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

Emeric Bouin (1, 2) , Vincent Calvez (1, 2) , Grégoire Nadin (3)

1 UMPA-ENSL - Unité de Mathématiques Pures et Appliquées
2 NUMED - Numerical Medicine
3 LJLL - Laboratoire Jacques-Louis Lions

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 $t^{3/2}$.

Mots clés: Kinetic equations; travelling waves; dispersion relation; superlinear spreading

Domaines: Equations aux dérivées partielles [math.AP]

Emeric Bouin, Vincent Calvez, Grégoire Nadin. Propagation in a kinetic reaction-transport equation: travelling waves and accelerating fronts. Archive for Rational Mechanics and Analysis, 2015, 217 (2), pp.571-617. ⟨hal-00849405v2⟩