Abstract
We study global bifurcation curves and the exact multiplicity of positive solutions for the two-point boundary value problem arising in combustion theory:
where \({\lambda > 0}\) is the Frank–Kamenetskii parameter and a > 0 is the activation energy parameter. We prove that there exists a critical bifurcation value a 0 \({\approx 4.069}\) such that, on the \({(\lambda,||u||_{\infty })}\)-plane, the bifurcation curve is S-shaped for \({a > a_{0}}\) and is monotone increasing for \({0 < a \leqq a_{0}}\). That is, we prove the long-standing conjecture for the one-dimensional perturbed Gelfand problem. We also study, in the \({(a,\lambda, \left \Vert u\right\Vert _{\infty})}\)-space, the shape and structure of the bifurcation surface.
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Communicated by P. Rabinowitz
This work is partially supported by the National Science Council of the Republic of China under Grant No. NSC 101-2115-M-007-001-MY2.
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Huang, SY., Wang, SH. Proof of a Conjecture for the One-Dimensional Perturbed Gelfand Problem from Combustion Theory. Arch Rational Mech Anal 222, 769–825 (2016). https://doi.org/10.1007/s00205-016-1011-1
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DOI: https://doi.org/10.1007/s00205-016-1011-1