Artículo
A Stochastic Pitchfork Bifurcation in a Reaction-Diffusion Equation
Autor/es | Caraballo Garrido, Tomás
Langa Rosado, José Antonio Robinson, James C. |
Departamento | Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico |
Fecha de publicación | 2001 |
Fecha de depósito | 2015-04-08 |
Publicado en |
|
Resumen | We study in some detail the structure of the random attractor for the Chafee{Infante
reaction{di¬usion equation perturbed by a multiplicative white noise,
du = (¢u + u ¡ u3) dt + ¼ u ¯ dWt; x 2 D » Rm:
First we prove, ... We study in some detail the structure of the random attractor for the Chafee{Infante reaction{di¬usion equation perturbed by a multiplicative white noise, du = (¢u + u ¡ u3) dt + ¼ u ¯ dWt; x 2 D » Rm: First we prove, for m 65, a lower bound on the dimension of the random attractor, which is of the same order in as the upper bound we derived in an earlier paper, and is the same as that obtained in the deterministic case. Then we show, for m = 1, that as passes through ¶ 1 (the rst eigenvalue of the negative Laplacian) from below, the system undergoes a stochastic bifurcation of pitchfork type. We believe that this is the rst example of such a stochastic bifurcation in an in nite-dimensional setting. Central to our approach is the existence of a random unstable manifold. |
Ficheros | Tamaño | Formato | Ver | Descripción |
---|---|---|---|---|
file_1.pdf | 254.1Kb | [PDF] | Ver/ | |