Traveling waves for a reaction–diffusion–advection system with interior or boundary losses

Thomas Giletti

doi:10.1016/j.crma.2011.04.002

Partial Differential Equations

Traveling waves for a reaction–diffusion–advection system with interior or boundary losses

Presented by: Haïm Brezis

Thomas Giletti ¹

¹ Université Aix-Marseille III, LATP, faculté des sciences et techniques, avenue Escadrille Normandie–Niemen, 13397 Marseille cedex 20, France

Comptes Rendus. Mathématique, Volume 349 (2011) no. 9-10, pp. 535-539.

Abstract
Résumé

This Note deals with the existence and qualitative properties of traveling wave solutions of a nonlinear reaction–diffusion system with losses inside the domain. In particular, we show the existence of a continuum of admissible speeds of traveling waves. Lastly, by considering losses concentrated near the boundary of the domain, these results are compared with those already known in the case of losses on the boundary.

Cette Note a pour objet lʼexistence et les propriétés des solutions de type front progressif pour un système de réaction–diffusion non linéaire avec pertes à lʼintérieur du domaine. Nous montrons en particulier lʼexistence dʼun continuum de vitesses admissibles pour les fronts. Enfin, en considérant des pertes localisées près du bord, ces résultats sont comparés avec ceux déjà connus pour des pertes à la frontière du domaine.

Received: 2010-08-11
Accepted: 2011-04-11
Published online: 2011-05-01

DOI: 10.1016/j.crma.2011.04.002

Author's affiliations:

Thomas Giletti ¹

¹ Université Aix-Marseille III, LATP, faculté des sciences et techniques, avenue Escadrille Normandie–Niemen, 13397 Marseille cedex 20, France

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     author = {Thomas Giletti},
     title = {Traveling waves for a reaction{\textendash}diffusion{\textendash}advection system with interior or boundary losses},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {535--539},
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     language = {en},
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Thomas Giletti. Traveling waves for a reaction–diffusion–advection system with interior or boundary losses. Comptes Rendus. Mathématique, Volume 349 (2011) no. 9-10, pp. 535-539. doi : 10.1016/j.crma.2011.04.002. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2011.04.002/

References
Cited by

[1] H. Berestycki, F. Hamel, Reaction–Diffusion Equations and Propagation Phenomena, Springer-Verlag, in press.

[2] H. Berestycki; F. Hamel; A. Kiselev; L. Ryzhik Quenching and propagation in KPP reaction–diffusion equations with a heat loss, Arch. Ration. Mech. Anal., Volume 178 (2005), pp. 57-80

[3] H. Berestycki; L. Nirenberg Traveling wave in cylinders, Ann. Inst. H. Poincaré Anal. Non Linéaire, Volume 9 (1992), pp. 497-572

[4] T. Giletti KPP reaction–diffusion equations with a non-linear loss inside a cylinder, Nonlinearity, Volume 23 (2010), p. 2307

[5] T. Giletti, KPP reaction–diffusion system with loss inside a cylinder: convergence toward the problem with Robin boundary conditions, preprint.

[6] F. Hamel; L. Ryzhik Non-adiabatic KPP fronts with an arbitrary Lewis number, Nonlinearity, Volume 18 (2005), pp. 2881-2902

[7] J.D. Murray Mathematical Biology, Springer, 2003

[8] J. Xin Analysis and modelling of front propagation in heterogeneous media, SIAM Rev., Volume 42 (2000), pp. 161-230

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