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

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 Ito 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