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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.

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
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