Pathwise upper semi-continuity of random pullback attractors along the time axis

Cui, Hongyong; Kloeden, Peter E.; Wu, Fuke

doi:10.1016/j.physd.2018.03.002

Cited by 47 publications

(36 citation statements)

References 27 publications

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“…We can prove an abstract result that a periodic random attractor is backward compact if and only if it is local compact. A random attractor may not be locally compact (see [10]), although a deterministic attractor is automatically locally compact (see [27]).…”

Section: Renhai Wang and Yangrong LImentioning

confidence: 99%

Backward compactness and periodicity of random attractors for stochastic wave equations with varying coefficients

Wang¹,

Li²

2019

Discrete &Amp; Continuous Dynamical Systems - B

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We study some time-related properties of the random attractor for the stochastic wave equation on an unbounded domain with time-varying coefficient and force. We assume that the coefficient is bounded and the timedependent force is backward tempered, backward complement-small, backward tail-small, and then prove both existence and backward compactness of a random attractor on the universe of all backward tempered sets. By using the Egoroff and Lusin theorems, we show the measurability of the absorbing set although it is the union of some random sets over an uncountable index set. Moreover, we obtain the backward compactness of the attractor if the force is periodic, and obtain the periodicity of the attractor if both force and coefficient are periodic.

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Section: Renhai Wang and Yangrong LImentioning

confidence: 99%

Backward compactness and periodicity of random attractors for stochastic wave equations with varying coefficients

Wang¹,

Li²

2019

Discrete &Amp; Continuous Dynamical Systems - B

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“…The existence of a forward controller relates to the forward compactness in Def.1.1, and also the local compactness (see [13,40]), which means ∪ r∈I A r (ω) is pre-compact in X for any compact interval I ⊂ R.…”

Section: Definition 22mentioning

confidence: 99%

“…Suppose Φ has a pullback attractor A = {A t (ω) : t ∈ R, ω ∈ Ω}. This type of attractors seems to be first introduced by Crauel et al [11] and independently by Wang [39] with developments in [1,13,14,26,27,41,44,45]. Only the compactness of the time-component A t was discussed in these papers.…”

mentioning

confidence: 99%

Forward controllability of a random attractor for the non-autonomous stochastic sine-Gordon equation on an unbounded domain

Yang¹,

Li²

2020

Evolution Equations &Amp; Control Theory

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A pullback random attractor is called forward controllable if its time-component is semi-continuous to a compact set in the future, and the minimum among all such compact limit-sets is called a forward controller. The existence of a forward controller closely relates to the forward compactness of the attractor, which is further argued by the forward-pullback asymptotic compactness of the system. The abstract results are applied to the non-autonomous stochastic sine-Gordon equation on an unbounded domain. The existence of a forward compact attractor is proved, which leads to the existence of a forward controller. The measurability of the attractor is proved by considering two different universes.

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“…The first aim in this paper is to establish a random attractor A δ for the problem (1.3)-(1.4). In view of both the non-autonomous and the random nature, the attractor is actually a bi-parametric set formulated by A δ = {A δ (τ , ω)} and called a pullback random attractor, which was first introduced by Crauel et al [8] and by Wang [32] independently, with developments [2,9,10,18,20,26,36,37,42,43,45].…”

Section: Introductionmentioning

confidence: 99%

Random attractors for Ginzburg–Landau equations driven by difference noise of a Wiener-like process

Wang

2019

Adv Differ Equ

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We consider a Wong-Zakai process, which is the difference of a Wiener-like process. We then prove that there are random attractors for non-autonomous Ginzburg-Landau equations driven by linear multiplicative noise in terms of Wong-Zakai process and Wiener-like process, respectively. Moreover, we establish the upper semi-continuity of random attractors as the size of difference noise tends to zero.

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Pathwise upper semi-continuity of random pullback attractors along the time axis

Cited by 47 publications

References 27 publications

Backward compactness and periodicity of random attractors for stochastic wave equations with varying coefficients

Backward compactness and periodicity of random attractors for stochastic wave equations with varying coefficients

Forward controllability of a random attractor for the non-autonomous stochastic sine-Gordon equation on an unbounded domain

Random attractors for Ginzburg–Landau equations driven by difference noise of a Wiener-like process

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