Figure 1.
ⅰ) monotone increasing. (ⅱ) $ \subset $ -shaped. (ⅲ) S-shaped
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June
2019, 39(6): 3443-3462.
doi: 10.3934/dcds.2019142
Bifurcation diagrams of positive solutions for one-dimensional Minkowski-curvature problem and its applications
Center for General Education, Ming Chi University of Technology, New Taipei City 24301, Taiwan |
In this paper, we study the classification and evolution of bifurcation curves of positive solutions for one-dimensional Minkowski-curvature problem
| $\left\{ \begin{array}{*{35}{l}} \begin{align} & -{{\left( {{u}^{\prime }}/\sqrt{1-{{u}^{\prime }}^{2}} \right)}^{\prime }}=\lambda f(u),\ \text{in }\left( -L,L \right), \\ & u(-L)=u(L)=0, \\ \end{align} \\\end{array} \right. $ |
where
| $ \lambda >0 $ |
| $ L>0 $ |
| $ f\in C[0, \infty )\cap C^{2}(0, \infty ) $ |
| $ \beta >0 $ |
| $ \left( \beta -z\right) f(z)>0 $ |
| $ z\neq \beta $ |
| $ S_{L} $ |
| $ L>0 $ |
| $ f(u)/u $ |
| $ \subset $ |
| $ L>0 $ |
| $ f(u)/u $ |
Mathematics Subject Classification:
34B15, 34B18, 34C23, 74G35.
Citation:
Shao-Yuan Huang. Bifurcation diagrams of positive solutions for one-dimensional Minkowski-curvature problem and its applications. Discrete and Continuous Dynamical Systems,
2019, 39
(6)
: 3443-3462.
doi: 10.3934/dcds.2019142
![]()
References:
| [1] |
R. Bartnik and L. Simon,
Spacelike hypersurfaces with prescribed boundary values and mean curvature, Comm. Math. Phys., 87 (1982), 131-152.
doi: 10.1007/BF01211061. |
| [2] |
D. Butler, R. Shivaji and A. Tuck, S-shaped bifurcation curves for logistic growth and weak Allee effect growth models with grazing on an interior patch, Proceedings of the Ninth MSU-UAB Conference on Differential Equations and Computational Simulations, 15–25, Electron. J. Differ. Equ. Conf., 20, Texas State Univ., San Marcos, TX, 2013. |
| [3] |
I. Coelho, C. Corsato, F. Obersnel and P. Omari,
Positive solutions of the Dirichlet problem for the one-dimensional Minkowski-curvature equation, Adv. Nonlinear Stud., 12 (2012), 621-638.
doi: 10.1515/ans-2012-0310. |
| [4] |
C. Corsato, Mathematical analysis of some differential models involving the Euclidean or the Minkowski mean curvature operator, PhD thesis, University of Trieste, 2015. Available at https://www.openstarts.units.it/bitstream/10077/11127/1/PhD_Thesis _Corsato.pdf. |
| [5] |
R. P. Feynman, R. B. Leighton and M. Sands, The Feynman Lectures on Physics, Vol. 2: Mainly electromagnetism and matter. Addison-Wesley Publishing Co., Inc., Reading, Mass.-London, 1964. |
| [6] |
S.-Y. Huang,
Classification and evolution of bifurcation curves for the one-dimensional Minkowski-curvature problem and its applications, J. Differential Equations, 264 (2018), 5977-6011.
doi: 10.1016/j.jde.2018.01.021. |
| [7] |
S.-Y. Huang,
Exact multiplicity and bifurcation curves of positive solutions of a one-dimensional Minkowski-curvature problem and its application, Commun. Pure Appl. Anal., 17 (2018), 1271-1294.
doi: 10.3934/cpaa.2018061. |
| [8] |
S.-Y. Huang and S.-H. Wang,
An evolutionary property of the bifurcation curves for a positone problem with cubic nonlinearity, Taiwanese J. Math., 20 (2016), 639-661.
doi: 10.11650/tjm.20.2016.6563. |
| [9] |
S.-Y. Huang, Some proofs in a paper "Bifurcation diagrams of one-dimensional Minkowski-curvature problem and its applications", available from http://mx.nthu.edu.tw/~symbol126sy-huang/Proofs2018. |
| [10] |
K.-C. Hung, S.-Y. Huang and S.-H. Wang,
A global bifurcation theorem for a positone multiparameter problem and its application, Discrete Contin. Dyn. Syst., 37 (2017), 5127-5149.
doi: 10.3934/dcds.2017222. |
| [11] |
K.-C. Hung and S.-H. Wang,
Global bifurcation and exact multiplicity of positive solutions for a positone problem with cubic nonlinearity and their applications, Trans. Amer. Math. Soc., 365 (2013), 1933-1956.
doi: 10.1090/S0002-9947-2012-05670-4. |
| [12] |
E. Lee, S. Sasi and R. Shivaji,
S-shaped bifurcation curves in ecosystems, J. Math. Anal. Appl., 381 (2011), 732-741.
doi: 10.1016/j.jmaa.2011.03.048. |
| [13] |
R. Ma and Y. Lu,
Multiplicity of positive solutions for second order nonlinear Dirichlet problem with one-dimension Minkowski-curvature operator, Adv. Nonlinear Stud., 15 (2015), 789-803.
doi: 10.1515/ans-2015-0403. |
| [14] |
P. M. McCabe, J. A. Leach and D. J. Needham,
The evolution of travelling waves in fractional order autocatalysis with decay. Ⅰ. Permanent from travelling waves, SIAM J. Appl. Math., 59 (1998), 870-899.
doi: 10.1137/S0036139996312594. |
| [15] |
T. Ouyang and J. Shi,
Exact multiplicity of positive solutions for a class of semilinear problems, Journal of Differential Equations, 146 (1998), 121-156.
doi: 10.1006/jdeq.1998.3414. |
| [16] |
E. Poole, B. Roberson and B. Stephenson,
Weak Allee effect, grazing, and S-shaped bifurcation curves, Involve, 5 (2012), 133-158.
doi: 10.2140/involve.2012.5.133. |
| [17] |
J. Shi and R. Shivaji,
Persistence in reaction diffusion models with weak Allee effect, J. Math. Biol., 52 (2006), 807-829.
doi: 10.1007/s00285-006-0373-7. |
| [18] |
C.-C. Tzeng, K.-C. Hung and S.-H. Wang,
Global bifurcation and exact multiplicity of positive solutions for a positone problem with cubic nonlinearity, J. Differential Equations, 252 (2012), 6250-6274.
doi: 10.1016/j.jde.2012.02.020. |
| [19] |
M.-H. Wang and M. Kot,
Speeds of invasion in a model with strong or weak Allee effects, Math. Biosci., 171 (2001), 83-97.
doi: 10.1016/S0025-5564(01)00048-7. |
| [20] |
J. Xin,
Front propagation in heterogeneous media, SIAM Rev., 42 (2000), 161-230.
doi: 10.1137/S0036144599364296. |
| [21] |
T.-S. Yeh,
S-shaped and broken S-shaped bifurcation curves for a multiparameter diffusive logistic problem with Holling type-Ⅲ functional response, Commun. Pure Appl. Anal., 16 (2017), 645-670.
doi: 10.3934/cpaa.2017032. |
| [22] |
T.-S. Yeh,
Bifurcation curves of positive steady-state solutions for a reaction-diffusion problem of lake eutrophication, J. Math. Anal. Appl., 449 (2017), 1708-1724.
doi: 10.1016/j.jmaa.2016.12.063. |
show all references
References:
| [1] |
R. Bartnik and L. Simon,
Spacelike hypersurfaces with prescribed boundary values and mean curvature, Comm. Math. Phys., 87 (1982), 131-152.
doi: 10.1007/BF01211061. |
| [2] |
D. Butler, R. Shivaji and A. Tuck, S-shaped bifurcation curves for logistic growth and weak Allee effect growth models with grazing on an interior patch, Proceedings of the Ninth MSU-UAB Conference on Differential Equations and Computational Simulations, 15–25, Electron. J. Differ. Equ. Conf., 20, Texas State Univ., San Marcos, TX, 2013. |
| [3] |
I. Coelho, C. Corsato, F. Obersnel and P. Omari,
Positive solutions of the Dirichlet problem for the one-dimensional Minkowski-curvature equation, Adv. Nonlinear Stud., 12 (2012), 621-638.
doi: 10.1515/ans-2012-0310. |
| [4] |
C. Corsato, Mathematical analysis of some differential models involving the Euclidean or the Minkowski mean curvature operator, PhD thesis, University of Trieste, 2015. Available at https://www.openstarts.units.it/bitstream/10077/11127/1/PhD_Thesis _Corsato.pdf. |
| [5] |
R. P. Feynman, R. B. Leighton and M. Sands, The Feynman Lectures on Physics, Vol. 2: Mainly electromagnetism and matter. Addison-Wesley Publishing Co., Inc., Reading, Mass.-London, 1964. |
| [6] |
S.-Y. Huang,
Classification and evolution of bifurcation curves for the one-dimensional Minkowski-curvature problem and its applications, J. Differential Equations, 264 (2018), 5977-6011.
doi: 10.1016/j.jde.2018.01.021. |
| [7] |
S.-Y. Huang,
Exact multiplicity and bifurcation curves of positive solutions of a one-dimensional Minkowski-curvature problem and its application, Commun. Pure Appl. Anal., 17 (2018), 1271-1294.
doi: 10.3934/cpaa.2018061. |
| [8] |
S.-Y. Huang and S.-H. Wang,
An evolutionary property of the bifurcation curves for a positone problem with cubic nonlinearity, Taiwanese J. Math., 20 (2016), 639-661.
doi: 10.11650/tjm.20.2016.6563. |
| [9] |
S.-Y. Huang, Some proofs in a paper "Bifurcation diagrams of one-dimensional Minkowski-curvature problem and its applications", available from http://mx.nthu.edu.tw/~symbol126sy-huang/Proofs2018. |
| [10] |
K.-C. Hung, S.-Y. Huang and S.-H. Wang,
A global bifurcation theorem for a positone multiparameter problem and its application, Discrete Contin. Dyn. Syst., 37 (2017), 5127-5149.
doi: 10.3934/dcds.2017222. |
| [11] |
K.-C. Hung and S.-H. Wang,
Global bifurcation and exact multiplicity of positive solutions for a positone problem with cubic nonlinearity and their applications, Trans. Amer. Math. Soc., 365 (2013), 1933-1956.
doi: 10.1090/S0002-9947-2012-05670-4. |
| [12] |
E. Lee, S. Sasi and R. Shivaji,
S-shaped bifurcation curves in ecosystems, J. Math. Anal. Appl., 381 (2011), 732-741.
doi: 10.1016/j.jmaa.2011.03.048. |
| [13] |
R. Ma and Y. Lu,
Multiplicity of positive solutions for second order nonlinear Dirichlet problem with one-dimension Minkowski-curvature operator, Adv. Nonlinear Stud., 15 (2015), 789-803.
doi: 10.1515/ans-2015-0403. |
| [14] |
P. M. McCabe, J. A. Leach and D. J. Needham,
The evolution of travelling waves in fractional order autocatalysis with decay. Ⅰ. Permanent from travelling waves, SIAM J. Appl. Math., 59 (1998), 870-899.
doi: 10.1137/S0036139996312594. |
| [15] |
T. Ouyang and J. Shi,
Exact multiplicity of positive solutions for a class of semilinear problems, Journal of Differential Equations, 146 (1998), 121-156.
doi: 10.1006/jdeq.1998.3414. |
| [16] |
E. Poole, B. Roberson and B. Stephenson,
Weak Allee effect, grazing, and S-shaped bifurcation curves, Involve, 5 (2012), 133-158.
doi: 10.2140/involve.2012.5.133. |
| [17] |
J. Shi and R. Shivaji,
Persistence in reaction diffusion models with weak Allee effect, J. Math. Biol., 52 (2006), 807-829.
doi: 10.1007/s00285-006-0373-7. |
| [18] |
C.-C. Tzeng, K.-C. Hung and S.-H. Wang,
Global bifurcation and exact multiplicity of positive solutions for a positone problem with cubic nonlinearity, J. Differential Equations, 252 (2012), 6250-6274.
doi: 10.1016/j.jde.2012.02.020. |
| [19] |
M.-H. Wang and M. Kot,
Speeds of invasion in a model with strong or weak Allee effects, Math. Biosci., 171 (2001), 83-97.
doi: 10.1016/S0025-5564(01)00048-7. |
| [20] |
J. Xin,
Front propagation in heterogeneous media, SIAM Rev., 42 (2000), 161-230.
doi: 10.1137/S0036144599364296. |
| [21] |
T.-S. Yeh,
S-shaped and broken S-shaped bifurcation curves for a multiparameter diffusive logistic problem with Holling type-Ⅲ functional response, Commun. Pure Appl. Anal., 16 (2017), 645-670.
doi: 10.3934/cpaa.2017032. |
| [22] |
T.-S. Yeh,
Bifurcation curves of positive steady-state solutions for a reaction-diffusion problem of lake eutrophication, J. Math. Anal. Appl., 449 (2017), 1708-1724.
doi: 10.1016/j.jmaa.2016.12.063. |
Figure 2.
Graphs of bifurcation curve $ S_{L} $ of (1) with varying $ L>0 $ . (ⅰ) (C2) and (H2) hold. (ⅱ) either (C4) or (C6) holds
Figure 3.
Graphs of bifurcation curve $S_{L}$ of (1) with varying $L>0$ . (ⅰ) $S_{\mathring{L}}$ is monotone increasing. (ⅱ) $S_{%
\mathring{L}}$ is not monotone increasing
Figure 4.
Graphs of bifurcation curve $ S_{L} $ of (1), (5). $ \phi _{1}, \phi _{2}\in C(\sqrt{27}
, \infty ) $ satisfy $ \phi _{1}(K)>\phi _{2}(K) $ and $ \Phi (K,
\phi _{1}(K)) = \Phi (K, \phi _{2}(K)) = 0. $
Figure 5.
Graphs of bifurcation curve SL of (1), (6)
Figure 6.
Graphs of bifurcation curve SL of (1), (7). (ⅰ) c = 0. (ⅱ) c > 0
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