Pushed traveling fronts in monostable equations with monotone delayed reaction

Abstract

We study the wavefront solutions of the scalar reaction-diffusion equations Delta t(t, x) = Delta u(t, x) - u(t, x) + g(u(t - h, x)); with monotone reaction term g : R+ -> R+ and h > 0. We are mostly interested in the situation when the graph of g is not dominated by its tangent line at zero, i.e. when the condition g(x) <= g'(0)x, x >= 0, is not satisfied. It is well known that, in such a case, a special type of rapidly decreasing wavefronts (pushed fronts) can appear in non-delayed equations (i.e. with h = 0). One of our main goals here is to establish a similar result for h > 0. To this end, we describe the asymptotics of all wavefronts (including critical and non-critical fronts) at -infinity. We also prove the uniqueness of wavefronts (up to a translation). In addition, a new uniqueness result for a class of nonlocal lattice equations is presented.