×

zbMATH — the first resource for mathematics

Propagation in Fisher-KPP type equations with fractional diffusion in periodic media. (English. Abridged French version) Zbl 1253.35198
Summary: 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.

MSC:
35R11 Fractional partial differential equations
35B40 Asymptotic behavior of solutions to PDEs
35K57 Reaction-diffusion equations
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Aronson, D.G.; Weinberger, H.F., Multidimensional nonlinear diffusion arising in population genetics, Advances in mathematics, 30, 33-76, (1978) · Zbl 0407.92014
[2] Berestycki, H.; Hamel, F.; Nadin, G., Asymptotic spreading in heterogeneous diffusive excitable media, Journal of functional analysis, 255, 2146-2189, (2008) · Zbl 1234.35033
[3] Berestycki, H.; Roquejoffre, J.-M.; Rossi, L., The periodic patch model for population dynamics with fractional diffusion, Discrete and continuous dynamical systems. series S, 4, 1-13, (2011) · Zbl 1207.35279
[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] Cabré, X.; Roquejoffre, J.-M., Front propagation in Fisher-KPP equations with fractional diffusion, C. R. acad. sci. Paris, ser. I, 347, 1361-1366, (2009) · Zbl 1182.35072
[6] X. Cabré, J.-M. Roquejoffre, The influence of fractional diffusion in Fisher-KPP equations, Communications in Mathematical Physics, submitted for publication, 2012, arXiv:1202.6072.
[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:1111.0408.
[8] Evans, L.C.; Souganidis, P.E., A PDE approach to certain large deviation problems for systems of parabolic equations, Annales de lʼinstitut Henri Poincaré, analyse non linéaire, 6, 229-258, (1989) · Zbl 0679.60040
[9] Garnier, J., Accelerating solutions in integro-differential equations, SIAM journal on mathematical analysis, 43, 1955-1974, (2011) · Zbl 1232.47058
[10] Gärtner, J.; Freidlin, M.I., On the propagation of concentration waves in periodic and random media, Doklady akademii nauk SSSR, 20, 1282-1286, (1979) · Zbl 0447.60060
[11] Hamel, F.; Roques, L., Fast propagation for KPP equations with slowly decaying initial conditions, Journal of differential equations, 249, 1726-1745, (2010) · Zbl 1213.35100
[12] Weinberger, H.F., On spreading speeds and traveling waves for growth and migration models in a periodic habitat, Journal of mathematical biology, 45, 511-548, (2002) · Zbl 1058.92036
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.