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

Abstract

We investigate the effect of perturbing the Chafee-Infante scalar reaction diffusion equation, u t − Δ u = β u − u 3 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.