Propagation in Fisher–KPP type equations with fractional diffusion in periodic media

Xavier Cabré; Anne-Charline Coulon; Jean-Michel Roquejoffre

doi:10.1016/j.crma.2012.10.007

Partial Differential Equations

Propagation in Fisher–KPP type equations with fractional diffusion in periodic media

Presented by: Haïm Brézis

Xavier Cabré ¹; Anne-Charline Coulon ²; Jean-Michel Roquejoffre ²

¹ ICREA and Universitat Politècnica de Catalunya, Dep. de Matemàtica Aplicada I, Av. Diagonal 647, 08028 Barcelone, Spain
² Institut de mathématiques, (UMR CNRS 5219), université Paul-Sabatier, 118, route de Narbonne, 31062 Toulouse cedex, France

Comptes Rendus. Mathématique, Volume 350 (2012) no. 19-20, pp. 885-890.

Abstract
Résumé

We are interested in the time asymptotic location of the level sets of solutions to Fisher–KPP reaction–diffusion equations with fractional diffusion in periodic media. We show that the speed of propagation is exponential in time, with a precise exponent depending on a periodic principal eigenvalue, and that it does not depend on the space direction. This is in contrast with the Freidlin–Gärtner formula for the standard Laplacian.

On sʼintéresse ici à la localisation asymptotique en temps des lignes de niveaux des solutions dʼequations de réaction–diffusion de type Fisher–KPP avec diffusion fractionnaire en milieu périodique. Nous montrons que la vitesse de propagation est exponentielle en temps, avec un exposant dépendant dʼune valeur propre principale périodique, et que cette vitesse ne dépend pas de la direction de propagation. Ceci est en contraste avec la formule de Freidlin–Gärtner pour le laplacien standard.

Received: 2012-09-26
Accepted: 2012-10-04
Published online: 2012-10-01

DOI: 10.1016/j.crma.2012.10.007

Author's affiliations:

Xavier Cabré ¹; Anne-Charline Coulon ²; Jean-Michel Roquejoffre ²

¹ ICREA and Universitat Politècnica de Catalunya, Dep. de Matemàtica Aplicada I, Av. Diagonal 647, 08028 Barcelone, Spain
² Institut de mathématiques, (UMR CNRS 5219), université Paul-Sabatier, 118, route de Narbonne, 31062 Toulouse cedex, France

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     author = {Xavier Cabr\'e and Anne-Charline Coulon and Jean-Michel Roquejoffre},
     title = {Propagation in {Fisher{\textendash}KPP} type equations with fractional diffusion in periodic media},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {885--890},
     publisher = {Elsevier},
     volume = {350},
     number = {19-20},
     year = {2012},
     doi = {10.1016/j.crma.2012.10.007},
     language = {en},
}

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Xavier Cabré; Anne-Charline Coulon; Jean-Michel Roquejoffre. Propagation in Fisher–KPP type equations with fractional diffusion in periodic media. Comptes Rendus. Mathématique, Volume 350 (2012) no. 19-20, pp. 885-890. doi : 10.1016/j.crma.2012.10.007. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2012.10.007/

References
Cited by

[1] D.G. Aronson; H.F. Weinberger Multidimensional nonlinear diffusion arising in population genetics, Advances in Mathematics, Volume 30 (1978), pp. 33-76

[2] H. Berestycki; F. Hamel; G. Nadin Asymptotic spreading in heterogeneous diffusive excitable media, Journal of Functional Analysis, Volume 255 (2008), pp. 2146-2189

[3] H. Berestycki; J.-M. Roquejoffre; L. Rossi The periodic patch model for population dynamics with fractional diffusion, Discrete and Continuous Dynamical Systems. Series S, Volume 4 (2011), pp. 1-13

[4] X. Cabré, A.-C. Coulon, J.-M. Roquejoffre, Fisher–KPP type equations with fractional diffusion in periodic media: propagation of fronts, forthcoming paper.

[5] X. Cabré; J.-M. Roquejoffre Front propagation in Fisher–KPP equations with fractional diffusion, C. R. Acad. Sci. Paris, Ser. I, Volume 347 (2009), pp. 1361-1366

[6] X. Cabré, J.-M. Roquejoffre, The influence of fractional diffusion in Fisher–KPP equations, Communications in Mathematical Physics, submitted for publication, 2012, . | arXiv

[7] A.-C. Coulon, J.-M. Roquejoffre, Transition between linear and exponential propagation in Fisher–KPP type reaction-diffusion equations, Communications in Partial Differential Equations, submitted for publication, 2011, . | arXiv

[8] L.C. Evans; P.E. Souganidis A PDE approach to certain large deviation problems for systems of parabolic equations, Annales de lʼInstitut Henri Poincaré, Analyse Non Linéaire, Volume 6 (1989), pp. 229-258

[9] J. Garnier Accelerating solutions in integro-differential equations, SIAM Journal on Mathematical Analysis, Volume 43 (2011), pp. 1955-1974

[10] J. Gärtner; M.I. Freidlin On the propagation of concentration waves in periodic and random media, Doklady Akademii Nauk SSSR, Volume 20 (1979), pp. 1282-1286

[11] F. Hamel; L. Roques Fast propagation for KPP equations with slowly decaying initial conditions, Journal of Differential Equations, Volume 249 (2010), pp. 1726-1745

[12] H.F. Weinberger On spreading speeds and traveling waves for growth and migration models in a periodic habitat, Journal of Mathematical Biology, Volume 45 (2002), pp. 511-548

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