Abstract
The authors study the large time asymptotics of a solution of the Fisher-KPP reaction-diffusion equation, with an initial condition that is a compact perturbation of a step function. A well-known result of Bramson states that, in the reference frame moving as 2t−(3/2)log t+x ∞, the solution of the equation converges as t → +∞ to a translate of the traveling wave corresponding to the minimal speed c * = 2. The constant x ∞ depends on the initial condition u(0, x). The proof is elaborate, and based on probabilistic arguments. The purpose of this paper is to provide a simple proof based on PDE arguments.
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Aïdékon, E., Berestycki, J., Brunet, É. and Shi, Z., Branching Brownian motion seen from its tip, Probab. Theory Relat. Fields, 157, 2013, 405–451.
Arguin, L.-P., Bovier, A. and Kistler, N., Poissonian statistics in the extremal process of branching Brownian motion, Ann. Appl. Probab., 22, 2012, 1693–1711.
Arguin, L.-P., Bovier, A. and Kistler, N., The extremal process of branching Brownian motion, Probab. Theory Relat. Fields, 157, 2013, 535–574.
Berestycki, H. and Hamel, F., Front propagation in periodic excitable media, Comm. Pure Appl. Math., 55, 2002, 949–1032.
Bramson, M. D., Maximal displacement of branching Brownian motion, Comm. Pure Appl. Math., 31, 1978, 531–581.
Bramson, M. D., Convergence of solutions of the Kolmogorov equation to travelling waves, Mem. Amer. Math. Soc., 44, 1983.
Brunet, E. and Derrida, B., A branching random walk seen from the tip, Journal of Statistical Physics., 143, 2011, 420–446.
Brunet, E. and Derrida, B., Statistics at the tip of a branching random walk and the delay of traveling waves, Eur. Phys. Lett., 87, 60010, 2009.
Fang, M. and Zeitouni, O., Slowdown for time inhomogeneous branching Brownian motion, J. Stat. Phys., 149, 2012, 1–9.
Fisher, R. A., The wave of advance of advantageous genes, Ann. Eugenics, 7, 1937, 353–369.
Hamel, F., Nolen, J., Roquejoffre, J.-M. and Ryzhik, L., A short proof of the logarithmic Bramson correction in Fisher-KPP equations, Netw. Het. Media, 8, 2013, 275–289.
Hamel, F., Nolen, J., Roquejoffre, J.-M., and Ryzhik, L., The logarithmic time delay of KPP fronts in a periodic medium, J. Europ. Math. Soc., 18, 2016, 465–505.
Hamel, F. and Roques, L., Uniqueness and stability properties of monostable pulsating fronts, J. Europ. Math. Soc., 13, 2011, 345–390.
Henderson, C., Population stabilization in branching Brownian motion with absorption and drift, Comm. Math. Sci., 14, 2016, 973–985.
Kolmogorov, A. N., Petrovskii, I. G. and Piskunov, N. S., Étude de l’équation de la diffusion avec croissance de la quantité dematière et son application à un problème biologique, Bull. Univ. État Moscou, Sér. Inter. A, 1, 1937, 1–26.
Lalley, S. P. and Sellke, T., A conditional limit theorem for the frontier of a branching Brownian motion, Annals of Probability, 15, 1987, 1052–1061.
Lau, K.-S., On the nonlinear diffusion equation of Kolmogorov, Petrovskii and Piskunov, J. Diff. Eqs., 59, 1985, 44–70.
Maillard, P. and Zeitouni, O., Slowdown in branching Brownian motion with inhomogeneous variance, Ann. IHP Prob. Stat., to appear.
McKean, H. P., Application of Brownian motion to the equation of Kolmogorov-Petrovskii-Piskunov, Comm. Pure Appl. Math., 28 1975, 323–331.
Nolen, J., Roquejoffre, J.-M. and Ryzhik, L., Power-like delay in time inhomogeneous Fisher-KPP equations, Comm. Partial Diff. Equations, 40, 2015, 475–505.
Nolen, J., Roquejoffre, J.-M. and Ryzhik, L., Refined large-time asymptotics for the Fisher-KPP equation, 2016, preprint. arXiv: 1607.08802
Roberts, M., A simple path to asymptotics for the frontier of a branching Brownian motion, Ann. Prob., 41, 2013, 3518–3541.
Roquejoffre, J.-M., Eventual monotonicity and convergence to travelling fronts for the solutions of parabolic equations in cylinders, Ann. Inst. H. Poincaré Anal. Non Linéaire, 14, 1997, 499–552.
Acknowledgments
Lenya Ryzhik and Jean-Michel Roquejoffre thank the Labex CIMI for a PDE-probability quarter in Toulouse, in Winter 2014, out of which the idea of this paper grew and which provided a stimulating scientific environment for this project.
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Dedicated to Haim Brezis, with admiration and respect
This work was supported by NSF grant DMS-1351653, NSF grant DMS-1311903 and the European Union’s Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement n. 321186-ReaDi-“Reaction-Diffusion Equations, Propagation and Modelling”, as well as the ANR project NONLOCAL ANR-14-CE25-0013.
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Nolen, J., Roquejoffre, JM. & Ryzhik, L. Convergence to a single wave in the Fisher-KPP equation. Chin. Ann. Math. Ser. B 38, 629–646 (2017). https://doi.org/10.1007/s11401-017-1087-4
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DOI: https://doi.org/10.1007/s11401-017-1087-4