On the stability of the state 1 in the non-local Fisher-KPP equation in bounded domains

TL;DR

This paper proves that the homogeneous steady state 1 in a non-local Fisher-KPP equation on bounded domains is globally asymptotically stable under certain kernel conditions, using a Lyapunov function approach.

Contribution

It introduces a Lyapunov-based method to establish global stability of the steady state in the non-local Fisher-KPP equation with Neumann boundary conditions.

Findings

Homogeneous steady state 1 is globally asymptotically stable.

Stability depends on kernel conditions linked to traveling wave existence.

Provides conditions aligning with existing literature on wave linking 0 to 1.

Abstract

We consider the non-local Fisher-KPP equation on a bounded domain with Neu-mann boundary conditions. Thanks to a Lyapunov function, we prove that under a general hypothesis on the Kernel involved in the non-local term, the homogenous steady state 1 is globally asymptotically stable. This assumption happens to be linked to some conditions given in the literature, which ensure that travelling waves link 0 to 1.