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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, 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.

[2]

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

[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.

[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.

[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.

[6]

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

[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.

[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.

[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.

[10]

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

[11]

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

[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.

[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.

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.

[2]

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

[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.

[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.

[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.

[6]

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

[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.

[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.

[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.

[10]

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

[11]

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

[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.

[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.

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