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

Kuo-Chih Hung; Shin-Hwa Wang

doi:10.3934/cpaa.2020281

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2021, Volume 20, Issue 2: 559-582. Doi: 10.3934/cpaa.2020281

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

Kuo-Chih Hung^1, , and
Shin-Hwa Wang^2,

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
^* Corresponding author

Received: April 30, 2020

Revised: August 31, 2020

Early access: December 2020

Published: February 2021

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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Abstract

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.

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

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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 }. $

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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

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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}] $

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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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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.
[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.
[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.
[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.
[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.
[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.
[7]	K. Scott, The Smouldering of Peat, Ph.D. Dissertation, University of Manchester, England, (2013), 178 pp.
[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.
[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.