Pushed traveling fronts in monostable equations with monotone delayed reaction

TL;DR

This paper investigates the existence, uniqueness, and asymptotic behavior of pushed traveling wavefronts in monostable reaction-diffusion equations with monotone delayed reactions, extending known results to cases with delay and nonlocal lattice equations.

Contribution

It establishes the existence of pushed fronts for delayed equations when the reaction term's graph is not dominated by its tangent at zero, and proves their uniqueness and asymptotics.

Findings

Existence of minimal propagation speed for pushed fronts.

Uniqueness of wavefronts up to translation.

Asymptotic description of wavefronts at negative infinity.

Abstract

We study the existence and uniqueness of wavefronts to the scalar reaction-diffusion equations $u_{t}(t,x) = \Delta u(t,x) - u(t,x) + g(u(t-h,x)),$ with monotone delayed reaction term $g: \R_+ \to \R_+$ and $h >0$. We are mostly interested in the situation when the graph of $g$ is not dominated by its tangent line at zero, i.e. when the condition $g(x) \leq g'(0)x,$ $x \geq 0$, is not satisfied. It is well known that, in such a case, a special type of rapidly decreasing wavefronts (pushed fronts) can appear in non-delayed equations (i.e. with $h=0$). One of our main goals here is to establish a similar result for $h>0$. We prove the existence of the minimal speed of propagation, the uniqueness of wavefronts (up to a translation) and describe their asymptotics at $-\infty$. We also present a new uniqueness result for a class of nonlocal lattice equations.