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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

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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