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
[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. |