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A new proof of the boundedness results for stable solutions to semilinear elliptic equations
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 |
$ \partial_t u = \Delta u +u-u^2 $ |
$ N $ |
$ s^\infty $ |
$ u(t, x) $ |
$ t $ |
$ U_{c_*}\bigg(|x|-c_*t + \frac{N+2}{c_*} \mathrm{ln}t + s^\infty\Big(\frac{x}{|x|}\Big)\bigg), $ |
$ U_{c*} $ |
$ c_* = 2 $ |
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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. |
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. |
[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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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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