Critical length for the spreading-vanishing dichotomy in higher dimensions

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