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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), 505-565. 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, Tulane Univ., New Orleans, La., 1974), 5-49. Lecture Notes in Math., Vol. 446, Springer, Berlin, 1975.  Google Scholar

[3]

D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusions arising in population genetics, Adv. Math., 30 (1978), 33-76.  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), 835-858.  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 Sér. I Math., 319 (1994), 679-684.  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), 1-34.  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), 549-554.  Google Scholar

[9]

X. Chen, Generation and propagation of interfaces for reaction-diffusion equations, J. Differential Equations, 96 (1992), 116-141.  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), 141-172.  Google Scholar

[11]

E. Feireisl, Front propagation for degenerate parabolic equations, Nonlinear Anal., 35 (1999), Ser. A: Theory Methods, 735-746.  Google Scholar

[12]

R. A. Fisher, The wave of advance of advantageous genes, Ann. of Eugenics, 7 (1937), 355-369. Google Scholar

[13]

M. I. Freidlin, Limit theorems for large deviations and reaction-diffusion equations, Ann. Probab., 13 (1985), 639-675.  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), 2872-2889.  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 Université d'Etat Moscou, Bjul. Moskowskogo Gos. Univ., 1937, 1-26. 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, Chapman $&$ Hall/CRC Press, Boca Raton, FL, 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, Trento, June 21-25, 2004), Springer, Dordrecht, 128 p. 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), Ser. A: Theory Methods, 757-776.  Google Scholar

[20]

A. Volpert, V. Volpert, V. Volpert, "Travelling Wave Solutions of Parabolic Systems," Translations of Mathematical Monographs, vol. 140, AMS Providence, RI, 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), 505-565. 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, Tulane Univ., New Orleans, La., 1974), 5-49. Lecture Notes in Math., Vol. 446, Springer, Berlin, 1975.  Google Scholar

[3]

D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusions arising in population genetics, Adv. Math., 30 (1978), 33-76.  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), 835-858.  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 Sér. I Math., 319 (1994), 679-684.  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), 1-34.  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), 549-554.  Google Scholar

[9]

X. Chen, Generation and propagation of interfaces for reaction-diffusion equations, J. Differential Equations, 96 (1992), 116-141.  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), 141-172.  Google Scholar

[11]

E. Feireisl, Front propagation for degenerate parabolic equations, Nonlinear Anal., 35 (1999), Ser. A: Theory Methods, 735-746.  Google Scholar

[12]

R. A. Fisher, The wave of advance of advantageous genes, Ann. of Eugenics, 7 (1937), 355-369. Google Scholar

[13]

M. I. Freidlin, Limit theorems for large deviations and reaction-diffusion equations, Ann. Probab., 13 (1985), 639-675.  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), 2872-2889.  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 Université d'Etat Moscou, Bjul. Moskowskogo Gos. Univ., 1937, 1-26. 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, Chapman $&$ Hall/CRC Press, Boca Raton, FL, 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, Trento, June 21-25, 2004), Springer, Dordrecht, 128 p. 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), Ser. A: Theory Methods, 757-776.  Google Scholar

[20]

A. Volpert, V. Volpert, V. Volpert, "Travelling Wave Solutions of Parabolic Systems," Translations of Mathematical Monographs, vol. 140, AMS Providence, RI, 1994.  Google Scholar

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