# American Institute of Mathematical Sciences

May  2013, 33(5): 2169-2187. doi: 10.3934/dcds.2013.33.2169

## Pushed traveling fronts in monostable equations with monotone delayed reaction

 1 Department of Differential Equations, National Technical University, Kyiv, Ukraine 2 Departamento de Matemáticas, Facultad de Ciencias, Universidad de Chile, Casilla 653, Santiago, Chile 3 Instituto de Matemática y Física, Universidad de Talca, Casilla 747, Talca

Received  November 2011 Revised  August 2012 Published  December 2012

We study the wavefront solutions of the scalar reaction-diffusion equations $u_{t}(t,x) = \Delta u(t,x) - u(t,x) + g(u(t-h,x)),$ with monotone reaction term $g: \mathbb{R}_{+} → \mathbb{R}_+$ 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 $g(x) \leq g'(0)x,$ $x \geq 0$, 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 $-\infty$. 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.
Citation: Elena Trofimchuk, Manuel Pinto, Sergei Trofimchuk. Pushed traveling fronts in monostable equations with monotone delayed reaction. Discrete & Continuous Dynamical Systems, 2013, 33 (5) : 2169-2187. doi: 10.3934/dcds.2013.33.2169
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