American Institute of Mathematical Sciences

Advanced
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 and 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.

[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.

[3]

D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusions arising in population genetics, Adv. Math., 30 (1978), 33-76.

[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.

[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.

[6]

H. Berestycki and F. Hamel, On the general definition of transition waves and their properties,, preprint., (). 

[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.

[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.

[9]

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

[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.

[11]

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

[12]

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

[13]

M. I. Freidlin, Limit theorems for large deviations and reaction-diffusion equations, Ann. Probab., 13 (1985), 639-675.

[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.

[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.

[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.

[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.

[18]

N. Shigesada and K. Kawasaki, "Biological Invasion: Theory and Practise," Oxford University Press, 1997.

[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.

[20]

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

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.

[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.

[3]

D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusions arising in population genetics, Adv. Math., 30 (1978), 33-76.

[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.

[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.

[6]

H. Berestycki and F. Hamel, On the general definition of transition waves and their properties,, preprint., (). 

[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.

[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.

[9]

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

[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.

[11]

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

[12]

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

[13]

M. I. Freidlin, Limit theorems for large deviations and reaction-diffusion equations, Ann. Probab., 13 (1985), 639-675.

[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.

[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.

[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.

[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.

[18]

N. Shigesada and K. Kawasaki, "Biological Invasion: Theory and Practise," Oxford University Press, 1997.

[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.

[20]

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

[1]

Matthieu Alfaro, Arnaud Ducrot. Sharp interface limit of the Fisher-KPP equation when initial data have slow exponential decay. Discrete and Continuous Dynamical Systems - B, 2011, 16 (1) : 15-29. doi: 10.3934/dcdsb.2011.16.15
[2]

Patrick Martinez, Judith Vancostenoble. Lipschitz stability for the growth rate coefficients in a nonlinear Fisher-KPP equation. Discrete and Continuous Dynamical Systems - S, 2021, 14 (2) : 695-721. doi: 10.3934/dcdss.2020362
[3]

Hiroshi Matsuzawa. A free boundary problem for the Fisher-KPP equation with a given moving boundary. Communications on Pure and Applied Analysis, 2018, 17 (5) : 1821-1852. doi: 10.3934/cpaa.2018087
[4]

Lina Wang, Xueli Bai, Yang Cao. Exponential stability of the traveling fronts for a viscous Fisher-KPP equation. Discrete and Continuous Dynamical Systems - B, 2014, 19 (3) : 801-815. doi: 10.3934/dcdsb.2014.19.801
[5]

Christian Kuehn, Pasha Tkachov. Pattern formation in the doubly-nonlocal Fisher-KPP equation. Discrete and Continuous Dynamical Systems, 2019, 39 (4) : 2077-2100. doi: 10.3934/dcds.2019087
[6]

Aaron Hoffman, Matt Holzer. Invasion fronts on graphs: The Fisher-KPP equation on homogeneous trees and Erdős-Réyni graphs. Discrete and Continuous Dynamical Systems - B, 2019, 24 (2) : 671-694. doi: 10.3934/dcdsb.2018202
[7]

Gregoire Nadin. How does the spreading speed associated with the Fisher-KPP equation depend on random stationary diffusion and reaction terms?. Discrete and Continuous Dynamical Systems - B, 2015, 20 (6) : 1785-1803. doi: 10.3934/dcdsb.2015.20.1785
[8]

Benjamin Contri. Fisher-KPP equations and applications to a model in medical sciences. Networks and Heterogeneous Media, 2018, 13 (1) : 119-153. doi: 10.3934/nhm.2018006
[9]

François Hamel, James Nolen, Jean-Michel Roquejoffre, Lenya Ryzhik. A short proof of the logarithmic Bramson correction in Fisher-KPP equations. Networks and Heterogeneous Media, 2013, 8 (1) : 275-289. doi: 10.3934/nhm.2013.8.275
[10]

Wenxian Shen, Zhongwei Shen. Transition fronts in nonlocal Fisher-KPP equations in time heterogeneous media. Communications on Pure and Applied Analysis, 2016, 15 (4) : 1193-1213. doi: 10.3934/cpaa.2016.15.1193
[11]

Matt Holzer. A proof of anomalous invasion speeds in a system of coupled Fisher-KPP equations. Discrete and Continuous Dynamical Systems, 2016, 36 (4) : 2069-2084. doi: 10.3934/dcds.2016.36.2069
[12]

Margarita Arias, Juan Campos, Cristina Marcelli. Fastness and continuous dependence in front propagation in Fisher-KPP equations. Discrete and Continuous Dynamical Systems - B, 2009, 11 (1) : 11-30. doi: 10.3934/dcdsb.2009.11.11
[13]

Jean-Michel Roquejoffre, Luca Rossi, Violaine Roussier-Michon. Sharp large time behaviour in $ N $-dimensional Fisher-KPP equations. Discrete and Continuous Dynamical Systems, 2019, 39 (12) : 7265-7290. doi: 10.3934/dcds.2019303
[14]

Zhenzhen Wang, Tianshou Zhou. Asymptotic behaviors and stochastic traveling waves in stochastic Fisher-KPP equations. Discrete and Continuous Dynamical Systems - B, 2021, 26 (9) : 5023-5045. doi: 10.3934/dcdsb.2020323
[15]

Thomas I. Seidman. Interface conditions for a singular reaction-diffusion system. Discrete and Continuous Dynamical Systems - S, 2009, 2 (3) : 631-643. doi: 10.3934/dcdss.2009.2.631
[16]

Jianping Gao, Shangjiang Guo, Wenxian Shen. Persistence and time periodic positive solutions of doubly nonlocal Fisher-KPP equations in time periodic and space heterogeneous media. Discrete and Continuous Dynamical Systems - B, 2021, 26 (5) : 2645-2676. doi: 10.3934/dcdsb.2020199
[17]

Patrick Martinez, Jean-Michel Roquejoffre. The rate of attraction of super-critical waves in a Fisher-KPP type model with shear flow. Communications on Pure and Applied Analysis, 2012, 11 (6) : 2445-2472. doi: 10.3934/cpaa.2012.11.2445
[18]

Aijun Zhang. Traveling wave solutions with mixed dispersal for spatially periodic Fisher-KPP equations. Conference Publications, 2013, 2013 (special) : 815-824. doi: 10.3934/proc.2013.2013.815
[19]

Denghui Wu, Zhen-Hui Bu. Multidimensional stability of pyramidal traveling fronts in degenerate Fisher-KPP monostable and combustion equations. Electronic Research Archive, 2021, 29 (6) : 3721-3740. doi: 10.3934/era.2021058
[20]

Grégory Faye, Thomas Giletti, Matt Holzer. Asymptotic spreading for Fisher-KPP reaction-diffusion equations with heterogeneous shifting diffusivity. Discrete and Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021146

2020 Impact Factor: 1.916

Tools

Metrics

  • PDF downloads (80)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy