The asymptotics of stochastically perturbed reaction-diffusion equations and front propagation

Pierre-Louis Lions,Panagiotis E Souganidis

doi:10.5802/crmath.117

Abstract

We study the asymptotics of Allen-Cahn-type bistable reaction-diffusion equations which are additively perturbed by a stochastic forcing (time white noise). The conclusion is that the long time, large space behavior of the solutions is governed by an interface moving with curvature dependent normal velocity which is additively perturbed by time white noise. The result is global in time and does not require any regularity assumptions on the evolving front. The main tools are (i)~the notion of stochastic (pathwise) solution for nonlinear degenerate parabolic equations with multiplicative rough (stochastic) time dependence, which has been developed by the authors, and (ii)~the theory of generalized front propagation put forward by the second author and collaborators to establish the onset of moving fronts in the asymptotics of reaction-diffusion equations.

Highlights

We investigate the onset of fronts in the long time and large space asymptotics of bistable reaction-diffusion equations, the prototype being the Allen–Cahn equation, which are additively perturbed by small relatively smooth stochastic in time forcing
The interfaces evolve with curvature dependent normal velocity which is additively perturbed by time white noise
No regularity assumptions are made about the fronts

Summary

Introduction

We investigate the onset of fronts in the long time and large space asymptotics of bistable reaction-diffusion equations, the prototype being the Allen–Cahn equation, which are additively perturbed by small relatively smooth (mild) stochastic in time forcing. The properties of (8) are used here to adapt the approach introduced in Evans, Soner and Souganidis [5], Barles, Soner and Souganidis [2], and Barles and Souganidis [3] to study the onset of moving fronts in the asymptotic limit of reaction-diffusion equations and interacting particle systems with long rage interactions This methodology allows to prove global in time asymptotic results and is not restricted to smoothly evolving fronts. Theorem 1, which was already announced in [10], is new It provides a complete characterization of the asymptotic behavior of the Allen–Cahn equation perturbed by mild approximations of the time white noise. A recent observation of the authors shows that the general conjecture cannot be mathematically correct It is shown in [8] that the formally conjectured interfaces, which should move by mean curvature additively perturbed with space-time white noise, are not well defined.

Stochastic viscosity solutions and generalized front propagation

Findings

The asymptotics of the Allen–Cahn equation