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October  2016, 36(10): 5627-5655. doi: 10.3934/dcds.2016047

Exact multiplicity of stationary limiting problems of a cell polarization model

Tatsuki Mori 1, , Kousuke Kuto 2, , Tohru Tsujikawa 3, and  Shoji Yotsutani 4,

1. 

Center for PDE, East China Normal University, Minhang, Shanghai 200241, China
2. 

Department of Communication Engineering and Informatics, The University of Electro-Communications, 1-5-1 Chofugaoka, Chofu, Tokyo, 182-8585, Japan
3. 

Faculty of Engineering, University of Miyazaki, Miyazaki, 889-2192
4. 

Department of Applied Mathematics and Informatics, Ryukoku University, Seta, Otsu, Shiga 520-2194

Received  September 2015 Revised  December 2015 Published  July 2016

We show existence, nonexistence, and exact multiplicity for stationary limiting problems of a cell polarization model proposed by Y. Mori, A. Jilkine and L. Edelstein-Keshet. It is a nonlinear boundary value problem with total mass constraint. We obtain exact multiplicity results by investigating a global bifurcation sheet which we constructed by using complete elliptic integrals in a previous paper.
Keywords: exact solution, nonlocal term, Reaction diffusion model, elliptic integral., bifurcation, level set analysis.
Mathematics Subject Classification: 34B10, 35B32, 35K5.
Citation: Tatsuki Mori, Kousuke Kuto, Tohru Tsujikawa, Shoji Yotsutani. Exact multiplicity of stationary limiting problems of a cell polarization model. Discrete & Continuous Dynamical Systems, 2016, 36 (10) : 5627-5655. doi: 10.3934/dcds.2016047
References:
[1]

J. Carr, M. E. Gurtin and M. Semrod, Structured phase transitions on a finite interval, Arch. Rational Mech. Anal., 86 (1984), 317-351. doi: 10.1007/BF00280031.  Google Scholar

[2]

S. Kosugi Y. Morita and S. Yotsutani, Stationary solutions to the one-dimensional Cahn-Hilliard equation: Proof by the complete elliptic integrals, Discrete and Continuous Dynamical Systems, 19 (2007), 609-629. doi: 10.3934/dcds.2007.19.609.  Google Scholar

[3]

K. Kuto and T. Tsujikawa, Stationary patterns for an adsorbate-induced phase transition model I, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 1105-1117. doi: 10.3934/dcdsb.2010.14.1105.  Google Scholar

[4]

K. Kuto and T. Tsujikawa, Stationary patterns for an adsorbate-induced phase transition model II, Nonlinearity, 26 (2013), 1313-1343. doi: 10.1088/0951-7715/26/5/1313.  Google Scholar

[5]

K. Kuto and T. Tsujikawa, Bifurcation structure of steady-states for bistable equations with nonlocal constraint, Discrete and Continuous Dynamical Systems Supplement, (2013), 467-476. doi: 10.3934/proc.2013.2013.467.  Google Scholar

[6]

Y. Lou, W.-M. Ni and S. Yotsutani, On a limiting system in the Lotka-Voltera competition with cross diffusion, Discrete Contin. Dyn. Syst., 10 (2004), 435-458. doi: 10.3934/dcds.2004.10.435.  Google Scholar

[7]

T.Mori, K.Kuto, T.Tujikawa, M.Nagayama and S.Yotsutani, Global bifurcation sheet and diagrams of wave-pinning in a reaction-diffusion model for cell polarization, Dynamical Systems, Differential Equations and Applications AIMS Proceedings, 2015, 861-877. doi: 10.3934/proc.2015.0861.  Google Scholar

[8]

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. Google Scholar

[9]

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. doi: 10.1137/10079118X.  Google Scholar

[10]

J. Smoller, Shock Waves and Reaction Diffusion Equations, Springer, 1994. doi: 10.1007/978-1-4612-0873-0.  Google Scholar

[11]

J. Smoller and A. Wasserman, Global bifurcation of steady-state solutions, J. Differential Equations, 39 (1981), 269-290. doi: 10.1016/0022-0396(81)90077-2.  Google Scholar

show all references

References:
[1]

J. Carr, M. E. Gurtin and M. Semrod, Structured phase transitions on a finite interval, Arch. Rational Mech. Anal., 86 (1984), 317-351. doi: 10.1007/BF00280031.  Google Scholar

[2]

S. Kosugi Y. Morita and S. Yotsutani, Stationary solutions to the one-dimensional Cahn-Hilliard equation: Proof by the complete elliptic integrals, Discrete and Continuous Dynamical Systems, 19 (2007), 609-629. doi: 10.3934/dcds.2007.19.609.  Google Scholar

[3]

K. Kuto and T. Tsujikawa, Stationary patterns for an adsorbate-induced phase transition model I, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 1105-1117. doi: 10.3934/dcdsb.2010.14.1105.  Google Scholar

[4]

K. Kuto and T. Tsujikawa, Stationary patterns for an adsorbate-induced phase transition model II, Nonlinearity, 26 (2013), 1313-1343. doi: 10.1088/0951-7715/26/5/1313.  Google Scholar

[5]

K. Kuto and T. Tsujikawa, Bifurcation structure of steady-states for bistable equations with nonlocal constraint, Discrete and Continuous Dynamical Systems Supplement, (2013), 467-476. doi: 10.3934/proc.2013.2013.467.  Google Scholar

[6]

Y. Lou, W.-M. Ni and S. Yotsutani, On a limiting system in the Lotka-Voltera competition with cross diffusion, Discrete Contin. Dyn. Syst., 10 (2004), 435-458. doi: 10.3934/dcds.2004.10.435.  Google Scholar

[7]

T.Mori, K.Kuto, T.Tujikawa, M.Nagayama and S.Yotsutani, Global bifurcation sheet and diagrams of wave-pinning in a reaction-diffusion model for cell polarization, Dynamical Systems, Differential Equations and Applications AIMS Proceedings, 2015, 861-877. doi: 10.3934/proc.2015.0861.  Google Scholar

[8]

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. Google Scholar

[9]

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. doi: 10.1137/10079118X.  Google Scholar

[10]

J. Smoller, Shock Waves and Reaction Diffusion Equations, Springer, 1994. doi: 10.1007/978-1-4612-0873-0.  Google Scholar

[11]

J. Smoller and A. Wasserman, Global bifurcation of steady-state solutions, J. Differential Equations, 39 (1981), 269-290. doi: 10.1016/0022-0396(81)90077-2.  Google Scholar

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