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Classification and evolution of bifurcation curves for a porous-medium combustion problem with large activation energy

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This work is partially supported by the Ministry of Science and Technology of the Republic of China under grant No. MOST 103-2115-M-167-002

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  • We study the classification and evolution of bifurcation curves for the porous-medium combustion problem

    $ \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*} $

    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.

    Mathematics Subject Classification: Primary: 34B18, 74G35.

    Citation:

    \begin{equation} \\ \end{equation}
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  • Figure 1.  Four different types of S-shaped bifurcation curves $ S $ of (1.1). (i). Type 1: $ \lambda _{\ast }< \lambda ^{\ast }<\bar{ \lambda} = \infty . $ (ii). Type 2: $ \lambda _{\ast }< \lambda ^{\ast }<\bar{ \lambda}<\infty . $ (iii). Type 3: $ \lambda _{\ast }<\bar{ \lambda} = \lambda ^{\ast }. $ (iv). Type 4: $ \lambda _{\ast }<\bar{ \lambda}< \lambda ^{\ast }. $

    Figure 2.  Classification of bifurcation curves $ S $ for (1.1) with $ d>0 $ and $ a\geq 0 $. $ d_{3} $ $ (\approx 1.170)<d_{2} $ $ (\approx 1.401) $ $ <d_{1} $ $ (\approx 2.225). $ The bifurcation curves $ S $ for the region bounded between curves $ A_{4}(d) $, $ A_{5}(d) $ and $ A_{1}(d) $ are all S-shaped

    Figure 3.  Graph of $ H_{d, a}(u) $ with $ H_{d, a}(u_{0})\leq 0 $ for some $ u_{0}\in (0, \gamma _{d, a}] $

    Figure 4.  Graphs of functions $ A_{4}(d) $ and $ A_{5}(d) $ for $ 0<d\leq d_{3} $ $ (\approx 1.170). $

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