Pulsating travelling fronts: Asymptotics and homogenization regimes

Abstract

This paper is concerned with some non-linear propagation phenomena for reaction–advection–diffusion equations with Kolmogrov–Petrovsky–Piskunov (KPP)-type non-linearities in general periodic domains or in infinite cylinders with oscillating boundaries. Having a variational formula for the minimal speed of propagation involving eigenvalue problems (proved in Berestycki, H., Hamel, F. & Nadirashvili, N. (2005) The speed of propagation for KPP type problems (periodic framework). J. Eur. Math. Soc. 7, 173–213), we consider the minimal speed of propagation as a function of diffusion factors, reaction factors and periodicity parameters. There we study the limits, the asymptotic behaviours and the variations of the considered functions with respect to these parameters. One of the sections deals with homogenization problem as an application of the results in the previous sections in order to find the limit of the minimal speed when the periodicity cell is very small.