Sharp interface limit of the Fisher-KPP equation

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January 2012, 11(1): 1-18. doi: 10.3934/cpaa.2012.11.1

Sharp interface limit of the Fisher-KPP equation

Matthieu Alfaro ^1, and Arnaud Ducrot ^2,

1.	I3M, Université de Montpellier 2, CC051, Place Eugène Bataillon, 34095 Montpellier Cedex 5
2.	UMR CNRS 5251, I.M.B. and INRIA Bordeaux Sud-ouest Anubis, case 36, UFR Sciences et Modélisation, Université Victor Segalen Bordeaux 2, 3 ter, place de la Victoire - 33076 Bordeaux cedex

Received January 2010 Revised July 2010 Published September 2011

Abstract
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We investigate the singular limit, as $\varepsilon\to 0$, of the Fisher equation $\partial_t u=\varepsilon\Delta u + \varepsilon^{-1}u(1-u)$ in the whole space. We consider initial data with compact support plus, possibly, perturbations very small as $||x|| \to \infty$. By proving both generation and motion of interface properties, we show that the sharp interface limit moves by a constant speed, which is the minimal speed of some related one-dimensional travelling waves. Moreover, we obtain a new estimate of the thickness of the transition layers. We also exhibit initial data "not so small" at infinity which do not allow the interface phenomena.

Keywords: singular perturbation, Fisher-KPP equation, motion of interface., generation of interface.

Mathematics Subject Classification: Primary: 35K57, 35B25; Secondary: 92D2.

Citation: Matthieu Alfaro, Arnaud Ducrot. Sharp interface limit of the Fisher-KPP equation. Communications on Pure & Applied Analysis, 2012, 11 (1) : 1-18. doi: 10.3934/cpaa.2012.11.1

References:

[1]	M. Alfaro, D. Hilhorst and H. Matano, The singular limit of the Allen-Cahn equation and the FitzHugh-Nagumo system,, J. Differential Equations, 245 (2008), 505. Google Scholar
[2]	D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation,, Partial differential equations and related topics (Program, (1974), 5. Google Scholar
[3]	D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusions arising in population genetics,, Adv. Math., 30 (1978), 33. Google Scholar
[4]	G. Barles, L. C. Evans and P. E. Souganidis, Wavefront propagation for reaction-diffusion systems of PDE,, Duke Math. J., 61 (1990), 835. Google Scholar
[5]	G. Barles and P. E. Souganidis, A remark on the asymptotic behavior of the solution of the KPP equation,, C. R. Acad. Sci. Paris S\'er. I Math., 319 (1994), 679. Google Scholar
[6]	H. Berestycki and F. Hamel, On the general definition of transition waves and their properties,, preprint., (). Google Scholar
[7]	H. Berestycki, F. Hamel and N. Nadirashvili, The speed of propagation for KPP type problems. II. General domains,, J. Amer. Math. Soc., 23 (2010), 1. Google Scholar
[8]	H. Berestycki, F. Hamel and L. Roques, Équations de réaction-diffusion et modèles d'invasions biologiques dans les milieux périodiques,, C. R. Math. Acad. Sci. Paris, 339 (2004), 549. Google Scholar
[9]	X. Chen, Generation and propagation of interfaces for reaction-diffusion equations,, J. Differential Equations, 96 (1992), 116. Google Scholar
[10]	L. C. Evans and P. E. Souganidis, A PDE approach to geometric optics for certain semilinear parabolic equations,, Indiana Univ. Math. J., 38 (1989), 141. Google Scholar
[11]	E. Feireisl, Front propagation for degenerate parabolic equations,, Nonlinear Anal., 35 (1999), 735. Google Scholar
[12]	R. A. Fisher, The wave of advance of advantageous genes,, Ann. of Eugenics, 7 (1937), 355. Google Scholar
[13]	M. I. Freidlin, Limit theorems for large deviations and reaction-diffusion equations,, Ann. Probab., 13 (1985), 639. Google Scholar
[14]	D. Hilhorst, R. Kersner, E. Logak and M. Mimura, Interface dynamics of the Fisher equation with degenerate diffusion,, J. Differential Equations, 244 (2008), 2872. Google Scholar
[15]	A. N. Kolmogorov, I. G. Petrovsky and N. S. Piskunov, Etude de l'équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique,, Bulletin Universit\'e d'Etat Moscou, (1937), 1. Google Scholar
[16]	H. Malchow, S. V. Petrovskii and E. Venturino, "Spatiotemporal Patterns in Ecology and Epidemiology. Theory, Models, and Simulations,", Mathematical and Computational Biology Series, (2008). Google Scholar
[17]	S. V. Petrovskii and H. Malchow, eds. (2005), "Biological Invasions in a Mathematical Perspective,", (A special issue of Biological Invasions: Proceedings of Computational and Mathematical Population Dynamics, (2004), 21. Google Scholar
[18]	N. Shigesada and K. Kawasaki, "Biological Invasion: Theory and Practise,", Oxford University Press, (1997). Google Scholar
[19]	S. Vakulenko and V. Volpert, Generalized travelling waves for perturbed monotone reaction-diffusion systems,, Nonlinear Anal., 46 (2001), 757. Google Scholar
[20]	A. Volpert, V. Volpert, V. Volpert, "Travelling Wave Solutions of Parabolic Systems,", Translations of Mathematical Monographs, (1994). Google Scholar

show all references

References:

[1]	M. Alfaro, D. Hilhorst and H. Matano, The singular limit of the Allen-Cahn equation and the FitzHugh-Nagumo system,, J. Differential Equations, 245 (2008), 505. Google Scholar
[2]	D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation,, Partial differential equations and related topics (Program, (1974), 5. Google Scholar
[3]	D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusions arising in population genetics,, Adv. Math., 30 (1978), 33. Google Scholar
[4]	G. Barles, L. C. Evans and P. E. Souganidis, Wavefront propagation for reaction-diffusion systems of PDE,, Duke Math. J., 61 (1990), 835. Google Scholar
[5]	G. Barles and P. E. Souganidis, A remark on the asymptotic behavior of the solution of the KPP equation,, C. R. Acad. Sci. Paris S\'er. I Math., 319 (1994), 679. Google Scholar
[6]	H. Berestycki and F. Hamel, On the general definition of transition waves and their properties,, preprint., (). Google Scholar
[7]	H. Berestycki, F. Hamel and N. Nadirashvili, The speed of propagation for KPP type problems. II. General domains,, J. Amer. Math. Soc., 23 (2010), 1. Google Scholar
[8]	H. Berestycki, F. Hamel and L. Roques, Équations de réaction-diffusion et modèles d'invasions biologiques dans les milieux périodiques,, C. R. Math. Acad. Sci. Paris, 339 (2004), 549. Google Scholar
[9]	X. Chen, Generation and propagation of interfaces for reaction-diffusion equations,, J. Differential Equations, 96 (1992), 116. Google Scholar
[10]	L. C. Evans and P. E. Souganidis, A PDE approach to geometric optics for certain semilinear parabolic equations,, Indiana Univ. Math. J., 38 (1989), 141. Google Scholar
[11]	E. Feireisl, Front propagation for degenerate parabolic equations,, Nonlinear Anal., 35 (1999), 735. Google Scholar
[12]	R. A. Fisher, The wave of advance of advantageous genes,, Ann. of Eugenics, 7 (1937), 355. Google Scholar
[13]	M. I. Freidlin, Limit theorems for large deviations and reaction-diffusion equations,, Ann. Probab., 13 (1985), 639. Google Scholar
[14]	D. Hilhorst, R. Kersner, E. Logak and M. Mimura, Interface dynamics of the Fisher equation with degenerate diffusion,, J. Differential Equations, 244 (2008), 2872. Google Scholar
[15]	A. N. Kolmogorov, I. G. Petrovsky and N. S. Piskunov, Etude de l'équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique,, Bulletin Universit\'e d'Etat Moscou, (1937), 1. Google Scholar
[16]	H. Malchow, S. V. Petrovskii and E. Venturino, "Spatiotemporal Patterns in Ecology and Epidemiology. Theory, Models, and Simulations,", Mathematical and Computational Biology Series, (2008). Google Scholar
[17]	S. V. Petrovskii and H. Malchow, eds. (2005), "Biological Invasions in a Mathematical Perspective,", (A special issue of Biological Invasions: Proceedings of Computational and Mathematical Population Dynamics, (2004), 21. Google Scholar
[18]	N. Shigesada and K. Kawasaki, "Biological Invasion: Theory and Practise,", Oxford University Press, (1997). Google Scholar
[19]	S. Vakulenko and V. Volpert, Generalized travelling waves for perturbed monotone reaction-diffusion systems,, Nonlinear Anal., 46 (2001), 757. Google Scholar
[20]	A. Volpert, V. Volpert, V. Volpert, "Travelling Wave Solutions of Parabolic Systems,", Translations of Mathematical Monographs, (1994). Google Scholar

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