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CRITICAL LENGTH FOR THE SPREADING–VANISHING DICHOTOMY IN HIGHER DIMENSIONS

Part of: Equations of mathematical physics and other areas of application

Published online by Cambridge University Press:  19 June 2020

MATTHEW J. SIMPSON Open the ORCID record for MATTHEW J. SIMPSON [Opens in a new window]
MATTHEW J. SIMPSON*
Affiliation:
School of Mathematical Sciences, Queensland University of Technology, Brisbane, Queensland 4001, Australia email [email protected]
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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.

Keywords

reaction–diffusionFisher–Kolmogorov modelFisher’s equationStefan conditionmoving boundary problemtravelling wavesinvasionextinction

MSC classification

Primary: 35Q79: PDEs in connection with classical thermodynamics and heat transfer
Type
Research Article
Information
The ANZIAM Journal , Volume 62 , Issue 1 , January 2020 , pp. 3 - 17
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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