American Institute of Mathematical Sciences

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March  2013, 8(1): 275-289. doi: 10.3934/nhm.2013.8.275

A short proof of the logarithmic Bramson correction in Fisher-KPP equations

François Hamel 1, , James Nolen 2, , Jean-Michel Roquejoffre 3, and  Lenya Ryzhik 4,

1. 

Université d'Aix-Marseille, LATP, 39 rue F. Joliot-Curie, 13453 Marseille Cedex 13, France
2. 

Department of Mathematics, Duke University, Box 90320, Durham, NC, 27708-0320
3. 

Institut de Mathématiques, Université Paul Sabatier, 118 route de Narbonne, F-31062 Toulouse Cedex 4
4. 

Department of Mathematics, Stanford University, Stanford, CA 94305

Received  May 2012 Revised  November 2012 Published  April 2013

In this paper, we explain in simple PDE terms a famous result of Bramson about the logarithmic delay of the position of the solutions $u(t,x)$ of Fisher-KPP reaction-diffusion equations in $\mathbb{R}$, with respect to the position of the travelling front with minimal speed. Our proof is based on the comparison of $u$ to the solutions of linearized equations with Dirichlet boundary conditions at the position of the minimal front, with and without the logarithmic delay. Our analysis also yields the large-time convergence of the solutions $u$ along their level sets to the profile of the minimal travelling front.
Keywords: logarithmic correction., Fisher-KPP equations, travelling fronts.
Mathematics Subject Classification: Primary: 35K57, 35C07; Secondary: 35B4.
Citation: 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
References:
[1]

D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusions arising in population genetics, Adv. Math., 30 (1978), 33-76. doi: 10.1016/0001-8708(78)90130-5.

[2]

H. Berestycki and F. Hamel, Generalized travelling waves for reaction-diffusion equations, in "Perspectives in Nonlinear Partial Differential Equations, in Honor of H. Brezis", Amer. Math. Soc., Contemp. Math., (2007), 101-123. doi: 10.1090/conm/446/08627.

[3]

M. D. Bramson, Maximal displacement of branching Brownian motion, Comm. Pure Appl. Math., 31 (1978), 531-581. doi: 10.1002/cpa.3160310502.

[4]

M. D. Bramson, "Convergence of Solutions of the Kolmogorov Equation to Travelling Waves," Mem. Amer. Math. Soc., 1983.

[5]

E. Brunet and B. Derrida, Shift in the velocity of a front due to a cutoff, Phys. Rev. E, 56 (1997), 2597-2604. doi: 10.1103/PhysRevE.56.2597.

[6]

C. Cuesta and J. King, Front propagation in a heterogeneous Fisher equation: The homogeneous case is non-generic, Quart. J. Mech. Appl. Math., 63 (2010), 521-571. doi: 10.1093/qjmam/hbq017.

[7]

J.-P. Eckmann and T. Gallay, Front solutions for the Ginzburg-Landau equation, Comm. Math. Phys., 152 (1993), 221-248. doi: 10.1007/BF02098298.

[8]

U. Ebert and W. van Saarloos, Front propagation into unstable states: Universal algebraic convergence towards uniformly translating pulled fronts, Phys. D, 146 (2000), 1-99. doi: 10.1016/S0167-2789(00)00068-3.

[9]

U. Ebert, W. van Saarloos and B. Peletier, Universal algebraic convergence in time of pulled fronts: The common mechanism for difference-differential and partial differential equations, European J. Appl. Math., 13 (2002), 53-66. doi: 10.1017/S0956792501004673.

[10]

P. C. Fife, "Mathematical Aspects of Reacting and Diffusing Systems," Lecture Notes in Biomathematics, Springer Verlag, 1979.

[11]

R. A. Fisher, The wave of advance of advantageous genes, Ann. Eugenics, 7 (1937), 353-369. doi: 10.1111/j.1469-1809.1937.tb02153.x.

[12]

F. Hamel, J. Nolen, J.-M. Roquejoffre and L. Ryzhik, The logarithmic delay of KPP fronts in a periodic medium,, preprint., (). 

[13]

K.-S. Lau, On the nonlinear diffusion equation of Kolmogorov, Petrovskii and Piskunov, J. Diff. Eqs., 59 (1985), 44-70. doi: 10.1016/0022-0396(85)90137-8.

[14]

A. N. Kolmogorov, I. G. Petrovsky and N. S. Piskunov, Étude de l'équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique, Bull. Univ. État Moscou, Sér. Inter. A, 1 (1937), 1-26.

[15]

J. D. Murray, "Mathematical Biology," Springer-Verlag, 2003. doi: 10.1007/b98869.

[16]

F. Rothe, Convergence to travelling fronts in semilinear parabolic equations, Proc. Royal Soc. Edinburgh A, 80 (1978), 213-234. doi: 10.1017/S0308210500010258.

[17]

D. H. Sattinger, Weighted norms for the stability of traveling waves, J. Diff. Eqs., 25 (1977), 130-144. doi: 10.1016/0022-0396(77)90185-1.

[18]

K. Uchiyama, The behavior of solutions of some nonlinear diffusion equations for large time, J. Math. Kyoto Univ., 18 (1978), 453-508.

[19]

J. Xin, "An Introduction to Fronts in Random Media," Surveys and Tutorials in the Applied Mathematical Sciences, Springer, New York, 2009. doi: 10.1007/978-0-387-87683-2.

show all references

References:
[1]

D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusions arising in population genetics, Adv. Math., 30 (1978), 33-76. doi: 10.1016/0001-8708(78)90130-5.

[2]

H. Berestycki and F. Hamel, Generalized travelling waves for reaction-diffusion equations, in "Perspectives in Nonlinear Partial Differential Equations, in Honor of H. Brezis", Amer. Math. Soc., Contemp. Math., (2007), 101-123. doi: 10.1090/conm/446/08627.

[3]

M. D. Bramson, Maximal displacement of branching Brownian motion, Comm. Pure Appl. Math., 31 (1978), 531-581. doi: 10.1002/cpa.3160310502.

[4]

M. D. Bramson, "Convergence of Solutions of the Kolmogorov Equation to Travelling Waves," Mem. Amer. Math. Soc., 1983.

[5]

E. Brunet and B. Derrida, Shift in the velocity of a front due to a cutoff, Phys. Rev. E, 56 (1997), 2597-2604. doi: 10.1103/PhysRevE.56.2597.

[6]

C. Cuesta and J. King, Front propagation in a heterogeneous Fisher equation: The homogeneous case is non-generic, Quart. J. Mech. Appl. Math., 63 (2010), 521-571. doi: 10.1093/qjmam/hbq017.

[7]

J.-P. Eckmann and T. Gallay, Front solutions for the Ginzburg-Landau equation, Comm. Math. Phys., 152 (1993), 221-248. doi: 10.1007/BF02098298.

[8]

U. Ebert and W. van Saarloos, Front propagation into unstable states: Universal algebraic convergence towards uniformly translating pulled fronts, Phys. D, 146 (2000), 1-99. doi: 10.1016/S0167-2789(00)00068-3.

[9]

U. Ebert, W. van Saarloos and B. Peletier, Universal algebraic convergence in time of pulled fronts: The common mechanism for difference-differential and partial differential equations, European J. Appl. Math., 13 (2002), 53-66. doi: 10.1017/S0956792501004673.

[10]

P. C. Fife, "Mathematical Aspects of Reacting and Diffusing Systems," Lecture Notes in Biomathematics, Springer Verlag, 1979.

[11]

R. A. Fisher, The wave of advance of advantageous genes, Ann. Eugenics, 7 (1937), 353-369. doi: 10.1111/j.1469-1809.1937.tb02153.x.

[12]

F. Hamel, J. Nolen, J.-M. Roquejoffre and L. Ryzhik, The logarithmic delay of KPP fronts in a periodic medium,, preprint., (). 

[13]

K.-S. Lau, On the nonlinear diffusion equation of Kolmogorov, Petrovskii and Piskunov, J. Diff. Eqs., 59 (1985), 44-70. doi: 10.1016/0022-0396(85)90137-8.

[14]

A. N. Kolmogorov, I. G. Petrovsky and N. S. Piskunov, Étude de l'équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique, Bull. Univ. État Moscou, Sér. Inter. A, 1 (1937), 1-26.

[15]

J. D. Murray, "Mathematical Biology," Springer-Verlag, 2003. doi: 10.1007/b98869.

[16]

F. Rothe, Convergence to travelling fronts in semilinear parabolic equations, Proc. Royal Soc. Edinburgh A, 80 (1978), 213-234. doi: 10.1017/S0308210500010258.

[17]

D. H. Sattinger, Weighted norms for the stability of traveling waves, J. Diff. Eqs., 25 (1977), 130-144. doi: 10.1016/0022-0396(77)90185-1.

[18]

K. Uchiyama, The behavior of solutions of some nonlinear diffusion equations for large time, J. Math. Kyoto Univ., 18 (1978), 453-508.

[19]

J. Xin, "An Introduction to Fronts in Random Media," Surveys and Tutorials in the Applied Mathematical Sciences, Springer, New York, 2009. doi: 10.1007/978-0-387-87683-2.

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