On S-shaped bifurcation curves for a two-point boundary value problem arising in a theory of thermal explosion

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October 2015, 35(10): 4839-4858. doi: 10.3934/dcds.2015.35.4839

On S-shaped bifurcation curves for a two-point boundary value problem arising in a theory of thermal explosion

Shao-Yuan Huang ^1, and Shin-Hwa Wang ^1,

1.	Department of Mathematics, National Tsing Hua University, Hsinchu 300, Taiwan, Taiwan

Received January 2014 Revised February 2015 Published April 2015

Abstract
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We study the bifurcation curve and exact multiplicity of positive solutions of a two-point boundary value problem arising in a theory of thermal explosion \begin{equation*} \left\{ \begin{array}{l} u^{\prime\prime}(x) + \lambda \exp ( \frac{au}{a+u}) =0, -1 < x < 1, \\ u(-1)=u(1)=0, \end{array} \right. \end{equation*} where $\lambda >0$ is the Frank--Kamenetskii parameter and $a>0$ is the activation energy parameter. By developing some new time-map techniques and applying Sturm's theorem, we prove that, if $a\geq a^{\ast \ast }\approx 4.107$, the bifurcation curve is S-shaped on the $(\lambda ,\Vert u \Vert _{\infty })$-plane. Our result improves one of the main results in Hung and Wang (J. Differential Equations 251 (2011) 223--237).

Keywords: turning point, Positive solution, S-shaped bifurcation curve, exact multiplicity, Sturm's theorem..

Mathematics Subject Classification: 34B18, 74G3.

Citation: 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 and Continuous Dynamical Systems, 2015, 35 (10) : 4839-4858. doi: 10.3934/dcds.2015.35.4839

References:

[1]	J. Bebernes and D. Eberly, Mathematical Problems from Combustion Theory, Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4612-4546-9.
[2]	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. Ser. A Math. Phys. Eng. Sci., 368 (1979), 441-461. doi: 10.1098/rspa.1979.0140.
[3]	K. J. Brown, M. M. A. Ibrahim and R. Shivaji, S-shaped bifurcation curves, Nonlinear Anal., 5 (1981), 475-486. doi: 10.1016/0362-546X(81)90096-1.
[4]	P. M. Cohn, Basic Algebra: Groups, Rings and Fields, Springer-Verlag, London, 2003. doi: 10.1007/978-0-85729-428-9.
[5]	Y. Du and Y. Lou, Proof of a conjecture for the perturbed Gelfand equation from combustion theory, J. Differential Equations, 173 (2001), 213-230. doi: 10.1006/jdeq.2000.3932.
[6]	J. Forde and P. Nelson, Applications of Sturm sequences to bifurcation analysis of delay differential equation models, J. Math. Anal. Appl., 300 (2004), 273-284. doi: 10.1016/j.jmaa.2004.02.063.
[7]	P. V. Gordon, E. Ko and R. Shivaji, Multiplicity and uniqueness of positive solutions for elliptic equations with nonlinear boundary conditions arising in a theory of thermal explosion, Nonlinear Anal. Real World Appl., 15 (2014), 51-57. doi: 10.1016/j.nonrwa.2013.05.005.
[8]	S.-Y. Huang and S.-H. Wang, Tasks in computations. Available from: http://oz.nthu.edu.tw/~d9621801/works.htm.
[9]	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.
[10]	K.-C. Hung and S.-H. Wang, Global bifurcation and exact multiplicity of positive solutions for a positone problem with cubic nonlinearity and their applications, Trans. Amer. Math. Soc., 365 (2013), 1933-1956. doi: 10.1090/S0002-9947-2012-05670-4.
[11]	A. K. Kapila and B. J. Matkowsky, Reactive-diffuse systems with Arrhenius kinetics: Multiple solutions, ignition and extinction, SIAM J. Appl. Math., 36 (1979), 373-389. doi: 10.1137/0136028.
[12]	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.
[13]	P. Korman, Y. Li and T. Ouyang, Computing the location and the direction of bifurcation, Math. Res. Lett., 12 (2005), 933-944. doi: 10.4310/MRL.2005.v12.n6.a13.
[14]	T. Laetsch, The number of solutions of a nonlinear two point boundary value problem, Indiana Univ. Math. J., 20 (1970), 1-13. doi: 10.1512/iumj.1971.20.20001.
[15]	A. Prestel and C. N. Delzell, Positive Polynomials: From Hilbert's 17th Problem to Real Algebra, Springer-Verlag, Berlin, 2001. doi: 10.1007/978-3-662-04648-7.
[16]	J. Shi, Persistence and bifurcation of degenerate solutions, J. Funct. Anal., 169 (1999), 494-531. doi: 10.1006/jfan.1999.3483.
[17]	R. Shivaji, Remarks on an S-shaped bifurcation curve, J. Math. Anal. Appl., 111 (1985), 374-387. doi: 10.1016/0022-247X(85)90223-9.
[18]	Ya. B. Zeldovich, G. I. Barenblatt, V. B. Librovich and G. M. Makhviladze, The Mathematical Theory of Combustion and Explosions, Consultants Bureau [Plenum], New York, 1985. doi: 10.1007/978-1-4613-2349-5.
[19]	M. Zhang and J. Deng, Number of zeros of interval polynomials, J. Comput. Appl. Math., 237 (2013), 102-110. doi: 10.1016/j.cam.2012.07.011.

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References:

[1]	J. Bebernes and D. Eberly, Mathematical Problems from Combustion Theory, Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4612-4546-9.
[2]	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. Ser. A Math. Phys. Eng. Sci., 368 (1979), 441-461. doi: 10.1098/rspa.1979.0140.
[3]	K. J. Brown, M. M. A. Ibrahim and R. Shivaji, S-shaped bifurcation curves, Nonlinear Anal., 5 (1981), 475-486. doi: 10.1016/0362-546X(81)90096-1.
[4]	P. M. Cohn, Basic Algebra: Groups, Rings and Fields, Springer-Verlag, London, 2003. doi: 10.1007/978-0-85729-428-9.
[5]	Y. Du and Y. Lou, Proof of a conjecture for the perturbed Gelfand equation from combustion theory, J. Differential Equations, 173 (2001), 213-230. doi: 10.1006/jdeq.2000.3932.
[6]	J. Forde and P. Nelson, Applications of Sturm sequences to bifurcation analysis of delay differential equation models, J. Math. Anal. Appl., 300 (2004), 273-284. doi: 10.1016/j.jmaa.2004.02.063.
[7]	P. V. Gordon, E. Ko and R. Shivaji, Multiplicity and uniqueness of positive solutions for elliptic equations with nonlinear boundary conditions arising in a theory of thermal explosion, Nonlinear Anal. Real World Appl., 15 (2014), 51-57. doi: 10.1016/j.nonrwa.2013.05.005.
[8]	S.-Y. Huang and S.-H. Wang, Tasks in computations. Available from: http://oz.nthu.edu.tw/~d9621801/works.htm.
[9]	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.
[10]	K.-C. Hung and S.-H. Wang, Global bifurcation and exact multiplicity of positive solutions for a positone problem with cubic nonlinearity and their applications, Trans. Amer. Math. Soc., 365 (2013), 1933-1956. doi: 10.1090/S0002-9947-2012-05670-4.
[11]	A. K. Kapila and B. J. Matkowsky, Reactive-diffuse systems with Arrhenius kinetics: Multiple solutions, ignition and extinction, SIAM J. Appl. Math., 36 (1979), 373-389. doi: 10.1137/0136028.
[12]	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.
[13]	P. Korman, Y. Li and T. Ouyang, Computing the location and the direction of bifurcation, Math. Res. Lett., 12 (2005), 933-944. doi: 10.4310/MRL.2005.v12.n6.a13.
[14]	T. Laetsch, The number of solutions of a nonlinear two point boundary value problem, Indiana Univ. Math. J., 20 (1970), 1-13. doi: 10.1512/iumj.1971.20.20001.
[15]	A. Prestel and C. N. Delzell, Positive Polynomials: From Hilbert's 17th Problem to Real Algebra, Springer-Verlag, Berlin, 2001. doi: 10.1007/978-3-662-04648-7.
[16]	J. Shi, Persistence and bifurcation of degenerate solutions, J. Funct. Anal., 169 (1999), 494-531. doi: 10.1006/jfan.1999.3483.
[17]	R. Shivaji, Remarks on an S-shaped bifurcation curve, J. Math. Anal. Appl., 111 (1985), 374-387. doi: 10.1016/0022-247X(85)90223-9.
[18]	Ya. B. Zeldovich, G. I. Barenblatt, V. B. Librovich and G. M. Makhviladze, The Mathematical Theory of Combustion and Explosions, Consultants Bureau [Plenum], New York, 1985. doi: 10.1007/978-1-4613-2349-5.
[19]	M. Zhang and J. Deng, Number of zeros of interval polynomials, J. Comput. Appl. Math., 237 (2013), 102-110. doi: 10.1016/j.cam.2012.07.011.

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