American Institute of Mathematical Sciences

Advanced
  • Previous Article
    General decay for a viscoelastic Kirchhoff equation with Balakrishnan-Taylor damping, dynamic boundary conditions and a time-varying delay term
  • EECT Home
  • This Issue
  • Next Article
    Optimal control for a hyperbolic problem in composites with imperfect interface: A memory effect
June  2017, 6(2): 219-237. doi: 10.3934/eect.2017012

Existence of periodic solution for a Cahn-Hilliard/Allen-Cahn equation in two space dimensions

Changchun Liu 1,, and  Hui Tang 2,

1. 

Department of Mathematics, Jilin University, Changchun 130012, China
2. 

School of Science, Changchun University, Changchun 130022, China

* Corresponding author:Changchun Liu

Received  June 2015 Revised  July 2016 Published  April 2017

Fund Project: This work is supported by the Jilin Scientific and Technological Development Program (no. 20170101143JC) and the National Science Foundation of China (no. 11471164)

In this paper, we discuss the existence of the periodic solutions of a Cahn-Hillard/Allen-Cahn equation which is introduced as a simplification of multiple microscopic mechanisms model in cluster interface evolution. Based on the Schauder type a priori estimates, which here will be obtained by means of a modified Campanato space, we prove the existence of time-periodic solutions in two space dimensions. The uniqueness of solutions is also discussed.

Keywords: Cahn-Hilliard/Allen-Cahn equation, existence, uniqueness, periodic solution.
Mathematics Subject Classification: Primary: 35B10, 35K25; Secondary: 35K55.
Citation: Changchun Liu, Hui Tang. Existence of periodic solution for a Cahn-Hilliard/Allen-Cahn equation in two space dimensions. Evolution Equations & Control Theory, 2017, 6 (2) : 219-237. doi: 10.3934/eect.2017012
References:
[1]

Y. Fu and B. Guo, Time periodic solution of the viscous Camassa-Holm equation, J. Math. Anal. Appl., 313 (2006), 311-321.  doi: 10.1016/j.jmaa.2005.08.073.  Google Scholar

[2]

M. Giaquinta and M. Struwe, On the partial regularity of weak solutions of nonlinear parabolic systems, Math. Z., 179 (1982), 437-451.  doi: 10.1007/BF01215058.  Google Scholar

[3]

G. Karali and T. Ricciardi, On the convergence of a fourth order evolution equation to the Allen-Cahn equation, Nonlinear Analysis, 72 (2010), 4271-4281.  doi: 10.1016/j.na.2010.02.003.  Google Scholar

[4]

G. Karali and M. A. Katsoulakis, The role of multiple microscopic mechanisms in cluster interface evolution, J. Differential Equations, 235 (2007), 418-438.  doi: 10.1016/j.jde.2006.12.021.  Google Scholar

[5]

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

[6]

C. Liu and Z. Wang, Time periodic solutions for a sixth order nonlinear parabolic equation in two space dimensions, Commun. Pure Appl. Anal., 13 (2014), 1087-1104.  doi: 10.3934/cpaa.2014.13.1087.  Google Scholar

[7]

R. Wang, The Schauder theory of the boundary value problem for parabolic problem equations, Acta Sci. Nature Univ. Jilin., 2 (1964), 35-64.   Google Scholar

[8]

Y. Wang and Y. Zhang, Time-periodic solutions to a nonlinear parabolic type equation of higher order, Acta Math. Appl. Sin., Engl. Ser., 24 (2008), 129-140.  doi: 10.1007/s10255-006-6174-3.  Google Scholar

[9]

L. Yin, Y. Li, R. Huang and J. Yin, Time periodic solutions for a Cahn-Hilliard type equation, Mathematical and Computer Modelling, 48 (2008), 11-18.  doi: 10.1016/j.mcm.2007.09.001.  Google Scholar

[10]

J. Yin, Y. Li and R. Huang, The Cahn-Hilliard type equations with periodic potentials and sources, Appl. Math. Comput., 211 (2009), 211-221.  doi: 10.1016/j.amc.2009.01.038.  Google Scholar

show all references

References:
[1]

Y. Fu and B. Guo, Time periodic solution of the viscous Camassa-Holm equation, J. Math. Anal. Appl., 313 (2006), 311-321.  doi: 10.1016/j.jmaa.2005.08.073.  Google Scholar

[2]

M. Giaquinta and M. Struwe, On the partial regularity of weak solutions of nonlinear parabolic systems, Math. Z., 179 (1982), 437-451.  doi: 10.1007/BF01215058.  Google Scholar

[3]

G. Karali and T. Ricciardi, On the convergence of a fourth order evolution equation to the Allen-Cahn equation, Nonlinear Analysis, 72 (2010), 4271-4281.  doi: 10.1016/j.na.2010.02.003.  Google Scholar

[4]

G. Karali and M. A. Katsoulakis, The role of multiple microscopic mechanisms in cluster interface evolution, J. Differential Equations, 235 (2007), 418-438.  doi: 10.1016/j.jde.2006.12.021.  Google Scholar

[5]

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

[6]

C. Liu and Z. Wang, Time periodic solutions for a sixth order nonlinear parabolic equation in two space dimensions, Commun. Pure Appl. Anal., 13 (2014), 1087-1104.  doi: 10.3934/cpaa.2014.13.1087.  Google Scholar

[7]

R. Wang, The Schauder theory of the boundary value problem for parabolic problem equations, Acta Sci. Nature Univ. Jilin., 2 (1964), 35-64.   Google Scholar

[8]

Y. Wang and Y. Zhang, Time-periodic solutions to a nonlinear parabolic type equation of higher order, Acta Math. Appl. Sin., Engl. Ser., 24 (2008), 129-140.  doi: 10.1007/s10255-006-6174-3.  Google Scholar

[9]

L. Yin, Y. Li, R. Huang and J. Yin, Time periodic solutions for a Cahn-Hilliard type equation, Mathematical and Computer Modelling, 48 (2008), 11-18.  doi: 10.1016/j.mcm.2007.09.001.  Google Scholar

[10]

J. Yin, Y. Li and R. Huang, The Cahn-Hilliard type equations with periodic potentials and sources, Appl. Math. Comput., 211 (2009), 211-221.  doi: 10.1016/j.amc.2009.01.038.  Google Scholar

[1]

Georgia Karali, Yuko Nagase. On the existence of solution for a Cahn-Hilliard/Allen-Cahn equation. Discrete & Continuous Dynamical Systems - S, 2014, 7 (1) : 127-137. doi: 10.3934/dcdss.2014.7.127
[2]

Christopher P. Grant. Grain sizes in the discrete Allen-Cahn and Cahn-Hilliard equations. Discrete & Continuous Dynamical Systems - A, 2001, 7 (1) : 127-146. doi: 10.3934/dcds.2001.7.127
[3]

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

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
[5]

Alain Miranville, Wafa Saoud, Raafat Talhouk. On the Cahn-Hilliard/Allen-Cahn equations with singular potentials. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3633-3651. doi: 10.3934/dcdsb.2018308
[6]

Alain Miranville, Ramon Quintanilla, Wafa Saoud. Asymptotic Behavior of a Cahn-Hilliard/Allen-Cahn System with Temperature. Communications on Pure & Applied Analysis, 2020, 19 (4) : 2257-2288. doi: 10.3934/cpaa.2020099
[7]

Ahmad Makki, Alain Miranville. Existence of solutions for anisotropic Cahn-Hilliard and Allen-Cahn systems in higher space dimensions. Discrete & Continuous Dynamical Systems - S, 2016, 9 (3) : 759-775. doi: 10.3934/dcdss.2016027
[8]

Cristina Pocci. On singular limit of a nonlinear $p$-order equation related to Cahn-Hilliard and Allen-Cahn evolutions. Evolution Equations & Control Theory, 2013, 2 (3) : 517-530. doi: 10.3934/eect.2013.2.517
[9]

Dimitra Antonopoulou, Georgia Karali. Existence of solution for a generalized stochastic Cahn-Hilliard equation on convex domains. Discrete & Continuous Dynamical Systems - B, 2011, 16 (1) : 31-55. doi: 10.3934/dcdsb.2011.16.31
[10]

Irena Pawłow. Thermodynamically consistent Cahn-Hilliard and Allen-Cahn models in elastic solids. Discrete & Continuous Dynamical Systems - A, 2006, 15 (4) : 1169-1191. doi: 10.3934/dcds.2006.15.1169
[11]

Gianni Gilardi. On an Allen-Cahn type integrodifferential equation. Discrete & Continuous Dynamical Systems - S, 2013, 6 (3) : 703-709. doi: 10.3934/dcdss.2013.6.703
[12]

Alain Miranville. Existence of solutions for Cahn-Hilliard type equations. Conference Publications, 2003, 2003 (Special) : 630-637. doi: 10.3934/proc.2003.2003.630
[13]

Desheng Li, Xuewei Ju. On dynamical behavior of viscous Cahn-Hilliard equation. Discrete & Continuous Dynamical Systems - A, 2012, 32 (6) : 2207-2221. doi: 10.3934/dcds.2012.32.2207
[14]

Laurence Cherfils, Alain Miranville, Sergey Zelik. On a generalized Cahn-Hilliard equation with biological applications. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 2013-2026. doi: 10.3934/dcdsb.2014.19.2013
[15]

Álvaro Hernández, Michał Kowalczyk. Rotationally symmetric solutions to the Cahn-Hilliard equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (2) : 801-827. doi: 10.3934/dcds.2017033
[16]

Isabeau Birindelli, Enrico Valdinoci. On the Allen-Cahn equation in the Grushin plane: A monotone entire solution that is not one-dimensional. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 823-838. doi: 10.3934/dcds.2011.29.823
[17]

Xinlong Feng, Yinnian He. On uniform in time $H^2$-regularity of the solution for the 2D Cahn-Hilliard equation. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5387-5400. doi: 10.3934/dcds.2016037
[18]

Fausto Cavalli, Giovanni Naldi. A Wasserstein approach to the numerical solution of the one-dimensional Cahn-Hilliard equation. Kinetic & Related Models, 2010, 3 (1) : 123-142. doi: 10.3934/krm.2010.3.123
[19]

Hongmei Cheng, Rong Yuan. Multidimensional stability of disturbed pyramidal traveling fronts in the Allen-Cahn equation. Discrete & Continuous Dynamical Systems - B, 2015, 20 (4) : 1015-1029. doi: 10.3934/dcdsb.2015.20.1015
[20]

Xinlong Feng, Huailing Song, Tao Tang, Jiang Yang. Nonlinear stability of the implicit-explicit methods for the Allen-Cahn equation. Inverse Problems & Imaging, 2013, 7 (3) : 679-695. doi: 10.3934/ipi.2013.7.679

2018 Impact Factor: 1.048

Tools

Metrics

  • PDF downloads (26)
  • HTML views (4)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences