Limiting behavior of dynamics for stochastic reaction-diffusion equations with additive noise on thin domains

Li, Dingshi; Lu, Kening; Wang, Bixiang; Wang, Xiaohu

doi:10.3934/dcds.2018009

Cited by 46 publications

(26 citation statements)

References 33 publications

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“…If we take the attracted universe D 0 by all of tempered sets as usual (see [2,9,20,21,28]), we cannot prove that the D 0 -pullback asymptotic compactness is uniform in the past.…”

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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“…If we take the attracted universe D 0 by all of tempered sets as usual (see [2,9,20,21,28]), we cannot prove that the D 0 -pullback asymptotic compactness is uniform in the past.…”

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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“…Using the Lyapunov function methods and combining the inequality techniques, some sufficient criteria on the exponential ultimate boundedness have been presented for the systems. The method presented in this paper may be applied to some other kinds of stochastic systems such as stochastic systems with exogenous disturbances [40], semi-Markov switched stochastic systems [41] and stochastic systems with additive noise [42].…”

Section: Discussionmentioning

confidence: 99%

Boundedness analysis of stochastic integro-differential systems with Lévy noise

2019

Journal of Taibah University for Science

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This paper is concerned with the pth moment globally exponential ultimate boundedness of stochastic integro-differential systems with Lévy noise. With the help of the Lyapunov function methods and the inequality techniques, several sufficient criteria on the exponential ultimate boundedness are presented for the systems. The results show that the boundedness was determined by the coefficients in the estimation of the Itô operator LV of the energy function V along the trajectories of the addressed systems. ARTICLE HISTORYwhere the scalar λ satisfies 0 < λ < λ 0 and is determined by (25). This concludes the proof of the result. Corollary 3.3:Suppose that the hypotheses of Theorem 3.1 hold. If 5 = 0, then system (1) is pth moment globally exponentially stable (p-GES).Proof: This result follows directly from Theorems 3.1.Corollary 3.4: Suppose that the Assumption 3.1 with K 3 = K 6 = K 9 = K 12 = 0 holds. Ifˆ 3 >ˆ 4 > 0, then system (1) is p-GES.Proof: This result follows directly from Theorems 3.2.

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“…The environmental noise is an intrinsic effect in a variety of settings and spatial scales. It is worth mentioning that the ergodicity of stochastic 3D Navier-Stokes equations in a thin domain was recently investigated in [17,18], the synchronization of semilinear parabolic stochastic equations in thin bounded tubular domains was studied in [11], and the upper semicontinuity of random attractors for reaction-diffusion equations in thin domains was established in [32,34]. However, as far as the author is aware, the limiting dynamics for stochastic retarded equations on thin domains are not well studied, even for deterministic retarded equations on thin domains.…”

Section: Dingshi LI Kening Lu Bixiang Wang and Xiaohu Wangmentioning

confidence: 99%

Limiting dynamics for non-autonomous stochastic retarded reaction-diffusion equations on thin domains

Li¹,

Lu²,

Wang³

et al. 2019

Discrete &Amp; Continuous Dynamical Systems - A

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A system of stochastic retarded reaction-diffusion equations with multiplicative noise and deterministic non-autonomous forcing on thin domains is considered. Relations between the asymptotic behavior for the stochastic retarded equations defined on thin domains in R n+1 and an equation on a domain in R n are investigated. We first show the existence and uniqueness of tempered random attractors for these equations. Then, we analyze convergence properties of the solutions as well as the attractors.

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Limiting behavior of dynamics for stochastic reaction-diffusion equations with additive noise on thin domains

Cited by 46 publications

References 33 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

Boundedness analysis of stochastic integro-differential systems with Lévy noise

Limiting dynamics for non-autonomous stochastic retarded reaction-diffusion equations on thin domains

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