# American Institute of Mathematical Sciences

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

 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).
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. [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. [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. [4] P. M. Cohn, Basic Algebra: Groups, Rings and Fields,, Springer-Verlag, (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. 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. 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. doi: 10.1016/j.nonrwa.2013.05.005. [8] S.-Y. Huang and S.-H. Wang, Tasks in computations., Available from: , (). [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. [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. [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. [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. [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. [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. [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. [16] J. Shi, Persistence and bifurcation of degenerate solutions,, J. Funct. Anal., 169 (1999), 494. doi: 10.1006/jfan.1999.3483. [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. [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. [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.

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. [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. [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. [4] P. M. Cohn, Basic Algebra: Groups, Rings and Fields,, Springer-Verlag, (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. 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. 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. doi: 10.1016/j.nonrwa.2013.05.005. [8] S.-Y. Huang and S.-H. Wang, Tasks in computations., Available from: , (). [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. [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. [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. [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. [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. [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. [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. [16] J. Shi, Persistence and bifurcation of degenerate solutions,, J. Funct. Anal., 169 (1999), 494. doi: 10.1006/jfan.1999.3483. [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. [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. [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.
 [1] 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 [2] 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 [3] 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 [4] 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 [5] 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 [6] Jagmohan Tyagi, Ram Baran Verma. Positive solution to extremal Pucci's equations with singular and gradient nonlinearity. Discrete & Continuous Dynamical Systems - A, 2019, 39 (5) : 2637-2659. doi: 10.3934/dcds.2019110 [7] 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 [8] Rabah Amir, Igor V. Evstigneev. On Zermelo's theorem. Journal of Dynamics & Games, 2017, 4 (3) : 191-194. doi: 10.3934/jdg.2017011 [9] John Hubbard, Yulij Ilyashenko. A proof of Kolmogorov's theorem. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 367-385. doi: 10.3934/dcds.2004.10.367 [10] Bernold Fiedler, Carlos Rocha, Matthias Wolfrum. Sturm global attractors for $S^1$-equivariant parabolic equations. Networks & Heterogeneous Media, 2012, 7 (4) : 617-659. doi: 10.3934/nhm.2012.7.617 [11] Hahng-Yun Chu, Se-Hyun Ku, Jong-Suh Park. Conley's theorem for dispersive systems. Discrete & Continuous Dynamical Systems - S, 2015, 8 (2) : 313-321. doi: 10.3934/dcdss.2015.8.313 [12] Sergei Ivanov. On Helly's theorem in geodesic spaces. Electronic Research Announcements, 2014, 21: 109-112. doi: 10.3934/era.2014.21.109 [13] Olga Kharlampovich and Alexei Myasnikov. Tarski's problem about the elementary theory of free groups has a positive solution. Electronic Research Announcements, 1998, 4: 101-108. [14] Christopher Bose, Rua Murray. The exact rate of approximation in Ulam's method. Discrete & Continuous Dynamical Systems - A, 2001, 7 (1) : 219-235. doi: 10.3934/dcds.2001.7.219 [15] Paweł Góra, Abraham Boyarsky. Stochastic perturbations and Ulam's method for W-shaped maps. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 1937-1944. doi: 10.3934/dcds.2013.33.1937 [16] V. Niţicâ. Journé's theorem for $C^{n,\omega}$ regularity. Discrete & Continuous Dynamical Systems - A, 2008, 22 (1&2) : 413-425. doi: 10.3934/dcds.2008.22.413 [17] Jacques Féjoz. On "Arnold's theorem" on the stability of the solar system. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3555-3565. doi: 10.3934/dcds.2013.33.3555 [18] Dmitry Kleinbock, Barak Weiss. Dirichlet's theorem on diophantine approximation and homogeneous flows. Journal of Modern Dynamics, 2008, 2 (1) : 43-62. doi: 10.3934/jmd.2008.2.43 [19] Lena Noethen, Sebastian Walcher. Tikhonov's theorem and quasi-steady state. Discrete & Continuous Dynamical Systems - B, 2011, 16 (3) : 945-961. doi: 10.3934/dcdsb.2011.16.945 [20] Fatiha Alabau-Boussouira, Piermarco Cannarsa. A constructive proof of Gibson's stability theorem. Discrete & Continuous Dynamical Systems - S, 2013, 6 (3) : 611-617. doi: 10.3934/dcdss.2013.6.611

2017 Impact Factor: 1.179