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July  2019, 24(7): 3077-3088. doi: 10.3934/dcdsb.2018301

On the steady state bifurcation of the Cahn-Hilliard/Allen-Cahn system

School of Data Sciences, Zhejiang University of Finance & Economics, Hangzhou, 310018, China

* Corresponding author E-mail: dmyan@zufe.edu.cn(Dongming Yan)

Received  March 2018 Published  October 2018

In this paper, the steady state bifurcation of the Cahn-Hilliard/Allen-Cahn system is investigated. By using the Lyapunov-Schmidt method, combining with the implicit function theorem, we prove that this system bifurcates from the trivial solution to the nontrivial solution branch as parameter crosses certain critical value. The expression of bifurcated solution is also obtained.

Citation: 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
References:
[1]

D. BrochetD. Hilhorst and A. Novick-Cohen, Finite-dimensional exponential attractor for a model for order-disorder and phase separation, Appl. Math. Lett., 7 (1994), 83-87.  doi: 10.1016/0893-9659(94)90118-X.  Google Scholar

[2]

J. Cahn and A. Novick-Cohen, Evolution equations for phase separation and ordering in binary alloys, J. Statist. Phys., 76 (1994), 877-909.   Google Scholar

[3]

H. Chan and J. Wei, Traveling wave solutions for bistable fractional Allen-Cahn equations with a pyramidal front, J. Differential Equations, 262 (2017), 4567-4609.  doi: 10.1016/j.jde.2016.12.010.  Google Scholar

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P. Frank and J. Wei, Stable solutions of the Allen-Cahn equation in dimension 8 and minimal cones, Journal of Functional Analysis, 264 (2013), 1131-1167.  doi: 10.1016/j.jfa.2012.03.010.  Google Scholar

[5]

M. Gokieli and A. Ito, Global attractor for the Cahn-Hilliard/Allen-Cahn system, Nonlinear Analysis, 52 (2003), 1821-1841.  doi: 10.1016/S0362-546X(02)00303-6.  Google Scholar

[6]

M. Gokieli and L. Marcinkowski, Modelling phase transitions in alloys, Nonlinear Analysis, 63 (2005), e1143-e1153.   Google Scholar

[7]

M. Kubo, The Cahn-Hilliard equation with time-dependent constraint, Nonlinear Analysis, 75 (2012), 5672-5685.  doi: 10.1016/j.na.2012.05.015.  Google Scholar

[8]

C. Laurence and M. Alain, Finite-dimensional attractors for a model of Allen-Cahn equation based on a microforce balance, Comptes Rendus de l'Academie des Sciences-Series I-Mathematics, 329 (1999), 1109-1114.  doi: 10.1016/S0764-4442(00)88483-9.  Google Scholar

[9]

D. Li and C. Zhong, Global attractor for the Cahn-Hilliard system with fast growing nonlinearity, J. Differential Equations, 149 (1998), 191-210.  doi: 10.1006/jdeq.1998.3429.  Google Scholar

[10]

T. Ma and S. Wang, Phase Transition Dynamics, Springer-Verlag, New York, 2014. doi: 10.1007/978-1-4614-8963-4.  Google Scholar

[11]

T. Ma and S. Wang, Bifurcation Theory and Applications, World Scientific, Singapore, 2005. Google Scholar

[12]

A. Novick-Cohen, Triple-junction motion for an Allen-Cahn/Cahn-Hilliard system, Physica D, 137 (2000), 1-24.  doi: 10.1016/S0167-2789(99)00162-1.  Google Scholar

[13]

L. SongY. Zhang and T. Ma, Global attractor of the Cahn-Hilliard equation in Hk spaces, J. Math. Anal. Appl, 355 (2009), 53-62.  doi: 10.1016/j.jmaa.2009.01.035.  Google Scholar

show all references

References:
[1]

D. BrochetD. Hilhorst and A. Novick-Cohen, Finite-dimensional exponential attractor for a model for order-disorder and phase separation, Appl. Math. Lett., 7 (1994), 83-87.  doi: 10.1016/0893-9659(94)90118-X.  Google Scholar

[2]

J. Cahn and A. Novick-Cohen, Evolution equations for phase separation and ordering in binary alloys, J. Statist. Phys., 76 (1994), 877-909.   Google Scholar

[3]

H. Chan and J. Wei, Traveling wave solutions for bistable fractional Allen-Cahn equations with a pyramidal front, J. Differential Equations, 262 (2017), 4567-4609.  doi: 10.1016/j.jde.2016.12.010.  Google Scholar

[4]

P. Frank and J. Wei, Stable solutions of the Allen-Cahn equation in dimension 8 and minimal cones, Journal of Functional Analysis, 264 (2013), 1131-1167.  doi: 10.1016/j.jfa.2012.03.010.  Google Scholar

[5]

M. Gokieli and A. Ito, Global attractor for the Cahn-Hilliard/Allen-Cahn system, Nonlinear Analysis, 52 (2003), 1821-1841.  doi: 10.1016/S0362-546X(02)00303-6.  Google Scholar

[6]

M. Gokieli and L. Marcinkowski, Modelling phase transitions in alloys, Nonlinear Analysis, 63 (2005), e1143-e1153.   Google Scholar

[7]

M. Kubo, The Cahn-Hilliard equation with time-dependent constraint, Nonlinear Analysis, 75 (2012), 5672-5685.  doi: 10.1016/j.na.2012.05.015.  Google Scholar

[8]

C. Laurence and M. Alain, Finite-dimensional attractors for a model of Allen-Cahn equation based on a microforce balance, Comptes Rendus de l'Academie des Sciences-Series I-Mathematics, 329 (1999), 1109-1114.  doi: 10.1016/S0764-4442(00)88483-9.  Google Scholar

[9]

D. Li and C. Zhong, Global attractor for the Cahn-Hilliard system with fast growing nonlinearity, J. Differential Equations, 149 (1998), 191-210.  doi: 10.1006/jdeq.1998.3429.  Google Scholar

[10]

T. Ma and S. Wang, Phase Transition Dynamics, Springer-Verlag, New York, 2014. doi: 10.1007/978-1-4614-8963-4.  Google Scholar

[11]

T. Ma and S. Wang, Bifurcation Theory and Applications, World Scientific, Singapore, 2005. Google Scholar

[12]

A. Novick-Cohen, Triple-junction motion for an Allen-Cahn/Cahn-Hilliard system, Physica D, 137 (2000), 1-24.  doi: 10.1016/S0167-2789(99)00162-1.  Google Scholar

[13]

L. SongY. Zhang and T. Ma, Global attractor of the Cahn-Hilliard equation in Hk spaces, J. Math. Anal. Appl, 355 (2009), 53-62.  doi: 10.1016/j.jmaa.2009.01.035.  Google Scholar

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