Admissible speeds of transition fronts for non-autonomous monostable equations

TL;DR

This paper investigates the possible speeds of transition fronts in a time-dependent Fisher-KPP reaction-diffusion equation, characterizing their asymptotic behavior and profiles as time approaches infinity or negative infinity.

Contribution

It establishes the set of admissible asymptotic speeds for non-autonomous monostable equations and describes the profiles of non-critical fronts at infinity.

Findings

Characterization of admissible asymptotic speeds.

Existence of two asymptotic speeds for non-critical fronts.

Description of asymptotic profiles as t approaches ±∞.

Abstract

We consider a reaction-diffusion equation with a nonlinear term of the Fisher-KPP type, depending on time $t$ and admitting two limits as $t\to\pm\infty$. We derive the set of admissible asymptotic past and future speeds of transition fronts for such equation. We further show that any transition front which is non-critical as $t\to-\infty$ always admits two asymptotic past and future speeds. We finally describe the asymptotic profiles of the non-critical fronts as $t\to\pm\infty$.