American Institute of Mathematical Sciences

doi: 10.3934/cpaa.2020281

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

 1 Fundamental General Education Center, National Chin-Yi University of Technology, Taichung 411, Taiwan 2 Department of Mathematics, National Tsing Hua University, Hsinchu 300, Taiwan

* Corresponding author

Received  May 2020 Revised  September 2020 Published  December 2020

Fund Project: 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 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
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.
Citation: Kuo-Chih Hung, Shin-Hwa Wang. Classification and evolution of bifurcation curves for a porous-medium combustion problem with large activation energy. Communications on Pure & Applied Analysis, doi: 10.3934/cpaa.2020281
References:
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References:
 [1] A. Friedman and A. E. Tzavaras, Combustion in a porous medium, SIAM J. Math. Anal., 19 (1988), 509–519. doi: 10.1137/0519036.  Google Scholar [2] 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. Differ. Equ., 251 (2011), 223–237. doi: 10.1016/j.jde.2011.03.017.  Google Scholar [3] 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.1970.20.20001.  Google Scholar [4] P. Nistri, Positive solutions of a nonlinear eigenvalue problem with discontinuous nonlinearity, Proc. Roy. Soc. Edinburgh Sect. A, 83 (1979), 133–145. doi: 10.1017/S0308210500011458.  Google Scholar [5] J. Norbury and A. M. Stuart, Parabolic free boundary problems arising in porous medium combustion, IMA J. Appl. Math., 39 (1987), 241–257. doi: 10.1093/imamat/39.3.241.  Google Scholar [6] J. Norbury and A. M. Stuart, A model for porous-medium combustion, Quart. J. Mech. Appl. Math., 42 (1989), 159–178. doi: 10.1093/qjmam/42.1.159.  Google Scholar [7] K. Scott, The Smouldering of Peat, Ph.D. Dissertation, University of Manchester, England, (2013), 178 pp. Google Scholar [8] S. H. Wang, Bifurcation of an equation arising in porous-medium combustion, IMA J. Appl. Math., 56 (1996), 219–234. doi: 10.1093/imamat/56.3.219.  Google Scholar [9] S. H. Wang and T. S. Yeh, A complete classification of bifurcation diagrams of a Dirichlet problem with concave-convex nonlinearities, J. Math. Anal. Appl., 291 (2004), 128–153. doi: 10.1016/j.jmaa.2003.10.021.  Google Scholar
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: $\lambda _{\ast }<\bar{ \lambda}< \lambda ^{\ast }.$
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
Graph of $H_{d, a}(u)$ with $H_{d, a}(u_{0})\leq 0$ for some $u_{0}\in (0, \gamma _{d, a}]$
Graphs of functions $A_{4}(d)$ and $A_{5}(d)$ for $0<d\leq d_{3}$ $(\approx 1.170).$
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