# American Institute of Mathematical Sciences

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May  2014, 34(5): 1775-1791. doi: 10.3934/dcds.2014.34.1775

## 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, France

Received  March 2013 Revised  August 2013 Published  October 2013

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.
Citation: Matthieu Alfaro, Jérôme Coville, Gaël Raoul. Bistable travelling waves for nonlocal reaction diffusion equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 1775-1791. doi: 10.3934/dcds.2014.34.1775
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