The speed of propagation for KPP type problems. II: General domains

Berestycki, Henri; Hamel, François; Nadirashvili, Nikolai

doi:10.1090/S0894-0347-09-00633-X

The speed of propagation for KPP type problems. II: General domains
HTML articles powered by AMS MathViewer

by Henri Berestycki, François Hamel and Nikolai Nadirashvili

J. Amer. Math. Soc. 23 (2010), 1-34

DOI: https://doi.org/10.1090/S0894-0347-09-00633-X

Published electronically: July 6, 2009

PDF | Request permission

Abstract:

This paper is devoted to nonlinear propagation phenomena in general unbounded domains of $\mathbb {R}^N$, for reaction-diffusion equations with Kolmogorov-Petrovsky-Piskunov (KPP) type nonlinearities. This article is the second in a series of two and it is the follow-up of the paper The speed of propagation for KPP type problems. I - Periodic framework, by the authors, which dealt which the case of periodic domains. This paper is concerned with general domains, and we give various definitions of the spreading speeds at large times for solutions with compactly supported initial data. We study the relationships between these new notions and analyze their dependence on the geometry of the domain and on the initial condition. Some a priori bounds are proved for large classes of domains. The case of exterior domains is also discussed in detail. Lastly, some domains which are very thin at infinity and for which the spreading speeds are infinite are exhibited; the construction is based on some new heat kernel estimates in such domains.

References

D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Adv. in Math. 30 (1978), no. 1, 33–76. MR 511740, DOI 10.1016/0001-8708(78)90130-5
B. Audoly, H. Berestycki, Y. Pomeau, Réaction-diffusion en écoulement stationnaire rapide, C. R. Acad. Sci. Paris 328 II (2000), pp. 255-262.
H. Berestycki, The influence of advection on the propagation of fronts in reaction-diffusion equations, In: Nonlinear PDE’s in Condensed Matter and Reactive Flows, H. Berestycki and Y. Pomeau, eds., Kluwer Academic Publ., 2002, pp. 1-45.
Henri Berestycki and François Hamel, Front propagation in periodic excitable media, Comm. Pure Appl. Math. 55 (2002), no. 8, 949–1032. MR 1900178, DOI 10.1002/cpa.3022
Henri Berestycki and François Hamel, Generalized travelling waves for reaction-diffusion equations, Perspectives in nonlinear partial differential equations, Contemp. Math., vol. 446, Amer. Math. Soc., Providence, RI, 2007, pp. 101–123. MR 2373726, DOI 10.1090/conm/446/08627
Henri Berestycki, François Hamel, and Nikolai Nadirashvili, Elliptic eigenvalue problems with large drift and applications to nonlinear propagation phenomena, Comm. Math. Phys. 253 (2005), no. 2, 451–480. MR 2140256, DOI 10.1007/s00220-004-1201-9
Henri Berestycki, François Hamel, and Nikolai Nadirashvili, The speed of propagation for KPP type problems. I. Periodic framework, J. Eur. Math. Soc. (JEMS) 7 (2005), no. 2, 173–213. MR 2127993, DOI 10.4171/JEMS/26
Henri Berestycki and Louis Nirenberg, Travelling fronts in cylinders, Ann. Inst. H. Poincaré C Anal. Non Linéaire 9 (1992), no. 5, 497–572 (English, with English and French summaries). MR 1191008, DOI 10.1016/S0294-1449(16)30229-3
Peter Constantin, Alexander Kiselev, Adam Oberman, and Leonid Ryzhik, Bulk burning rate in passive-reactive diffusion, Arch. Ration. Mech. Anal. 154 (2000), no. 1, 53–91. MR 1778121, DOI 10.1007/s002050000090
E. B. Davies, Heat kernels and spectral theory, Cambridge Tracts in Mathematics, vol. 92, Cambridge University Press, Cambridge, 1989. MR 990239, DOI 10.1017/CBO9780511566158
Paul C. Fife and J. B. McLeod, The approach of solutions of nonlinear diffusion equations to travelling front solutions, Arch. Rational Mech. Anal. 65 (1977), no. 4, 335–361. MR 442480, DOI 10.1007/BF00250432
R.A. Fisher, The advance of advantageous genes, Ann. Eugenics 7 (1937), pp. 335-369.
Mark I. Freidlin, On wavefront propagation in periodic media, Stochastic analysis and applications, Adv. Probab. Related Topics, vol. 7, Dekker, New York, 1984, pp. 147–166. MR 776979
Ju. Gertner and M. I. Freĭdlin, The propagation of concentration waves in periodic and random media, Dokl. Akad. Nauk SSSR 249 (1979), no. 3, 521–525 (Russian). MR 553200
Th. Gallay, Local stability of critical fronts in nonlinear parabolic partial differential equations, Nonlinearity 7 (1994), no. 3, 741–764. MR 1275528, DOI 10.1088/0951-7715/7/3/003
Alexander Grigor′yan, Gaussian upper bounds for the heat kernel on arbitrary manifolds, J. Differential Geom. 45 (1997), no. 1, 33–52. MR 1443330
Manfred Gruber, Harnack inequalities for solutions of general second order parabolic equations and estimates of their Hölder constants, Math. Z. 185 (1984), no. 1, 23–43. MR 724044, DOI 10.1007/BF01214972
K. P. Hadeler and F. Rothe, Travelling fronts in nonlinear diffusion equations, J. Math. Biol. 2 (1975), no. 3, 251–263. MR 411693, DOI 10.1007/BF00277154
François Hamel, Formules min-max pour les vitesses d’ondes progressives multidimensionnelles, Ann. Fac. Sci. Toulouse Math. (6) 8 (1999), no. 2, 259–280 (French, with English and French summaries). MR 1751443, DOI 10.5802/afst.932
François Hamel, Qualitative properties of monostable pulsating fronts: exponential decay and monotonicity, J. Math. Pures Appl. (9) 89 (2008), no. 4, 355–399 (English, with English and French summaries). MR 2401143, DOI 10.1016/j.matpur.2007.12.005
F. Hamel, L. Roques, Uniqueness and stability properties of monostable pulsating fronts, preprint.
Steffen Heinze, Diffusion-advection in cellular flows with large Peclet numbers, Arch. Ration. Mech. Anal. 168 (2003), no. 4, 329–342. MR 1994746, DOI 10.1007/s00205-003-0256-7
S. Heinze, Large convection limits for KPP fronts, preprint.
S. Heinze, G. Papanicolaou, and A. Stevens, Variational principles for propagation speeds in inhomogeneous media, SIAM J. Appl. Math. 62 (2001), no. 1, 129–148. MR 1857539, DOI 10.1137/S0036139999361148
W. Hudson and B. Zinner, Existence of traveling waves for reaction diffusion equations of Fisher type in periodic media, Boundary value problems for functional-differential equations, World Sci. Publ., River Edge, NJ, 1995, pp. 187–199. MR 1375475
Christopher K. R. T. Jones, Spherically symmetric solutions of a reaction-diffusion equation, J. Differential Equations 49 (1983), no. 1, 142–169. MR 704268, DOI 10.1016/0022-0396(83)90023-2
Ja. I. Kanel′, Stabilization of the solutions of the equations of combustion theory with finite initial functions, Mat. Sb. (N.S.) 65 (107) (1964), 398–413 (Russian). MR 0177209
Alexander Kiselev and Leonid Ryzhik, Enhancement of the traveling front speeds in reaction-diffusion equations with advection, Ann. Inst. H. Poincaré C Anal. Non Linéaire 18 (2001), no. 3, 309–358 (English, with English and French summaries). MR 1831659, DOI 10.1016/S0294-1449(01)00068-3
A.N. Kolmogorov, I.G. Petrovsky, N.S. Piskunov, Étude de l’équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique, Bulletin Université d’Etat à Moscou, Sér. Internationale A 1 (1937), pp. 1-26.
N. V. Krylov and M. V. Safonov, A property of the solutions of parabolic equations with measurable coefficients, Izv. Akad. Nauk SSSR Ser. Mat. 44 (1980), no. 1, 161–175, 239 (Russian). MR 563790
Gary M. Lieberman, Second order parabolic differential equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. MR 1465184, DOI 10.1142/3302
Roger Lui, Biological growth and spread modeled by systems of recursions. I. Mathematical theory, Math. Biosci. 93 (1989), no. 2, 269–295. MR 984281, DOI 10.1016/0025-5564(89)90026-6
Andrew J. Majda and Panagiotis E. Souganidis, Large-scale front dynamics for turbulent reaction-diffusion equations with separated velocity scales, Nonlinearity 7 (1994), no. 1, 1–30. MR 1260130, DOI 10.1088/0951-7715/7/1/001
Jean-François Mallordy and Jean-Michel Roquejoffre, A parabolic equation of the KPP type in higher dimensions, SIAM J. Math. Anal. 26 (1995), no. 1, 1–20. MR 1311879, DOI 10.1137/S0036141093246105
H. Matano, Oral communication.
Jean-Michel Roquejoffre, Eventual monotonicity and convergence to travelling fronts for the solutions of parabolic equations in cylinders, Ann. Inst. H. Poincaré C Anal. Non Linéaire 14 (1997), no. 4, 499–552 (English, with English and French summaries). MR 1464532, DOI 10.1016/S0294-1449(97)80137-0
Lenya Ryzhik and Andrej Zlatoš, KPP pulsating front speed-up by flows, Commun. Math. Sci. 5 (2007), no. 3, 575–593. MR 2352332, DOI 10.4310/CMS.2007.v5.n3.a4
N. Shigesada, K. Kawasaki, Biological invasions: theory and practice, Oxford Series in Ecology and Evolution, Oxford: Oxford UP, 1997.
Aizik I. Volpert, Vitaly A. Volpert, and Vladimir A. Volpert, Traveling wave solutions of parabolic systems, Translations of Mathematical Monographs, vol. 140, American Mathematical Society, Providence, RI, 1994. Translated from the Russian manuscript by James F. Heyda. MR 1297766, DOI 10.1090/mmono/140
Hans F. Weinberger, On spreading speeds and traveling waves for growth and migration models in a periodic habitat, J. Math. Biol. 45 (2002), no. 6, 511–548. MR 1943224, DOI 10.1007/s00285-002-0169-3
Jack X. Xin, Existence of planar flame fronts in convective-diffusive periodic media, Arch. Rational Mech. Anal. 121 (1992), no. 3, 205–233. MR 1188981, DOI 10.1007/BF00410613
A. Zlatos, Sharp asymptotics for KPP pulsating front speed-up and diffusion enhancement by flows, preprint.