# American Institute of Mathematical Sciences

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

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

 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.
Citation: Jean-Michel Roquejoffre, Luca Rossi, Violaine Roussier-Michon. Sharp large time behaviour in $N$-dimensional Fisher-KPP equations. Discrete and 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. [2] H. Berestycki and F. Hamel, Reaction-diffusion Equations and Propagation Phenomena, Applied Mathematical Sciences, 2014. [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). [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. [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. [6] A.-C. Chalmin and J.-M. Roquejoffre,, in preparation. [7] Y. Du, F. Quiros and M. Zhou,, Logarithmic corrections in Fisher-KPP type Porous Medium equations, arXiv: 1806.02022. [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. [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. [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. [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. [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. [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. [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. [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. [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. [17] D. Henry, Geometric Theory of Semlinear Parabolic Equations, Lecture Notes in Mathematics, 840. Springer-Verlag, Berlin-New York, 1981. [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. [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. [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. [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. [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. [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. [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. [25] L. Rossi, Symmetrization and anti-symmetrization in parabolic equations, Proc. Amer. Math. Soc., 145 (2017), 2527-2537.  doi: 10.1090/proc/13391. [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. [27] B. Shabani, Univ. Stanford PhD thesis, Paper in preparation. [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. [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.

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##### 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. [2] H. Berestycki and F. Hamel, Reaction-diffusion Equations and Propagation Phenomena, Applied Mathematical Sciences, 2014. [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). [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. [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. [6] A.-C. Chalmin and J.-M. Roquejoffre,, in preparation. [7] Y. Du, F. Quiros and M. Zhou,, Logarithmic corrections in Fisher-KPP type Porous Medium equations, arXiv: 1806.02022. [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. [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. [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. [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. [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. [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. [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. [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. [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. [17] D. Henry, Geometric Theory of Semlinear Parabolic Equations, Lecture Notes in Mathematics, 840. Springer-Verlag, Berlin-New York, 1981. [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. [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. [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. [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. [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. [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. [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. [25] L. Rossi, Symmetrization and anti-symmetrization in parabolic equations, Proc. Amer. Math. Soc., 145 (2017), 2527-2537.  doi: 10.1090/proc/13391. [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. [27] B. Shabani, Univ. Stanford PhD thesis, Paper in preparation. [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. [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.
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