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Spreading in kinetic reaction–transport equations in higher velocity dimensions

Published online by Cambridge University Press:  07 February 2018

EMERIC BOUIN  and
NILS CAILLERIE
EMERIC BOUIN
Affiliation:
CEREMADE – Université Paris-Dauphine, UMR CNRS 7534, Place du Maréchal de Lattre de Tassigny, 75775 Paris Cedex 16, France email: bouin@ceremade.dauphine.fr
NILS CAILLERIE
Affiliation:
Institut Camille Jordan (ICJ), Université Claude Bernard Lyon 1, 43 boulevard du 11 novembre 1918, 69622 Villeurbanne Cedex, France email: caillerie@math.univ-lyon1.fr
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Abstract

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.

Keywords

Kinetic equationstravelling wavesdispersion relation35Q9245K0535C07
Type
Papers
Information
European Journal of Applied Mathematics , Volume 30 , Issue 2 , April 2019 , pp. 219 - 247
Copyright
Copyright © Cambridge University Press 2018 

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Footnotes

EB and NC acknowledge the support of the ERC Grant MATKIT (ERC-2011-StG). NC has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (Grant agreement no. 639638).

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