Elsevier

Nonlinear Analysis

Volume 128, November 2015, Pages 303-324
Nonlinear Analysis

Existence and upper semicontinuity of bi-spatial pullback attractors for smoothing cocycles

https://doi.org/10.1016/j.na.2015.08.009Get rights and content

Abstract

In this work, we establish several criteria for the existence as well as the upper semi-continuity of bi-spatial attractors under a closedness condition, which dramatically weakens the usual requirement on the continuity of the cocycle. It is also shown that, though the continuity plays a less important role in the study of attractors, it is impossible to establish an existence criteria for common attractors for systems without any continuity-like properties. However, for such “bad” systems, one can expect a mini attractor, which is shown adequate well to depict the asymptotic behavior of non-continuous systems. Finally, we study the (L2,H01)-pullback attractor for a stochastic complex Ginzburg–Landau equation. A spectrum decomposition method is employed to overcome the lack of Sobolev compactness embeddings in H01.

Introduction

An infinite-dimensional dynamical system is usually generated by an evolution equation which models some dynamical phenomena in the real world, see Temam  [27] for autonomous dynamical systems and Carvalho et al.  [7] for non-autonomous ones. If such an evolution equation is well-defined, that is, for every initial value u0X endowed at the initial time τ there is a unique solution u(t)X, tτ, satisfying the equation, then the solution operator mapping initial data to their solutions defines a dynamical system.

Very often, the system can have a smoothing property (as described by Goubet and Rosa  [11]), in the sense that the solution can belong to Y(X) (probably after a time, and Y a Banach space more regular than X) though the data is merely in X, such as systems generated by parabolic equations. Actually, every system can be regarded as smoothing if the case X=Y is not excluded (as we do in this paper).

To understand the asymptotic dynamical properties of infinite-dimensional systems, the theory of attractors is a powerful tool. On one hand, it depicts all the possible longtime behavior of the system; on the other hand, it is small (compact) itself and is often finite-dimensional though the phase space is infinite-dimensional. For smoothing systems, there is a concept of bi-spatial attractors better curving the long time behavior. By definition, an (X,Y)-attractor is a compact subset of Y attracting some class of sets in X in the topology of Y. Thus, compared with single-spatial attractors, a bi-spatial attractor has higher regularity (compact in Y) and stronger attracting ability (in the topology of Y) and thereby it is of significance to study bi-spatial attractors for smoothing systems.

Most recently, Li et al.  [18], [17] studied the existence and the upper semi-continuity of bi-spatial attractors for random dynamical systems. They obtained a general criteria for a bi-spatial attractor to exist and be upper semi-continuous. It was somewhat surprising that, though a bi-spatial attractor has much nicer topological properties, it requires no extra continuity conditions if the system is continuous in the phase space X.

In this paper, we go further into the study of bi-spatial attractors for (X,Y)-smoothing cocycles. We restrict ourselves to pullback behaviors of cocyles (for an interesting discussion on this kind of behavior we refer the readers to Kloeden et al.  [15]). Such cocycles will cover two parameterized random dynamical systems which can be regarded as a stochastic perturbation of a deterministic non-autonomous dynamical systems, and thereby will have more applications. Moreover, in the view of analysis, by introducing two parameters many interesting features of pullback attractors for deterministic non-autonomous dynamical systems are successfully studied under stochastic perturbations, such as the characterization of the structure of the attractor by complete orbits, complete quasi-solutions and periodic complete orbits, etc., see  [7], [30], [31].

Firstly in this paper, by replacing the continuity condition with a closedness condition we strengthen the existence result of  [18] to obtain a sufficient and necessary condition for the unique existence of bi-spatial attractors for smoothing cocycles. Such a closedness condition is much weaker than norm-to-weak continuity  [35] and even quasi-continuity  [19], [10]. Our first conclusion is that, roughly speaking, if an (X,Y)-smoothing cocycle is closed in X, i.e., the solution operator is a closed operator in X, then it has a unique (X,Y)-pullback attractor A iff it is (X,Y)-asymptotic compactness and has an (X,X)-absorbing set.

It is worth pointing out that, compared with the well-known result for single-spatial attractors, though the bi-spatial attractor A has better topological properties, the condition ensuring its existence only have higher requirements in the dynamical compactness; the continuity stays still. In other words, we have the result: if a smoothing cocycle ϕ has an attractor A in X, then A is the (X,Y)-attractor iff Φ is (X,Y)-asymptotically compact. This result shows that the only factor influencing the regularity of an attractor is the dynamical compactness of the system. This observation inspires us an interesting question: is it possible to establish a general criteria for the existence of an attractor without any continuity-like conditions?

The second aim of this paper is studying the attractors for cocycles without any continuity properties. We first introduce an concept of mini attractors, which are defined as the minimal compact sets with the attracting property. Then it is found that a mini attractor coincides with a standard attractor when the cocycle is continuous. Moreover, a mini attractor exists iff the cocycle has the dynamical compactness and absorbing set, even though the cocycle is not continuous at all. But unfortunately, an example shows that the existence of a mini attractor cannot imply that of a standard one. Thus standard attractors may not be expectable for non-continuous cocycles, and mini attractors are more adequate to them.

We next study the stability problem of a bi-spatial attractor A. That is, suppose ϕϵ is a perturbed system of ϕ (where ϵ denotes the perturbation intensity) with an (X,Y)-attractor Aϵ, then under what conditions it can hold the relation distY(Aϵ,A)0 as the perturbation intensity ϵ tends to zero. Such a stability subject, also known as the upper semi-continuity of attractors, has been presented by many publications, such as  [6], [4], [5], [25], [29] for single-spatial attractors, and recently,  [18], [17] for bi-spatial attractors. In both cases, the central assumption of all the above publications to ensure the stability is the so-called pathwise convergence condition (see Section  3, also Robinson  [25]). In this paper, we loosen this condition to a pathwise closedness condition. We also find that the stability can be robust under higher topology if the union of attractors has stronger compactness (see Theorem 3.3).

To better show the difference involving in the study of bi-spatial attractors from single-spatial attractors, we take the following stochastic Ginzburg–Landau (GL) equation as a model: du=[(λ+iα(t))u(κ+iβ(t))|u|2u+δu+g(x,t)]dt+ϵh(x)dW(t), where the dispersion coefficients α, β and the external force g are all time-dependent. W is a standard two-sides real-valued Brown motion and ϵ is the perturbation intensity. The unknown u is a complex-valued function defined on a bounded interval I of R.

The (single-spatial) dynamical properties of GL systems have been widely studied (see, e.g.,  [27], [22], [28], [13]). Remarkably, the authors of  [22], [13], [28] studied the long time behavior of GL equations defined on unbounded domains by introducing weighted functions, which helps to obtain the asymptotic compactness of the system in weighted spaces. For stochastic cases, Yang  [33] investigated the random attractor for stochastic GL equation with multiplicative noise, while Wang et al.  [32] studied stochastic GL equation with additive noise.

Unlike papers mentioned above working on the dynamical properties only in the phase space, Lü et al.  [20] investigated a 3D autonomous GL equation and showed that, under some assumptions, the attractor can be much more regular than the initial data. This result indicates a strong asymptotic smoothing property of the system in the sense that, though the initial data has low regularity, the solution can go closer and closer to high regularity attractors in the topology of the phase space.

In this paper, we study the bi-spatial attractor for the stochastic GL system. We prove that, under some classical conditions, the stochastic system possesses an (L2,H01)-pullback attractor. Moreover, this attractor is upper semi-continuous in H01 as the perturbation decays (see Theorem 4.4).

The main difficulty here arises in verifying the (L2,H01)-type dynamical compactness as predicted since this is the central part in the analysis of bi-spatial attractors. As we will see, when the initial data are only supposed to be in L2, the solution will at most belong to H01. Thus Sobolev compactness embeddings are not available in H01. To overcome this difficulty, we first make some complementary estimates uniform in tails of solutions, and then employ an idea of spectrum decomposition method to obtain an (L2,H01)-type flattening property from which the desired (L2,H01)-asymptotic compactness follows.

In the next section, we establish some criteria for the existence of bi-spatial attractors when the cocycle has poor continuity. Some relations between a bi-spatial attractor and a single-spatial attractor are highlighted through direct comparisons. We also introduce the concept of a mini attractor in this section for non-continuous cocycles, and an example is made to show the role played by the mini attractor in describing the long-time behavior of a non-continuous system. The stability of bi-spatial attractors is studied in Section  3. The application to a stochastic complex Ginzburg–Landau system is placed in Section  4.

Section snippets

Bi-spacial pullback attractors for smoothing cocycles

We now study the existence of a bi-spatial attractor, (X,Y)-type attractor, for smoothing cocycles. Let us begin with a preliminary setting for the two spaces X and Y. Throughout this paper, we let (X,dX)  and  (Y,dY)  be two complete metric spaces with  YX  continuously except otherwise stated.1

Upper semi-continuity of bi-spacial pullback attractors

Since an (X,Y)-attractor is a compact subset of Y, it is natural to consider in what conditions it can be stable in Y (not only in X). In this section, we let {ϵn}nN{0}R be a set of parameters with ϵnϵ0 as n.

Definition 3.1

Let {ϕϵn}n=1 be a family of (X,Y)-smoothing cocycles with a family {Aϵn}n=1 of corresponding (X,Y)-DX-pullback attractors, ϕϵ0 an (X,Y)-smoothing cocycle with an (X,Y)-DX0-pullback attractor Aϵ0(ω1,ω2), where DX0DX. Then the family {Aϵn} is called upper semi-continuous at ϵ0 in Y

An application to stochastic complex Ginzburg–Landau equations

In this section, we apply the relations between bi-spatial attractors and single-spatial attractors presented by Theorem 2.11, Theorem 3.3 to a non-autonomous complex Ginzburg–Landau equation perturbed by additive noises. It shows that, under some standard assumptions, the complex Ginzburg–Landau equation admits an (L2,H01)-pullback attractor, which strengthens the result by Cui and Li  [16] where the (L2,L2)-pullback attractor was established.

Given τR and tτ, we consider the stochastic

Acknowledgments

The authors would like to thank the anonymous reviewers for carefully reading the manuscript and for their valuable comments. This work was supported by the NSFC Grants 11571283 and 11371183.

References (35)

  • G. Wang et al.

    The asymptotic behavior of the stochastic Ginzburg–Landau equation with additive noise

    Appl. Math. Comput.

    (2008)
  • C. Zhong et al.

    The existence of global attractors for the norm-to-weak continuous semigroup and application to the nonlinear reaction–diffusion equations

    J. Differential Equations

    (2006)
  • L. Arnold

    Random Dynamical Systems

    (1998)
  • J.P. Aubin et al.

    Set-Valued Analysis

    (1990)
  • T. Caraballo et al.

    On the upper semicontinuity of cocycle attractors for non-autonomous and random dynamical systems

    Dyn. Contin. Discrete Impuls. Syst. Ser. A

    (2003)
  • T. Caraballo et al.

    Upper semicontinuity of attractors for small random perturbations of dynamical systems

    Comm. Partial Differential Equations

    (1998)
  • T. Caraballo et al.

    Stability and random attractors for a reaction–diffusion equation with multiplicative noise

    Discrete Contin. Dyn. Syst.

    (2000)
  • Cited by (28)

    • Local equi-attraction of pullback attractor sections

      2021, Journal of Mathematical Analysis and Applications
      Citation Excerpt :

      The study of regularity in more regular spaces for attractors is of certain interest. Generally, more functional analysis is involved and more careful techniques are often required, e.g., the measure of noncompactness [32], flattening and squeezing [38,22], see also [4,29,8,9], etc. In this paper we shall use an effective but technical joint bi-spatial continuity method.

    • Regularity and structure of pullback attractors for reaction-diffusion type systems without uniqueness

      2016, Nonlinear Analysis, Theory, Methods and Applications
      Citation Excerpt :

      Such a dynamical compactness property is also termed as pullback asymptotically upper semi-compact in the literature, see [21,18]. Slightly modifying the proof one can establish the previous lemma also for m-NDS not upper semi-continuous but with closed graph, see [6] and also [7]. The structure of attractors in terms of complete trajectories for multi-valued dynamical systems was also studied by [20,9] for uniform attractors and [14,8] for multi-valued semigroups.

    View all citing articles on Scopus
    View full text