RANDOM ATTRACTORS FOR NON-AUTONOMOUS FRACTIONAL STOCHASTIC GINZBURG-LANDAU EQUATIONS ON UNBOUNDED DOMAINS∗

This paper deals with the dynamical behavior of solutions for nonautonomous stochastic fractional Ginzburg-Landau equations driven by additive noise with α ∈ (0, 1). We prove the existence and uniqueness of tempered pullback random attractors for the equations in L(R). In addition, we also obtain the upper semicontinuity of random attractors when the intensity of noise approaches zero. The main difficulty here is the noncompactness of Sobolev embeddings on unbounded domains. To solve this, we establish the pullback asymptotic compactness of solutions in L(R) by the tail-estimates of solutions.

The generalized complex Ginzburg-Landau equation is one of the most important equations in mathematical physics, which can describe turbulent dynamics and has a long history in physics as a generic amplitude equation near the onset of instabilities in fluid mechanical systems, as well as in the theory of phase transitions and superconductivity [2,9]. The fractional Ginzburg-Landau equation describes the dynamical processes in a medium with fractal dispersion and the fractional generalization of Ginzburg-Landau equation from variational Euler-Lagrange equation for the fractal media is derived in [45]. The global existence and long behavior of Ginzburg-Landau equation were studied in [9,11,19,25,26]. In fractional case, the well-posedness and dynamical behavior were proved in [27,36]. For stochastic fractional Ginzburg-Landau equation with α ∈ ( 1 2 , 1), the existence of the random attractor in L 2 and H 1 were respectively discussed in [28,41,44]. We note that in [28], the authors derived the estimates of solutions for H 1+α and the tailestimates of solutions in H 1 instead of H α . This may be caused by the definition of fractional Laplace operator. In this paper, we will furthermore consider uniform a priori estimates of solutions in H α and the tail-estimates of solutions in L 2 by introducing another definition of fractional Laplace operator, which is different from [28].
As we know, there are some results on random attractors for stochastic fractional Ginzburg-Landau equation, but few results for the fractional case with α ∈ (0, 1)(see [23,24,60]). In this paper, motivated by [4,55], we explore the random attractors for non-autonomous stochastic fractional Ginzburg-Landau equation with additive noise for α ∈ (0, 1). However, there are several difficulties to overcome. Firstly, the fractional Laplace operator (−∆) α is non-local and thus deriving uniform estimates on the solutions of (1.1) is much more involved than the standard Laplacian −∆. Secondly, the domain is unbounded, so the Sobolev embedding H α (R 3 ) → L 2 (R 3 ) with α ∈ (0, 1) is not compact. Thirdly, since the Ginzburg-Landau equation is a complex equation, the condition of nonlinearity and uniform estimates of solutions in L 2 are slightly different from the real equation such as reaction-diffusion equation [55], thus we need to develop some different technologies to solve these problems. Lastly, due to α ∈ (0, 1) instead of α ∈ ( 1 2 , 1), the methods in [28] are not suitable, so we have to deal with (1.1) by some different methods. We mention that, in this paper, since considering fractional Laplacian operator with α ∈ (0, 1) and estimates of solutions for H α , the result of random attractors is a generalization in some sense for the results of [28] with α ∈ ( 1 2 , 1). When we prove the existence and uniqueness of random attractors in H α (R 3 ), the nonlinearity f (u) is special form, i.e.,f (u) = −(1+iµ)|u| 2 u with µ ∈ R, which is consistent with the general physical background for the Ginzburg-Landau equation [2,9]. In addition, we apply equivalent representations of the fractional Laplace operator (−∆) α to derive the uniform estimates on solutions of (1.1) in H α (R 3 ) and carefully treat all terms involved. To overcome the lack of compactness of Sobolev embeddings on unbounded domains, we apply the idea of uniform estimates on the tails of solutions and prove the solutions are asymptotically null when t and x tend to infinity, which is slightly different from the fractional case [28] with α ∈ ( 1 2 , 1), and the standard Laplacian case [4] in H 1 (R 3 ).
This paper is organized as follows. In Section 2, the working function space, some basic concepts related to the non-autonomous random dynamical system, upper semicontinuity of random attractors, the fractional derivative and Sobolev space are introduced. In Section 3, we transform the stochastic equation into a random equation which solutions generate a random dynamical system, then give the existence and uniqueness of solutions for non-autonomous stochastic fractional Ginzburg-Landau equation. In Section 4, we derive uniform estimates for solutions and the pullback asymptotic compactness, then the existence of a pullback random attractor is proved. In Section 5, we establish the upper semicontinuity of random attractors when the coefficient δ approaches zero.

Preliminaries
In this section, we first present some basic notions about random attractors and non-autonomous random dynamical systems, which can be found in [1, 6,38].
At last, we review some concepts and notations of the fractional derivative and fractional Sobolev space(see [34]) for details). Let S be the Schwartz space of rapidly decaying C ∞ functions on R 3 , then for 0 < α < 1, the fractional Laplace operator (−∆) α is given by, for u ∈ S, where C(α) is a positive constant depending on α as given by In particular, it follows from [34] that for any u ∈ S, where F is the Fourier transform defined by From now on, we write the norm and the inner product of L 2 (R 3 ) as . and (., .), respectively. We also write the Gagliardo semi-norm of is a Hilbert space with inner product given by In terms of (2.3), one can verify (see [34]):

The stochastic fractional Ginzburg-Landau equation with additive noise
In this section, we will give the existence and uniqueness of solutions of problem (1.1)-(1.2) which generates a continuous random dynamical system.
The standard probability space (Ω, F, P ) will be used in this paper where Ω = {ω ∈ C(R, R) : ω(0) = 0}, and F is the Borel σ-algebra induced by the compactopen topology of Ω, and P is the Wiener measure on (Ω, F). Given t ∈ R, define θ t : Ω → Ω by Then (Ω, F, P, (θ t ) t∈R ) is a parametric dynamical system. Let y : Ω → R be a random variable given by: y(ω) = −γ 0 −∞ e γτ ω(τ )dτ for ω ∈ Ω. Then y(t) is the unique stationary solution of the stochastic equation In addition, it follows from [1], that there exists a θ t -invariant set of full measure (still denoted by Ω) such that y(θ t ω) is pathwise continuous for each fixed ω ∈ Ω and there exists a tempered function r(ω) > 0 such that where r(ω) satisfies, for P.a.e.ω ∈ Ω, From above, we obtain that, for P − a.e.ω ∈ Ω, We now transform the stochastic equation (1.1) into a pathwise deterministic one by using the random variable z.
Next we will first give the existence and uniqueness of solutions for problem (3.5)-(3.6), and then obtain the solutions of (1.1)-(1.2) by the transform (3.4).
To give the existence of solutions, we also need the space By the standard Galerkin method and compactness argument, as shown in [24], we can prove that in the case of a bounded domain with Dirichet boundary conditions, for P .a.e.ω ∈ Ω and for all This is similar to [9,36,55]. Then, following the approach in [33], we take the domain to be a sequence of balls with radii approaching ∞ to deduce the existence of a weak solution of equation (3.5) on R 3 . Furthermore, we can get that v(t, τ, ω, v τ ) is unique and continuous with respect to v τ in H α (R 3 ) for all t ≥ τ . Now by the solution v of (3.5)-(3.6) and the transform (3.4), we get a solution u of the stochastic equation (1.1)-(1.2) which is given by with u τ = v τ + δz(θ τ ω). We note that u(t, τ, ω, u τ ) is both continuous in t ∈ [τ, ∞) and in u τ ∈ H. Moreover, u(t, τ, ., u τ ) : Ω → H is measurable. Then we can define a continuous cocycle in H associated with the solutions of problem (1.1)-(1.2). Let Φ : R + × R × Ω × H → H be a mapping given by, for every t ∈ R + , τ ∈ R, ω ∈ Ω and u τ ∈ H, where v τ = u τ − δz(θ τ ω). In later sections, we will prove the existence and upper semicontinuity of tempered random attractors for Φ in H.
where the norm B of set B in H is given by B = sup u∈B u . From now on, we will use D to denote the collection of all tempered families of bounded nonempty subsets of H: When deriving uniform estimates, for simplicity, we assume that γ > 0, and also assume that for every τ ∈ R, 0 −∞ e γs g(s + τ, .) 2 ds < ∞. (3.8) Sometimes, we also assume g is tempered in the following sense: for every c > 0, Note that these conditions do not require g to be bounded in H when t → ∞. Through this paper, c denotes a positive constant which may be different from the context.

Random attractors
In this section, we will derive uniform estimates on the solutions of non-autonomous stochastic fractional Ginzburg-Landau equation in H and V . These estimates are necessary for proving the existence of random attractors. Then we prove the existence and uniqueness of pullback random attractors. We first derive uniform estimates of solutions in H.
Proof. Taking the inner product of (3.5) with v, and taking the real part, we have We now estimate all terms in (4.1). For the last term, we have On the other hand, we find It follows from (4.1)-(4.5) that 4 . Therefore, the first term on the right-hand side of (4.6) can be bounded by By (3.3), we find that for P − a.e.ω ∈ Ω Multiplying (4.6) with e γt , then integrating the inequality on ( Replacing After change of variables, we obtain Now we estimate the first term on the right-hand side of (4.8).
Proof. Taking the inner product of (3.5) with ρ( |x| l )v and taking the real part, we have Now we estimate each term on the right side of (4.32). For the first term, from (4.31), we get For the second term, we have For the last term, we get For the nonlinear term, we find It follows from (4.32)-(4.36) that for all l ≥ L 1 (ω) ≥ 1 Given t ∈ R + , τ ∈ R and ω ∈ Ω, multiplying (4) with e γt , then integrating the result on (τ − t, τ ), we get for all l ≥ L 1 (ω) Then replacing ω by θ −τ ω we have for all l ≥ L 1 (ω) After change of variables, for all l ≥ L 1 (ω) ≥ 1 we obtain Now we estimate the first term on the right-hand side of (4.38).
For the second term, from Lemma 4.1 with σ = τ , there exists T 2 = T 2 (τ, ω, B, ε) > 0 such that for all t ≥ T 2 , we have  By (4.38)-(4.45) we obtain, for all l ≥ L 3 and t ≥ T 2 , From (4.46), the desired estimates follow immediately. In this following, we give the existence of tempered pullback absorbing set in H, and the asymptotic compactness of (1.1)-(1.2) in H.
Proof. We first prove that K δ absorbs every member B of D. By (3.7) we have where B 1 (τ, ω) is the same as in (4.13). It follows from (4.47)-(4.48) and (4.13)-(4.14) that for all t ≥ T , On the other hand, by (3.7) we have which along with (4.50) shows that Φ(t, τ − t, θ −t ω, u τ −t ) ∈ K δ for all t ≥ T , and hence K δ absorbs all elements of D. We now prove K δ is tempered, i.e., K δ ∈ D.
By (3.7) and r(ω) is tempered, for every ε > 0, τ ∈ R and ω ∈ Ω, there exists T = T (τ, ω, ε) > 0 such that for all t > T , From (5.4), we now prove the following uniform estimates on the tails of functions in random attractors.