Sharp interface limit of the Fisher-KPP equation

Matthieu Alfaro; Arnaud Ducrot

doi:10.3934/cpaa.2012.11.1

Article Contents

2012, Volume 11, Issue 1: 1-18. Doi: 10.3934/cpaa.2012.11.1

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Sharp interface limit of the Fisher-KPP equation

Matthieu Alfaro^1, and
Arnaud Ducrot^2,

1.
I3M, Université de Montpellier 2, CC051, Place Eugène Bataillon, 34095 Montpellier Cedex 5
2.
UMR CNRS 5251, I.M.B. and INRIA Bordeaux Sud-ouest Anubis, case 36, UFR Sciences et Modélisation, Université Victor Segalen Bordeaux 2, 3 ter, place de la Victoire - 33076 Bordeaux cedex

Received: December 31, 2009

Revised: June 30, 2010

Published: December 2011

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Abstract

We investigate the singular limit, as $\varepsilon\to 0$, of the Fisher equation $\partial_t u=\varepsilon\Delta u + \varepsilon^{-1}u(1-u)$ in the whole space. We consider initial data with compact support plus, possibly, perturbations very small as $||x|| \to \infty$. By proving both generation and motion of interface properties, we show that the sharp interface limit moves by a constant speed, which is the minimal speed of some related one-dimensional travelling waves. Moreover, we obtain a new estimate of the thickness of the transition layers. We also exhibit initial data "not so small" at infinity which do not allow the interface phenomena.

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Mathematics Subject Classification: Primary: 35K57, 35B25; Secondary: 92D25.

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