Critical length for the spreading-vanishing dichotomy in higher dimensions

Abstract

We consider an extension of the classical Fisher–Kolmogorov equation, called the “Fisher–Stefan” model, which is a moving boundary problem on \(0<x<L(t)\) . A key property of the Fisher–Stefan model is the “spreading–vanishing dichotomy”, where solutions with \(L(t) > L_{\textrm{c}}\) will eventually spread as \(t \to \infty\) , whereas solutions where \(L(t) \ngtr L_{\textrm{c}}\) will vanish as \(t \to \infty\) . In one dimension it is well known that the critical length is \(L_{\textrm{c}} = \pi/2\) . In this work, we re-formulate the Fisher–Stefan model in higher dimensions and calculate \(L_{\textrm{c}}\) as a function of spatial dimensions in a radially symmetric coordinate system. Our results show how \(L_{\textrm{c}}\) depends upon the dimension of the problem, and numerical solutions of the governing partial differential equation are consistent with our calculations.

doi: 10.1017/S1446181120000103