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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 |
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. |
show all references
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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