American Institute of Mathematical Sciences

Advanced
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  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 & Applied Analysis, 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.  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

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.  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

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 Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

[1]

Nicolas Dirr, Hubertus Grillmeier, Günther Grün. On stochastic porous-medium equations with critical-growth conservative multiplicative noise. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020388
[2]

Yan'e Wang, Nana Tian, Hua Nie. Positive solution branches of two-species competition model in open advective environments. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2021006
[3]

Maika Goto, Kazunori Kuwana, Yasuhide Uegata, Shigetoshi Yazaki. A method how to determine parameters arising in a smoldering evolution equation by image segmentation for experiment's movies. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 881-891. doi: 10.3934/dcdss.2020233
[4]

Michal Beneš, Pavel Eichler, Jakub Klinkovský, Miroslav Kolář, Jakub Solovský, Pavel Strachota, Alexandr Žák. Numerical simulation of fluidization for application in oxyfuel combustion. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 769-783. doi: 10.3934/dcdss.2020232
[5]

Jérôme Lohéac, Chaouki N. E. Boultifat, Philippe Chevrel, Mohamed Yagoubi. Exact noise cancellation for 1d-acoustic propagation systems. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020055
[6]

Eduard Marušić-Paloka, Igor Pažanin. Homogenization and singular perturbation in porous media. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020279
[7]

Yangjian Sun, Changjian Liu. The Poincaré bifurcation of a SD oscillator. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1565-1577. doi: 10.3934/dcdsb.2020173
[8]

Gabrielle Nornberg, Delia Schiera, Boyan Sirakov. A priori estimates and multiplicity for systems of elliptic PDE with natural gradient growth. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3857-3881. doi: 10.3934/dcds.2020128
[9]

Yuxi Zheng. Absorption of characteristics by sonic curve of the two-dimensional Euler equations. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 605-616. doi: 10.3934/dcds.2009.23.605
[10]

Takiko Sasaki. Convergence of a blow-up curve for a semilinear wave equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1133-1143. doi: 10.3934/dcdss.2020388
[11]

Nalin Fonseka, Jerome Goddard II, Ratnasingham Shivaji, Byungjae Son. A diffusive weak Allee effect model with U-shaped emigration and matrix hostility. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020356
[12]

Xiyou Cheng, Zhitao Zhang. Structure of positive solutions to a class of Schrödinger systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020461
[13]

Kung-Ching Chang, Xuefeng Wang, Xie Wu. On the spectral theory of positive operators and PDE applications. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3171-3200. doi: 10.3934/dcds.2020054
[14]

Bernold Fiedler. Global Hopf bifurcation in networks with fast feedback cycles. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 177-203. doi: 10.3934/dcdss.2020344
[15]

Riccarda Rossi, Ulisse Stefanelli, Marita Thomas. Rate-independent evolution of sets. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 89-119. doi: 10.3934/dcdss.2020304
[16]

Olivier Pironneau, Alexei Lozinski, Alain Perronnet, Frédéric Hecht. Numerical zoom for multiscale problems with an application to flows through porous media. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 265-280. doi: 10.3934/dcds.2009.23.265
[17]

Xuefei He, Kun Wang, Liwei Xu. Efficient finite difference methods for the nonlinear Helmholtz equation in Kerr medium. Electronic Research Archive, 2020, 28 (4) : 1503-1528. doi: 10.3934/era.2020079
[18]

Juliana Fernandes, Liliane Maia. Blow-up and bounded solutions for a semilinear parabolic problem in a saturable medium. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1297-1318. doi: 10.3934/dcds.2020318
[19]

Josselin Garnier, Knut Sølna. Enhanced Backscattering of a partially coherent field from an anisotropic random lossy medium. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 1171-1195. doi: 10.3934/dcdsb.2020158
[20]

Hong Fu, Mingwu Liu, Bo Chen. Supplier's investment in manufacturer's quality improvement with equity holding. Journal of Industrial & Management Optimization, 2021, 17 (2) : 649-668. doi: 10.3934/jimo.2019127

2019 Impact Factor: 1.105

Tools

Metrics

  • PDF downloads (15)
  • HTML views (64)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences