DCDS
Random attractor for stochastic non-autonomous damped wave equation with critical exponent
Zhaojuan Wang Shengfan Zhou
Discrete & Continuous Dynamical Systems - A 2017, 37(1): 545-573 doi: 10.3934/dcds.2017022

In this paper, we prove the existence of random attractor and obtainan upper bound of fractal dimension of random attractor forstochastic non-autonomous damped wave equation with criticalexponent and additive white noise. We first prove the existence of arandom attractor by carefully splitting the positivity of the linearoperator in the corresponding random evolution equation of the firstorder in time and by carefully decomposing the solutions of systemthrough two different modes, and we show the boundedness of randomattractor in a higher regular space by a recurrence method. Then weestablish a criterion to bound the fractal dimension of a randominvariant set for a cocycle and applied these conditions to get anupper bound of fractal dimension of the random attractor ofconsidered system.

keywords: Stochastic damped wave equation random attractor fractal dimension critical exponent
DCDS
Random attractor and random exponential attractor for stochastic non-autonomous damped cubic wave equation with linear multiplicative white noise
Zhaojuan Wang Shengfan Zhou
Discrete & Continuous Dynamical Systems - A 2018, 38(9): 4767-4817 doi: 10.3934/dcds.2018210

In this paper, we first establish some sufficient conditions for the existence and construction of a random exponential attractor for a continuous cocycle on a separable Banach space. Then we mainly consider the random attractor and random exponential attractor for stochastic non-autonomous damped wave equation driven by linear multiplicative white noise with small coefficient when the nonlinearity is cubic. First step, we prove the existence of a random attractor for the cocycle associated with the considered system by carefully decomposing the solutions of system in two different modes and estimating the bounds of solutions. Second step, we consider an upper semicontinuity of random attractors as the coefficient of random term tends zero. Third step, we show the regularity of random attractor in a higher regular space through a recurrence method. Fourth step, we prove the existence of a random exponential attractor for the considered system, which implies the finiteness of fractal dimension of random attractor. Finally we remark that the stochastic non-autonomous damped cubic wave equation driven by additive white noise also has a random exponential attractor.

keywords: Stochastic damped wave equation random attractor random exponential attractor multiplicative white noise upper semicontinuity fractal dimension regularity
DCDS
Existence and upper semicontinuity of random attractors for non-autonomous stochastic strongly damped wave equation with multiplicative noise
Zhaojuan Wang Shengfan Zhou
Discrete & Continuous Dynamical Systems - A 2017, 37(5): 2787-2812 doi: 10.3934/dcds.2017120

In this paper we study the asymptotic behavior of solutions of the non-autonomous stochastic strongly damped wave equation driven by multiplicative noise defined on unbounded domains. We first introduce a continuous cocycle for the equation. Then we consider the existence of a tempered pullback random attractor for the cocycle. Finally we establish the upper semicontinuity of random attractors as the coefficient of the white noise term tends to zero.

keywords: Stochastic strongly damped wave equation unbounded domains random attractor upper semicontinuity

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