Existence of a double S-shaped bifurcation curve with six positive solutions for a combustion problem
type: Item
instance of: scholarly article
title: Existence of a double S-shaped bifurcation curve with six positive solutions for a combustion problem
zbMATH Open document ID: 1315.34035
author: Kuo-Chih Hung
author: Shin-Hwa Wang
author: Chien-Hsien Yu
published in: Journal of Mathematical Analysis and Applications
publication date: 2012-06-08T00:00:00Z
reviewed by: Bo Yang
Mathematics Subject Classification ID: 34B18
Mathematics Subject Classification ID: 34B15
Mathematics Subject Classification ID: 34B60
Mathematics Subject Classification ID: 34C23
Mathematics Subject Classification ID: 80A25
zbMATH DE Number: 6043603
zbMATH Keywords: bifurcation
zbMATH Keywords: multiplicity
zbMATH Keywords: positive solution
zbMATH Keywords: combustion problem
zbMATH Keywords: double S-shaped bifurcation curve
zbMATH Keywords: nonlinear boundary condition
zbMATH Keywords: time map
MaRDI profile type: Publication
full work available at URL: j.jmaa.2012.02.036
OpenAlex ID: W1965861163
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DOI: 10.1016/J.JMAA.2012.02.036
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review text: The authors study the bifurcation curve of positive solutions of the boundary value problem NEWLINE\[NEWLINE -u''(x) = \lambda \exp\left(\frac{\beta u}{\beta+u}\right), \quad 0<x<1, NEWLINE\]NEWLINE NEWLINE\[NEWLINE u(0)=0, \quad \frac{u(1)}{u(1)+1} u'(1) + \left[1-\frac{u(1)}{u(1)+1}\right] u(1)=0, NEWLINE\]NEWLINE where \(\lambda,\beta>0\) are parameters. They prove that, for \(\beta>\beta_1\approx 6.459\) for some constant \(\beta_1\), the bifurcation curve is double S-shaped on the \((\lambda,\|u\|_\infty)\)-plane and the problem has at least six positive solutions for a certain range of positive \(\lambda\). They also provide proofs for some other bifurcation phenomena of the problem.
description: scientific article; zbMATH DE number 6043603