American Institute of Mathematical Sciences

Advanced
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

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 & 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,, \emph{Combust. Flame}, 48 (1982), 303.   Google Scholar

[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,, \emph{Proc. R. Soc. Lond. A}, 390 (1983), 247.   Google Scholar

[3]

T. Boddington, C.-G. Feng and P. Gray, Thermal explosion and the theory of its initiation by steady intense light,, \emph{Proc. R. Soc. Lond. A}, 390 (1983), 265.   Google Scholar

[4]

T. Boddington, P. Gray and C. Robinson, Thermal explosion and the disappearance of criticality at small activation energies: exact results for the slab,, \emph{Proc. R. Soc. Lond. A}, 368 (1979), 441.   Google Scholar

[5]

M. G. Crandall and P. H. Rabinowitz, Bifurcation, perturbation of simple eigenvalues and linearized stability,, \emph{Arch. Rational Mech. Anal.}, 52 (1973), 161.   Google Scholar

[6]

Y. Du, Exact multiplicity and S-shaped bifurcation curve for some semilinear elliptic problems from combustion theory,, \emph{SIAM J. Math. Anal.}, 32 (2000), 707.  doi: 10.1137/S0036141098343586.  Google Scholar

[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,, \emph{J. Differential Equations}, 251 (2011), 223.  doi: 10.1016/j.jde.2011.03.017.  Google Scholar

[8]

P. Korman and Y. Li, On the exactness of an S-shaped bifurcation curve,, \emph{Proc. Amer. Math. Soc.}, 127 (1999), 1011.  doi: 10.1090/S0002-9939-99-04928-X.  Google Scholar

[9]

G. P. Miller, The structure of a stoichiometric CCI4-CH4-air flat flame,, \emph{Combust. Flame}, 101 (1995), 101.   Google Scholar

[10]

M. Mimura and K. Sakamoto, Multi-dimensional transition layers for an exothermic reaction-diffusion system in long cylindrical domains,, \emph{J. Math. Sci. Univ. Tokyo}, 3 (1996), 109.   Google Scholar

[11]

A. L. Sánchez, A. Liñán and F. A. Williams, Chain-branching explosions in mixing layers,, \emph{SIAM J. Appl. Math.}, 59 (1999), 1335.  doi: 10.1137/S003613999732648X.  Google Scholar

[12]

K. Taira, Semilinear elliptic boundary-value problems in combustion theory,, \emph{Proc. Roy. Soc. Edinburgh Sect. A}, 132 (2002), 1453.   Google Scholar

[13]

S.-H. Wang, On S-shaped bifurcation curves,, \emph{Nonlinear Anal.}, 22 (1994), 1475.  doi: 10.1016/0362-546X(94)90183-X.  Google Scholar

[14]

S.-H. Wang, Rigorous analysis and estimates of S-shaped bifurcation curves in a combustion problem with general Arrhenius reaction-rate laws,, \emph{Proc. R. Soc. Lond. A}, 454 (1998), 1031.   Google Scholar

show all references

References:
[1]

T. Boddington, C.-G. Feng and P. Gray, Disappearance of criticality in thermal explosion under Frank-Kamenetskii boundary conditions,, \emph{Combust. Flame}, 48 (1982), 303.   Google Scholar

[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,, \emph{Proc. R. Soc. Lond. A}, 390 (1983), 247.   Google Scholar

[3]

T. Boddington, C.-G. Feng and P. Gray, Thermal explosion and the theory of its initiation by steady intense light,, \emph{Proc. R. Soc. Lond. A}, 390 (1983), 265.   Google Scholar

[4]

T. Boddington, P. Gray and C. Robinson, Thermal explosion and the disappearance of criticality at small activation energies: exact results for the slab,, \emph{Proc. R. Soc. Lond. A}, 368 (1979), 441.   Google Scholar

[5]

M. G. Crandall and P. H. Rabinowitz, Bifurcation, perturbation of simple eigenvalues and linearized stability,, \emph{Arch. Rational Mech. Anal.}, 52 (1973), 161.   Google Scholar

[6]

Y. Du, Exact multiplicity and S-shaped bifurcation curve for some semilinear elliptic problems from combustion theory,, \emph{SIAM J. Math. Anal.}, 32 (2000), 707.  doi: 10.1137/S0036141098343586.  Google Scholar

[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,, \emph{J. Differential Equations}, 251 (2011), 223.  doi: 10.1016/j.jde.2011.03.017.  Google Scholar

[8]

P. Korman and Y. Li, On the exactness of an S-shaped bifurcation curve,, \emph{Proc. Amer. Math. Soc.}, 127 (1999), 1011.  doi: 10.1090/S0002-9939-99-04928-X.  Google Scholar

[9]

G. P. Miller, The structure of a stoichiometric CCI4-CH4-air flat flame,, \emph{Combust. Flame}, 101 (1995), 101.   Google Scholar

[10]

M. Mimura and K. Sakamoto, Multi-dimensional transition layers for an exothermic reaction-diffusion system in long cylindrical domains,, \emph{J. Math. Sci. Univ. Tokyo}, 3 (1996), 109.   Google Scholar

[11]

A. L. Sánchez, A. Liñán and F. A. Williams, Chain-branching explosions in mixing layers,, \emph{SIAM J. Appl. Math.}, 59 (1999), 1335.  doi: 10.1137/S003613999732648X.  Google Scholar

[12]

K. Taira, Semilinear elliptic boundary-value problems in combustion theory,, \emph{Proc. Roy. Soc. Edinburgh Sect. A}, 132 (2002), 1453.   Google Scholar

[13]

S.-H. Wang, On S-shaped bifurcation curves,, \emph{Nonlinear Anal.}, 22 (1994), 1475.  doi: 10.1016/0362-546X(94)90183-X.  Google Scholar

[14]

S.-H. Wang, Rigorous analysis and estimates of S-shaped bifurcation curves in a combustion problem with general Arrhenius reaction-rate laws,, \emph{Proc. R. Soc. Lond. A}, 454 (1998), 1031.   Google Scholar

[1]

Shao-Yuan Huang, Shin-Hwa Wang. On S-shaped bifurcation curves for a two-point boundary value problem arising in a theory of thermal explosion. Discrete & Continuous Dynamical Systems - A, 2015, 35 (10) : 4839-4858. doi: 10.3934/dcds.2015.35.4839
[2]

Tzung-shin Yeh. S-shaped and broken s-shaped bifurcation curves for a multiparameter diffusive logistic problem with holling type-Ⅲ functional response. Communications on Pure & Applied Analysis, 2017, 16 (2) : 645-670. doi: 10.3934/cpaa.2017032
[3]

Sabri Bensid, Jesús Ildefonso Díaz. Stability results for discontinuous nonlinear elliptic and parabolic problems with a S-shaped bifurcation branch of stationary solutions. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1757-1778. doi: 10.3934/dcdsb.2017105
[4]

Shao-Yuan Huang. Exact multiplicity and bifurcation curves of positive solutions of a one-dimensional Minkowski-curvature problem and its application. Communications on Pure & Applied Analysis, 2018, 17 (3) : 1271-1294. doi: 10.3934/cpaa.2018061
[5]

Mauro Garavello. Boundary value problem for a phase transition model. Networks & Heterogeneous Media, 2016, 11 (1) : 89-105. doi: 10.3934/nhm.2016.11.89
[6]

Shao-Yuan Huang. Global bifurcation and exact multiplicity of positive solutions for the one-dimensional Minkowski-curvature problem with sign-changing nonlinearity. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3267-3284. doi: 10.3934/cpaa.2019147
[7]

Xue Dong He, Roy Kouwenberg, Xun Yu Zhou. Inverse S-shaped probability weighting and its impact on investment. Mathematical Control & Related Fields, 2018, 8 (3&4) : 679-706. doi: 10.3934/mcrf.2018029
[8]

Patricio Cerda, Leonelo Iturriaga, Sebastián Lorca, Pedro Ubilla. Positive radial solutions of a nonlinear boundary value problem. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1765-1783. doi: 10.3934/cpaa.2018084
[9]

Francesca Marcellini. Existence of solutions to a boundary value problem for a phase transition traffic model. Networks & Heterogeneous Media, 2017, 12 (2) : 259-275. doi: 10.3934/nhm.2017011
[10]

Chan-Gyun Kim, Yong-Hoon Lee. A bifurcation result for two point boundary value problem with a strong singularity. Conference Publications, 2011, 2011 (Special) : 834-843. doi: 10.3934/proc.2011.2011.834
[11]

Michael E. Filippakis, Nikolaos S. Papageorgiou. Existence and multiplicity of positive solutions for nonlinear boundary value problems driven by the scalar $p$-Laplacian. Communications on Pure & Applied Analysis, 2004, 3 (4) : 729-756. doi: 10.3934/cpaa.2004.3.729
[12]

Piotr Kowalski. The existence of a solution for Dirichlet boundary value problem for a Duffing type differential inclusion. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2569-2580. doi: 10.3934/dcdsb.2014.19.2569
[13]

Shaoyong Lai, Yong Hong Wu, Xu Yang. The global solution of an initial boundary value problem for the damped Boussinesq equation. Communications on Pure & Applied Analysis, 2004, 3 (2) : 319-328. doi: 10.3934/cpaa.2004.3.319
[14]

John R. Graef, Bo Yang. Multiple positive solutions to a three point third order boundary value problem. Conference Publications, 2005, 2005 (Special) : 337-344. doi: 10.3934/proc.2005.2005.337
[15]

John R. Graef, Lingju Kong, Bo Yang. Positive solutions of a nonlinear higher order boundary-value problem. Conference Publications, 2009, 2009 (Special) : 276-285. doi: 10.3934/proc.2009.2009.276
[16]

John R. Graef, Bo Yang. Positive solutions of a third order nonlocal boundary value problem. Discrete & Continuous Dynamical Systems - S, 2008, 1 (1) : 89-97. doi: 10.3934/dcdss.2008.1.89
[17]

Guglielmo Feltrin. Existence of positive solutions of a superlinear boundary value problem with indefinite weight. Conference Publications, 2015, 2015 (special) : 436-445. doi: 10.3934/proc.2015.0436
[18]

Christina A. Hollon, Jeffrey T. Neugebauer. Positive solutions of a fractional boundary value problem with a fractional derivative boundary condition. Conference Publications, 2015, 2015 (special) : 615-620. doi: 10.3934/proc.2015.0615
[19]

John R. Graef, Johnny Henderson, Bo Yang. Positive solutions to a fourth order three point boundary value problem. Conference Publications, 2009, 2009 (Special) : 269-275. doi: 10.3934/proc.2009.2009.269
[20]

Eric R. Kaufmann. Existence and nonexistence of positive solutions for a nonlinear fractional boundary value problem. Conference Publications, 2009, 2009 (Special) : 416-423. doi: 10.3934/proc.2009.2009.416

2019 Impact Factor: 1.105

Tools

Metrics

  • PDF downloads (17)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences