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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

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 & Continuous Dynamical Systems - A, 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, (1989). doi: 10.1007/978-1-4612-4546-9. Google Scholar

[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. doi: 10.1098/rspa.1979.0140. Google Scholar

[3]

K. J. Brown, M. M. A. Ibrahim and R. Shivaji, S-shaped bifurcation curves,, Nonlinear Anal., 5 (1981), 475. doi: 10.1016/0362-546X(81)90096-1. Google Scholar

[4]

P. M. Cohn, Basic Algebra: Groups, Rings and Fields,, Springer-Verlag, (2003). doi: 10.1007/978-0-85729-428-9. Google Scholar

[5]

Y. Du and Y. Lou, Proof of a conjecture for the perturbed Gelfand equation from combustion theory,, J. Differential Equations, 173 (2001), 213. doi: 10.1006/jdeq.2000.3932. Google Scholar

[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. doi: 10.1016/j.jmaa.2004.02.063. Google Scholar

[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. doi: 10.1016/j.nonrwa.2013.05.005. Google Scholar

[8]

S.-Y. Huang and S.-H. Wang, Tasks in computations., Available from: , (). Google Scholar

[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. doi: 10.1016/j.jde.2011.03.017. Google Scholar

[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. doi: 10.1090/S0002-9947-2012-05670-4. Google Scholar

[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. doi: 10.1137/0136028. Google Scholar

[12]

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

[13]

P. Korman, Y. Li and T. Ouyang, Computing the location and the direction of bifurcation,, Math. Res. Lett., 12 (2005), 933. doi: 10.4310/MRL.2005.v12.n6.a13. Google Scholar

[14]

T. Laetsch, The number of solutions of a nonlinear two point boundary value problem,, Indiana Univ. Math. J., 20 (1970), 1. doi: 10.1512/iumj.1971.20.20001. Google Scholar

[15]

A. Prestel and C. N. Delzell, Positive Polynomials: From Hilbert's 17th Problem to Real Algebra,, Springer-Verlag, (2001). doi: 10.1007/978-3-662-04648-7. Google Scholar

[16]

J. Shi, Persistence and bifurcation of degenerate solutions,, J. Funct. Anal., 169 (1999), 494. doi: 10.1006/jfan.1999.3483. Google Scholar

[17]

R. Shivaji, Remarks on an S-shaped bifurcation curve,, J. Math. Anal. Appl., 111 (1985), 374. doi: 10.1016/0022-247X(85)90223-9. Google Scholar

[18]

Ya. B. Zeldovich, G. I. Barenblatt, V. B. Librovich and G. M. Makhviladze, The Mathematical Theory of Combustion and Explosions,, Consultants Bureau [Plenum], (1985). doi: 10.1007/978-1-4613-2349-5. Google Scholar

[19]

M. Zhang and J. Deng, Number of zeros of interval polynomials,, J. Comput. Appl. Math., 237 (2013), 102. doi: 10.1016/j.cam.2012.07.011. Google Scholar

show all references

References:
[1]

J. Bebernes and D. Eberly, Mathematical Problems from Combustion Theory,, Springer-Verlag, (1989). doi: 10.1007/978-1-4612-4546-9. Google Scholar

[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. doi: 10.1098/rspa.1979.0140. Google Scholar

[3]

K. J. Brown, M. M. A. Ibrahim and R. Shivaji, S-shaped bifurcation curves,, Nonlinear Anal., 5 (1981), 475. doi: 10.1016/0362-546X(81)90096-1. Google Scholar

[4]

P. M. Cohn, Basic Algebra: Groups, Rings and Fields,, Springer-Verlag, (2003). doi: 10.1007/978-0-85729-428-9. Google Scholar

[5]

Y. Du and Y. Lou, Proof of a conjecture for the perturbed Gelfand equation from combustion theory,, J. Differential Equations, 173 (2001), 213. doi: 10.1006/jdeq.2000.3932. Google Scholar

[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. doi: 10.1016/j.jmaa.2004.02.063. Google Scholar

[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. doi: 10.1016/j.nonrwa.2013.05.005. Google Scholar

[8]

S.-Y. Huang and S.-H. Wang, Tasks in computations., Available from: , (). Google Scholar

[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. doi: 10.1016/j.jde.2011.03.017. Google Scholar

[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. doi: 10.1090/S0002-9947-2012-05670-4. Google Scholar

[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. doi: 10.1137/0136028. Google Scholar

[12]

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

[13]

P. Korman, Y. Li and T. Ouyang, Computing the location and the direction of bifurcation,, Math. Res. Lett., 12 (2005), 933. doi: 10.4310/MRL.2005.v12.n6.a13. Google Scholar

[14]

T. Laetsch, The number of solutions of a nonlinear two point boundary value problem,, Indiana Univ. Math. J., 20 (1970), 1. doi: 10.1512/iumj.1971.20.20001. Google Scholar

[15]

A. Prestel and C. N. Delzell, Positive Polynomials: From Hilbert's 17th Problem to Real Algebra,, Springer-Verlag, (2001). doi: 10.1007/978-3-662-04648-7. Google Scholar

[16]

J. Shi, Persistence and bifurcation of degenerate solutions,, J. Funct. Anal., 169 (1999), 494. doi: 10.1006/jfan.1999.3483. Google Scholar

[17]

R. Shivaji, Remarks on an S-shaped bifurcation curve,, J. Math. Anal. Appl., 111 (1985), 374. doi: 10.1016/0022-247X(85)90223-9. Google Scholar

[18]

Ya. B. Zeldovich, G. I. Barenblatt, V. B. Librovich and G. M. Makhviladze, The Mathematical Theory of Combustion and Explosions,, Consultants Bureau [Plenum], (1985). doi: 10.1007/978-1-4613-2349-5. Google Scholar

[19]

M. Zhang and J. Deng, Number of zeros of interval polynomials,, J. Comput. Appl. Math., 237 (2013), 102. doi: 10.1016/j.cam.2012.07.011. Google Scholar

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