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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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The effect of noise on the Chafee-Infante equation: A nonlinear case study
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by Tomás Caraballo, Hans Crauel, José A. Langa and James C. Robinson PDF
Proc. Amer. Math. Soc. 135 (2007), 373-382 Request permission

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
  • ORCID: 0000-0003-4697-898X
  • 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
  • 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
  • © Copyright 2006 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 135 (2007), 373-382
  • MSC (2000): Primary 37L55, 35K57; Secondary 60H15
  • DOI: https://doi.org/10.1090/S0002-9939-06-08593-5
  • MathSciNet review: 2255283