Random attractor and random exponential attractor for stochastic non-autonomous damped cubic wave equation with linear multiplicative white noise

Zhaojuan Wang; Shengfan Zhou

doi:10.3934/dcds.2018210

Article Contents

2018, Volume 38, Issue 9: 4767-4817. Doi: 10.3934/dcds.2018210

This issue Previous Article Local and global strong solution to the stochastic 3-D incompressible anisotropic Navier-Stokes equations Next Article

Random attractor and random exponential attractor for stochastic non-autonomous damped cubic wave equation with linear multiplicative white noise

Zhaojuan Wang^1, and
Shengfan Zhou^2, ,

1.
School of Mathematical Science, Huaiyin Normal University, Huaian 223300, China
2.
Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China

* Corresponding author: Shengfan Zhou
* Corresponding author: Shengfan Zhou

Received: August 31, 2016

Revised: May 31, 2017

Published: August 2018

This work is supported by the National Natural Science Foundation of China (under Grant Nos. 11471290, 11326114, 11401244) and Natural Science Research Project of Ordinary Universities in Jiangsu Province (grant No. 14KJB110003)

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Abstract

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.

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Mathematics Subject Classification: Primary: 37L55; Secondary: 35B41, 35B40.

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