Abstract

This paper concerns the semi-wavefronts (i.e. bounded solutions u = phi(x.v+ct) >0, -v- = 1, satisfying phi(-infinity) = 0) to the delayed KPP-Fisher equation u(t)(t, x) = x) u(t, x)(1-u(t -tau,x)), u >= 0, x is an element of R-m First, we show that the profile phi of each semi-wavefront should be either monotone or eventually sine-like slowly oscillating around the positive equilibrium. Then a solution to the problem of existence of semi-wavefronts is provided. Next, we prove that the semi-wavefronts are in fact wavefronts (i.e. additionally phi(+infinity) = 1) if c >= 2 and tau <= 1; our proof uses dynamical properties of an auxiliary one-dimensional map with the negative Schwarzian. However, we also show that, for c >= 2 and tau >= 1.87, each semi-wavefront profile phi(t) should develop non-decaying oscillations around 1 as t ->+infinity.