American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Existence of sliding motions for nonlinear evolution equations in Banach spaces
  • PROC Home
  • This Issue
  • Next Article
    Analytical approach of one-dimensional solute transport through inhomogeneous semi-infinite porous domain for unsteady flow: Dispersion being proportional to square of velocity
2013, 2013(special): 467-476. doi: 10.3934/proc.2013.2013.467

Bifurcation structure of steady-states for bistable equations with nonlocal constraint

Kousuke Kuto 1, and  Tohru Tsujikawa 2,

1. 

Department of Communication Engineering and Informatics, The University of Electro-Communications, 1-5-1 Chofugaoka, Chofu-shi, Tokyo 182-8585
2. 

Department of Applied Physics, University of Miyazaki, Miyazaki, 889-2192

Received  September 2012 Revised  April 2013 Published  November 2013

This paper studies the 1D Neumann problem of bistable equations with nonlocal constraint. We obtain the global bifurcation structure of solutions by a level set analysis for the associate integral mapping. This structure implies that solutions can form a saddle-node bifurcation curve connecting boundary-layer states with internal-layer states. Furthermore, we exhibit the applications of our result to a couple of shadow systems arising in surface chemistry and physiology.
Keywords: saddle-node bifurcation, shadow system., level set, Allen-Cahn equation, nonlocal constraint.
Mathematics Subject Classification: Primary: 34B18; Secondary: 34C23, 34E20, 37G1.
Citation: Kousuke Kuto, Tohru Tsujikawa. Bifurcation structure of steady-states for bistable equations with nonlocal constraint. Conference Publications, 2013, 2013 (special) : 467-476. doi: 10.3934/proc.2013.2013.467
References:
[1]

M. Hildebrand, "Selbstorganisierte nanostrukturen in katakyschen oberflächenreaktionen,", D. dissertation, (1999).   Google Scholar

[2]

M. Hildebrand, M. Kuperman, H. Wio, A. S. Mikhailov and G. Ertl, Self-organized chemical nanoscale microreactors,, Phys. Rev. Lett., 83 (1999), 1475.   Google Scholar

[3]

K. Kuto and T. Tsujikawa, Stationary patterns for an adsorbate-induced phase transition model: I. Existence,, Discrete Continuous Dynam. Systems - B, 14 (2010), 1105.   Google Scholar

[4]

K. Kuto and T. Tsujikawa, Stationary patterns for an adsorbate-induced phase transition model: II. Shadow system,, Nonlinearity, 26 (2013), 1313.   Google Scholar

[5]

K. Kuto and T. Tsujikawa, Bifurcation structure of steady-states for generalized Allen-Cahn equations with nonlocal constraint,, preprint., ().   Google Scholar

[6]

Y. Mori, A. Jilkine and L. Edelstein-Keshet, Wave-pinning and cell polarity from a bistable reaction-diffusion system,, Biophys. J., 94 (2008), 3684.   Google Scholar

[7]

Y. Mori, A. Jilkine and L. Edelstein-Keshet, Asymptotic and bifurcation analysis of wave-pinning in a reaction-diffusion model for cell polarization,, SIAM J. Appl. Math., 71 (2011), 1401.   Google Scholar

[8]

R. Schaaf, "Global solution branches of two-point boundary value problems,", Lecture Notes in Mathematics, (1458).   Google Scholar

[9]

J. Shi, Semilinear Neumann boundary value problems on a rectangle,, Trans. Amer. Math. Soc., 354 (2002), 3117.   Google Scholar

[10]

J. Smoller and A. Wasserman, Global bifurcation of steady-state solutions,, J. Differential Equations, 39 (1981), 269.   Google Scholar

show all references

References:
[1]

M. Hildebrand, "Selbstorganisierte nanostrukturen in katakyschen oberflächenreaktionen,", D. dissertation, (1999).   Google Scholar

[2]

M. Hildebrand, M. Kuperman, H. Wio, A. S. Mikhailov and G. Ertl, Self-organized chemical nanoscale microreactors,, Phys. Rev. Lett., 83 (1999), 1475.   Google Scholar

[3]

K. Kuto and T. Tsujikawa, Stationary patterns for an adsorbate-induced phase transition model: I. Existence,, Discrete Continuous Dynam. Systems - B, 14 (2010), 1105.   Google Scholar

[4]

K. Kuto and T. Tsujikawa, Stationary patterns for an adsorbate-induced phase transition model: II. Shadow system,, Nonlinearity, 26 (2013), 1313.   Google Scholar

[5]

K. Kuto and T. Tsujikawa, Bifurcation structure of steady-states for generalized Allen-Cahn equations with nonlocal constraint,, preprint., ().   Google Scholar

[6]

Y. Mori, A. Jilkine and L. Edelstein-Keshet, Wave-pinning and cell polarity from a bistable reaction-diffusion system,, Biophys. J., 94 (2008), 3684.   Google Scholar

[7]

Y. Mori, A. Jilkine and L. Edelstein-Keshet, Asymptotic and bifurcation analysis of wave-pinning in a reaction-diffusion model for cell polarization,, SIAM J. Appl. Math., 71 (2011), 1401.   Google Scholar

[8]

R. Schaaf, "Global solution branches of two-point boundary value problems,", Lecture Notes in Mathematics, (1458).   Google Scholar

[9]

J. Shi, Semilinear Neumann boundary value problems on a rectangle,, Trans. Amer. Math. Soc., 354 (2002), 3117.   Google Scholar

[10]

J. Smoller and A. Wasserman, Global bifurcation of steady-state solutions,, J. Differential Equations, 39 (1981), 269.   Google Scholar

[1]

Rui Dilão, András Volford. Excitability in a model with a saddle-node homoclinic bifurcation. Discrete & Continuous Dynamical Systems - B, 2004, 4 (2) : 419-434. doi: 10.3934/dcdsb.2004.4.419
[2]

Ping Liu, Junping Shi, Yuwen Wang. A double saddle-node bifurcation theorem. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2923-2933. doi: 10.3934/cpaa.2013.12.2923
[3]

Eleonora Cinti. Saddle-shaped solutions for the fractional Allen-Cahn equation. Discrete & Continuous Dynamical Systems - S, 2018, 11 (3) : 441-463. doi: 10.3934/dcdss.2018024
[4]

Shixing Li, Dongming Yan. On the steady state bifurcation of the Cahn-Hilliard/Allen-Cahn system. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3077-3088. doi: 10.3934/dcdsb.2018301
[5]

Flaviano Battelli. Saddle-node bifurcation of homoclinic orbits in singular systems. Discrete & Continuous Dynamical Systems - A, 2001, 7 (1) : 203-218. doi: 10.3934/dcds.2001.7.203
[6]

Gianni Gilardi. On an Allen-Cahn type integrodifferential equation. Discrete & Continuous Dynamical Systems - S, 2013, 6 (3) : 703-709. doi: 10.3934/dcdss.2013.6.703
[7]

Grégory Faye. Multidimensional stability of planar traveling waves for the scalar nonlocal Allen-Cahn equation. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2473-2496. doi: 10.3934/dcds.2016.36.2473
[8]

Georgia Karali, Yuko Nagase. On the existence of solution for a Cahn-Hilliard/Allen-Cahn equation. Discrete & Continuous Dynamical Systems - S, 2014, 7 (1) : 127-137. doi: 10.3934/dcdss.2014.7.127
[9]

Matthieu Alfaro, Hiroshi Matano. On the validity of formal asymptotic expansions in Allen-Cahn equation and FitzHugh-Nagumo system with generic initial data. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 1639-1649. doi: 10.3934/dcdsb.2012.17.1639
[10]

Yan Hu. Layer solutions for an Allen-Cahn type system driven by the fractional Laplacian. Communications on Pure & Applied Analysis, 2016, 15 (3) : 947-964. doi: 10.3934/cpaa.2016.15.947
[11]

Mohammad Hassan Farshbaf-Shaker, Takeshi Fukao, Noriaki Yamazaki. Singular limit of Allen--Cahn equation with constraint and its Lagrange multiplier. Conference Publications, 2015, 2015 (special) : 418-427. doi: 10.3934/proc.2015.0418
[12]

Hongmei Cheng, Rong Yuan. Multidimensional stability of disturbed pyramidal traveling fronts in the Allen-Cahn equation. Discrete & Continuous Dynamical Systems - B, 2015, 20 (4) : 1015-1029. doi: 10.3934/dcdsb.2015.20.1015
[13]

Xinlong Feng, Huailing Song, Tao Tang, Jiang Yang. Nonlinear stability of the implicit-explicit methods for the Allen-Cahn equation. Inverse Problems & Imaging, 2013, 7 (3) : 679-695. doi: 10.3934/ipi.2013.7.679
[14]

Christos Sourdis. On the growth of the energy of entire solutions to the vector Allen-Cahn equation. Communications on Pure & Applied Analysis, 2015, 14 (2) : 577-584. doi: 10.3934/cpaa.2015.14.577
[15]

Paul H. Rabinowitz, Ed Stredulinsky. On a class of infinite transition solutions for an Allen-Cahn model equation. Discrete & Continuous Dynamical Systems - A, 2008, 21 (1) : 319-332. doi: 10.3934/dcds.2008.21.319
[16]

Ciprian G. Gal, Maurizio Grasselli. The non-isothermal Allen-Cahn equation with dynamic boundary conditions. Discrete & Continuous Dynamical Systems - A, 2008, 22 (4) : 1009-1040. doi: 10.3934/dcds.2008.22.1009
[17]

Zhuoran Du, Baishun Lai. Transition layers for an inhomogeneous Allen-Cahn equation in Riemannian manifolds. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1407-1429. doi: 10.3934/dcds.2013.33.1407
[18]

Charles-Edouard Bréhier, Ludovic Goudenège. Analysis of some splitting schemes for the stochastic Allen-Cahn equation. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 4169-4190. doi: 10.3934/dcdsb.2019077
[19]

Ale Jan Homburg, Todd Young. Intermittency and Jakobson's theorem near saddle-node bifurcations. Discrete & Continuous Dynamical Systems - A, 2007, 17 (1) : 21-58. doi: 10.3934/dcds.2007.17.21
[20]

W.-J. Beyn, Y.-K Zou. Discretizations of dynamical systems with a saddle-node homoclinic orbit. Discrete & Continuous Dynamical Systems - A, 1996, 2 (3) : 351-365. doi: 10.3934/dcds.1996.2.351

 Impact Factor: 

Tools

Metrics

  • PDF downloads (24)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences