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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

 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.

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, 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, 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, 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, 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, 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, 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, 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, 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, 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.

show all references

##### References:
 [1] Y. Fu, 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, 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, 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, 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, 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, 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, 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, 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, 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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