### AIMS Mathematics

2018, Issue 2: 263-287. doi: 10.3934/Math.2018.2.263
Research article

# Time periodic solutions of Cahn-Hilliard systems with dynamic boundary conditions

• Received: 20 February 2018 Accepted: 03 April 2018 Published: 25 May 2018
• MSC : 35K25, 35A01, 35B10, 35D30

• The existence problem for Cahn-Hilliard systems with dynamic boundary conditions and time periodic conditions is discussed. We apply the abstract theory of evolution equations with viscosity approach and the Schauder fixed point theorem in the level of approximate problems. One of the key points is the assumption for maximal monotone graphs with respect to their e ective domains. Thanks to this, we obtain the existence result of periodic solutions by using the passage to the limit.

Citation: Taishi Motoda. Time periodic solutions of Cahn-Hilliard systems with dynamic boundary conditions[J]. AIMS Mathematics, 2018, 3(2): 263-287. doi: 10.3934/Math.2018.2.263

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• The existence problem for Cahn-Hilliard systems with dynamic boundary conditions and time periodic conditions is discussed. We apply the abstract theory of evolution equations with viscosity approach and the Schauder fixed point theorem in the level of approximate problems. One of the key points is the assumption for maximal monotone graphs with respect to their e ective domains. Thanks to this, we obtain the existence result of periodic solutions by using the passage to the limit.
 [1] T. Aiki, Two-phase Stefan problems with dynamic boundary conditions, Adv. Math. Sci. Appl., 2 (1993), 253-270. [2] T. Aiki, Multi-dimensional Stefan problems with dynamic boundary conditions, Appl. Anal., 56 (1995), 71-94. [3] T. Aiki, Periodic stability of solutions to some degenerate parabolic equations with dynamic boundary conditions, J. Math. Soc. Japan, 48 (1996), 37-59. [4] G. Akagi and U. Stefanelli, Periodic solutions for doubly nonlinear evolution equations, J. Di er. Equations, 251 (2011), 1790-1812. [5] V. Barbu, Nonlinear di erential equations of monotone types in Banach spaces, Springer, London, 2010. [6] H. Brézis, Opérateurs maximaux monotones et semi-groupes de contractions dans les especes de Hilbert, North-Holland, Amsterdam, 1973. [7] F. Brezzi and G. Gilardi, Chapters 1-3 in Finite element handbook, H. Kardestuncer and D. H. Norrie (Eds. ), McGraw-Hill Book Co., New York, 1987. [8] J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system Ⅰ. interfacial free energy, J. Chem. Phys., 2 (1958), 258-267. [9] L. Calatroni and P. Colli, Global solution to the Allen-Cahn equation with singular potentials and dynamic boundary conditions, Nonlinear Anal., 79 (2013), 12-27. [10] L. Cherfils, A. Miranville and S. Zelik, The Cahn-Hilliard equation with logarithmic potentials, Milan J. Math., 79 (2011), 561-596. [11] P. Colli and T. Fukao, Equation and dynamic boundary condition of Cahn-Hilliard type with singular potentials, Nonlinear Anal., 127 (2015), 413-433. [12] P. Colli, G. Gilardi and J. Sprekels, On the Cahn-Hilliard equation with dynamic boundary conditions and a dominating boundary potential, J. Math. Anal. Appl., 419 (2014), 972-994. [13] A. Damlamian and N. Kenmochi, Evolution equations associated with non-isothermal phase separation: a subdi erential approach, Ann. Mat. Pura Appl., 176 (1999), 167-190. [14] T. Fukao, Convergence of Cahn-Hilliard systems to the Stefan problem with dynamic boundary conditions, Asymptot. Anal., 99 (2016), 1-21. [15] T. Fukao, Cahn-Hilliard approach to some degenerate parabolic equations with dynamic boundary conditions, In: L. Bociu, J-A. Désidéri and A. Habbal (Eds. ), System Modeling and Optimization, Springer, Switzerland, 2016, pp. 282-291. [16] T. Fukao and T. Motoda, Abstract approach to degenerate parabolic equations with dynamic boundary conditions, to appear in Adv. Math. Sci. Appl., 27 (2018), 29-44. [17] C. Gal, A Cahn-Hilliard model in bounded domains with permeable walls, Math. Methods Appl. Sci., 29 (2006), 2009-2036. [18] G. Gilardi, A. Miranville and G. Schimperna, On the Cahn-Hilliard equation with irregular potentials and dynamic boundary conditions, Commun. Pure Appl. Anal., 8 (2009), 881-912. [19] G. R. Goldstein and A. Miranville, A Cahn-Hilliard-Gurtin model with dynamic boundary conditions, Discrete Contin. Dyn. Syst. Ser. S, 6 (2013), 387-400. [20] G. R. Goldstein, A. Miranville and G. Schimperna, A Cahn-Hilliard model in a domain with non-permeable walls, Phys. D, 240 (2011), 754-766. [21] A. Grigor'yan, Heat kernel and analysis on manifolds, American Mathematical Society, International Press, Boston, 2009. [22] N. Kajiwara, Maximal Lp regularity of a Cahn-Hilliard equation in bounded domains with permeable and non-permeable walls, preprint, 2017. [23] N. Kenmochi, Monotonicity and compactness methods for nonlinear variational inequalities, M. Chipot (Ed. ), Handbook of di erential equations: Stationary partial di erential equations, North-Holland, Amsterdam, 4 (2007), 203-298. [24] N. Kenmochi, M. Niezgódka and I. Pawłow, Subdi erential operator approach to the Cahn-Hilliard equation with constraint, J. Di erential Equations, 117 (1995), 320-354. [25] M. Kubo, The Cahn-Hilliard equation with time-dependent constraint, Nonlinear Anal., 75 (2012), 5672-5685. [26] Y. Li and Y. Jingxue, The viscous Cahn-Hilliard equation with periodic potentials and sources, J. Fixed Point Theory Appl., 9 (2011), 63-84. [27] Y. Li, L. Yinghua, H. Rui and Y. Jingxue, Time periodic solutions for a Cahn-Hilliard type equation, Math. Comput. Modelling, 48 (2008), 11-18. [28] J. Liu, Y. Wang and J. Zheng, Periodic solutions of a multi-dimensional Cahn-Hilliard equation, Electron. J. Di erential Equations, 2016 (2016), 1-23. [29] C. Liu and H. Wu, An energetic variational approach for the Cahn-Hilliard equation with dynamic boundary conditions: derivation and analysis, preprint arXiv: 1710. 08318v1[math. AP], 2017, pp. 1-68. [30] J. Simon, Compact sets in the spaces Lp(0; T; B), Ann. Mat. Pura. Appl., 146 (1987), 65-96. [31] Y. Wang and J. Zheng, Periodic solutions to the Cahn-Hilliard equation with constraint, Math. Methods Apply. Sci., 39 (2016), 649-660.

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