Abstract
We study the Fisher-KPP equation where the Laplacian is replaced by the generator of a Feller semigroup with power decaying kernel, an important example being the fractional Laplacian. In contrast with the case of the standard Laplacian where the stable state invades the unstable one at constant speed, we prove that with fractional diffusion, generated for instance by a stable Lévy process, the front position is exponential in time. Our results provide a mathematically rigorous justification of numerous heuristics about this model.
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Aronson D.G., Weinberger H.F.: Multidimensional nonlinear diffusion arising in population genetics. Adv. Math. 30, 33–76 (1978)
Berestycki H., Roquejoffre J.-M., Rossi L.: The periodic patch model for population dynamics with fractional diffusion. Disc. Cont. Dyn. Syst. Ser. S 4, 1–13 (2011)
Bony J.-M., Courrège P., Priouret P.: Semi-groupes de Feller sur une variété à bord compacte et problèmes aux limites intégro-différentiels du second ordre donnant lieu au principe du maximum. Ann. Inst. Fourier 18, 369–521 (1968)
Bramson, M.: Convergence of solutions of the Kolmogorov equation to travelling waves. Memoirs of the AMS 44, Providence, RI: Amer. Math. Soc., 1983
Cabré X., Roquejoffre J.-M.: Propagation de fronts dans les équations de Fisher-KPP avec diffusion fractionnaire. C.R. Acad. Sci. Paris 347, 1361–1366 (2009)
Cazenave, T., Haraux, A.: An Introduction to Semilinear Evolution Equations. Oxford Lecture Series in Mathematics and its Applications 13. New York: Oxford University Press, 1998
Del-Castillo-Negrete D., Carreras B.A., Lynch V.E.: Front propagation and segregation in a reaction-diffusion model with cross-diffusion. Phys. D 168/169, 45–60 (2002)
Del-Castillo-Negrete D., Carreras B.A., Lynch V.E.: Front Dynamics in Reaction-Diffusion Systems with Levy Flights: A Fractional Diffusion Approach. Phys. Rev. Lett. 91(1), 018302 (2003)
Engler, H.: On the speed of spread for fractional reaction-diffusion equations. Int. J. Diff. Eq. 2010 Art. ID 315421, 16 pp. (2010)
Garnier J.: Accelerating solutions in integro-differential equations. SIAM J. Math. Anal. 43, 1955–1974 (2011)
Hamel F., Roques L.: Fast propagation for KPP equations with slowly decaying initial conditions. J. Diff. Eqs. 249, 1726–1745 (2010)
Jones C.K.R.T.: Asymptotic behavior of a reaction-diffusion equation in higher space dimensions. Rocky Mountain J. Math. 13, 355–364 (1983)
Kolmogorov A.N., Petrovskii I.G., Piskunov N.S.: Etude de l’équation de diffusion avec accroissement de la quantité de matière, et son application à un problème biologique. Bjul. Moskowskogo Gos. Univ. 17, 1–26 (1937)
Kolokoltsov V.N.: Symmetric stable laws and stable-like jump-diffusions. London Math. Soc. 80, 725–768 (2000)
Lamperti J.: Semi-stable stochastic processes. Trans. Amer. Math. Soc. 104, 62–78 (1962)
Mancinelli R., Vergni D., Vulpiani A.: Front propagation in reactive systems with anomalous diffusion. Phys. D 185, 175–195 (2003)
Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Applied Mathematical Sciences, 44. New York: Springer-Verlag, 1983
Taira, K.: Diffusion Processes and Partial Differential Equations. Boston, MA: Academic Press, Inc., 1988
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Communicated by P. Constantin
The first author was supported by grants MICINN MTM2008-06349-C03-01/FEDER, MINECO MTM2011-27739-C04-01, and GENCAT 2009SGR-345.
The second author is supported by the ANR grant PREFERED.
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Cabré, X., Roquejoffre, JM. The Influence of Fractional Diffusion in Fisher-KPP Equations. Commun. Math. Phys. 320, 679–722 (2013). https://doi.org/10.1007/s00220-013-1682-5
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DOI: https://doi.org/10.1007/s00220-013-1682-5