We study the classification and evolution of bifurcation curves for the porous-medium combustion problem \begin{document}$ \begin{equation*} \begin{cases} u^{\prime \prime }(x)+\lambda \dfrac{1+au}{1+e^{d(1-u)}} = 0, \ -1<x<1, \\ u(-1) = u(1) = 0, \end{cases} \end{equation*} $\end{document} where $ u $ is the solid temperature, parameters $ \lambda >0 $, $ a\geq 0 $, and the activation energy parameter $ d>0 $ is large. We mainly prove that, on the $ (\lambda , ||u||_{\infty }) $-plane, the bifurcation curve is S-shaped with exactly two turning points for any$ \ (d, a)\in \Omega \equiv \left \{ (d, a):(0<d<d_{1}, \text{ }a\geq A_{1}(d))\text{ or }(d\geq d_{1}, \text{ }a\geq 0)\right \} $ for some positive number $ d_{1}\approx 2.225 $ and a nonnegative, strictly decreasing function $ A_{1}(d) $ defined on $ (0, d_{1}]. $ Furthermore, for any$ \ (d, a)\in \Omega , $ we give a classification and evolution of totally four different S-shaped bifurcation curves. In addition, for any $ d>0 $ and $ a\geq \tilde{a}\approx 1.704 $ for some positive $ \tilde{a}, $ then the bifurcation curve $ S $ is type 4 S-shaped on the $ (\lambda , \left \Vert u\right \Vert _{\infty }) $-plane.
Read full abstract