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CRITICAL LENGTH FOR THE SPREADING–VANISHING DICHOTOMY IN HIGHER DIMENSIONS

Published online by Cambridge University Press:  19 June 2020

MATTHEW J. SIMPSON*
Affiliation:
School of Mathematical Sciences, Queensland University of Technology, Brisbane, Queensland 4001, Australia email [email protected]

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 $L(t)>L_{\text{c}}$ 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.

Type
Research Article
Copyright
© 2020 Australian Mathematical Society

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Footnotes

*

This is a contribution to the series of invited papers by past Tuck medallists (Editorial, Issue 62(1)). Matthew J. Simpson was awarded the 2020 Tuck medal.

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