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December  2019, 39(12): 7265-7290. doi: 10.3934/dcds.2019303

Sharp large time behaviour in $ N $-dimensional Fisher-KPP equations

Jean-Michel Roquejoffre 1,, , Luca Rossi 2, and  Violaine Roussier-Michon 3,

1. 

Institut de Mathématiques de Toulouse; UMR 5219, Université de Toulouse; CNRS, Université Toulouse Ⅲ, 118 route de Narbonne, 31062 Toulouse, France
2. 

Centre d'Analyse et de Mathématique Sociales; UMR 8557, Paris Sciences et Lettres; CNRS, EHESS, 54 Bv. Raspail, 75006 Paris, France
3. 

Institut de Mathématiques de Toulouse; UMR 5219, Université de Toulouse; CNRS, INSA Toulouse, 135 av. Rangueil, 31077 Toulouse, France

* Corresponding author

Dedicated to L. Caffarelli, as a sign of friendship, admiration and respect

Received  February 2019 Revised  August 2019 Published  September 2019

Fund Project: The first and second authors are supported by the European Union's Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement n. 321186 - ReaDi - "Reaction-Diffusion Equations, Propagation and Modelling". The third author is supported by the ANR project NONLOCAL ANR-14-CE25-0013.

We study the large time behaviour of the Fisher-KPP equation
$ \partial_t u = \Delta u +u-u^2 $
 in spatial dimension
$ N $
, when the initial datum is compactly supported. We prove the existence of a Lipschitz function
$ s^\infty $
 of the unit sphere, such that
$ u(t, x) $
 approaches, as
$ t $
 goes to infinity, the function
$ U_{c_*}\bigg(|x|-c_*t + \frac{N+2}{c_*} \mathrm{ln}t + s^\infty\Big(\frac{x}{|x|}\Big)\bigg), $
where
$ U_{c*} $
 is the 1D travelling front with minimal speed
$ c_* = 2 $
. This extends an earlier result of Gärtner.
Keywords: Fisher-KPP equation, logarithmc delay, large time behaviour.
Mathematics Subject Classification: Primary: 35K57.
Citation: Jean-Michel Roquejoffre, Luca Rossi, Violaine Roussier-Michon. Sharp large time behaviour in $ N $-dimensional Fisher-KPP equations. Discrete & Continuous Dynamical Systems, 2019, 39 (12) : 7265-7290. doi: 10.3934/dcds.2019303
References:
[1]

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

[2]

H. Berestycki and F. Hamel, Reaction-diffusion Equations and Propagation Phenomena, Applied Mathematical Sciences, 2014. Google Scholar

[3]

H. Berestycki, The inluence of advection on the propagation of fronts in reaction-diffusion equations, in: Nonlinear PDE's in Condensed Matter and Reactive Flows, eds. H. Berestycki, Y. Pomeau, NATO Science Series C, Mathematical and Physical Sciences, Kluwer Acad. Publ., Dordrecht, NL, 569 (2002). Google Scholar

[4]

J. Berestycki, E. Brunet and J. Derrida, A new approach to computing the asymptotics of the position of Fisher-KPP fronts, J. Phys. A, 51 (2018), 035204, 21 pp, https://arXiv.org/pdf/1802.03262.pdf. doi: 10.1088/1751-8121/aa899f.  Google Scholar

[5]

M. D. Bramson, Convergence of solutions of the Kolmogorov equation to travelling waves, Mem. Amer. Math. Soc., 44 (1983), iv+190 pp. doi: 10.1090/memo/0285.  Google Scholar

[6]

A.-C. Chalmin and J.-M. Roquejoffre,, in preparation. Google Scholar

[7]

Y. Du, F. Quiros and M. Zhou,, Logarithmic corrections in Fisher-KPP type Porous Medium equations, arXiv: 1806.02022. Google Scholar

[8]

A. Ducrot, On the large time behaviour of the multi-dimensional Fisher-KPP equation with compactly supported initial data, Nonlinearity, 28 (2015), 1043-1076.  doi: 10.1088/0951-7715/28/4/1043.  Google Scholar

[9]

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.  Google Scholar

[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.  doi: 10.1512/iumj.1989.38.38007.  Google Scholar

[11]

P. C. Fife and B. McLeod, The approach of solutions of nonlinear diffusion equations to travelling front solutions, Arch. Ration. Mech. Anal., 65 (1977), 335-361.  doi: 10.1007/BF00250432.  Google Scholar

[12]

J. Gärtner, Location of wave fronts for the multidimensional KPP equation and Brownian first exit densities, Math. Nachr., 105 (1982), 317-351.  doi: 10.1002/mana.19821050117.  Google Scholar

[13]

J. Gärtner and M. I. Freidlin, The propagation of concentration waves in periodic and random media, Dokl. Akad. Nauk SSSR, 249 (1979), 521-525.   Google Scholar

[14]

C. Graham, Precise asymptotics for Fisher-KPP fronts, Nonlinearity, 32 (2019), 1967–1998, https://arXiv.org/abs/1712.02472. doi: 10.1088/1361-6544/aaffe8.  Google Scholar

[15]

F. Hamel, J. Nolen, J.-M. Roquejoffre and L. Ryzhik, A short proof of the logarithmic Bramson correction in Fisher-KPP equations, Netw. Heterog. Media, 8 (2013), 275-289.  doi: 10.3934/nhm.2013.8.275.  Google Scholar

[16]

F. Hamel, J. Nolen, J.-M. Roquejoffre and L. Ryzhik, The logarithmic delay of KPP fronts in a periodic medium, Journal of the European Mathematical Society, 18 (2016), 465-505.  doi: 10.4171/JEMS/595.  Google Scholar

[17]

D. Henry, Geometric Theory of Semlinear Parabolic Equations, Lecture Notes in Mathematics, 840. Springer-Verlag, Berlin-New York, 1981.  Google Scholar

[18]

C. K. R. T. Jones, Asymptotic behaviour of a reaction-diffusion equation in higher space dimensions, Rocky Mountain J. Math., 13 (1983), 355-364.  doi: 10.1216/RMJ-1983-13-2-355.  Google Scholar

[19]

A. N. Kolmogorov, I. G. Petrovskii 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.   Google Scholar

[20]

J. Nolen, J.-M. Roquejoffre and L. Ryzhik, Convergence to a single wave in the Fisher-KPP equation, Chinese Ann. Math. Ser. B (special issue in honour of H. Brezis), 38 (2017), 629–646. doi: 10.1007/s11401-017-1087-4.  Google Scholar

[21]

J. Nolen, J.-M. Roquejoffre and L. Ryzhik, Refined long time asymptotics for the Fisher-KPP fronts, Comm. Contemp. Math., 2018. doi: 10.1142/S0219199718500724.  Google Scholar

[22]

J.-M. Roquejoffre and V. Roussier-Michon, Nontrivial large-time behaviour in bistable reaction-diffusion equations, Annali Mat. Pura Appl., 188 (2009), 207-233.  doi: 10.1007/s10231-008-0072-7.  Google Scholar

[23]

J.-M. Roquejoffre and V. Roussier-Michon, Nontrivial dynamics beyond the logarithmic shift in two-dimensional Fisher-KPP equations, Nonlinearity, 31 (2018), 3284-3307.  doi: 10.1088/1361-6544/aaba3b.  Google Scholar

[24]

L. Rossi, The Freidlin-Gärtner formula for general reaction terms, Adv. Math., 317 (2017), 267-298.  doi: 10.1016/j.aim.2017.07.002.  Google Scholar

[25]

L. Rossi, Symmetrization and anti-symmetrization in parabolic equations, Proc. Amer. Math. Soc., 145 (2017), 2527-2537.  doi: 10.1090/proc/13391.  Google Scholar

[26]

V. Roussier, Stability of radially symmetric travelling waves in reaction-diffusion equations, Ann. Inst. Henri Poincaré, Analyse non linéaire, 21 (2004), 341-379.  doi: 10.1016/S0294-1449(03)00042-8.  Google Scholar

[27]

B. Shabani, Univ. Stanford PhD thesis, Paper in preparation. Google Scholar

[28]

H. F. Weinberger, On spreading speeds and traveling waves for growth and migration models in a periodic habitat, J. Math. Biol., 45 (2002), 511-548.  doi: 10.1007/s00285-002-0169-3.  Google Scholar

[29]

H. Yagisita, Nearly spherically symmetric expanding fronts in a bistable reaction-diffusion equation, J. Dynam. Differential Equations, 13 (2001), 323-353.  doi: 10.1023/A:1016632124792.  Google Scholar

show all references

References:
[1]

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

[2]

H. Berestycki and F. Hamel, Reaction-diffusion Equations and Propagation Phenomena, Applied Mathematical Sciences, 2014. Google Scholar

[3]

H. Berestycki, The inluence of advection on the propagation of fronts in reaction-diffusion equations, in: Nonlinear PDE's in Condensed Matter and Reactive Flows, eds. H. Berestycki, Y. Pomeau, NATO Science Series C, Mathematical and Physical Sciences, Kluwer Acad. Publ., Dordrecht, NL, 569 (2002). Google Scholar

[4]

J. Berestycki, E. Brunet and J. Derrida, A new approach to computing the asymptotics of the position of Fisher-KPP fronts, J. Phys. A, 51 (2018), 035204, 21 pp, https://arXiv.org/pdf/1802.03262.pdf. doi: 10.1088/1751-8121/aa899f.  Google Scholar

[5]

M. D. Bramson, Convergence of solutions of the Kolmogorov equation to travelling waves, Mem. Amer. Math. Soc., 44 (1983), iv+190 pp. doi: 10.1090/memo/0285.  Google Scholar

[6]

A.-C. Chalmin and J.-M. Roquejoffre,, in preparation. Google Scholar

[7]

Y. Du, F. Quiros and M. Zhou,, Logarithmic corrections in Fisher-KPP type Porous Medium equations, arXiv: 1806.02022. Google Scholar

[8]

A. Ducrot, On the large time behaviour of the multi-dimensional Fisher-KPP equation with compactly supported initial data, Nonlinearity, 28 (2015), 1043-1076.  doi: 10.1088/0951-7715/28/4/1043.  Google Scholar

[9]

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.  Google Scholar

[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.  doi: 10.1512/iumj.1989.38.38007.  Google Scholar

[11]

P. C. Fife and B. McLeod, The approach of solutions of nonlinear diffusion equations to travelling front solutions, Arch. Ration. Mech. Anal., 65 (1977), 335-361.  doi: 10.1007/BF00250432.  Google Scholar

[12]

J. Gärtner, Location of wave fronts for the multidimensional KPP equation and Brownian first exit densities, Math. Nachr., 105 (1982), 317-351.  doi: 10.1002/mana.19821050117.  Google Scholar

[13]

J. Gärtner and M. I. Freidlin, The propagation of concentration waves in periodic and random media, Dokl. Akad. Nauk SSSR, 249 (1979), 521-525.   Google Scholar

[14]

C. Graham, Precise asymptotics for Fisher-KPP fronts, Nonlinearity, 32 (2019), 1967–1998, https://arXiv.org/abs/1712.02472. doi: 10.1088/1361-6544/aaffe8.  Google Scholar

[15]

F. Hamel, J. Nolen, J.-M. Roquejoffre and L. Ryzhik, A short proof of the logarithmic Bramson correction in Fisher-KPP equations, Netw. Heterog. Media, 8 (2013), 275-289.  doi: 10.3934/nhm.2013.8.275.  Google Scholar

[16]

F. Hamel, J. Nolen, J.-M. Roquejoffre and L. Ryzhik, The logarithmic delay of KPP fronts in a periodic medium, Journal of the European Mathematical Society, 18 (2016), 465-505.  doi: 10.4171/JEMS/595.  Google Scholar

[17]

D. Henry, Geometric Theory of Semlinear Parabolic Equations, Lecture Notes in Mathematics, 840. Springer-Verlag, Berlin-New York, 1981.  Google Scholar

[18]

C. K. R. T. Jones, Asymptotic behaviour of a reaction-diffusion equation in higher space dimensions, Rocky Mountain J. Math., 13 (1983), 355-364.  doi: 10.1216/RMJ-1983-13-2-355.  Google Scholar

[19]

A. N. Kolmogorov, I. G. Petrovskii 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.   Google Scholar

[20]

J. Nolen, J.-M. Roquejoffre and L. Ryzhik, Convergence to a single wave in the Fisher-KPP equation, Chinese Ann. Math. Ser. B (special issue in honour of H. Brezis), 38 (2017), 629–646. doi: 10.1007/s11401-017-1087-4.  Google Scholar

[21]

J. Nolen, J.-M. Roquejoffre and L. Ryzhik, Refined long time asymptotics for the Fisher-KPP fronts, Comm. Contemp. Math., 2018. doi: 10.1142/S0219199718500724.  Google Scholar

[22]

J.-M. Roquejoffre and V. Roussier-Michon, Nontrivial large-time behaviour in bistable reaction-diffusion equations, Annali Mat. Pura Appl., 188 (2009), 207-233.  doi: 10.1007/s10231-008-0072-7.  Google Scholar

[23]

J.-M. Roquejoffre and V. Roussier-Michon, Nontrivial dynamics beyond the logarithmic shift in two-dimensional Fisher-KPP equations, Nonlinearity, 31 (2018), 3284-3307.  doi: 10.1088/1361-6544/aaba3b.  Google Scholar

[24]

L. Rossi, The Freidlin-Gärtner formula for general reaction terms, Adv. Math., 317 (2017), 267-298.  doi: 10.1016/j.aim.2017.07.002.  Google Scholar

[25]

L. Rossi, Symmetrization and anti-symmetrization in parabolic equations, Proc. Amer. Math. Soc., 145 (2017), 2527-2537.  doi: 10.1090/proc/13391.  Google Scholar

[26]

V. Roussier, Stability of radially symmetric travelling waves in reaction-diffusion equations, Ann. Inst. Henri Poincaré, Analyse non linéaire, 21 (2004), 341-379.  doi: 10.1016/S0294-1449(03)00042-8.  Google Scholar

[27]

B. Shabani, Univ. Stanford PhD thesis, Paper in preparation. Google Scholar

[28]

H. F. Weinberger, On spreading speeds and traveling waves for growth and migration models in a periodic habitat, J. Math. Biol., 45 (2002), 511-548.  doi: 10.1007/s00285-002-0169-3.  Google Scholar

[29]

H. Yagisita, Nearly spherically symmetric expanding fronts in a bistable reaction-diffusion equation, J. Dynam. Differential Equations, 13 (2001), 323-353.  doi: 10.1023/A:1016632124792.  Google Scholar

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