Spreading in kinetic reaction-transport equations in higher velocity dimensions

Emeric Bouin (1) , Nils Caillerie (2)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 MMCS - Modélisation mathématique, calcul scientifique

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.

Mots clés: Key-words: Kinetic equations; travelling waves; dispersion relation AMS Class No: 35Q92; 45K05; 35C07

Domaines: Equations aux dérivées partielles [math.AP]

Emeric Bouin, Nils Caillerie. Spreading in kinetic reaction-transport equations in higher velocity dimensions. European Journal of Applied Mathematics, 2019, 30 (2), pp.219-247. ⟨10.1017/S0956792518000037⟩. ⟨hal-01518398v2⟩