July  2011, 16(1): 15-29. doi: 10.3934/dcdsb.2011.16.15

Sharp interface limit of the Fisher-KPP equation when initial data have slow exponential decay

1. 

I3M, Université de Montpellier 2, CC051, Place Eugène Bataillon, 34095 Montpellier Cedex 5, France

2. 

UMR CNRS 5251 IMB and INRIA Bordeaux Sud-Ouest ANUBIS, Université de Bordeaux, 3, Place de la Victoire, 33000 Bordeaux, France

Received  March 2010 Revised  January 2011 Published  April 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 perturbations with slow exponential decay, therefore completing the analysis presented in [1]. We prove that the sharp interface limit moves with a constant speed depending dramatically on the tails of the initial data. We make a thorough analysis of both the generation and motion of interface, thus providing a new estimate of the thickness of the transition layers.
Citation: Matthieu Alfaro, Arnaud Ducrot. Sharp interface limit of the Fisher-KPP equation when initial data have slow exponential decay. Discrete & Continuous Dynamical Systems - B, 2011, 16 (1) : 15-29. doi: 10.3934/dcdsb.2011.16.15
References:
[1]

M. Alfaro and A. Ducrot, Sharp interface limit of the Fisher-KPP equation,, to appear in Comm. Pure Appl. Anal., ().   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, 446 (1974), 5.   Google Scholar

[3]

G. Barles, L. C. Evans and P. E. Souganidis, Wavefront propagation for reaction-diffusion systems of PDE,, Duke Math. J., 61 (1990), 835.  doi: 10.1215/S0012-7094-90-06132-0.  Google Scholar

[4]

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\'erie I, 319 (1994), 679.   Google Scholar

[5]

M. Bramson, Convergence of solutions of the Kolmogorov equation to travelling waves,, Mem. Amer. Math. Soc., 44 (1983).   Google Scholar

[6]

X. Chen, Generation and propagation of interfaces for reaction-diffusion equations,, J. Differential equations, 96 (1992), 116.  doi: 10.1016/0022-0396(92)90146-E.  Google Scholar

[7]

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.  doi: 10.1512/iumj.1989.38.38007.  Google Scholar

[8]

R. A. Fisher, The wave of advance of advantageous genes,, Ann. of Eugenics, 7 (1937), 355.  doi: 10.1111/j.1469-1809.1937.tb02153.x.  Google Scholar

[9]

M. I. Freidlin, Limit theorems for large deviations and reaction-diffusion equations,, Ann. Probab., 13 (1985), 639.  doi: 10.1214/aop/1176992901.  Google Scholar

[10]

F. Hamel and L. Roques, Fast propagation for KPP equations with slowly decaying initial conditions,, J. Differential equations, 249 (2010), 1726.  doi: 10.1016/j.jde.2010.06.025.  Google Scholar

[11]

D. Hilhorst, R. Kersner, E. Logak and M. Mimura, Interface dynamics of the Fisher equation with degenerate diffusion,, J. Differential Equations, 244 (2008), 2872.  doi: 10.1016/j.jde.2008.02.018.  Google Scholar

[12]

R. Huang, Stability of travelling fronts of the Fisher-KPP equation in $R^N$,, Nonlinear Diff. Equ. Appl., 15 (2008), 599.  doi: 10.1007/s00030-008-7041-0.  Google Scholar

[13]

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, (): 1.   Google Scholar

[14]

D. A. Larson, Transient bounds and time-asymptotic behavior of solutions to nonlinear equations of Fisher type,, SIAM J. Appl. Math., 34 (1978), 93.  doi: 10.1137/0134008.  Google Scholar

[15]

K. S. Lau, On the nonlinear diffusion equation of Kolmogorov, Petrovsky, and Piscounov,, J. Differential Equations, 59 (1985), 44.  doi: 10.1016/0022-0396(85)90137-8.  Google Scholar

[16]

H. P. McKean, Application of Brownian motion to the equation of Kolmogorov-Petrovskii-Piskunov,, Comm. Pure Appl. Math., 28 (1975), 323.  doi: 10.1002/cpa.3160280302.  Google Scholar

[17]

F. Rothe, Convergence to travelling fronts in semilinear parabolic equations,, Proc. Roy. Soc. Edinburgh Sect. A, 80 (1978), 213.   Google Scholar

[18]

K. Uchiyama, The behavior of solutions of some nonlinear diffusion equations for large time,, J. Math. Kyoto Univ., 18 (1978), 453.   Google Scholar

[19]

A. Volpert, V. Volpert and V. Volpert, "Travelling Wave Solutions of Parabolic Systems,", Translations of Mathematical Monographs, 140 (1994).   Google Scholar

show all references

References:
[1]

M. Alfaro and A. Ducrot, Sharp interface limit of the Fisher-KPP equation,, to appear in Comm. Pure Appl. Anal., ().   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, 446 (1974), 5.   Google Scholar

[3]

G. Barles, L. C. Evans and P. E. Souganidis, Wavefront propagation for reaction-diffusion systems of PDE,, Duke Math. J., 61 (1990), 835.  doi: 10.1215/S0012-7094-90-06132-0.  Google Scholar

[4]

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\'erie I, 319 (1994), 679.   Google Scholar

[5]

M. Bramson, Convergence of solutions of the Kolmogorov equation to travelling waves,, Mem. Amer. Math. Soc., 44 (1983).   Google Scholar

[6]

X. Chen, Generation and propagation of interfaces for reaction-diffusion equations,, J. Differential equations, 96 (1992), 116.  doi: 10.1016/0022-0396(92)90146-E.  Google Scholar

[7]

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.  doi: 10.1512/iumj.1989.38.38007.  Google Scholar

[8]

R. A. Fisher, The wave of advance of advantageous genes,, Ann. of Eugenics, 7 (1937), 355.  doi: 10.1111/j.1469-1809.1937.tb02153.x.  Google Scholar

[9]

M. I. Freidlin, Limit theorems for large deviations and reaction-diffusion equations,, Ann. Probab., 13 (1985), 639.  doi: 10.1214/aop/1176992901.  Google Scholar

[10]

F. Hamel and L. Roques, Fast propagation for KPP equations with slowly decaying initial conditions,, J. Differential equations, 249 (2010), 1726.  doi: 10.1016/j.jde.2010.06.025.  Google Scholar

[11]

D. Hilhorst, R. Kersner, E. Logak and M. Mimura, Interface dynamics of the Fisher equation with degenerate diffusion,, J. Differential Equations, 244 (2008), 2872.  doi: 10.1016/j.jde.2008.02.018.  Google Scholar

[12]

R. Huang, Stability of travelling fronts of the Fisher-KPP equation in $R^N$,, Nonlinear Diff. Equ. Appl., 15 (2008), 599.  doi: 10.1007/s00030-008-7041-0.  Google Scholar

[13]

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, (): 1.   Google Scholar

[14]

D. A. Larson, Transient bounds and time-asymptotic behavior of solutions to nonlinear equations of Fisher type,, SIAM J. Appl. Math., 34 (1978), 93.  doi: 10.1137/0134008.  Google Scholar

[15]

K. S. Lau, On the nonlinear diffusion equation of Kolmogorov, Petrovsky, and Piscounov,, J. Differential Equations, 59 (1985), 44.  doi: 10.1016/0022-0396(85)90137-8.  Google Scholar

[16]

H. P. McKean, Application of Brownian motion to the equation of Kolmogorov-Petrovskii-Piskunov,, Comm. Pure Appl. Math., 28 (1975), 323.  doi: 10.1002/cpa.3160280302.  Google Scholar

[17]

F. Rothe, Convergence to travelling fronts in semilinear parabolic equations,, Proc. Roy. Soc. Edinburgh Sect. A, 80 (1978), 213.   Google Scholar

[18]

K. Uchiyama, The behavior of solutions of some nonlinear diffusion equations for large time,, J. Math. Kyoto Univ., 18 (1978), 453.   Google Scholar

[19]

A. Volpert, V. Volpert and V. Volpert, "Travelling Wave Solutions of Parabolic Systems,", Translations of Mathematical Monographs, 140 (1994).   Google Scholar

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