A Stochastic Pitchfork Bifurcation in a Reaction-Diffusion Equation

Abstract

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.