On the nonlocal Fisher-KPP equation: steady states, spreading speed and global bounds

TL;DR

This paper studies a nonlocal Fisher-KPP equation, establishing conditions for periodic solutions, and providing bounds on solutions and their spreading speeds, advancing understanding of nonlocal reaction-diffusion models.

Contribution

It introduces new conditions for the existence of periodic solutions and derives uniform bounds and spreading rate estimates for solutions of the nonlocal Fisher-KPP equation.

Findings

Existence of non-constant periodic solutions under certain interaction conditions

Uniform upper bounds for solutions of the Cauchy problem

Bounds on the spreading rate for solutions with compact initial support

Abstract

We consider the Fisher-KPP equation with a non-local interaction term. We establish a condition on the interaction that allows for existence of non-constant periodic solutions, and prove uniform upper bounds for the solutions of the Cauchy problem, as well as upper and lower bounds on the spreading rate of the solutions with compactly supported initial data.