On a model of a population with variable motility

TL;DR

This paper analyzes a reaction-diffusion model for populations with variable motility, establishing bounds and long-term behavior, and linking the dynamics to solutions of a Hamilton-Jacobi equation.

Contribution

It introduces a novel analysis of a nonlocal reaction-diffusion equation with variable motility, connecting population dynamics to Hamilton-Jacobi solutions.

Findings

Established a global supremum bound for solutions.

Proved convergence of rescaled solutions to zero or positivity regions.

Linked population behavior to viscosity solutions of a Hamilton-Jacobi equation.

Abstract

We study a reaction-diffusion equation with a nonlocal reaction term that models a population with variable motility. We establish a global supremum bound for solutions of the equation. We investigate the asymptotic (long-time and long-range) behavior of the population. We perform a certain rescaling and prove that solutions of the rescaled problem converge locally uniformly to zero in a certain region and stay positive (in some sense) in another region. These regions are determined by two viscosity solutions of a related Hamilton-Jacobi equation.