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CRITICAL LENGTH FOR THE SPREADING–VANISHING DICHOTOMY IN HIGHER DIMENSIONS
- Part of
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- Journal:
- The ANZIAM Journal / Volume 62 / Issue 1 / January 2020
- Published online by Cambridge University Press:
- 19 June 2020, pp. 3-17
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- Article
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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 
$L(t)>L_{\text{c}}$ will eventually spread as 
$t\rightarrow \infty$, whereas solutions where 
$L(t)\ngtr L_{\text{c}}$ will vanish as 
$t\rightarrow \infty$. In one dimension it is well known that the critical length is 
$L_{\text{c}}=\unicode[STIX]{x1D70B}/2$. 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.
