The Bramson delay in the non-local Fisher-KPP equation

Emeric Bouin (1) , Christopher Henderson (2) , Lenya Ryzhik (3)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 Department of Mathematics [Chicago]
3 Department of Mathematics [Stanford]

We consider the non-local Fisher-KPP equation modeling a population with individuals competing with each other for resources with a strength related to their distance, and obtain the asymptotics for the position of the invasion front starting from a localized population. Depending on the behavior of the competition kernel at infinity, the location of the front is either $2t - ({3}/2)\log t + O(1)$, as in the local case, or $2t - O(t^\beta)$ for some explicit $\beta \in (0,1)$. Our main tools here are a local-in-time Harnack inequality and an analysis of the linearized problem with a suitable moving Dirichlet boundary condition. Our analysis also yields, for any $\beta\in(0,1)$, examples of Fisher-KPP type non-linearities $f_\beta$ such that the front for the local Fisher-KPP equation with reaction term $f_\beta$ is at $2t - O(t^\beta)$.

Mots clés: Reaction-diffusion equations; Logarithmic delay; Parabolic Harnack inequality AMS Class No: 35K57; 35Q92; 45K05; 35C07

Mots clés: Key-Words: Reaction-diffusion equations; Logarithmic delay; Parabolic Harnack inequality AMS Class No: 35K57; 35Q92; 45K05; 35C07

Domaines: Equations aux dérivées partielles [math.AP]

Emeric Bouin, Christopher Henderson, Lenya Ryzhik. The Bramson delay in the non-local Fisher-KPP equation. 2019. ⟨hal-01613495v2⟩