# American Institute of Mathematical Sciences

September  2014, 34(9): 3511-3533. doi: 10.3934/dcds.2014.34.3511

## Slowly oscillating wavefronts of the KPP-Fisher delayed equation

 1 Mathematical Institute, Silesian University, 746 01 Opava, Czech Republic 2 Instituto de Matemática y Física, Universidad de Talca, Casilla 747, Talca

Received  June 2013 Revised  November 2013 Published  March 2014

This paper concerns the semi-wavefronts (i.e. bounded solutions $u=\phi(x\cdot\nu +ct) >0,$ $|\nu|=1,$ satisfying $\phi(-\infty)=0$) to the delayed KPP-Fisher equation $u_t(t,x) = \Delta u(t,x) + u(t,x)(1-u(t-\tau,x)), \ u \geq 0,\ x \in \mathbb{R}^m.$ 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 $\phi(+\infty)=1$) if $c \geq 2$ and $\tau \leq 1$; our proof uses dynamical properties of an auxiliary one-dimensional map with the negative Schwarzian. However, we also show that, for $c \geq 2$ and $\tau \geq 1.87$, each semi-wavefront profile $\phi(t)$ should develop non-decaying oscillations around $1$ as $t \to +\infty$.
Citation: Karel Hasik, Sergei Trofimchuk. Slowly oscillating wavefronts of the KPP-Fisher delayed equation. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3511-3533. doi: 10.3934/dcds.2014.34.3511
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