Behavior of solutions of a nonlinear ordinary differential equation
type: Item
instance of: scholarly article
title: Behavior of solutions of a nonlinear ordinary differential equation
zbMATH Open document ID: 0898.34009
author: Keng Deng
author: Mingxi Xu
published in: Journal of Mathematical Analysis and Applications
publication date: 1998-11-02T00:00:00Z
review text: The authors consider positive solutions to initial value problems \[ (\omega^{-\beta} \omega')'- {y\over\beta +2}\omega' +{1\over \beta+2} \omega=0,\;y>0 \tag{*} \] \[ \omega(0) =\eta,\quad \omega' (0)=1, \tag{**} \] where \(0<\beta <\infty\) and \(\eta\) is a positive parameter. It is shown that the existence and nonexistence of global solutions to (*) and (**) strongly depend on \(\eta\) and the asymptotic behaviour of solutions varies according to different choices of \(\beta\). Such problems arise in the study of the quenching rate for solutions to a singular diffusion equation \[ u_t= (u^{-\beta} u_r)_r +{n-1 \over r} u^{-\beta} u_r, \quad 0<r <R,\;t>0, \] \[ u_r(0,t)=0, \quad u_r(R,t)= -1,\;t>0 \] \[ u(r,0)= u_0(r), \quad 0<r <R. \]
Mathematics Subject Classification ID: 34A34
Mathematics Subject Classification ID: 34C11
Mathematics Subject Classification ID: 35B25
Mathematics Subject Classification ID: 35K57
zbMATH DE Number: 1123523
zbMATH Keywords: positive solutions
zbMATH Keywords: initial value problems
zbMATH Keywords: global solutions
zbMATH Keywords: quenching rate
zbMATH Keywords: singular diffusion equation
MaRDI profile type: Publication
full work available at URL: jmaa.1997.5777
OpenAlex ID: W2088305744
Wikidata QID: Q115395403
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DOI: 10.1006/JMAA.1997.5777
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description: scientific article; zbMATH DE number 1123523