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We study the existence and uniqueness of wavefronts to the scalarreaction-diffusion equations $u_{t}(t,x) = \Delta u(t,x) - u(t,x) +g(u(t-h,x)),$ with monotone delayed reaction term $g: \R_+ \to \R_+$ and $h>0$. We are mostly interested in the situation when the graph of $g$ is notdominated by its tangent line at zero, i.e. when the condition $g(x) \leqg'(0)x,$ $x \geq 0$, is not satisfied. It is well known that, in such a case, aspecial type of rapidly decreasing wavefronts (pushed fronts) can appear innon-delayed equations (i.e. with $h=0$). One of our main goals here is toestablish a similar result for $h>0$. We prove the existence of the minimalspeed of propagation, the uniqueness of wavefronts (up to a translation) anddescribe their asymptotics at $-\infty$. We also present a new uniquenessresult for a class of nonlocal lattice equations.

Pushed traveling fronts in monostable equations with monotone delayed reaction