Singular perturbations and dead core behavior
type: Item
instance of: scholarly article
title: Singular perturbations and dead core behavior
zbMATH Open document ID: 0619.34058
DOI: 10.1016/0022-247X(87)90030-8
author: L. E. Bobisud
published in: Journal of Mathematical Analysis and Applications
publication date: 1987-01-01T00:00:00Z
review text: Let u(x) be a smooth function satisfying u''(x)\(\geq 0\) on [0,1] and consider the Dirichlet problem \[ (*)\quad \epsilon y''=f(y-u(x)),\quad x\in (0,1),\quad y(0)\geq u(0),y(1)\geq u(1); \] here \(\epsilon\) is a small positive parameter, \(f(0)=0\), and f is continuous and increasing. It is shown that there exists a unique solution \(y_{\epsilon}\) of (*), and \(y_{\epsilon}(x)\to u(x)\) uniformly on compact subsets of (0,1). If additionally \(u''=0\), necessary and sufficient conditions are given for \(y_{\epsilon}(x)\) to coincide with u(x) for x in some compact subset \([p_{\epsilon},q_{\epsilon}]\subset [0,1]\), and estimates on \(p_{\epsilon}\) and \(1-q_{\epsilon}\) are given. Generalizations to equations of more general form \(\epsilon y''=f(x,y)\) are given, as are generalizations to less smooth u.
Mathematics Subject Classification ID: 34E15
zbMATH DE Number: 4004685
zbMATH Keywords: singular perturbation
zbMATH Keywords: dead core
zbMATH Keywords: angular behavior
zbMATH Keywords: second order differential equation
zbMATH Keywords: Dirichlet problem
zbMATH Keywords: small positive parameter
MaRDI profile type: Publication
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full work available at URL: 0022-247x(87)90030-8
OpenAlex ID: W2083554234
description: scientific article; zbMATH DE number 4004685