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The speed of propagation for KPP type problems. I: Periodic framework. (English) Zbl 1142.35464

Summary: This paper is devoted to some nonlinear propagation phenomena in periodic and more general domains, for reaction-diffusion equations with Kolmogorov-Petrovsky-Piskunov (KPP) type nonlinearities. The case of periodic domains with periodic underlying excitable media is a follow-up of the first two authors’ article [Commun. Pure Appl. Math. 55, No. 8, 949–1032 (2002; Zbl 1024.37054)]. It is proved that the minimal speed of pulsating fronts is given by a variational formula involving linear eigenvalue problems. Some consequences concerning the influence of the geometry of the domain, of the reaction, advection and diffusion coefficients are given. The last section deals with the notion of asymptotic spreading speed. The main properties of the spreading speed are given. Some of them are based on some new Liouville type results for nonlinear elliptic equations in unbounded domains.

MSC:

35K57 Reaction-diffusion equations
35K55 Nonlinear parabolic equations
35P15 Estimates of eigenvalues in context of PDEs
35B10 Periodic solutions to PDEs
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35B50 Maximum principles in context of PDEs
35K15 Initial value problems for second-order parabolic equations

Citations:

Zbl 1024.37054
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References:

[1] Aronson, D. G., Weinberger, H. F.: Nonlinear diffusion in population genetics, combustion and nerve propagation. In: Partial Differential Equations and Related Topics, Lecture Notes in Math. 446, Springer, New York, 5-49 (1975) · Zbl 0325.35050
[2] Aronson, D. G., Weinberger, H. F.: Multidimensional nonlinear diffusions arising in popula- tion genetics. Adv. Math. 30 , 33-76 (1978) · Zbl 0407.92014 · doi:10.1016/0001-8708(78)90130-5
[3] Audoly, B., Berestycki, H., Pomeau, Y.: Réaction-diffusion en écoulement stationnaire rapide. C. R. Acad. Sci. Paris Sér. II 328 , 255-262 (2000) · Zbl 0992.76097
[4] Bates, P. W., Fife, P. C., Ren, X., Wang, X.: Traveling waves in a convolution model for phase transitions. Arch. Rat. Mech. Anal. 138 , 105-136 (1997) · Zbl 0889.45012 · doi:10.1007/s002050050037
[5] Benguria, R. D., Depassier, M. C.: Variational characterization of the speed of propagation of fronts for the nonlinear diffusion equation. Comm. Math. Phys. 175 , 221-227 (1996) · Zbl 0856.35058 · doi:10.1007/BF02101631
[6] Berestycki, H.: 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, 1-45 (2002)
[7] Berestycki, H., Hamel, F.: Front propagation in periodic excitable media. Comm. Pure Appl. Math. 55 , 949-1032 (2002) · Zbl 1024.37054 · doi:10.1002/cpa.3022
[8] Berestycki, H., Hamel, F.: Gradient estimates for elliptic regularizations of semilinear parabolic and degenerate elliptic equations. Comm. Partial Differential Equations 30 , 139- 156 (2005) · Zbl 1142.35432 · doi:10.1081/PDE-200044478
[9] Berestycki, H., Hamel, F., Nadirashvili, N.: The principal eigenvalue of elliptic operators with large drift and applications to nonlinear propagation phenomena. Comm. Math. Phys. 253 , 451-480 (2005) · Zbl 1123.35033 · doi:10.1007/s00220-004-1201-9
[10] Berestycki, H., Hamel, F., Nadirashvili, N.: The speed of propagation for KPP type problems. I: General domains. Preprint · Zbl 1197.35073 · doi:10.1090/S0894-0347-09-00633-X
[11] Berestycki, H., Hamel, F., Roques, L.: Analysis of the periodically fragmented environment model. I: Influence of periodic heterogeneous environment on species persistence. J. Math. Biology, to appear · Zbl 1066.92047 · doi:10.1007/s00285-004-0313-3
[12] Berestycki, H., Hamel, F., Roques, L.: Analysis of the periodically fragmented environment model. II: Biological invasions and pulsating travelling fronts. J. Math. Pures Appl., to appear (2005) · Zbl 1083.92036 · doi:10.1016/j.matpur.2004.10.006
[13] Berestycki, H., Larrouturou, B., Lions, P.-L.: Multidimensional traveling-wave solutions of a flame propagation model. Arch. Rat. Mech. Anal. 111 , 33-49 (1990) · Zbl 0711.35066 · doi:10.1007/BF00375699
[14] Berestycki, H., Larrouturou, B., Roquejoffre, J.-M.: Stability of traveling fronts in a curved flame model, Part I, Linear analysis. Arch. Rat. Mech. Anal. 117 , 97-117 (1992) · Zbl 0763.76033 · doi:10.1007/BF00387761
[15] Berestycki, H., Nicolaenko, B., Scheurer, B.: Traveling wave solutions to combustion mod- els and their singular limits. SIAM J. Math. Anal. 16 , 1207-1242 (1985) · Zbl 0596.76096 · doi:10.1137/0516088
[16] Berestycki, H., Nirenberg, L.: On the method of moving planes and the sliding method. Bol. Soc. Bras. Mat. 22 , 1-37 (1991) · Zbl 0784.35025 · doi:10.1007/BF01244896
[17] Berestycki, H., Nirenberg, L.: Travelling fronts in cylinders. Ann. Inst. H. Poincaré Anal. Non Linéaire 9 , 497-572 (1992) · Zbl 0799.35073
[18] Berestycki, H., Nirenberg, L., Varadhan, S. R. S.: The principal eigenvalue and maximum principle for second order elliptic operators in general domains. Comm. Pure Appl. Math. 47 , 47-92 (1994) · Zbl 0806.35129 · doi:10.1002/cpa.3160470105
[19] Bonnet, A., Hamel, F.: Existence of non-planar solutions of a simple model of premixed Bun- sen flames. SIAM J. Math. Anal. 31 , 80-118 (1999) · Zbl 0942.35072 · doi:10.1137/S0036141097316391
[20] Bramson, M.: Convergence of solutions of the Kolmogorov equation to travelling waves. Mem. Amer. Math. Soc. 44 (1983) · Zbl 0517.60083
[21] Caffarelli, L. A., Lee, K.-A., Mellet, A.: Singular limit and homogenization for flame propa- gation in periodic excitable media. Preprint · Zbl 1058.76070 · doi:10.1007/s00205-003-0299-9
[22] Capdeboscq, Y.: Homogenization of a neutronic critical diffusion problem with drift. Proc. Roy. Soc. Edinburgh Sect. A 132 , 567-594 (2002) · Zbl 1066.82530 · doi:10.1017/S0308210500001785
[23] Conca, C., Vanninathan, M.: Homogenization of periodic structures via Bloch decomposition. SIAM J. Appl. Math. 57 , 1639-1659 (1997) · Zbl 0990.35019 · doi:10.1137/S0036139995294743
[24] Constantin, P., Kiselev, A., Oberman, A., Ryzhik, L.: Bulk burning rate in passive-reactive diffusion. Arch. Rat. Mech. Anal. 154 , 53-91 (2000) · Zbl 0979.76093 · doi:10.1007/s002050000090
[25] Constantin, P., Kiselev, A., Ryzhik, L.: Quenching of flames by fluid advection. Comm. Pure Appl. Math. 54 , 1320-1342 (2001) · Zbl 1032.35087 · doi:10.1002/cpa.3000
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