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

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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.

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Authors and Affiliations

  1. Ecole Normale Supérieure de Lyon, UMR CNRS 5669 ‘UMPA’, and INRIA Alpes, project-team NUMED, 46 allée d’Italie, 69364, Lyon cedex 07, France

    Emeric Bouin & Vincent Calvez

  2. Université Pierre et Marie Curie-Paris 6, UMR CNRS 7598 ‘LJLL’, BC187, 4 place de Jussieu, 75252, Paris cedex 05, France

    Grégoire Nadin

Authors
  1. Emeric Bouin
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  2. Vincent Calvez
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  3. Grégoire Nadin
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Correspondence to Emeric Bouin.

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Communicated by L. Saint-Raymond

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Bouin, E., Calvez, V. & Nadin, G. Propagation in a Kinetic Reaction-Transport Equation: Travelling Waves And Accelerating Fronts. Arch Rational Mech Anal 217, 571–617 (2015). https://doi.org/10.1007/s00205-014-0837-7

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