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The effect of noise on the Chafee-Infante equation: A nonlinear case study |
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| Authors: | Tomá s Caraballo Hans Crauel José A Langa James C Robinson |
| Institution: | Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Apdo. de Correos 1160, 41080 Sevilla, Spain ; Institut für Mathematik, Technische Universität Ilmenau, Weimarer Straße 25, 98693 Ilmenau, Germany ; Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Apdo. de Correos 1160, 41080 Sevilla, Spain ; Mathematics Institute, University of Warwick, Coventry, CV4 7AL United Kingdom |
| Abstract: | We investigate the effect of perturbing the Chafee-Infante scalar reaction diffusion equation,  , 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. |
| Keywords: | Chafee-Infante equation stochastic stabilisation random attractor random equilibrium one-point attractor attractor collapse |
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