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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.

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.

In this paper, we study the long-term dynamical behavior of stochastic Boisso nade systems with time-dependent deterministic forces, additive white noise and multiplicative white noise. We first prove the existence of random attrac tor for the considered systems. And then we establish the upper semi-continui ty of random attractors for the systems as the coefficient of quadratic term tends to zero and intensities of the noises approach zero, respectively. At last, we obtain an upper bound of fractal dimension of the random attractors for both systems without quadratic term.

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