The speed of propagation for KPP type problems. I: Periodic framework

Abstract

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 article [7]. 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.

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

“…Propagation phenomena in a homogeneous framework are well understood, and we will recall below the main results. This article is the second in a series of two, and it is the follow-up of the article [7] (part I). Both papers deal with heterogeneous problems.…”

“…In other words, the presence of holes or of an undulating boundary always hinder the progression or the spreading. Moreover, we proved in [7] that the speeds c * (e) are not in general monotone with respect to the size of the perforations. The inequality w * (e) ≤ c * (e) always works.…”

“…The inequality w * (e) ≤ c * (e) always works. The equality w * (e) = c * (e) (= 2 f (0)) holds in the homogeneous framework (1.1) in R N , but the inequality w * (e) ≤ c * (e) may be strict in general (see Remark 1.12 in [7]). …”

“…Both the coefficients of the equation, namely the diffusion matrix A(x), the drift q(x) and the reaction term f (x, s), as well as the geometry of the underlying connected open set Ω, were and w * (e) = 2 f (0) if and only if Ω is invariant in the direction e (straight cylinder in the direction e, with bounded or unbounded section). Notice that this geometrical condition is also necessary for the equality c * (e) = 2 f (0) to hold (see [7]). In other words, the presence of holes or of an undulating boundary always hinder the progression or the spreading.…”

“…Namely, if u 0 is a nonnegative continuous function in R N with compact support and u 0 ≡ 0, then the solution u(t, x) of (1.1) with initial condition u 0 at time t = 0 spreads with the speed c * in all directions for large times: as t → +∞, max |x|≤ct |u(t, x) − 1| → 0 for each c ∈ [0, c * ) and max |x|≥ct u(t, x) → 0 for each c > c * . In Part I of [7] and in an earlier paper [4], we introduced a general heterogeneous periodic framework extending (1.1). The types of equations which were considered there were (1.3) u t − ∇ · (A(x)∇u) + q(x) · ∇u = f (x, u) in Ω, ν · A∇u = 0 on ∂Ω,…”