Pullback attractor in $H^{1}$ for nonautonomous stochastic reaction-diffusion equations on $\mathbb{R}^n$

Linfang Liu; Xianlong Fu; Yuncheng You

doi:10.3934/dcdsb.2017143

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2017, Volume 22, Issue 10: 3629-3651. Doi: 10.3934/dcdsb.2017143

This issue Previous Article Next Article A PDE model of intraguild predation with cross-diffusion

Pullback attractor in $H^{1}$ for nonautonomous stochastic reaction-diffusion equations on $\mathbb{R}^n$

1.
Department of Mathematics, Shanghai Key Laboratory of PMMP, East China Normal University, Shanghai 200241, China
2.
Department of Mathematics and Statistics, Uniersity of South Florida, Tampa, FL 33620, USA

* Corresponding author: Yuncheng You
* Corresponding author: Yuncheng You

Received: September 30, 2016

Revised: December 31, 2016

Published: September 2017

The second author is supported by NSF grant of China (Nos. 11671142 and 11371087), Science and Technology Commission of Shanghai Municipality (STCSM) (grant No. 13dz2260400) and Shanghai Leading Academic Discipline Project (No. B407).

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Abstract

In this paper we study the asymptotic dynamics of the weak solutions of nonautonomous stochastic reaction-diffusion equations driven by a time-dependent forcing term and the multiplicative noise. By conducting the uniform estimates we show that the cocycle generated by this SRDE has a pullback $(L^2, H^1)$ absorbing set and it is pullback asymptotically compact through the pullback flattening approach. The existence of a pullback $(L^2, H^1)$ random attractor for this random dynamical system in space $H^{1}(\mathbb{R}^{n})$ is proved.

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Mathematics Subject Classification: 35B40, 35B41, 35R60, 37L30.

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