Abstract

Long time behavior of stochastic Ginzburg-Landau equations with nonautonomous deterministic external forces, dispersion coefficients, and nonautonomous perturbations is studied. The domain is taken as a bounded interval in . By making use of Sobolev embeddings and Gialiardo-Nirenberg inequality we obtain the existence and upper semicontinuity of the pullback attractor in for the equation. The upper semicontinuity shows the stability of attractors under perturbations.

1. Introduction

Consider the following stochastic complex Ginzburg-Landau equation with nonautonomous deterministic external force and dispersion coefficients: where is unknown and complex-valued; ; dispersion coefficients , and external force are all time-dependent; , , , and are positive real constants; is a given complex-valued function; is supposed to be a real-valued Wiener process on a complete probability space. (For detail of conditions see Lemma 12.)

As a kind of Schrödinger type equation, complex Ginzburg-Landau (G.-L.) equations arise in various areas of physics and chemistry as pointed out by Temam [1]. Due to their rich mathematical properties, Ginzburg-Landau equations have drawn much attention of mathematicians and many related articles appear in literature, such as [17]. In [15], existence and attractors for deterministic G.-L. equations are studied and in [6, 7], the authors take perturbations into consideration and have investigated the existence of attractors of solutions.

This paper deals with a nonautonomous case, equation . Note that, except for perturbations, external force and dispersions and are all time-dependent. To investigate the long time behavior of solutions for such stochastic equations with nonautonomous deterministic terms, Wang [8, 9] generalized the theory on existence and upper semicontinuity of attractors for random dynamical systems (RDS), which provides us with abstract criteria (see Lemmas 10 and 11) to investigate system .

Since the spatial domain is bounded and the G.-L. equation has smoothing effect, Sobolev embeddings are applied to obtain the asymptotic compactness of the system, as well as constructing a compact random absorbing set, which is needed to investigate not only the existence, but also the upper semicontinuity (or the stability at ) of pullback attractors for G.-L. system . Gialiardo-Nirenberg inequality also contributes to estimation of the nonlinear term.

The main result of this paper contains Theorems 20 and 22.

This paper is organized as follows. In Section 2 we recall some concepts and abstract results on attractors of RDS and define a cocycle for nonautonomous system . Uniform estimates of solutions are established in Section 3. The main result of this paper is concluded in the last section.

Notations. In this paper, any function space should be considered complex-valued except that there are particular explanations. We denote by the norm of any (complex- or real-valued) Banach space . The norm of a -times Lebesgue integrable space , , is denoted by for short and . We often write and for convenience and denote the inner product of by with , for all . To investigate the stability of attractors, we use subscript “” or superscript “” to indicate the dependence on when necessary. Letter stands for positive constants independent of and may change its value from line to line; indicates the exclusive dependence on the parameters in the bracket.

2. Preliminary Result and the Cocycle

2.1. Pullback Attractors for Cocycles

In this part we recall some theory of pullback attractors for RDS with nonautonomous deterministic terms. The reader is referred to [810] for more details.

Suppose we are given a nonempty unbounded subset of , a probability space and a Banach space with Borel -algebra .

Definition 1. A nonautonomous random set of is a set-valued mapping: which is measurable with respect to in . That is, the value of every is a closed nonempty subset of and the mapping is -measurable for each fixed and .

Definition 2. Suppose there are two groups and acting on and , respectively. and are called two parametric dynamical systems if (i) is a mapping such that is the identity operator on ;(ii) for all ;(i′) is a -measurable mapping such that is the identity operator on ;(ii′) for all and for all .

Definition 3. Class is defined for any fixed as the collection of all nonempty and bounded nonautonomous random sets of , of which every member satisfies where .

Definition 4. Let and be two parametric dynamical systems. A mapping is called a cocycle on over and if for all , , and , it satisfies that (i) is -measurable;(ii) is the identity operator on ;(iii). If, in addition, is always a continuous mapping on , is called a continuous cocycle.

Let in the sequel be a continuous cocycle on a Banach space over and .

Definition 5. Suppose . Then is called a -pullback absorbing set for if for all , -a.e. and, for every , there exists a such that If, in addition, for each and -a.e. , is a closed subset of and is measurable with respect to in , then we say is a closed measurable -pullback absorbing set for .

Definition 6. is said to be -pullback asymptotically compact in if for all and -a.e. , sequence has a convergent subsequence in whenever with and .

Definition 7. A random variable is called tempered with respect to if where applies only to two-sided time.

Proposition 8. By Definition 7 the following properties hold true. (i)If a random variable is tempered, then (ii)If random variables and are both tempered, so are and .

Definition 9. A nonautonomous random set of is called a -pullback attractor for if the following conditions are satisfied for every and -a.e. . Consider (i) and is compact in ;(ii) is invariant; that is, (iii) attracts every set in ; that is, for every , where denotes the Hausdorff semimetric in defined as

The following results are borrowed from Wang [811].

Lemma 10. Let be a continuous cocycle on over and . Then has a unique -pullback attractor in if and only if is -pullback asymptotically compact in and has a closed measurable -pullback absorbing set in .

Given a metric space . For every , let be a continuous cocycle over and .

Lemma 11. Suppose for each , has a -pullback attractor , particularly, ; then for every and , if provided (i)for every , , , and with , , with , it holds that (ii)there exists a mapping such that the family (iii), has a -pullback absorbing set such that for all and , where is as in (ii) and , for all ;(iv)for every and ,

2.2. Nonautonomous Stochastic G.-L. Equations

Given and , consider equations defined for , with initial-boundary value conditions where the unknown is a complex-valued function; dispersion coefficients , , and external force are all time-dependent and real-valued functions; , , , and are positive constants; ; is supposed to be a two-sided real-valued Wiener process on a complete probability space , where is Borel -algebra induced by the compact open topology of , and is the corresponding Wiener measure on .

To define a proper cocycle for system (13), we first define the related parametric dynamic systems as follows. Let be the group acting on satisfying and the group acting on such that where we have identified with Then by Definition 2 it is evident that and are two parametric dynamical systems.

We now define a continuous cocycle over and for system (13). Consider the one-dimensional Ornstein-Uhlenbeck equation of which a stationary solution is provided by It is known that there exists a -invariant set with such that is continuous in for every , and the random variable is tempered (see, e.g., [8, 10, 12, 13], and hereafter we will not distinguish from ). Therefore, by Proposition 8 and [12, Proposition ] (see also [1416]), there exists a tempered variable such that where satisfies Let , , and Then if solves (13)-(14), should satisfy, by (18), (22), and , with conditions for all and . Since (23)-(24) is a deterministic problem, by the “standard” Galërkin method as in [17] or similar arguments of [2] (see also [1, 5, 6] for autonomous G.-L. equations), we have the following well-possessedness result.

Lemma 12. Assume that (i), , , , ;(ii) and ;(iii) and are such that Then, for each , the initial-boundary value problem (23)-(24) has a unique weak solution Besides, is -measurable in and continuous in with respect to the norm of for each .

Let Then under assumptions of Lemma 12 it is evident that solves problem (13)-(14) and is -measurable in , continuous in both , and . Consider the mapping with where . By the property of solution trajectories of well-possessed nonautonomous dynamical systems one can readily check according to Definition 4 that (28) defines a continuous cocycle for problem (13)-(14) on over and , where and are given by (15) and (16), respectively.

To investigate the (-) pullback attractor in for system (13), hereafter in this paper, we let, for an arbitrarily fixed , be the class satisfying Definition 3 with .

3. Uniform Estimates of Solutions

In this section we estimate the solution of problem (13)-(14) to establish a -random absorbing set, as well as to obtain the -pullback asymptotic compactness for the cocycle under assumptions of Lemma 12. We begin with two useful lemmas.

Lemma 13 (Young’s inequality). Let , . Then for every , satisfying , , , it holds that

Lemma 14 (Gagliardo-Nirenberg’s inequality). Let , , , . Then for , , there exists a constant such that where .

By Lemma 14 we derive the following estimates for later convenience.

Corollary 15. Let . Then it holds true for every well-defined that

Lemma 16. Let assumptions of Lemma 12 hold. Then for every , , and , there exists a and a positive constant , which depends on but is independent of , , , and , such that the solution with of (23)-(24) satisfies, for all , that where is the tempered random variable given by (20) and (21).

Proof. Taking the inner product of (23) with in and taking the real part, we get By conditions , , and we derive that where . Since similarly we have where ; from (34)–(36) it follows that which implies that where . Multiply (38) by and integrate over , , to get, for each , Notice that . Therefore, replacing in (39) with and by (20)-(21) we obtain where is a positive constant depending on but independent of , , , and . Since , by Definition 3 there exists a such that which along with (40) and (20)-(21) completes the proof.

Lemma 17. Let assumptions of Lemma 12 hold. Then for every , , and , there exists a and a positive constant , which depends on but is independent of , , , and , such that the solution with of (23)-(24) satisfies, for all , where is the tempered random variable given by (20) and (21).

Proof. Notice that for all . Hence, by (33) we have for all , which concludes the lemma.

Lemma 18. Let assumptions of Lemma 12 hold. Then for every , , and , there exists a and a positive constant , which depends on but is independent of , , , and , such that the solution with of (23)-(24) satisfies, for all , where is the tempered random variable given by (20) and (21).

Proof. Taking the inner product of (23) with in and taking the real part, we have Estimate the first term in the right hand side of (45) to get (52). Since , we have By the condition , for all , we have where denotes the conjugate transpose of matrix and Recall the Agmon inequality that By Lemma 13, , and (49) we estimate the second term in the right hand side of (46) to obtain where . Similarly, for the last term of (46) we have From (46)–(51) and Lemma 13 it follows that For the last term of (45), by Lemma 13 again we get where . Since, by (31) and Lemma 13 again, from (45), (52), and (53) we conclude that where is a positive constant independent of , , and . Given , , , and , integrating (55) over and by (20) we find that where is a positive constant independent of , , and . Integrating (56) with respect to over and replacing with , by (21) we derive that where depends on but is independent of , , , and . Let be the same as in Lemma 17. Then from (57) and (42) it follows that which completes the proof.

To derive uniform estimates on the solutions of (13)-(14), recall from (28) that where . Hence, for or , we have Moreover, by the temperance of it is evident that comes from a nonautonomous random set in provided, and so does . Therefore, Lemmas 16 and 18 imply the following lemma.

Lemma 19. Let assumptions of Lemma 12 hold. Then for every , , and , there exists a and a positive constant , which depends on but is independent of , , , and , such that the solution with of (13)-(14) satisfies, for all , that and that where is the tempered random variable given by (20) and (21).

4. Pullback Attractors for Stochastic G.-L. Equations

4.1. Existence

In this part, for each we establish the existence of the -pullback attractor for system (13).

Consider the nonautonomous random set with for each and , where is given by where is the constant found out by Lemma 19. It is evident that is -measurable for each . Moreover, by the temperance of and assumption (iii) of Lemma 12 one can readily verify that which indicates by Definition 3 that . Therefore, by Definition 5, definition (28), and Lemma 19 (62), is a closed (and moreover, compact by Sobolev compactness embeddings) measurable -pullback absorbing set for . In addition, it is evident that is -pullback asymptotically compact in . Hence, by Lemma 10 we obtain the following existence result.

Theorem 20. Let assumptions of Lemma 12 hold. Then for each the cocycle associated with problem (13)-(14) has a unique -pullback attractor in .

4.2. Upper Semicontinuity

In the sequel we denote by the cocycle corresponding to the solution with of (13) (to indicate the dependence of ), and let be the solution operator generated by the deterministic nonautonomous system and given by where with solves the problem (66)-(67). Analogously to cocycles , under assumptions of Lemma 12   is a continuous operator from to , and moreover, it has a -pullback attractor .

Lemma 21. Let assumptions of Lemma 12 hold. Then for every , , , and , with as , it holds that

Proof. Let with and write . Then by (23) and (66), satisfies where . Since, for each fixed , there exists a such that, for the fixed ,
Taking the inner product of (70) with in and taking the real part we have In the sequel we consider . By the condition , for all , we have where depends on but is independent of . Similarly by Lemma 13 we have Then from (73)–(75) it follows that where is a positive constant independent of and is given by Let be an arbitrarily fixed time. By Agmon inequality (49) and (62) we have Therefore, replacing with in (76) and by Gronwall lemma we have Since , from (71) and (79) it follows that which completes the proof.

Theorem 22. Let assumptions of Lemma 12 hold. Then for every and ,

Proof. We prove that the result verifies the four conditions of Lemma 11. First, it is evident that condition (i) is actually proved by Lemma 21. To verify the rest, let where is the constant as in (63). Since for each defined by (63) is a compact -pullback absorbing set for in , conditions (ii) and (iii) of Lemma 11 hold and condition (iv) follows from the fact that Then the theorem is concluded.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to express their thanks to anonymous reviewers for their valuable comments. This work was partially supported by the NSFC Grants 11071199 and 11371183.