Existence of fast positive wavefronts for a non-local delayed reaction–diffusion equation

We establish the existence of a continuous family of fast positive wavefronts u(t,x)=ϕ(x+ct)u(t,x)=ϕ(x+ct), ϕ(−∞)=0ϕ(−∞)=0, ϕ(+∞)=κϕ(+∞)=κ, for the non-local delayed reaction–diffusion equation ut(t,x)=uxx(t,x)−u(t,x)+∫RK(x−w)g(u(t−h,w))dw. Here 0 and κ>0κ>0 are fixed points of g∈C2(R+,R+)g∈C2(R+,R+) and the non-negative KK is such that ∫RK(w)eλwdw is finite for every real λλ. We also prove that the fast wavefronts are non-monotone if g′(κ)heh+1<−1.