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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 |
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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