Bistable travelling waves for nonlocal reaction diffusion equations

Matthieu Alfaro; Jérôme Coville; Gaël Raoul

doi:10.3934/dcds.2014.34.1775

Article Contents

2014, Volume 34, Issue 5: 1775-1791. Doi: 10.3934/dcds.2014.34.1775

This issue Previous Article Elliptic problems with nonlinear terms depending on the gradient and singular on the boundary: Interaction with a Hardy-Leray potential Next Article Kolmogorov-Sinai entropy via separation properties of order-generated $\sigma$-algebras

Bistable travelling waves for nonlocal reaction diffusion equations

1.
Univ. Montpellier 2, I3M, UMR CNRS 5149, CC051, Place Eugène Bataillon, 34095 Montpellier Cedex 5
2.
INRA, Equipe BIOSP, Centre de Recherche d'Avignon, Domaine Saint Paul, Site Agroparc, 84914 Avignon cedex 9
3.
CEFE, UMR 5175, CNRS, 1919 Route de Mende, 34293 Montpellier

Received: March 2013

Revised: August 2013

Published: May 2014

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Abstract

We are concerned with travelling wave solutions arising in a reaction diffusion equation with bistable and nonlocal nonlinearity, for which the comparison principle does not hold. Stability of the equilibrium $u\equiv 1$ is not assumed. We construct a travelling wave solution connecting 0 to an unknown steady state, which is "above and away", from the intermediate equilibrium. For focusing kernels we prove that, as expected, the wave connects 0 to 1. Our results also apply readily to the nonlocal ignition case.

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Mathematics Subject Classification: Primary: 45K05, 35C07.

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References

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