[Propagation dans les equations de type Fisher–KPP avec diffusion fractionnaire en milieux périodiques]
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.
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.
Accepté le :
Publié le :
@article{CRMATH_2012__350_19-20_885_0,
author = {Cabr\'e, Xavier and Coulon, Anne-Charline and Roquejoffre, Jean-Michel},
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},
url = {http://www.numdam.org/articles/10.1016/j.crma.2012.10.007/}
}
TY - JOUR AU - Cabré, Xavier AU - Coulon, Anne-Charline AU - Roquejoffre, Jean-Michel TI - Propagation in Fisher–KPP type equations with fractional diffusion in periodic media JO - Comptes Rendus. Mathématique PY - 2012 SP - 885 EP - 890 VL - 350 IS - 19-20 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2012.10.007/ DO - 10.1016/j.crma.2012.10.007 LA - en ID - CRMATH_2012__350_19-20_885_0 ER -
%0 Journal Article %A Cabré, Xavier %A Coulon, Anne-Charline %A Roquejoffre, Jean-Michel %T Propagation in Fisher–KPP type equations with fractional diffusion in periodic media %J Comptes Rendus. Mathématique %D 2012 %P 885-890 %V 350 %N 19-20 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2012.10.007/ %R 10.1016/j.crma.2012.10.007 %G en %F CRMATH_2012__350_19-20_885_0
Cabré, Xavier; Coulon, Anne-Charline; Roquejoffre, Jean-Michel. Propagation in Fisher–KPP type equations with fractional diffusion in periodic media. Comptes Rendus. Mathématique, Tome 350 (2012) no. 19-20, pp. 885-890. doi : 10.1016/j.crma.2012.10.007. http://www.numdam.org/articles/10.1016/j.crma.2012.10.007/
[1] Multidimensional nonlinear diffusion arising in population genetics, Advances in Mathematics, Volume 30 (1978), pp. 33-76
[2] Asymptotic spreading in heterogeneous diffusive excitable media, Journal of Functional Analysis, Volume 255 (2008), pp. 2146-2189
[3] 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] 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] 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] Accelerating solutions in integro-differential equations, SIAM Journal on Mathematical Analysis, Volume 43 (2011), pp. 1955-1974
[10] On the propagation of concentration waves in periodic and random media, Doklady Akademii Nauk SSSR, Volume 20 (1979), pp. 1282-1286
[11] Fast propagation for KPP equations with slowly decaying initial conditions, Journal of Differential Equations, Volume 249 (2010), pp. 1726-1745
[12] 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
Cité par Sources :
