Front propagation in a kinetic reaction-transport equation

In this paper, we study the existence and stability of travelling wave solutions of a kinetic reactiontransport 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 suband supersolutions and prove their stability in a weighted L space. In case of an unbounded velocity set, we prove a superlinear spreading and give partial results concerning the rate of spreading associated to particular initial data. It appears that the rate of spreading depends strongly on the decay at infinity of the stationary Maxwellian. Key-words: Kinetic equations, travelling waves, dispersion relation, superlinear spreading.