Asymptotic behavior of a class of stochastic nonlinear wave equations with dispersive and dissipative terms

Jones, Robert Stephen; Wang, Bixiang

doi:10.1016/j.nonrwa.2012.09.019

Cited by 26 publications

(13 citation statements)

References 27 publications

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“…The existence of an attractor for the wave equation had been investigated in various special cases. For examples, the deterministic case (ε = 0) was discussed in [1,5,6,15,29,30], the stochastic case was studied in [14,32,34,35,36,39,40]. Comparing with (1), the nearest equation was given by Wang [33], in which β(t) is a constant.…”

mentioning

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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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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“…By the tempered property of y, one can verify that Q 3,3 is finite and thus Q 3 is finite. Finally, by using F + φ 3 ≥ 0 (see (12)), it follows from (42) that…”

Section: Collective Tail-estimatesmentioning

confidence: 99%

“…The terms −∆u t and −∆u tt in Eq. (1) are called the viscosity dissipative and the dispersive terms respectively (see [12]). The well-posedness and dynamics for deterministic equation were widely investigated in [19,27] and [3,4,23] respectively.…”

Section: Introductionmentioning

confidence: 99%

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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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“…[2][3][4][8][9][10][17][18][19]21,26,28,29,32,35] and the references therein. The random attractors for autonomous stochastic wave equations have been explored in [13,16,20,27,30,34].…”

Section: Introductionmentioning

confidence: 99%

Random attractors and averaging for non-autonomous stochastic wave equations with nonlinear damping

You

2015

Journal of Differential Equations

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This paper is devoted to the asymptotic behavior of solutions to a non-autonomous stochastic wave equation with nonlinear damping and multiplicative white noise defined on an unbounded domain. By showing the pullback asymptotic compactness of the cocycle in a certain parameter region, we prove the existence of a random attractor when the intensity of noise is sufficiently small. For the stochastic wave equation with rapidly oscillating external force we prove that the Hausdorff distance between the random attractor A of the original equation and the random attractor A 0 of the averaged equation is in the order of O( ).

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Asymptotic behavior of a class of stochastic nonlinear wave equations with dispersive and dissipative terms

Cited by 26 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

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

Random attractors and averaging for non-autonomous stochastic wave equations with nonlinear damping

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