Stability and random attractors for a reaction-diffusion equation with multiplicative noise

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October 2000, 6(4): 875-892. doi: 10.3934/dcds.2000.6.875

Stability and random attractors for a reaction-diffusion equation with multiplicative noise

Tomás Caraballo ^1, , José A. Langa ^1, and James C. Robinson ^2,

1.	Dpto. Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla, Spain, Spain
2.	Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom

Received July 1999 Revised July 2000 Published August 2000

Abstract
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We study the asymptotic behaviour of a reaction-diffusion equation, and prove that the addition of multiplicative white noise (in the sense of Itô) stabilizes the stationary solution $x\equiv 0$. We show in addition that this stochastic equation has a finite-dimensional random attractor, and from our results conjecture a possible bifurcation scenario.

Keywords: Random attractors, Chafee-Infante equation., stochastic stabilization, Hausdorff dimension.

Mathematics Subject Classification: 35B40, 35K57, 58F12, 60H1.

Citation: Tomás Caraballo, José A. Langa, James C. Robinson. Stability and random attractors for a reaction-diffusion equation with multiplicative noise. Discrete & Continuous Dynamical Systems - A, 2000, 6 (4) : 875-892. doi: 10.3934/dcds.2000.6.875

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