American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Isometric embedding with nonnegative Gauss curvature under the graph setting
  • DCDS Home
  • This Issue
  • Next Article
    Global existence of almost energy solution to the two-dimensional chemotaxis-Navier-Stokes equations with partial diffusion
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

Shao-Yuan Huang

Center for General Education, Ming Chi University of Technology, New Taipei City 24301, Taiwan

Received  August 2018 Published  February 2019

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 $
 is a bifurcation parameter,
$ L>0 $
 is an evolution parameter,
$ f\in C[0, \infty )\cap C^{2}(0, \infty ) $
 and there exists
$ \beta >0 $
 such that
$ \left( \beta -z\right) f(z)>0 $
 for
$ z\neq \beta $
. In particular, we find that the bifurcation curve
$ S_{L} $
 is monotone increasing for all
$ L>0 $
 when
$ f(u)/u $
 is of Logistic type, and is either
$ \subset $
-shaped or S-shaped for large
$ L>0 $
 when
$ f(u)/u $
 is of weak Allee effect type. Finally, we can apply these results to obtain the global bifurcation diagrams in some important applications including ecosystem model.
Keywords: Global bifurcation, multiplicity of positive solution, Minkowski-curvature problem.
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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

J. Xin, Front propagation in heterogeneous media, SIAM Rev., 42 (2000), 161-230.  doi: 10.1137/S0036144599364296.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

J. Xin, Front propagation in heterogeneous media, SIAM Rev., 42 (2000), 161-230.  doi: 10.1137/S0036144599364296.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

Figure 1.  ⅰ) monotone increasing. (ⅱ) $ \subset $-shaped. (ⅲ) S-shaped
Figure Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

Figure 5.  Graphs of bifurcation curve SL of (1), (6)
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 6.  Graphs of bifurcation curve SL of (1), (7). (ⅰ) c = 0. (ⅱ) c > 0
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Shao-Yuan Huang. Global bifurcation and exact multiplicity of positive solutions for the one-dimensional Minkowski-curvature problem with sign-changing nonlinearity. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3267-3284. doi: 10.3934/cpaa.2019147
[2]

Shao-Yuan Huang. Exact multiplicity and bifurcation curves of positive solutions of a one-dimensional Minkowski-curvature problem and its application. Communications on Pure & Applied Analysis, 2018, 17 (3) : 1271-1294. doi: 10.3934/cpaa.2018061
[3]

Alberto Boscaggin, Francesca Colasuonno, Benedetta Noris. Positive radial solutions for the Minkowski-curvature equation with Neumann boundary conditions. Discrete & Continuous Dynamical Systems - S, 2020, 13 (7) : 1921-1933. doi: 10.3934/dcdss.2020150
[4]

Ruyun Ma, Man Xu. Connected components of positive solutions for a Dirichlet problem involving the mean curvature operator in Minkowski space. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2701-2718. doi: 10.3934/dcdsb.2018271
[5]

Chiara Corsato, Franco Obersnel, Pierpaolo Omari, Sabrina Rivetti. On the lower and upper solution method for the prescribed mean curvature equation in Minkowski space. Conference Publications, 2013, 2013 (special) : 159-169. doi: 10.3934/proc.2013.2013.159
[6]

Anna Lisa Amadori. Global bifurcation for the Hénon problem. Communications on Pure & Applied Analysis, 2020, 19 (10) : 4797-4816. doi: 10.3934/cpaa.2020212
[7]

Kuan-Ju Huang, Yi-Jung Lee, Tzung-Shin Yeh. Classification of bifurcation curves of positive solutions for a nonpositone problem with a quartic polynomial. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1497-1514. doi: 10.3934/cpaa.2016.15.1497
[8]

Meiyue Jiang, Chu Wang. The Orlicz-Minkowski problem for polytopes. Discrete & Continuous Dynamical Systems - S, 2021, 14 (6) : 1917-1930. doi: 10.3934/dcdss.2021043
[9]

Guowei Dai, Alfonso Romero, Pedro J. Torres. Global bifurcation of solutions of the mean curvature spacelike equation in certain standard static spacetimes. Discrete & Continuous Dynamical Systems - S, 2020, 13 (11) : 3047-3071. doi: 10.3934/dcdss.2020118
[10]

Ling Mi. Asymptotic behavior for the unique positive solution to a singular elliptic problem. Communications on Pure & Applied Analysis, 2015, 14 (3) : 1053-1072. doi: 10.3934/cpaa.2015.14.1053
[11]

Libin Wang. Breakdown of $C^1$ solution to the Cauchy problem for quasilinear hyperbolic systems with characteristics with constant multiplicity. Communications on Pure & Applied Analysis, 2003, 2 (1) : 77-89. doi: 10.3934/cpaa.2003.2.77
[12]

Matthias Bergner, Lars Schäfer. Time-like surfaces of prescribed anisotropic mean curvature in Minkowski space. Conference Publications, 2011, 2011 (Special) : 155-162. doi: 10.3934/proc.2011.2011.155
[13]

Hongjie Ju, Jian Lu, Huaiyu Jian. Translating solutions to mean curvature flow with a forcing term in Minkowski space. Communications on Pure & Applied Analysis, 2010, 9 (4) : 963-973. doi: 10.3934/cpaa.2010.9.963
[14]

Kuo-Chih Hung, Shao-Yuan Huang, Shin-Hwa Wang. A global bifurcation theorem for a positone multiparameter problem and its application. Discrete & Continuous Dynamical Systems, 2017, 37 (10) : 5127-5149. doi: 10.3934/dcds.2017222
[15]

Yves Coudène, Barbara Schapira. Counterexamples in non-positive curvature. Discrete & Continuous Dynamical Systems, 2011, 30 (4) : 1095-1106. doi: 10.3934/dcds.2011.30.1095
[16]

Yukio Kan-On. Bifurcation structures of positive stationary solutions for a Lotka-Volterra competition model with diffusion II: Global structure. Discrete & Continuous Dynamical Systems, 2006, 14 (1) : 135-148. doi: 10.3934/dcds.2006.14.135
[17]

Alejandro Allendes, Alexander Quaas. Multiplicity results for extremal operators through bifurcation. Discrete & Continuous Dynamical Systems, 2011, 29 (1) : 51-65. doi: 10.3934/dcds.2011.29.51
[18]

Shaoyong Lai, Yong Hong Wu, Xu Yang. The global solution of an initial boundary value problem for the damped Boussinesq equation. Communications on Pure & Applied Analysis, 2004, 3 (2) : 319-328. doi: 10.3934/cpaa.2004.3.319
[19]

Zongming Guo, Xuefei Bai. On the global branch of positive radial solutions of an elliptic problem with singular nonlinearity. Communications on Pure & Applied Analysis, 2008, 7 (5) : 1091-1107. doi: 10.3934/cpaa.2008.7.1091
[20]

Olga Kharlampovich and Alexei Myasnikov. Tarski's problem about the elementary theory of free groups has a positive solution. Electronic Research Announcements, 1998, 4: 101-108.

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (148)
  • HTML views (138)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences