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

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.
Citation: François Hamel, James Nolen, Jean-Michel Roquejoffre, Lenya Ryzhik. A short proof of the logarithmic Bramson correction in Fisher-KPP equations. Networks & 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.  doi: 10.1016/0001-8708(78)90130-5.  Google Scholar

[2]

H. Berestycki and F. Hamel, Generalized travelling waves for reaction-diffusion equations,, in, (2007), 101.  doi: 10.1090/conm/446/08627.  Google Scholar

[3]

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

[4]

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

[5]

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

[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.  doi: 10.1093/qjmam/hbq017.  Google Scholar

[7]

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

[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.  doi: 10.1016/S0167-2789(00)00068-3.  Google Scholar

[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.  doi: 10.1017/S0956792501004673.  Google Scholar

[10]

P. C. Fife, "Mathematical Aspects of Reacting and Diffusing Systems,", Lecture Notes in Biomathematics, (1979).   Google Scholar

[11]

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

[12]

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

[13]

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

[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, 1 (1937), 1.   Google Scholar

[15]

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

[16]

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

[17]

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

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

show all references

References:
[1]

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

[2]

H. Berestycki and F. Hamel, Generalized travelling waves for reaction-diffusion equations,, in, (2007), 101.  doi: 10.1090/conm/446/08627.  Google Scholar

[3]

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

[4]

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

[5]

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

[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.  doi: 10.1093/qjmam/hbq017.  Google Scholar

[7]

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

[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.  doi: 10.1016/S0167-2789(00)00068-3.  Google Scholar

[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.  doi: 10.1017/S0956792501004673.  Google Scholar

[10]

P. C. Fife, "Mathematical Aspects of Reacting and Diffusing Systems,", Lecture Notes in Biomathematics, (1979).   Google Scholar

[11]

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

[12]

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

[13]

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

[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, 1 (1937), 1.   Google Scholar

[15]

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

[16]

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

[17]

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

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

[1]

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

[2]

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

[3]

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

[4]

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

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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

[10]

Yanni Zeng, Kun Zhao. On the logarithmic Keller-Segel-Fisher/KPP system. Discrete & Continuous Dynamical Systems - A, 2019, 39 (9) : 5365-5402. doi: 10.3934/dcds.2019220

[11]

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

[12]

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

[13]

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 & Applied Analysis, 2012, 11 (6) : 2445-2472. doi: 10.3934/cpaa.2012.11.2445

[14]

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

[15]

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

[16]

Feng Cao, Wenxian Shen. Spreading speeds and transition fronts of lattice KPP equations in time heterogeneous media. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 4697-4727. doi: 10.3934/dcds.2017202

[17]

Yaping Wu, Xiuxia Xing, Qixiao Ye. Stability of travelling waves with algebraic decay for $n$-degree Fisher-type equations. Discrete & Continuous Dynamical Systems - A, 2006, 16 (1) : 47-66. doi: 10.3934/dcds.2006.16.47

[18]

Yi Li, Yaping Wu. Stability of travelling waves with noncritical speeds for double degenerate Fisher-Type equations. Discrete & Continuous Dynamical Systems - B, 2008, 10 (1) : 149-170. doi: 10.3934/dcdsb.2008.10.149

[19]

James Nolen, Jack Xin. KPP fronts in a one-dimensional random drift. Discrete & Continuous Dynamical Systems - B, 2009, 11 (2) : 421-442. doi: 10.3934/dcdsb.2009.11.421

[20]

Karel Hasik, Sergei Trofimchuk. Slowly oscillating wavefronts of the KPP-Fisher delayed equation. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3511-3533. doi: 10.3934/dcds.2014.34.3511

2018 Impact Factor: 0.871

Metrics

  • PDF downloads (15)
  • HTML views (0)
  • Cited by (23)

[Back to Top]