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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, Mathematisch-Naturwissenschaftlichen Fakultät I, Humboldt-Universität, Berlin, 1999.

[2]

M. Hildebrand, M. Kuperman, H. Wio, A. S. Mikhailov and G. Ertl, Self-organized chemical nanoscale microreactors, Phys. Rev. Lett., 83 (1999), 1475-1478.

[3]

K. Kuto and T. Tsujikawa, Stationary patterns for an adsorbate-induced phase transition model: I. Existence, Discrete Continuous Dynam. Systems - B, 14 (2010), 1105-1117.

[4]

K. Kuto and T. Tsujikawa, Stationary patterns for an adsorbate-induced phase transition model: II. Shadow system, Nonlinearity, 26 (2013), 1313-1343.

[5]

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

[6]

Y. Mori, A. Jilkine and L. Edelstein-Keshet, Wave-pinning and cell polarity from a bistable reaction-diffusion system, Biophys. J., 94 (2008), 3684-3697.

[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), 1401-1427.

[8]

R. Schaaf, "Global solution branches of two-point boundary value problems," Lecture Notes in Mathematics, 1458. Springer-Verlag, Berlin, 1990.

[9]

J. Shi, Semilinear Neumann boundary value problems on a rectangle, Trans. Amer. Math. Soc., 354 (2002), 3117-3154.

[10]

J. Smoller and A. Wasserman, Global bifurcation of steady-state solutions, J. Differential Equations, 39 (1981), 269-290.

show all references

References:
[1]

M. Hildebrand, "Selbstorganisierte nanostrukturen in katakyschen oberflächenreaktionen," D. dissertation, Mathematisch-Naturwissenschaftlichen Fakultät I, Humboldt-Universität, Berlin, 1999.

[2]

M. Hildebrand, M. Kuperman, H. Wio, A. S. Mikhailov and G. Ertl, Self-organized chemical nanoscale microreactors, Phys. Rev. Lett., 83 (1999), 1475-1478.

[3]

K. Kuto and T. Tsujikawa, Stationary patterns for an adsorbate-induced phase transition model: I. Existence, Discrete Continuous Dynam. Systems - B, 14 (2010), 1105-1117.

[4]

K. Kuto and T. Tsujikawa, Stationary patterns for an adsorbate-induced phase transition model: II. Shadow system, Nonlinearity, 26 (2013), 1313-1343.

[5]

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

[6]

Y. Mori, A. Jilkine and L. Edelstein-Keshet, Wave-pinning and cell polarity from a bistable reaction-diffusion system, Biophys. J., 94 (2008), 3684-3697.

[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), 1401-1427.

[8]

R. Schaaf, "Global solution branches of two-point boundary value problems," Lecture Notes in Mathematics, 1458. Springer-Verlag, Berlin, 1990.

[9]

J. Shi, Semilinear Neumann boundary value problems on a rectangle, Trans. Amer. Math. Soc., 354 (2002), 3117-3154.

[10]

J. Smoller and A. Wasserman, Global bifurcation of steady-state solutions, J. Differential Equations, 39 (1981), 269-290.

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