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

TL;DR

This paper studies reaction-diffusion systems with stochastic boundary conditions, showing that in the fast diffusion limit, solutions can be approximated by simpler equations, revealing new effective reactions when boundary noise is large.

Contribution

It provides a rigorous approximation of stochastic boundary reaction-diffusion systems in the fast diffusion limit, including the emergence of additional reaction terms due to boundary noise.

Findings

Solutions approximate stochastic PDEs by simpler equations in the fast diffusion limit.

Large boundary noise can induce new effective reaction terms.

Applications demonstrated on nonlinear heat and auto-catalytic systems.

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, if the noise on the boundary does not change the averaging concentration, but is sufficiently large. Then surprising additional effective reaction terms appear. We focus on systems with polynomial nonlinearities only and give applications to the two dimensional nonlinear heat equation and the cubic auto-catalytic reaction between two chemicals.