Abstract

This paper deals with the stability of semi-wavefronts to the follRowing delay non-local monostable equation: (v) over dot(t, x) = Delta v(t, x) - v(t, x) + integral(Rd) K(y)g(v(t - h, x - y))dy, x is an element of R-d, t > 0; where h > 0 and d is an element of Z(+). We give two general results for d >= 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 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.