Spreading in kinetic reaction-transport equations in higher velocity dimensions

TL;DR

This paper investigates the propagation dynamics of particles in higher-dimensional velocity-transport models with bounded velocities, extending previous one-dimensional results and analyzing the existence of traveling waves and spreading speeds using Hamilton-Jacobi methods.

Contribution

It extends previous one-dimensional kinetic reaction-transport results to higher dimensions, analyzing spectral problems and minimal speeds in bounded velocity settings.

Findings

Established spreading results in higher dimensions

Proved existence of traveling wave solutions

Identified potential singularities affecting minimal speed

Abstract

In this paper, we extend and complement previous works about propagation in kinetic reaction-transport equations. The model we study describes particles moving according to a velocity-jump process, and proliferating according to a reaction term of monostable type. We focus on the case of bounded velocities, having dimension higher than one. We extend previous results obtained by the first author with Calvez and Nadin in dimension one. We study the large time/large scale hyperbolic limit via an Hamilton-Jacobi framework together with the half-relaxed limits method. We deduce spreading results and the existence of travelling wave solutions. A crucial difference with the mono-dimensional case is the resolution of the spectral problem at the edge of the front, that yields potential singular velocity distributions. As a consequence, the minimal speed of propagation may not be determined by a first order condition.