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 & Continuous Dynamical Systems - A, 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. HamelJ. NolenJ.-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. HamelJ. NolenJ.-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. KolmogorovI. 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. HamelJ. NolenJ.-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. HamelJ. NolenJ.-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. KolmogorovI. 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

[1]

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

[2]

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

[3]

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

[4]

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

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[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]

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

[15]

Liviu I. Ignat, Ademir F. Pazoto. Large time behaviour for a nonlocal diffusion - convection equation related with gas dynamics. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3575-3589. doi: 10.3934/dcds.2014.34.3575

[16]

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

[17]

Michiel Bertsch, Danielle Hilhorst, Hirofumi Izuhara, Masayasu Mimura, Tohru Wakasa. A nonlinear parabolic-hyperbolic system for contact inhibition and a degenerate parabolic fisher kpp equation. Discrete & Continuous Dynamical Systems - A, 2019, 0 (0) : 1-26. doi: 10.3934/dcds.2019226

[18]

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

[19]

Takanobu Okazaki. Large time behaviour of solutions of nonlinear ode describing hysteresis. Conference Publications, 2007, 2007 (Special) : 804-813. doi: 10.3934/proc.2007.2007.804

[20]

Jianfeng Feng, Mariya Shcherbina, Brunello Tirozzi. Dynamical behaviour of a large complex system. Communications on Pure & Applied Analysis, 2008, 7 (2) : 249-265. doi: 10.3934/cpaa.2008.7.249

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (60)
  • HTML views (45)
  • Cited by (0)

[Back to Top]