Propagation and its failure in a lattice delayed differential equation with global interaction
type: Item
instance of: scholarly article
title: Propagation and its failure in a lattice delayed differential equation with global interaction
zbMATH Open document ID: 1086.34064
author: Shiwang Ma
author: Xingfu Zou
published in: Journal of Differential Equations
publication date: 2005-05-04T00:00:00Z
review text: The authors study the following lattice delayed differential equation with global interaction \[ u_n^{\prime}=D[u_{n+1}+u_{n-1}-2u_n]-du_n+\sum\limits_{i=-\infty}^{\infty}J(i)b(u_{n-i}(t-r)), n=1,2,\dots \tag{1} \] Equation (1) describes the evolution of a single-species in one-dimensional patchy environment with infinite number of patches connected locally by diffusion. It is assumed in equation (1) that \(J(i)=J(-i)\geq 0, \sum_{i}J(i)=1\) and \(\sum_{i}| i| J(i)<+\infty,\) where \(\sum_{i}\) denotes the sum over \(i\in \mathbb{Z}\). It is also assumed that the birth function \(b\in C^1(\mathbb{R})\) and there exists a constant \(K>0\) such that \(b(0)=dK-b(K)=0.\) In the bistable case, under realistic assumptions on the birth function, it is proved that equation (1) admits a strictly monotone increasing travelling wave front. Moreover, if the wave speed does not vanish, then the wave front is unique (up to a translation) and globally asymptotically stable with phase shift. Of particular interest is the phenomenon of ``propagation failure'' or ``pinning'' (i.e., wave speed \(c=0\)), some criteria for pinning are given in this paper, too.
reviewed by: Rui Xu
Mathematics Subject Classification ID: 34K30
Mathematics Subject Classification ID: 34K20
zbMATH DE Number: 2164826
zbMATH Keywords: lattice delayed differential equation
zbMATH Keywords: global interaction
zbMATH Keywords: traveling wave front
zbMATH Keywords: existence
zbMATH Keywords: uniqueness
zbMATH Keywords: asymptotic stability
zbMATH Keywords: propagation failure
zbMATH Keywords: pinning
MaRDI profile type: Publication
full work available at URL: j.jde.2004.07.014
OpenAlex ID: W2164949167
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DOI: 10.1016/J.JDE.2004.07.014
description: scientific article; zbMATH DE number 2164826