# 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 and Applied Analysis, 2014, 13 (6) : 2589-2608. doi: 10.3934/cpaa.2014.13.2589
##### References:
 [1] T. Boddington, C.-G. Feng and P. Gray, Disappearance of criticality in thermal explosion under Frank-Kamenetskii boundary conditions, Combust. Flame, 48 (1982), 303-304. [2] T. Boddington, C.-G. Feng and P. Gray, Thermal explosion, criticality and the disappearance of criticality in systems with distributed temperatures. I. Arbitrary Biot number and general reaction-rate laws, Proc. R. Soc. Lond. A, 390 (1983), 247-264. [3] T. Boddington, C.-G. Feng and P. Gray, Thermal explosion and the theory of its initiation by steady intense light, Proc. R. Soc. Lond. A, 390 (1983), 265-281. [4] T. Boddington, P. Gray and C. Robinson, Thermal explosion and the disappearance of criticality at small activation energies: exact results for the slab, Proc. R. Soc. Lond. A, 368 (1979), 441-461. [5] M. G. Crandall and P. H. Rabinowitz, Bifurcation, perturbation of simple eigenvalues and linearized stability, Arch. Rational Mech. Anal., 52 (1973), 161-180. [6] Y. Du, Exact multiplicity and S-shaped bifurcation curve for some semilinear elliptic problems from combustion theory, SIAM J. Math. Anal., 32 (2000), 707-733. doi: 10.1137/S0036141098343586. [7] K.-C. Hung and S.-H. Wang, A theorem on S-shaped bifurcation curve for a positone problem with convex-concave nonlinearity and its applications to the perturbed Gelfand problem, J. Differential Equations, 251 (2011), 223-237. doi: 10.1016/j.jde.2011.03.017. [8] P. Korman and Y. Li, On the exactness of an S-shaped bifurcation curve, Proc. Amer. Math. Soc., 127 (1999), 1011-1020. doi: 10.1090/S0002-9939-99-04928-X. [9] G. P. Miller, The structure of a stoichiometric CCI4-CH4-air flat flame, Combust. Flame, 101 (1995), 101-112. [10] M. Mimura and K. Sakamoto, Multi-dimensional transition layers for an exothermic reaction-diffusion system in long cylindrical domains, J. Math. Sci. Univ. Tokyo, 3 (1996), 109-179. [11] A. L. Sánchez, A. Liñán and F. A. Williams, Chain-branching explosions in mixing layers, SIAM J. Appl. Math., 59 (1999), 1335-1355. doi: 10.1137/S003613999732648X. [12] K. Taira, Semilinear elliptic boundary-value problems in combustion theory, Proc. Roy. Soc. Edinburgh Sect. A, 132 (2002), 1453-1476. [13] S.-H. Wang, On S-shaped bifurcation curves, Nonlinear Anal., 22 (1994), 1475-1485. doi: 10.1016/0362-546X(94)90183-X. [14] S.-H. Wang, Rigorous analysis and estimates of S-shaped bifurcation curves in a combustion problem with general Arrhenius reaction-rate laws, Proc. R. Soc. Lond. A, 454 (1998), 1031-1048.

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##### References:
 [1] T. Boddington, C.-G. Feng and P. Gray, Disappearance of criticality in thermal explosion under Frank-Kamenetskii boundary conditions, Combust. Flame, 48 (1982), 303-304. [2] T. Boddington, C.-G. Feng and P. Gray, Thermal explosion, criticality and the disappearance of criticality in systems with distributed temperatures. I. Arbitrary Biot number and general reaction-rate laws, Proc. R. Soc. Lond. A, 390 (1983), 247-264. [3] T. Boddington, C.-G. Feng and P. Gray, Thermal explosion and the theory of its initiation by steady intense light, Proc. R. Soc. Lond. A, 390 (1983), 265-281. [4] T. Boddington, P. Gray and C. Robinson, Thermal explosion and the disappearance of criticality at small activation energies: exact results for the slab, Proc. R. Soc. Lond. A, 368 (1979), 441-461. [5] M. G. Crandall and P. H. Rabinowitz, Bifurcation, perturbation of simple eigenvalues and linearized stability, Arch. Rational Mech. Anal., 52 (1973), 161-180. [6] Y. Du, Exact multiplicity and S-shaped bifurcation curve for some semilinear elliptic problems from combustion theory, SIAM J. Math. Anal., 32 (2000), 707-733. doi: 10.1137/S0036141098343586. [7] K.-C. Hung and S.-H. Wang, A theorem on S-shaped bifurcation curve for a positone problem with convex-concave nonlinearity and its applications to the perturbed Gelfand problem, J. Differential Equations, 251 (2011), 223-237. doi: 10.1016/j.jde.2011.03.017. [8] P. Korman and Y. Li, On the exactness of an S-shaped bifurcation curve, Proc. Amer. Math. Soc., 127 (1999), 1011-1020. doi: 10.1090/S0002-9939-99-04928-X. [9] G. P. Miller, The structure of a stoichiometric CCI4-CH4-air flat flame, Combust. Flame, 101 (1995), 101-112. [10] M. Mimura and K. Sakamoto, Multi-dimensional transition layers for an exothermic reaction-diffusion system in long cylindrical domains, J. Math. Sci. Univ. Tokyo, 3 (1996), 109-179. [11] A. L. Sánchez, A. Liñán and F. A. Williams, Chain-branching explosions in mixing layers, SIAM J. Appl. Math., 59 (1999), 1335-1355. doi: 10.1137/S003613999732648X. [12] K. Taira, Semilinear elliptic boundary-value problems in combustion theory, Proc. Roy. Soc. Edinburgh Sect. A, 132 (2002), 1453-1476. [13] S.-H. Wang, On S-shaped bifurcation curves, Nonlinear Anal., 22 (1994), 1475-1485. doi: 10.1016/0362-546X(94)90183-X. [14] S.-H. Wang, Rigorous analysis and estimates of S-shaped bifurcation curves in a combustion problem with general Arrhenius reaction-rate laws, Proc. R. Soc. Lond. A, 454 (1998), 1031-1048.
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