Sharp large time behaviour in N -dimensional Fisher-KPP equations

Jean-Michel Roquejoffre (1) , Luca Rossi (2) , Violaine Roussier-Michon (1)

1 IMT - Institut de Mathématiques de Toulouse UMR5219
2 CNRS - Centre National de la Recherche Scientifique

We study the large time behaviour of the Fisher-KPP equation ∂ t u = ∆u + u − u 2 in spatial dimension N , when the initial datum is compactly supported. We prove the existence of a Lipschitz function s of the unit sphere, such that u(t, x) converges, as t goes to infinity, to U c * |x| − c * t + N + 2 c * lnt + s ∞ x |x| , where U c * is the 1D travelling front with minimal speed c * = 2. This extends an earlier result of Gärtner.

Domaines: Equations aux dérivées partielles [math.AP]

Jean-Michel Roquejoffre, Luca Rossi, Violaine Roussier-Michon. Sharp large time behaviour in N -dimensional Fisher-KPP equations. 2019. ⟨hal-02077817v1⟩