Slowly oscillating wavefronts of the KPP-Fisher delayed equation

Abstract

This paper concerns the semi-wavefronts (i.e. bounded solutions $u=\varphi (x\nu +ct)>0,$ $|\nu |=1,$ satisfying $\varphi (-\infty )=0$) to the delayed KPP-Fisher equation $$u_t(t,x) = Δu(t,x) + u(t,x)(1-u(t-τ,x)), \ u \geq 0,\ x \in \R^m. \eqno(*)$$ First, we show that each semi-wavefront should be either monotone or slowly oscillating. Then a complete solution to the problem of existence of semi-wavefronts is provided. We prove next that the semi-wavefronts are in fact wavefronts (i.e. additionally $\varphi (+\infty )=1$) if $c\ge 2$ and $\tau \le 1$; our proof uses dynamical properties of some auxiliary one-dimensional map with the negative Schwarzian. The analysis of the fronts' asymptotic expansions at infinity is another key ingredient of our approach. It allows to indicate the maximal domain ${\mathcal{D}}_n$ of $(\tau ,c)$ where the existence of non-monotone wavefronts can be expected. Here we show that the problem of wavefront's existence is closely related to the Wright's global stability conjecture.

25 pages, submitted

References in corpus (4)

• Global continuation of monotone wavefronts
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• A note on the existence of non-monotone non-oscillating wavefronts
• An extension of the Wright's 3/2-theorem for the KPP-Fisher delayed equation
• On the geometric diversity of wavefronts for the scalar Kolmogorov ecological equation
• An iterative estimation for disturbances of semi-wavefronts to the delayed Fisher-KPP equation