Slowly oscillating wavefronts of the KPP-Fisher delayed equation

This paper concerns the semi-wavefronts (i.e. bounded solutions $u=ϕ(x ν+ct) >0,$ $|ν|=1,$ satisfying $ϕ(-\infty)=0$) 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 $ϕ(+\infty)=1$) if $c \geq 2$ and $τ\leq 1$; 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 $(τ,c)$ 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.

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