Fast Diffusion Limit for Reaction-Diffusion Systems with Stochastic Neumann Boundary Conditions

Abstract

We consider a class of reaction-diffusion equations with a stochastic perturbation on the boundary. We show that in the limit of fast diffusion, one can rigorously approximate solutions of the system of PDEs with stochastic Neumann boundary conditions by the solution of a suitable stochastic/deterministic differential equation for the average concentration that involves reactions only. An interesting effect occurs in case the noise on the boundary does not change the averaging concentration but is sufficiently large. Here due to the presence of noise surprising new effective reaction terms may appear in the limit. To study this phenomenon we focus on systems with polynomial nonlinearities and illustrate it with simplified, somewhat artificial, examples, namely, a two-dimensional nonlinear heat equation and the cubic autocatalytic reaction between two chemicals.