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Optimal control for a hyperbolic problem in composites with imperfect interface: A memory effect
Existence of periodic solution for a Cahn-Hilliard/Allen-Cahn equation in two space dimensions
| 1. | Department of Mathematics, Jilin University, Changchun 130012, China |
| 2. | School of Science, Changchun University, Changchun 130022, China |
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.
References:
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Time periodic solution of the viscous Camassa-Holm equation, J. Math. Anal. Appl., 313 (2006), 311-321.
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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.
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G. Karali and M. A. Katsoulakis,
The role of multiple microscopic mechanisms in cluster interface evolution, J. Differential Equations, 235 (2007), 418-438.
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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. |
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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. |
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R. Wang,
The Schauder theory of the boundary value problem for parabolic problem equations, Acta Sci. Nature Univ. Jilin., 2 (1964), 35-64.
|
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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. |
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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. |
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J. Yin, Y. Li and R. Huang,
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doi: 10.1016/j.amc.2009.01.038. |
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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [7] |
R. Wang,
The Schauder theory of the boundary value problem for parabolic problem equations, Acta Sci. Nature Univ. Jilin., 2 (1964), 35-64.
|
| [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. |
| [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. |
| [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. |
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