• Previous Article
    Attractivity of saturated equilibria for Lotka-Volterra systems with infinite delays and feedback controls
  • DCDS-B Home
  • This Issue
  • Next Article
    Uniqueness of traveling front solutions for the Lotka-Volterra system in the weak competition case
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  July 2019 Early access  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 and 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.

[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.

[1]

Quan Wang, Dongming Yan. On the stability and transition of the Cahn-Hilliard/Allen-Cahn system. Discrete and Continuous Dynamical Systems - B, 2020, 25 (7) : 2607-2620. doi: 10.3934/dcdsb.2020024

[2]

Alain Miranville, Ramon Quintanilla, Wafa Saoud. Asymptotic behavior of a Cahn-Hilliard/Allen-Cahn system with temperature. Communications on Pure and Applied Analysis, 2020, 19 (4) : 2257-2288. doi: 10.3934/cpaa.2020099

[3]

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

[4]

Christopher P. Grant. Grain sizes in the discrete Allen-Cahn and Cahn-Hilliard equations. Discrete and Continuous Dynamical Systems, 2001, 7 (1) : 127-146. doi: 10.3934/dcds.2001.7.127

[5]

Jie Shen, Xiaofeng Yang. Numerical approximations of Allen-Cahn and Cahn-Hilliard equations. Discrete and Continuous Dynamical Systems, 2010, 28 (4) : 1669-1691. doi: 10.3934/dcds.2010.28.1669

[6]

Alain Miranville, Wafa Saoud, Raafat Talhouk. On the Cahn-Hilliard/Allen-Cahn equations with singular potentials. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 3633-3651. doi: 10.3934/dcdsb.2018308

[7]

Changchun Liu, Hui Tang. Existence of periodic solution for a Cahn-Hilliard/Allen-Cahn equation in two space dimensions. Evolution Equations and Control Theory, 2017, 6 (2) : 219-237. doi: 10.3934/eect.2017012

[8]

Irena Pawłow. Thermodynamically consistent Cahn-Hilliard and Allen-Cahn models in elastic solids. Discrete and Continuous Dynamical Systems, 2006, 15 (4) : 1169-1191. doi: 10.3934/dcds.2006.15.1169

[9]

Ahmad Makki, Alain Miranville. Existence of solutions for anisotropic Cahn-Hilliard and Allen-Cahn systems in higher space dimensions. Discrete and Continuous Dynamical Systems - S, 2016, 9 (3) : 759-775. doi: 10.3934/dcdss.2016027

[10]

Cristina Pocci. On singular limit of a nonlinear $p$-order equation related to Cahn-Hilliard and Allen-Cahn evolutions. Evolution Equations and Control Theory, 2013, 2 (3) : 517-530. doi: 10.3934/eect.2013.2.517

[11]

Juan Wen, Yaling He, Yinnian He, Kun Wang. Stabilized finite element methods based on multiscale enrichment for Allen-Cahn and Cahn-Hilliard equations. Communications on Pure and Applied Analysis, 2022, 21 (6) : 1873-1894. doi: 10.3934/cpaa.2021074

[12]

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

[13]

Elena Bonetti, Pierluigi Colli, Luca Scarpa, Giuseppe Tomassetti. A doubly nonlinear Cahn-Hilliard system with nonlinear viscosity. Communications on Pure and Applied Analysis, 2018, 17 (3) : 1001-1022. doi: 10.3934/cpaa.2018049

[14]

Pierluigi Colli, Gianni Gilardi, Paolo Podio-Guidugli, Jürgen Sprekels. An asymptotic analysis for a nonstandard Cahn-Hilliard system with viscosity. Discrete and Continuous Dynamical Systems - S, 2013, 6 (2) : 353-368. doi: 10.3934/dcdss.2013.6.353

[15]

Pierluigi Colli, Gianni Gilardi, Danielle Hilhorst. On a Cahn-Hilliard type phase field system related to tumor growth. Discrete and Continuous Dynamical Systems, 2015, 35 (6) : 2423-2442. doi: 10.3934/dcds.2015.35.2423

[16]

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

[17]

Heinz Schättler, Urszula Ledzewicz. Lyapunov-Schmidt reduction for optimal control problems. Discrete and Continuous Dynamical Systems - B, 2012, 17 (6) : 2201-2223. doi: 10.3934/dcdsb.2012.17.2201

[18]

Xufeng Xiao, Xinlong Feng, Jinyun Yuan. The stabilized semi-implicit finite element method for the surface Allen-Cahn equation. Discrete and Continuous Dynamical Systems - B, 2017, 22 (7) : 2857-2877. doi: 10.3934/dcdsb.2017154

[19]

Xiaofeng Yang. Error analysis of stabilized semi-implicit method of Allen-Cahn equation. Discrete and Continuous Dynamical Systems - B, 2009, 11 (4) : 1057-1070. doi: 10.3934/dcdsb.2009.11.1057

[20]

Ciprian G. Gal, Maurizio Grasselli. Longtime behavior of nonlocal Cahn-Hilliard equations. Discrete and Continuous Dynamical Systems, 2014, 34 (1) : 145-179. doi: 10.3934/dcds.2014.34.145

2021 Impact Factor: 1.497

Metrics

  • PDF downloads (379)
  • HTML views (552)
  • Cited by (0)

Other articles
by authors

[Back to Top]