A stochastic pitchfork bifurcation in a reaction-diffusion equation

We study in some detail the structure of the random attractor for the Chafee-Infante reaction-diffusion equation perturbed by a multiplicative white noise, du=( Δu+βu- u 3 ) dt+σuod W t ,x∈D⊂ R m First we prove, for m ⩽ 5, a lower bound on the dimension of the random attractor, which is of the same order in β 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 β passes through λ1 (the first eigenvalue of the negative Laplacian) from below, the system undergoes a stochastic bifurcation of pitchfork type. We believe that this is the first example of such a stochastic bifurcation in an infinite–dimensional setting. Central to our approach is the existence of a random unstable manifold.

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