The speed of propagation for KPP type problems. I - Periodic framework

Henri Berestycki (1) , Francois Hamel (2) , Nikolai Nadirashvili (3)

1 CAMS - Centre d'Analyse et de Mathématique sociales
2 LATP - Laboratoire d'Analyse, Topologie, Probabilités
3 Department of Mathematics [Chicago]

This paper is devoted to some nonlinear propagation phenomena in periodic and more general domains, for reaction-diffusion equations with Kolmogorov-Petrovsky-Piskunov (KPP) type nonlinearities. The case of periodic domains with periodic underlying excitable media is a follow-up of the article [H. Berestycki, F. Hamel, Comm. Pure Appl. Math. 55 (2002), 949-1032]. It is proved that the minimal speed of pulsating fronts is given by a variational formula involving linear eigenvalue problems. Some consequences concerning the influence of the geometry of the domain, of the reaction, advection and diffusion coefficients are given. The last section deals with the notion of asymptotic spreading speed. The main properties of the spreading speed are given. Some of them are based on some new Liouville type results for nonlinear elliptic equations in unbounded domains.

Mots clés: eigenvalue problems; propagation speeds; reaction-diffusion; Periodic equations

Domaines: Equations aux dérivées partielles [math.AP]

Henri Berestycki, Francois Hamel, Nikolai Nadirashvili. The speed of propagation for KPP type problems. I - Periodic framework. Journal of the European Mathematical Society, 2005, pp.173. ⟨hal-00003805⟩