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

Tomás Caraballo; José A. Langa; James C. Robinson

doi:10.3934/dcds.2000.6.875

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2000, Volume 6, Issue 4: 875-892. Doi: 10.3934/dcds.2000.6.875

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

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

Received: July 1999

Revised: July 2000

Published: October 2000

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Abstract

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.

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Mathematics Subject Classification: 35B40, 35K57, 58F12, 60H15.

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