Asymptotic convergence to pushed wavefronts in a monostable equation with delayed reaction

TL;DR

This paper proves the nonlinear stability and asymptotic convergence of pushed traveling wavefronts in a delayed monostable reaction-diffusion equation, establishing sharp spreading speed results.

Contribution

It demonstrates the nonlinear stability of pushed wavefronts with asymptotic phase and their role in attracting solutions with rapidly decaying initial data.

Findings

Pushed wavefronts are nonlinearly stable with asymptotic phase.

Solutions with rapid decay are attracted to these wavefronts.

The results provide a sharp characterization of spreading speeds.

Abstract

We study the asymptotic behaviour of solutions to the delayed monostable equation $(*)$: $u_{t}(t,x) = u_{xx}(t,x) - u(t,x) + g(u(t-h,x)),$ $x \in R,\ t >0,$ with monotone reaction term $g: R_+ \to R_+$. Our basic assumption is that this equation possesses pushed traveling fronts. First we prove that the pushed wavefronts are nonlinearly stable with asymptotic phase. Moreover, combinations of these waves attract, uniformly on $R$, every solution of equation $(*)$ with the initial datum sufficiently rapidly decaying at one (or at the both) infinities of the real line. These results provide a sharp form of the theory of spreading speeds for equation $(*)$.