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

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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

Chih-Yuan Chen ^1, , Shin-Hwa Wang ^2, and Kuo-Chih Hung ^3,

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

Abstract
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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.)

Keywords: S-shaped bifurcation curve, combustion problem, Semenov transitional value., Frank-Kamenetskii transition value, exact multiplicity, Positive solution.

Mathematics Subject Classification: 34B18, 74G3.

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.

show all references

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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