Critical exponents, special large-time behavior and oscillatory blow-up in nonlinear ODE's
type: Item
instance of: scholarly article
title: Critical exponents, special large-time behavior and oscillatory blow-up in nonlinear ODE's
zbMATH Open document ID: 1015.34038
author: Philippe Souplet
published in: Differential and Integral Equations
publication date: 2000-05-24T00:00:00Z
review text: The author gives a complete classification of the long-time behavior in the interval \(t\geq 0\) of nontrivial solutions to autonomous second-order ODEs of the form \(u'' + |u|^{p-1}u = |u'|^{q-1}u'\) with exponents \(p, q > 1\). He finds three subdomains of the parameter domain separated by the curves \(q = p\) and \(q = 2p/(p+1)\) with qualitatively different properties. A global solution exists (and is unique up to the sign and a time translation) if and only if \(q\geq p\); all the other solutions blow up in a finite time. The blow-up is oscillatory if and only if \(1<q\leq 2p/(p+1)\), and the blow-up rates are different in each subdomain.
Mathematics Subject Classification ID: 34D05
Mathematics Subject Classification ID: 34C11
zbMATH DE Number: 1452498
zbMATH Keywords: critical exponents
zbMATH Keywords: special large-time behavior
zbMATH Keywords: oscillatory blow-up
zbMATH Keywords: nontrivial solutions
zbMATH Keywords: global solution
reviewed by: Pavel Krejč{í }
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1452498