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