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

Cui, Hongyong; Li, Yangrong; Yin, Jinyan

doi:10.1016/j.na.2015.08.009

Cited by 28 publications

(37 citation statements)

References 28 publications

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“…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].…”

Section: Existence Of a Backwards Bounded Pullback Attractormentioning

confidence: 99%

“…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%

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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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“…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].…”

Section: Existence Of a Backwards Bounded Pullback Attractormentioning

confidence: 99%

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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“…A pullback attractor given in Def.3.1 involves two phases spaces, in this respect, the concept coincides with bi-spatial attractors (see e.g. [7,14,15,16,18,28,29]). However, a bi-spatial attractor is sufficiently regular, which means it is compact and attracting in the terminate space E (rather than E 0 only).…”

Section: Remarkmentioning

confidence: 96%

Backward controllability of pullback trajectory attractors with applications to multi-valued Jeffreys-Oldroyd equations

Li¹,

Wang²,

She³

2018

Evolution Equations &Amp; Control Theory

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This paper analyzes the time-dependence and backward controllability of pullback attractors for the trajectory space generated by a nonautonomous evolution equation without uniqueness. A pullback trajectory attractor is called backward controllable if the norm of its union over the past is controlled by a continuous function, and backward compact if it is backward controllable and pre-compact in the past on the underlying space. We then establish two existence theorems for such a backward compact trajectory attractor, which leads to the existence of a pullback attractor with the backward compactness and backward boundedness in two original phase spaces respectively. An essential criterion is the existence of an increasing, compact and absorbing brochette. Applying to the non-autonomous Jeffreys-Oldroyd equations with a backward controllable force, we obtain a backward compact trajectory attractor, and also a pullback attractor with backward compactness in the negative-exponent Sobolev space and backward boundedness in the Lebesgue space.2000 Mathematics Subject Classification. Primary: 35B40, 35B41; Secondary: 37B55.

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“…The wave equation without the dispersive term was also discussed in Wang [21] and Yang, Duan and Kloeden [24] for such additive noise and in Wang, Zhou and Gu [22] for usual multiplicative noise, i.e. Su = u, also see [8,9,11,13,17,18,20,25,26,32].…”

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

Cited by 28 publications

References 28 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

Backward controllability of pullback trajectory attractors with applications to multi-valued Jeffreys-Oldroyd equations

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

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