Pushed traveling fronts in monostable  equations withmonotone delayed reaction

Elena Trofimchuk; Manuel Pinto; Sergei Trofimchuk

doi:10.3934/dcds.2013.33.2169

Article Contents

2013, Volume 33, Issue 5: 2169-2187. Doi: 10.3934/dcds.2013.33.2169

This issue Previous Article Non-degeneracy and uniqueness of periodic solutions for $2n$-order differential equations Next Article Application of the subharmonic Melnikov method to piecewise-smooth systems

Pushed traveling fronts in monostable equations with monotone delayed reaction

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

Received: November 2011

Revised: August 2012

Published: May 2013

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Abstract

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.

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Mathematics Subject Classification: Primary: 34K12, 35K31; Secondary: 92D25.

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