Stability of non-monotone and backward waves for delay non-local reaction-diffusion equations

TL;DR

This paper investigates the stability of semi-wavefronts and backward waves in delay non-local reaction-diffusion equations, providing new stability results for non-monotone and non-symmetric cases across various function spaces.

Contribution

It introduces novel stability results for non-monotone semi-wavefronts and backward waves in delay non-local equations, including global stability in Sobolev spaces.

Findings

Global stability of semi-wavefronts in unbounded weighted spaces

Local stability of planar wavefronts in bounded weighted spaces

Global stability of critical wavefronts in Sobolev spaces

Abstract

This paper deals with the stability of semi-wavefronts to the following delay non-local monostable equation: $\dot{v}(t,x) = \Delta v(t,x) - v(t,x) + \int_{\R^d}K(y)g(v(t-h,x-y))dy, x \in \R^d,\ t >0;$ where $h>0$ and $d\in\Z_+$. We give two general results for $d\geq1$: on the global stability of semi-wavefronts in $L^p$-spaces with unbounded weights and the local stability of planar wavefronts in $L^p$-spaces with bounded weights. We also give a global stability result for $d=1$ which includes the global stability on Sobolev spaces. Here $g$ is not assumed to be monotone and the kernel $K$ is not assumed to be symmetric, therefore non-monotone semi-wavefronts and {\it backward traveling fronts} appear for which we show their stability. In particular, the global stability of critical wavefronts is stated.