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

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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

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

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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

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