Existence and continuity of bi-spatial random attractors and application to stochastic semilinear Laplacian equations

Li, Yangrong; Gu, Anhui; Li, Jia

doi:10.1016/j.jde.2014.09.021

Cited by 154 publications

(125 citation statements)

References 24 publications

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“…The first is the existence of the (H, V )-pullback attractor A = {A(τ )} τ ∈R for (1.1), which is a non-autonomous set pullback attracting bounded sets of H under the topology of V , where H := (L 2 (O)) d , V := (H 1 0 (O)) d . This is a study of bi-spatial attractors which attracted much attention these years due to their higher regularity and stronger attracting ability compared with usual attractor, see [7,13,12] for single-valued non-autonomous/random cocycles and [21,19] for multi-valued semi-groups and random cocycles, respectively. In this paper, we develop a study for m-NDS which can be regarded as an extension of the bi-spacial attractor theory on one hand, and is interesting because of the multi-valued feature on the other hand.…”

Section: Introductionmentioning

confidence: 99%

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

Cui

Langa

2016

Nonlinear Analysis

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In this paper, we study the pullback attractor for a general reaction-diffusion system for which the uniqueness of solutions is not assumed. We first establish some general results for a multi-valued dynamical system to have a bi-spatial pullback attractor, and then we find that the attractor can be backwards compact and composed of all the backwards bounded complete trajectories. As an application, a general reaction-diffusion system is proved to have an invariant (H, V )-pullback attractor A = {A(τ )} τ ∈R . This attractor is composed of all the backwards compact complete trajectories of the system, pullback attracts bounded subsets of H in the topology of V , and moreoverA non-autonomous Fitz-Hugh-Nagumo equation is studied as a specific example of the reactiondiffusion system.

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Section: Introductionmentioning

confidence: 99%

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

Cui

Langa

2016

Nonlinear Analysis

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“…By the abstract result given by [14,Theorem 4.1], it suffices to verify the following three aspects.…”

Section: Theoremmentioning

confidence: 99%

“…By applying the abstract result given in Li et al [14], we need to verify the uniform absorption (Section 3), the collectively limit-set compactness (Section 4) and the convergence (Section 5) of the random system. In particular, the novelties and difficulties come from verifying the limit-set compactness.…”

Section: Introductionmentioning

confidence: 99%

Probabilistic Robustness for Dispersive-Dissipative Wave Equations Driven by Small Laplace-Multiplier Noise

Wang¹,

Li²,

Li³

2018

DSA

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This paper is devoted to limit-dynamics for dispersive-dissipative wave equations on an unbounded domain. An interesting feature is that the stochastic term is multiplied by an unbounded Laplace operator. A random attractor in the Sobolev space is obtained when the density of noise is small and the growth rate of nonlinearity is subcritical. The random attractor is upper semicontinuous to the global attractor when the density of noise tends to zero. Both methods of spectrum and tail-estimate are combined to prove the collective limit-set compactness. Furthermore, a probabilistic method is used to show that the robustness of attractors is basically uniform in probability.

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“…All those assumptions are different from dealing with the pullback dynamics for other nonautonomous dissipative equations (see, e.g., [17,[21][22][23][24][25][26][27][28]). …”

Section: Introductionmentioning

confidence: 99%

Pullback-Forward Dynamics for Damped Schrödinger Equations with Time-Dependent Forcing

She

Wang

2018

Discrete Dynamics in Nature and Society

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This paper deals with pullback dynamics for the weakly damped Schrödinger equation with time-dependent forcing. An increasing, bounded, and pullback absorbing set is obtained if the forcing and its time-derivative are backward uniformly integrable. Also, we obtain the forward absorption, which is only used to deduce the backward compact-decay decomposition according to high and low frequencies. Based on a new existence theorem of a backward compact pullback attractor, we show that the nonautonomous Schrödinger equation has a pullback attractor which is compact in the past. The method of energy, high-low frequency decomposition, Sobolev embedding, and interpolation are quite involved in calculating a priori pullback or forward bound.

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Existence and continuity of bi-spatial random attractors and application to stochastic semilinear Laplacian equations

Cited by 154 publications

References 24 publications

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

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

Probabilistic Robustness for Dispersive-Dissipative Wave Equations Driven by Small Laplace-Multiplier Noise

Pullback-Forward Dynamics for Damped Schrödinger Equations with Time-Dependent Forcing

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