# American Institute of Mathematical Sciences

November  2014, 13(6): 2589-2608. doi: 10.3934/cpaa.2014.13.2589

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

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

Received  June 2012 Revised  May 2014 Published  July 2014

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.)
Citation: Chih-Yuan Chen, Shin-Hwa Wang, Kuo-Chih Hung. S-shaped bifurcation curves for a combustion problem with general arrhenius reaction-rate laws. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2589-2608. doi: 10.3934/cpaa.2014.13.2589
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