S-shaped bifurcation curves for a combustion problem with general arrhenius reaction-rate laws

Chih-Yuan Chen; Shin-Hwa Wang; Kuo-Chih Hung

doi:10.3934/cpaa.2014.13.2589

Article Contents

2014, Volume 13, Issue 6: 2589-2608. Doi: 10.3934/cpaa.2014.13.2589

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S-shaped bifurcation curves for a combustion problem with general arrhenius reaction-rate laws

1.
Department of Mathematics, National Tsing Hua University, Hsinchu 300
2.
Department of Mathematics, National Tsing Hua University, Hsinchu, Taiwan 300
3.
Fundamental General Education Center, National Chin-Yi University of Technology, Taichung 411

Received: May 31, 2012

Revised: April 30, 2014

Published: October 2014

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Abstract

We study the bifurcation curve and exact multiplicity of positive solutions of the combustion problem with general Arrhenius reaction-rate laws \begin{eqnarray} u^{\prime \prime }(x)+\lambda (1+\epsilon u)^{m}e^{\frac{u}{1+\epsilon u}}=0, -1 < x < 1, \\ u(-1)=u(1)=0, \end{eqnarray} where the bifurcation parameters $\lambda, \epsilon >0$ and $-\infty < m <1$. We prove that, for $(-4.103\approx)$ $\tilde{m}\leq m < 1$ for some constant $\tilde{m}$, the bifurcation curve is S-shaped on the $(\lambda, \|u\|_{\infty })$-plane if $0<\epsilon \leq \frac{6}{7}\epsilon _{\text{tr}}^{\text{Sem}}(m)$, where \begin{eqnarray} \epsilon _{\text{tr}}^{\text{Sem}}(m)=\left\{ \begin{array}{l} (\frac{1-\sqrt{1-m}}{m})^{2}\ \text{ for }-\infty < m < 1, m \neq 0, \\ \frac{1}{4}\ \text{for}\ m=0, \end{array}\right. \end{eqnarray} is the Semenov transitional value for general Arrhenius kinetics. In addition, for $-\infty < m < 1$, the bifurcation curve is S-like shaped if $0<\epsilon \leq \frac{8}{9} \epsilon _{\text{tr}}^{\text{Sem}}(m).$ Our results improve and extend those in Wang (Proc. Roy. Soc. London Sect. A, 454 (1998), 1031--1048.)

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Mathematics Subject Classification: 34B18, 74G35.

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