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arXiv · 1307.8325

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

Emeric Bouin·Vincent Calvez·Grégoire Nadin
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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 $t^{3/2}$.

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Emeric Bouin, Vincent Calvez, Grégoire Nadin. 2014-08-09. Propagation in a kinetic reaction-transport equation: travelling waves and accelerating fronts. https://arxiv.org/abs/1307.8325

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