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February  2021, 20(2): 559-582. doi: 10.3934/cpaa.2020281

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

Received  May 2020 Revised  September 2020 Published  February 2021 Early access  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<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.
Keywords: Porous-medium combustion, positive solution, exact multiplicity, evolution, S-shaped bifurcation curve.
Mathematics Subject Classification: Primary: 34B18, 74G35.
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 and Applied Analysis, 2021, 20 (2) : 559-582. doi: 10.3934/cpaa.2020281
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.

show all references

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.

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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