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

Caraballo, Tomás; Crauel, Hans; Langa, José; Robinson, James

doi:10.1090/S0002-9939-06-08593-5

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

Proc. Amer. Math. Soc. 135 (2007), 373-382

DOI: https://doi.org/10.1090/S0002-9939-06-08593-5

Published electronically: 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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