How travelling waves attract the solutions of KPP-type equations

Bages, Michaël; Martinez, Patrick; Roquejoffre, Jean-Michel

doi:10.1090/S0002-9947-2012-05554-1

How travelling waves attract the solutions of KPP-type equations
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by Michaël Bages, Patrick Martinez and Jean-Michel Roquejoffre PDF

Trans. Amer. Math. Soc. 364 (2012), 5415-5468 Request permission

Abstract:

We consider in this paper a general reaction-diffusion equation of the KPP (Kolmogorov, Petrovskiĭ, Piskunov) type, posed on an infinite cylinder. Such a model will have a family of pulsating waves of constant speed, larger than a critical speed $c_*$. The family of all supercritical waves attracts a large class of initial data, and we try to understand how. We describe in this paper the fate of an initial datum trapped between two supercritical waves of the same velocity: the solution will converge to a whole set of translates of the same wave, and we identify the convergence dynamics as that of an effective drift, around which an effective diffusion process occurs. In several nontrivial particular cases, we are able to describe the dynamics by an effective equation.

References

M. Bages, P. Martinez, Existence of pulsating waves for a thermo-diffusive system modelling solid combustion, preprint.
Michaël Bages, Patrick Martinez, and Jean-Michel Roquejoffre, Dynamique en grand temps pour une classe d’équations de type KPP en milieu périodique, C. R. Math. Acad. Sci. Paris 346 (2008), no. 19-20, 1051–1056 (French, with English and French summaries). MR 2462047, DOI 10.1016/j.crma.2008.07.028
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, Bernard Larrouturou, and Jean-Michel Roquejoffre, Stabilité des solutions d’ondes progressives d’un modèle de flamme plissée, C. R. Acad. Sci. Paris Sér. I Math. 311 (1990), no. 12, 769–774 (French, with English summary). MR 1082630
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
Maury Bramson, Convergence of solutions of the Kolmogorov equation to travelling waves, Mem. Amer. Math. Soc. 44 (1983), no. 285, iv+190. MR 705746, DOI 10.1090/memo/0285
P. Collet and J.-P. Eckmann, Space-time behaviour in problems of hydrodynamic type: a case study, Nonlinearity 5 (1992), no. 6, 1265–1302. MR 1192518
Peter Constantin, Komla Domelevo, Jean-Michel Roquejoffre, and Lenya Ryzhik, Existence of pulsating waves in a model of flames in sprays, J. Eur. Math. Soc. (JEMS) 8 (2006), no. 4, 555–584. MR 2262195, DOI 10.4171/JEMS/67
Ute Ebert and Wim van Saarloos, Front propagation into unstable states: universal algebraic convergence towards uniformly translating pulled fronts, Phys. D 146 (2000), no. 1-4, 1–99. MR 1787406, DOI 10.1016/S0167-2789(00)00068-3
R. A. Gardner, On the structure of the spectra of periodic travelling waves, J. Math. Pures Appl. (9) 72 (1993), no. 5, 415–439. MR 1239098
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
François Hamel and Nikolaï Nadirashvili, Travelling fronts and entire solutions of the Fisher-KPP equation in ${\Bbb R}^N$, Arch. Ration. Mech. Anal. 157 (2001), no. 2, 91–163. MR 1830037, DOI 10.1007/PL00004238
François Hamel and Lionel Roques, Uniqueness and stability properties of monostable pulsating fronts, J. Eur. Math. Soc. (JEMS) 13 (2011), no. 2, 345–390. MR 2746770, DOI 10.4171/JEMS/256
François Hamel and Lenya Ryzhik, Non-adiabatic KPP fronts with an arbitrary Lewis number, Nonlinearity 18 (2005), no. 6, 2881–2902. MR 2176963, DOI 10.1088/0951-7715/18/6/024
Daniel Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Mathematics, vol. 840, Springer-Verlag, Berlin-New York, 1981. MR 610244
A.N. Kolmogorov, I.G. Petrovskiĭ, N.S. Piskunov, A study of the equation of diffusion with increase in the quantity of matter, and its application to a biological problem, Bjul. Moskowskogo Gos. Univ. 17 (1937), pp. 1–26.
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
P. Martinez, J.-M. Roquejoffre, Rate of attraction of super-critical waves in a Fisher-KPP type equation with shear flow, preprint.
P. Martinez, J.-M. Roquejoffre, Convergence to critical waves in KPP equations, in preparation.
Grégoire Nadin, Traveling fronts in space-time periodic media, J. Math. Pures Appl. (9) 92 (2009), no. 3, 232–262 (English, with English and French summaries). MR 2555178, DOI 10.1016/j.matpur.2009.04.002
J. R. Norris, Long-time behaviour of heat flow: global estimates and exact asymptotics, Arch. Rational Mech. Anal. 140 (1997), no. 2, 161–195. MR 1482931, DOI 10.1007/s002050050063
Jaime H. Ortega and Enrique Zuazua, Large time behavior in $\textbf {R}^N$ for linear parabolic equations with periodic coefficients, Asymptot. Anal. 22 (2000), no. 1, 51–85. MR 1739518
Peter Poláčik and Eiji Yanagida, On bounded and unbounded global solutions of a supercritical semilinear heat equation, Math. Ann. 327 (2003), no. 4, 745–771. MR 2023315, DOI 10.1007/s00208-003-0469-y
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
Jean-Michel Roquejoffre and Violaine Roussier-Michon, Nontrivial large-time behaviour in bistable reaction-diffusion equations, Ann. Mat. Pura Appl. (4) 188 (2009), no. 2, 207–233. MR 2491801, DOI 10.1007/s10231-008-0072-7
L. Rossi, Generalized principal eigenvalue in unbounded domains and applications to nonlinear elliptic and parabolic problems, Ph.D. Thesis, December, 2006.
D. H. Sattinger, On the stability of waves of nonlinear parabolic systems, Advances in Math. 22 (1976), no. 3, 312–355. MR 435602, DOI 10.1016/0001-8708(76)90098-0
H. Bruce Stewart, Generation of analytic semigroups by strongly elliptic operators under general boundary conditions, Trans. Amer. Math. Soc. 259 (1980), no. 1, 299–310. MR 561838, DOI 10.1090/S0002-9947-1980-0561838-5
K\B{o}hei Uchiyama, The behavior of solutions of some nonlinear diffusion equations for large time, J. Math. Kyoto Univ. 18 (1978), no. 3, 453–508. MR 509494, DOI 10.1215/kjm/1250522506
J. L. Vázquez and E. Zuazua, Complexity of large time behaviour of evolution equations with bounded data, Chinese Ann. Math. Ser. B 23 (2002), no. 2, 293–310. Dedicated to the memory of Jacques-Louis Lions. MR 1924144, DOI 10.1142/S0252959902000274
Eiji Yanagida, Irregular behavior of solutions for Fisher’s equation, J. Dynam. Differential Equations 19 (2007), no. 4, 895–914. MR 2357530, DOI 10.1007/s10884-007-9096-8