Proceedings of the American Mathematical Society

Author(s): Tomás Caraballo; Hans Crauel; José A. Langa; James C. Robinson
Journal: Proc. Amer. Math. Soc. 135 (2007), 373-382.
MSC (2000): Primary 37L55, 35K57; Secondary 60H15
Posted: August 1, 2006
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Abstract: We investigate the effect of perturbing the Chafee-Infante scalar reaction diffusion equation, $u_t-\Delta u=\beta u-u^3$ , by noise. While a single multiplicative Itô noise of sufficient intensity will stabilise the origin, its Stratonovich counterpart leaves the dimension of the attractor essentially unchanged. We then show that a collection of multiplicative Stratonovich terms can make the origin exponentially stable, while an additive noise of sufficient richness reduces the random attractor to a single point.

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Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 37L55, 35K57, 60H15

Tomás Caraballo
Affiliation: Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Apdo. de Correos 1160, 41080 Sevilla, Spain
Email: caraball@us.es

Hans Crauel
Affiliation: Institut für Mathematik, Technische Universität Ilmenau, Weimarer Straße 25, 98693 Ilmenau, Germany
Email: hans.crauel@tu-ilmenau.de

José A. Langa
Affiliation: Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Apdo. de Correos 1160, 41080 Sevilla, Spain
Email: langa@us.es

James C. Robinson
Affiliation: Mathematics Institute, University of Warwick, Coventry, CV4 7AL United Kingdom
Email: jcr@maths.warwick.ac.uk

DOI: 10.1090/S0002-9939-06-08593-5
PII: S 0002-9939(06)08593-5
Keywords: Chafee-Infante equation, stochastic stabilisation, random attractor, random equilibrium, one-point attractor, attractor collapse
Received by editor(s): August 12, 2005
Posted: August 1, 2006
Additional Notes: The first and third authors were supported by Ministerio de Ciencia y Tecnología (Spain) and FEDER (European Community), project BFM2002-03068.
The fourth author is a Royal Society University Research Fellow, and would like to thank the Society for all their support.
Communicated by: Walter Craig
Copyright of article: Copyright 2006, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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