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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

The effect of noise on the Chafee-Infante equation: A nonlinear case study


Authors: Tomás Caraballo, Hans Crauel, José A. Langa and James C. Robinson
Journal: Proc. Amer. Math. Soc. 135 (2007), 373-382
MSC (2000): Primary 37L55, 35K57; Secondary 60H15
Published electronically: August 1, 2006
MathSciNet review: 2255283
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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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Additional Information

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: http://dx.doi.org/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
Published electronically: 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
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.