Propagation in a kinetic reaction-transport equation: travelling waves and accelerating fronts

TL;DR

This paper investigates the existence, stability, and spreading behavior of travelling wave solutions in a kinetic reaction-transport equation, revealing conditions for wave existence and how velocity distribution influences spreading rates.

Contribution

It establishes the necessary and sufficient condition for travelling wave existence based on velocity set boundedness and derives explicit minimal wave speeds.

Findings

Bounded velocity sets are crucial for positive travelling wave existence.

Explicit dispersion relation determines minimal wave speed.

Unbounded velocity sets lead to superlinear spreading, e.g., $t^{3/2}$ for Gaussian distributions.

Abstract

In this paper, we study the existence and stability of travelling wave solutions of a kinetic reaction-transport equation. The model describes particles moving according to a velocity-jump process, and proliferating thanks to a reaction term of monostable type. The boundedness of the velocity set appears to be a necessary and sufficient condition for the existence of positive travelling waves. The minimal speed of propagation of waves is obtained from an explicit dispersion relation. We construct the waves using a technique of sub- and supersolutions and prove their \eb{weak} stability in a weighted $L^2$ space. In case of an unbounded velocity set, we prove a superlinear spreading. It appears that the rate of spreading depends on the decay at infinity of the velocity distribution. In the case of a Gaussian distribution, we prove that the front spreads as $t^{3/2}$.