Bistable travelling waves for nonlocal reaction diffusion equations

Matthieu Alfaro (1) , Jérôme Coville (2) , Gael Raoul (3)

1 I3M - Institut de Mathématiques et de Modélisation de Montpellier
2 BioSP - Biostatistique et Processus Spatiaux
3 CEFE - Centre d’Ecologie Fonctionnelle et Evolutive

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 \lq\lq above and away\rq\rq\, 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.

Mots clés: travelling waves; nonlocal reaction-diffusion equation; Leray-Schauder topological degree; ignition case

Mots clés: bistable case

Domaines: Equations aux dérivées partielles [math.AP]

Matthieu Alfaro, Jérôme Coville, Gael Raoul. Bistable travelling waves for nonlocal reaction diffusion equations. Discrete and Continuous Dynamical Systems - Series A, 2014, 34 (5), pp.1775-1791. ⟨10.3934/dcds.2014.34.1775⟩. ⟨hal-00800918⟩