DYNAMICS OF NON-AUTONOMOUS FRACTIONAL GINZBURG-LANDAU EQUATIONS DRIVEN BY COLORED NOISE

. In this work, the existence and uniqueness of random attractors of a class of non-autonomous non-local fractional stochastic Ginzburg-Landau equation driven by colored noise with a nonlinear diﬀusion term is established. We comment that compared to white noise, the colored noise is much easier to handle in examining the pathwise dynamics of stochastic systems. Addi-tionally, we prove the attractors of the random fractional Ginzburg-Landau system driven by a linear multiplicative colored noise converge to those of the corresponding stochastic system driven by a linear multiplicative white noise.

The O-U process is a stationary Gaussian process with the mathematical expectation E(ζ δ ) = 0, and so far, the O-U process is the only existing Markovian Gaussian colored noise (see [6,26] for example). Furthermore, the O-U process is also called a colored noise due to the fact that its power spectrum is not flat compared to the white noise ( [2,9,26,30,32]).
As we know, one can choose the Wiener process W as a stochastic process to represent the position of the Brownian particle, however, the velocity of the particle cannot be obtained from the Wiener process due to the nowhere differentiability of the sample paths of W . However, the O-U process was originally constructed to approximately describe the stochastic behavior of the velocity ( [30,32]), therefore, one can further use it to determine the position of the particle. Moreover, as demonstrated in [26], in many complex systems, stochastic fluctuations are actually correlated and hence should be modeled by colored noise rather than white noise.
During studying stochastic dynamics, one of the most crucial issues arises from the modeling of random forcing. To study such random forcing, we need to consider both the time scale τ d of the deterministic system and the time scale τ r of the random forcing. The stochastic forcing is modeled in different ways based on the ratio of τ r /τ d . If τ r /τ d 1, the dynamical system is very slow with respect to the temporal variability of its random drivers, and hence the random forcing could be modeled as white noise. If τ r /τ d 1, then the dynamics of the system is sensitive to the autocorrelation of the random forcing, and hence the random forcing should be modeled by colored noise. Based on these considerations, the colored noise has been employed in many works to study the dynamics of physical and biological system (see [2,9,15,16,17,26,30,32] and the reference therein).
In this work, we will consider the dynamics of system (1) driven by colored noise. We will prove the random system (1) is pathwise well-posed in L 2 (I) and hence generates a continuous non-autonomous cocycle. Moreover, this cocycle possesses a unique tempered random attractor in L 2 (I). This is in contrast with the corresponding stochastic system driven by a white noise: ∂u ∂t +(1+iν)(−∆) α u+ρu = f (t, x, u)+g(t, x)+R(t, x, u)• dW dt , x ∈ I, t > τ, (2) where the symbol • indicates the system is understood in the sense of Stratonovich's integration. Currently, we can only define a random dynamical system for (2) when the diffusion term R(·, ·, u) is a linear function in u ∈ C. In other words, we are unable to define a random dynamical system for (2) with a general nonlinear function R and hence cannot investigate the dynamics of the stochastic equations by the random dynamical system approach. Therefore, there is no result available in the literature on the existence of random attractors for (2) with a nonlinear function R. The differential equations involving the fractional Laplacian have a wide range of applications in physics, biology, chemistry and other fields of science, see [1,8,11,12,13,14,18]. The solutions of fractional deterministic equations have been studied extensively ( [1,4,5,7,8,11,12,13,14,18,19,22,23,27,28,29], and the references therein). Our main interest in this work is to establish the existence of random attractors of the fractional stochastic Ginzburg-Landau system (1).
We point out that the deterministic fractional Ginzburg-Landau equation ( [19,22,23,25]) and the random attractors of stochastic equations with the Wiener process ( [20,21]) have already been studied. In particular, in [20,21], the author established the existence of random attractors for stochastic equations driven by white noise. However, there is no result available in the literature for the existence of random attractors for the fractional stochastic Ginzburg-Landau equation (1) driven by colored noise. The purpose of the present paper is to close this gap and prove system (1) driven by colored noise has a unique tempered random attractor for α ∈ ( 1 2 , 1).
The rest of the paper is organized as follows. In Section 2, we prove the existence and uniqueness of tempered random attractors for system (1) with a nonlinear diffusion term R. We then prove the existence of such attractors for the stochastic system (2) when R(t, x, u) ≡ u in Section 3. In Section 4, we prove the convergence of solutions and random attractors of system (1) with R(t, x, u) ≡ u, as δ → 0.
Hereafter, we will denote the inner product and norm of L 2 (I) by (·, ·) and · , respectively. The letter c and c i are used for positive constants whose values may change from line to line.

2.
Attractors of random fractional Ginzburg-Landau systems. In this section, we study the dynamics of the random fractional Ginzburg-Landau systems driven by a colored noise, which consists of two steps. In the first step, we define a continuous non-autonomous cocycle for the system. Secondly, we prove the existence of pullback random attractors in L 2 (I) for a wide class of nonlinear functions f and R.
Such a family is called tempered in L 2 (I). Let D be the collection of all tempered families of bounded nonempty subsets of L 2 (I), i.e.
Lemma 2.4. Let u belong to L q and its derivatives of order m, D m u, belong to L r , 1 ≤ q, r ≤ ∞. For the derivatives D j u, 0 ≤ j < m, the following inequalities hold where (the constant c depending only on n, m, j, q, r, θ), with the following exceptional case 1. If j = 0, rm < n, q = ∞ then we make the additional assumption that either u tends to zero at infinite or u ∈ Lq for some finiteq > 0. 2. If 1 < r < ∞, and m − j − n/r is a nonnegative integer then (23) holds only for θ satisfying j/m ≤ θ < 1.

2.2.
Existence of pullback random attractors. In this subsection, we prove the existence of tempered pullback random attractors for system (3)-(5). We first prove the existence of tempered absorbing sets and then show the pullback asymptotic compactness of solutions.
Proof. Taking the inner product of (3) with u in L 2 (I) and taking the real part, we obtain d dt Applying (6), (9) and Young's inequality, we deduce that and 2ζ δ (θ t ω)Re By (25)- (28), we obtain Applying Gronwall's Lemma to (29) over (r, ξ) with ξ ≥ r and for every ω ∈ Ω, we infer Now, replacing r by τ − t and ω by θ −τ ω in (30), we get Next, we estimate every term on the right-hand side of (31). For the first term, Thus, there exists T = T (τ, ω, D, ξ, δ) > 0 such that for all t ≥ T , For the second term, applying (21) we have Due to (14), the third term on the right-hand side of (31) is well-defined . Therefore, we obtain that for all t ≥ T , As a consequence of Lemma 2.5, we obtain that problem (3)-(5) has a D-pullback absorbing set in L 2 (I).
Proof. Let ξ = τ in Lemma 2.5. We obtain that there exists T = T (τ, ω, D, δ) > 0 such that for all t ≥ T , Next, we prove K ∈ D. Let β be an arbitrary positive constant and consider which along with (22) and Lemma 2.1 implies This completes the proof.
Also, we can obtain the following estimates from Lemma 2.5 for later purpose.
Proof. Taking the inner product of (3) with (−∆) α u in L 2 (I) and taking the real part, we obtain For the first term of the right-hand side of (37), by (7)- (8), Gagliardo-Nirenberg inequality and Young's inequality, we have For the second term of the right-hand side of (37), applying Young's inequality, we get
We now prove the existence of D-pullback attractors of Φ. Proof. Since Φ has a closed measurable D-pullback absorbing set K ∈ D by Corollary 2.6 and is D-pullback asymptotically compact in L 2 (I) by Lemma 2.9, then the existence and uniqueness of D-pullback attractor A of Φ follows immediately.
3. Attractors of stochastic fractional Ginzburg-Landau equations. In this section, we study the dynamics of stochastic fractional Ginzburg-Landau equations driven by a linear multiplicative noise. More precisely, we will prove the existence and uniqueness of tempered random attractors for the system. The results of this section will be used for studying the limiting behavior of solutions of the random system (3)-(5) when δ → 0.
Given τ ∈ R, consider the stochastic fractional Ginzburg-Landau equations with homogeneous Dirichlet boundary condition and initial condition where f and g are the same as in the previous section.
By Lemma 3.2 we obtain the following estimates.
where u τ −t ∈ D(τ − t, θ −t ω) and with L 1 being the same constant as in Lemma 3.2 which is independent of τ and ω.
Proof. The proof is similar to that of Lemma 2.9, and hence omitted here.
We now in the position to show the existence of D-pullback attractors of Ψ.

Convergence of random attractors.
In this section, we study the limiting behavior of solutions of the random system (3)-(5) when δ → 0. Under certain conditions, we will show the solutions and attractors of system (3)-(5) converge to that of the corresponding stochastic system when δ → 0. Consider the following random system ∂u δ ∂t (74) Note that system (72)-(74) is a special case of (3)-(5) and can be obtained by formally replacing W (t) by t 0 ζ δ (θ r ω)dr in (46). We will establish the relations between the solutions of systems (46)-(48) and (72)-(74) and show that the limiting behavior of system (72)-(74) is governed by the stochastic system (46)-(48) as δ → 0.
On the other hand, since D ∈ D, applying (15), we obtain that there exists T = T (τ, ω, D, δ) > 0 such that for all t ≥ T , which together with (12) and (87) completes the proof.
As an immediate consequence of Lemma 4.2, we obtain the following estimates.
where u δ,τ −t ∈ D(τ − t, θ −t ω) and with L 4 being the same positive constant as in Lemma 4.2 which is independent of τ, ω and δ.
Proof. Due to (75), we get which along with Lemma 4.2 implies the desired estimates.
To establish the uniform compactness of random attractors, we need the following estimates.
Recall that for each δ > 0, A δ is the unique D-pullback attractor of Φ δ in L 2 (I). To obtain the uniform compactness of these attractors with respect to δ, we need further estimates on Φ δ as given below.
We finally establish the upper semicontinuity of random attractors as δ → 0.