Coupled reaction-diffusion equations
type: Item
instance of: scholarly article
title: Coupled reaction-diffusion equations
zbMATH Open document ID: 0724.60086
published in: The Annals of Probability
publication date: 1991-01-01T00:00:00Z
review text: Consider the system of equations of the form: \[ \partial u^{\epsilon}_ k(t,x)/\partial t=\epsilon L_ ku^{\epsilon}_ k(t,x)+\epsilon^{-1}f_ k(x,u^{\epsilon}_ k)+\sum^{n}_{j=1}d_{kj}(u^{\epsilon}_ j-u^{\epsilon}_ k), \] (*) \[ x\in {\mathbb{R}}^ r,\quad t>0,\quad u^{\epsilon}_ k(0,x)=g_ k(x)\geq 0,\quad k=1,2,...,n, \] where \(L_ k=\sum^{r}_{i,j=1}a_ k^{ij}(x) \partial^ 2/\partial x_ i\partial x_ j\) are elliptic operators with smooth coefficients, \(d_{kj}>0\), \(f_ k(x,\cdot)\in {\mathcal F}=\{f\in C^ 1({\mathbb{R}}):\) \(f(u)>0\), \(0<u<1\), \(f(0)=f(1)=0\), \(f(u)<0\) for \(u\in [0,1]^ c\) and \(f'(0)=\sup_{u>0}f(u)/u\}\), for \(x\in {\mathbb{R}}^ r\) and the initial data \(g_ k\) are bounded and continuous except possibly on the boundaries of their supports. If \(\epsilon =1\), \(k=1\) and \(d_{1j}=0\), one has the usual reaction diffusion equation, and the above equations (*) are generalizations of the original Kolmogorov-Petrovskij-Piskunov equation. The asymptotic behavior of the solution of (*) uses probabilistic methods, especially the large deviation principle. This may be stated in the author's words as: ``We describe the behavior of \(\lim_{\epsilon \downarrow 0}u^{\epsilon}_ k(t,x)\) for the solutions of (*). In the case of systems, in general, the motion of the wave front will not be continuous, [and] can have jumps. Evolution of the wave front, in general, will not be Markovian. If \(\partial f_ k(x,0)/\partial u=c\) is independent of x and k, the motion of the front will be continuous, Markovian and can be described by the Huygens principle. The corresponding velocity field is homogeneous and isotropic not in a Riemannian metric, but in a Finsler metric. The main machinery is the Feynman-Kac formula and the large deviation principle.'' Some generalizations are also discussed. The material is technical and the reader should consult the paper for precise details which cannot be outlined here.
Mathematics Subject Classification ID: 60J60
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 60F10
zbMATH DE Number: 4194888
zbMATH Keywords: elliptic operators with smooth coefficients
zbMATH Keywords: Kolmogorov-Petrovskij- Piskunov equation
zbMATH Keywords: large deviation principle
zbMATH Keywords: Feynman-Kac formula
author: Mark Freidlin
reviewed by: Malempati M. Rao
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1176990535
OpenAlex ID: W2009748208
DOI: 10.1214/AOP/1176990535
description: scientific article; zbMATH DE number 4194888