R, ω ∈ Ω} in H s (R n ). Furthermore, we prove that such upper semi-continuity is uniform in a probability subspace Ω ε ⊆ Ω with P(Ω ε ) > 1 − ε for any small ε > 0, see Theorem 4.8.
To show the existence of random attractors in H s (R n ), we need to establish the pullback asymptotic compactness of the solutions in H s (R n ). The main difficulties for proving such compactness for (1) come from the following aspects: (i) The uniform estimates of the solutions for the fractional equations in H s (R n ) are more complicated than that for the equations with the standard Laplace operator.
(ii) The Sobolev embedding H r (R n ) → H s (R n ) with r > s is not compact.
(iii) Due to the term (−∆) s u t , the fractional equation (1) has no smoothing effect on its solutions.
(iv) Because the Wiener process is nowhere differentiable with respect to time, we cannot use the method by differentiating the equation with respect to t, as given in the deterministic case [5].
To overcome these difficulties, in the present paper, we will combine the methods of uniform tail-estimate and spectral decomposition to establish the desired pullback asymptotic compactness in H s (R n ), see Lemma 2.6 for more details.
This paper is organized as follows. In the next section, we show that problem (1) has a unique random attractor in H s (R n ) for a wide class of nonlinear functions f and h. In Section 3, we prove that problem (2) driven by additive noise has also a unique random attractor in H s (R n ). In the last section, we establish the upper semi-continuity of these attractors in H s (R n ) as δ approaches zero.
2. Attractors of fractional equations driven by colored noise. In this section, we study the existence and uniqueness of random attractors for the fractional nonclassical diffusion equations driven by colored noise on R n : where λ > 0, s ∈ (0, 1), g ∈ L 2 loc (R, L 2 (R n )). The nonlinear functions f, h : R × R n × R → R satisfy the following conditions: for all (t, x, u) ∈ R × R n × R, where Note that, by [28], there exists a {θ t } t∈R -invariant subset (still denoted by Ω) of full measure such that lim t→±∞ |ζ δ (θ t ω)| t = 0 for each δ ∈ (0, 1] and ω ∈ Ω. Next we 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 4094 RENHAI WANG, YANGRONG LI AND BIXIANG WANG 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 |x − y| n+2s dxdy < ∞ , which is a Hilbert space equipped with the inner product and the norm: For convenience, we use (·, ·)Ḣ s (R n ) and · Ḣs (R n ) to denote Then By [17], one can verify which means that It follows from [17,Prop. 6.5] and (11) that where p is the number in (5). In addition, by [42, Prop. 5.9 and 5.14], we have Note that (4) can be viewed as a deterministic random equation parameterized by ω ∈ Ω, then one can show that for each τ ∈ R, ω ∈ Ω and u τ ∈ H s (R n ), problem (4) has a unique solution u(·, τ, ω, u τ ) ∈ C([τ, ∞), H s (R n )) such that the solution continuously depends on u τ ∈ H s (R n ). In addition, one can show the (F, B(H s (R n )))-measurability of the solution. Hence the mapping Φ: is a continuous cocycle in the sense of [49, Def. 1.1].
Next we study the existence and uniqueness of D-pullback random attractors of Φ in H s (R n ), where D is a family of bi-parametric sets in H s (R n ) such that In this article, we let κ = min(λ, 1). We use · r (resp. · ) to denote the norm of L r (R n ) (resp. L 2 (R n )) and make the following assumptions: 2.1. Uniform estimates. This subsection is devoted to the uniform estimates of the solutions of problem (4), which are crucial for constructing D-pullback random absorbing sets for the cocycle Φ.

2.2.
Uniform tail-estimates. To derive uniform tail-estimates of solutions of (4), we need the following auxiliary estimate.
Next we derive uniform tail-estimates of the solutions. To this end, we let ρ k (x) = ρ( |x| k ) for x ∈ R n and k ∈ N, where ρ : R + → [0, 1] is a smooth function with ρ(t) ≡ 0 for all 0 ≤ t ≤ 1 2 and ρ(t) ≡ 1 for all t ≥ 1. By [27,Lemma 3.4], for every s ∈ (0, 1) and k ∈ N, Lemma 2.3. Let (5)- (10) and (17) be satisfied. Then for every τ ∈ R, ω ∈ Ω and where We calculate the second term as follows: By (36), we obtain the lower bound of I 2 : . Therefore, the second term on the left-hand side of (39) satisfies We then calculate the last term on the left-hand side of (39): By (36), I 4 has the following lower bound: . So, the last term on the left-hand side of (39) enjoys By (5) and (9), the first and second terms on the right-hand side of (39) are controlled by Moreover, the last term on the right-hand side of (39) is estimated by Substituting (40)-(43) into (39), we see from Lemma 2.2 that where I 3 is just defined as above, that is, Multiplying (44) by e κt and integrating over (τ − t, τ ) with t ≥ 0, then we replace ω by θ −τ ω in the resulting inequality to obtain

RENHAI WANG, YANGRONG LI AND BIXIANG WANG
where J i are defined and estimated as follows. By By Lemma 2.1, there exists T := T (τ, ω, D) > 0 such that for all t ≥ T , as k → ∞, By (17) and (12), as k → ∞, For the fourth term on the right-hand side of (45). We deduce from (17) that as k → ∞ By (12), the Lebesgue controlled convergence theorem gives that tends to zero as k → ∞. Therefore, by all estimates given in (46)-(50), we see from (45) that And thereby by (51), we obtain the first result (37). Moreover it follows from (52) that Interchanging x and y in (53) yields Combining (53) and (54), we obtain the second result (38). The proof is concluded.

Uniform estimates on bounded domains.
To verify the pullback asymptotic compactness of solutions in H s (R n ), we derive uniform estimates of solutions on bounded domains. For every x ∈ R n and k ∈ N, we put By the definition of (−∆) s , we can verify and similarly Multiplying (4) by ξ k (x) and substituting (56)-(57) into the obtained result, we find with initial-boundary conditions: Note that the following eigenvalue problem: in V form an orthonormal basis of H. Let P m : H → span{e 1 , e 2 · ··, e m } be the canonical projection.
2.4. Existence of pullback random attractors. In this section, we show that the cocycle Φ generated by the fractional nonclassical diffusion equations driven by colored noise has a unique D-pullback random attractors in H s (R n ). We first establish the existence of D-pullback absorbing sets in H s (R n ).
We then show the D-pullback asymptotic compactness of Φ.
Proof. Let ε > 0 be an arbitrary number, we show that the sequence has a finite open cover with radiis less then ε in H s (R n ) provided t n → +∞ and u 0,n ∈ D(τ − t n , θ −tn ω). By Lemma 2.3, there are N 1 = N 1 (τ, ω, D, ε) ≥ 1 and Then we see from (71) and (72) that for all n ≥ N 1 , By Lemma 2.4 with k = 2k 0 , there are N 2 ≥ N 1 and m 0 ≥ 1 such that for all n ≥ N 2 , By Lemma 2.1, there are N 3 ≥ N 2 and c 1 = c 1 (τ, ω) > 0 such that for all n ≥ N 3 , which further implies that there exists c 2 = c 2 (τ, ω) > 0 such that for all n ≥ N 3 , Then {P m0 ξ 2k0 u(τ, τ − t n , θ −τ , ω, u 0,n )} ∞ n=1 is pre-compact in P m0 H s (R n ) due to (75) and the finite-dimensional range of P m0 . By (74), n=1 has a finite ε-net in H s (R n ). The proof is completed.
Finally, we present the main result of this section as follows. Proof. Based on Lemmas 2.5 and 2.6, the result follows from [49] and [51] immediately.

Attractors of fractional equations driven by additive white noise.
In this section, we show the existence and uniqueness of random attractors for the following fractional nonclassical diffusion equations driven by additive noise: where λ > 0, s ∈ (0, 1), h ∈ L 2 (R n ), f and g are the same as in Section 2 and W is the Wiener process on the probability space (Ω, F, P). Denote by where z(θ t ω) = I + (−∆) s −1 hy(θ t ω) with y(θ t ω) = − 0 −∞ e s (θ t ω)(s)ds being the stationary solution of the one-dimensional Ornstein-Uhlenbeck equation dy + ydt = dW (t). By [16], we know lim t→±∞ y(θ t ω) t = 0 for every ω ∈ Ω.

3.2.
Uniform tail-estimates. The proof of the following auxiliary estimate is similar to Lemma 2.2 and so omitted.

3.4.
Existence of pullback random attractors. Now, we are in the position to present the main results of this section.
By Lemma 4.1, we establish the existence of D-pullback absorbing set K δ for Φ δ . Furthermore, we show the convergence of the absorbing radii of K δ as δ → 0.