Random attractors for non-autonomous stochastic
wave equations with multiplicative noise

Wang, Bixiang

doi:10.3934/dcds.2014.34.269

Cited by 217 publications

(154 citation statements)

References 39 publications

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“…In this part we recall some theory of pullback attractors for RDS with nonautonomous deterministic terms. The reader is referred to [8][9][10] for more details.…”

Section: Pullback Attractors Formentioning

confidence: 99%

“…It is known that there exists a -invariant setΩ ⊆ Ω with P(Ω) = 1 such that ( ) is continuous in for every ∈ Ω, and the random variable | ( )| is tempered (see, e.g., [8,10,12,13], and hereafter we will not distinguishΩ from Ω). Therefore, by Proposition 8 and [12, Proposition 4.3.3] (see also [14][15][16]), there exists a tempered variable ( ) > 0 such that…”

Section: Nonautonomous Stochastic G-l Equationsmentioning

confidence: 99%

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Pullback Attractor for Nonautonomous Ginzburg-Landau Equation with Additive Noise

Cui

2014

Abstract and Applied Analysis

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Long time behavior of stochastic Ginzburg-Landau equations with nonautonomous deterministic external forces, dispersion coefficients, and nonautonomous perturbations is studied. The domain is taken as a bounded interval I in R. By making use of Sobolev embeddings and Gialiardo-Nirenberg inequality we obtain the existence and upper semicontinuity of the pullback attractor in 2 (I) for the equation. The upper semicontinuity shows the stability of attractors under perturbations.

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“…In this part we recall some theory of pullback attractors for RDS with nonautonomous deterministic terms. The reader is referred to [8][9][10] for more details.…”

Section: Pullback Attractors Formentioning

confidence: 99%

Section: Nonautonomous Stochastic G-l Equationsmentioning

confidence: 99%

Pullback Attractor for Nonautonomous Ginzburg-Landau Equation with Additive Noise

Cui

2014

Abstract and Applied Analysis

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“…In comparison with the results recently published in [31], the novelty and the difficulties of this work are in three aspects: (i) The nonlinear damping h(u t ) in Eq. (1.1) and its treatment by using a new Ornstein-Uhlenbeck process which does not depend on the damping coefficients but depends on an adjustable parameter δ, which is substantially different from [31] in which the damping term is linear.…”

Section: Introductionmentioning

confidence: 98%

“…The major difficulty here is hard to prove the asymptotic compactness because Sobolev compact embedding is lost for unbounded domain, especially under the weakened and more general assumptions. We shall use the decomposition technique proposed in [6,30,31] and conduct new and more complicated uniform estimates on both the tails and the bounded truncations of solutions to prove the existence of a pullback random attractor of the stochastic damped wave equation with multiplicative white noise defined on R n .…”

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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“…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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Random attractors for non-autonomous stochastic wave equations with multiplicative noise

Cited by 217 publications

References 39 publications

Pullback Attractor for Nonautonomous Ginzburg-Landau Equation with Additive Noise

Pullback Attractor for Nonautonomous Ginzburg-Landau Equation with Additive Noise

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

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

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