Doubly nonlocal Fisher-KPP equation: Front propagation

TL;DR

This paper investigates the propagation behavior of solutions to a doubly nonlocal Fisher-KPP equation with anisotropic kernels, establishing conditions under which solutions propagate at most linearly in time across multiple directions.

Contribution

It provides new results on the linear propagation speed of solutions for anisotropic, doubly nonlocal Fisher-KPP equations under specific kernel and initial condition decay assumptions.

Findings

Solutions propagate at most linearly in time under certain kernel integrability conditions.

Linear propagation is established in all directions for anisotropic kernels and initial conditions.

The results extend understanding of wave propagation in nonlocal reaction-diffusion systems.

Abstract

We study propagation over $\mathbb{R}^d$ of the solution to a nonlocal nonlinear equation with anisotropic kernels, which can be interpretted as a doubly nonlocal reaction-diffusion equation of the Fisher--KPP-type. We prove that if the kernel of the nonlocal diffusion is exponentially integrable in a direction and if the initial condition decays in this direction faster than any exponential function, then the solution propagates at most linearly in time in that direction. Moreover, if both the kernel and the initial condition have the above properties in any direction (being, in general, anisotropic), then we prove linear in time propagation of the corresponding solution over $\mathbb{R}^d$.