Critical length for the spreading-vanishing dichotomy in higher dimensions

Abstract

We consider an extension of the classical Fisher-Kolmogorov equation, called the \textit{Fisher-Stefan} model, which is a moving boundary problem on $0 < x < L(t)$. A key property of the Fisher-Stefan model is the \textit{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 is it 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.