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

Abstract

This paper deals with the stability of semi-wavefronts to the following delay non-local monostable equation: $\dot{v}(t,x)=\Delta v(t,x)-v(t,x)+{\int }_{{\mathbb{R}}^d}K(y)g(v(t-h,x-y))dy,x\in {\mathbb{R}}^d,t>0;$ where $h>0$ and $d\in {\mathbb{Z}}_{+}$. We give two general results for $d\ge 1$: 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.