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Archive for Rational Mechanics and Analysis

, Volume 217, Issue 2, pp 571–617 | Cite as

Propagation in a Kinetic Reaction-Transport Equation: Travelling Waves And Accelerating Fronts

  • Emeric BouinEmail author
  • Vincent Calvez
  • Grégoire Nadin
Article

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 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.

Keywords

Cauchy Problem Dispersion Relation Travel Wave Solution Spreading Rate Nonlinear Stability 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Ecole Normale Supérieure de Lyon, UMR CNRS 5669 ‘UMPA’, and INRIA Alpes, project-team NUMEDLyon cedex 07France
  2. 2.Université Pierre et Marie Curie-Paris 6, UMR CNRS 7598 ‘LJLL’, BC187Paris cedex 05France

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