Improved bounds for reaction-diffusion propagation with a line of nonlocal diffusion

TL;DR

This paper refines the understanding of accelerating reaction-diffusion fronts with nonlocal diffusion on a line, providing explicit algebraic correction terms and revealing closer behavior to the linearized solution than classical models.

Contribution

It offers explicit algebraic correction estimates for the acceleration and compares the solution's behavior to the linearized equation, advancing the theoretical understanding of nonlocal diffusion models.

Findings

Explicit algebraic correction term derived

Solution mimics linearized equation more closely

Accelerated propagation confirmed with refined bounds

Abstract

We consider here a model of accelerating fronts, introduced in [2], consisting of one equation with nonlocal diffusion on a line, coupled via the boundary condition with a reaction-diffusion equation of the Fisher-KPP type in the upper half-plane. It was proved in [2] that the propagation is accelerated in the direction of the line exponentially fast in time. We make this estimate more precise by computing an explicit correction that is algebraic in time. Unexpectedly, the solution mimicks the behaviour of the solution of the equation linearised around the rest state 0 in a closer way than in the classical fractional Fisher-KPP model.