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: ˙ v(t,x) = ∆v(t,x) − v(t,x) +R Rd K(y)g(v(t−h,x−y))dy,x ∈Rd, t > 0; where h > 0 and d ∈Z+. We give two general results for d ≥ 1: on the global stability of semi-wavefronts in Lpspaces with unbounded weights and the local stability of planar wavefronts in Lp-spaces with bounded weights. We also give a global stability result for d = 1 which yields to the global stability in Sobolev spaces with bounded weights. Here g is not assumed to be monotone and the kernel K is not assumed to be symmetric, therefore non-monotone semi-wavefronts and backward semiwavefronts appear for which we show their stability. In particular, the global stability of critical wavefronts is stated.