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Article Contents

# 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

• 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:

• 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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