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A Cahn-Hilliard model in a domain with non-permeable walls. (English) Zbl 1228.35051
Let \(\Omega\) be a bounded domain of \(\mathbb R^3\) with smooth boundary \(\Gamma\) and consider the following Cahn-Hilliard equation
\[ \partial_t\rho - \Delta\mu = 0 , \quad \mu = -\Delta\rho + f(\rho) \quad\text{in }(0,\infty)\times \Omega \] with dynamic boundary conditions
\[ w\partial_t \rho - \delta \Delta_\Gamma \mu + \partial_n \mu = \partial_n\rho - \sigma\Delta_\Gamma \rho + g(\rho) - w\mu = 0 \quad \text{on } (0,\infty)\times\Gamma. \] Here \(\delta\) and \(\sigma\) are nonnegative real numbers, \(w\) is a positive bounded function such that \(1/w\) is bounded, \(\Delta_\Gamma\) denotes the Laplace-Beltrami operator on \(\Gamma\), \(f\) is the derivative of a (possibly non-smooth) double-well potential and \(g\) is a smooth function. The existence and uniqueness of a weak solution \(\rho\) are proved for a rather general class of data \(f\) and \(g\). Regularity for positive times of the solution is investigated according to the growth of \(f\) and \(g\). Existence of a bounded absorbing set and a compact attractor is shown as well. Finally, if \(f\) and \(g\) are also assumed to be real analytic functions, convergence of the solution to a steady state is proved with the help of the now classical Łojasiewicz-Simon inequality.

MSC:
35B40 Asymptotic behavior of solutions to PDEs
35K52 Initial-boundary value problems for higher-order parabolic systems
35B41 Attractors
80A22 Stefan problems, phase changes, etc.
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[1] Cahn, J.W., On spinodal decomposition, Acta metall., 9, 795-801, (1961)
[2] Cahn, J.W.; Hilliard, J.E., Free energy of a nonuniform system. I. interfacial free energy, J. chem. phys., 28, 258-267, (1958)
[3] Novick-Cohen, A., The cahn – hilliard equation, (), 201-228 · Zbl 1185.35001
[4] Fischer, H.P.; Maass, P.; Dieterich, W., Novel surface modes in spinodal decomposition, Phys. rev. lett., 79, 893-896, (1997)
[5] Fischer, H.P.; Maass, P.; Dieterich, W., Diverging time and length scales of spinodal decomposition modes in thin flows, Europhys. lett., 42, 49-54, (1998)
[6] Fischer, H.P.; Reinhard, J.; Dieterich, W.; Gouyet, J.-F.; Maass, P.; Majhofer, A.; Reinel, D., Time-dependent density functional theory and the kinetics of lattice gas systems in contact with a wall, J. chem. phys., 108, 3028-3037, (1998)
[7] Chill, R.; Fašangová, E.; Prüss, J., Convergence to steady states of solutions of the cahn – hilliard equation with dynamic boundary conditions, Math. nachr., 279, 1448-1462, (2006) · Zbl 1107.35058
[8] Gilardi, G.; Miranville, A.; Schimperna, G., On the cahn – hilliard equation with irregular potentials and dynamic boundary conditions, Commun. pure appl. anal., 8, 881-912, (2009) · Zbl 1172.35417
[9] Gilardi, G.; Miranville, A.; Schimperna, G., Long-time behavior of the cahn – hilliard equation with irregular potentials and dynamic boundary conditions, Chin. ann. math. ser. B, 31, 679-712, (2010) · Zbl 1223.35067
[10] Miranville, A.; Zelik, S., Exponential attractors for the cahn – hilliard equation with dynamic boundary conditions, Math. methods appl. sci., 28, 709-735, (2005) · Zbl 1068.35020
[11] Miranville, A.; Zelik, S., The cahn – hilliard equation with singular potentials and dynamic boundary conditions, Discrete contin. dyn. syst., 28, 275-310, (2010) · Zbl 1203.35046
[12] Prüss, J.; Racke, R.; Zheng, S., Maximal regularity and asymptotic behavior of solutions for the cahn – hilliard equation with dynamic boundary conditions, Ann. mat. pura appl. (4), 185, 627-648, (2006) · Zbl 1232.35081
[13] Racke, R.; Zheng, S., The cahn – hilliard equation with dynamic boundary conditions, Adv. differential equations, 8, 83-110, (2003) · Zbl 1035.35050
[14] Cherfils, L.; Gatti, S.; Miranville, A., Corrigendum to “existence of global solutions to the Caginalp phase-field system with dynamic boundary conditions and singular potentials” [J. math. anal. appl. 343 (2008) 557-566], J. math. anal. appl., 348, 1029-1030, (2008) · Zbl 1160.35433
[15] Cherfils, L.; Miranville, A., On the Caginalp system with dynamic boundary conditions and singular potentials, Appl. math., 54, 89-115, (2009) · Zbl 1212.35012
[16] Gal, C.G.; Grasselli, M., The nonisothermal allen – cahn equation with dynamic boundary conditions, Discrete contin. dyn. syst. A, 22, 1009-1040, (2008) · Zbl 1160.35353
[17] Gal, C.G.; Grasselli, M., On the asymptotic behavior of the Caginalp system with dynamic boundary conditions, Commun. pure appl. anal., 8, 689-710, (2009) · Zbl 1171.35337
[18] Gal, C.G.; Grasselli, M.; Miranville, A., Nonisothermal allen – cahn equations with coupled dynamic boundary conditions, (), 117-139 · Zbl 1178.35076
[19] Gatti, S.; Miranville, A., Asymptotic behavior of a phase-field system with dynamic boundary conditions, (), 149-170 · Zbl 1123.35310
[20] Grasselli, M.; Miranville, A.; Schimperna, G., The Caginalp phase-field system with coupled dynamic boundary conditions and singular potentials, Discrete contin. dyn. syst., 28, 67-98, (2010) · Zbl 1194.35074
[21] Gal, C.G., A cahn – hilliard model in bounded domains with permeable walls, Math. methods appl. sci., 29, 2009-2036, (2006) · Zbl 1109.35057
[22] Gal, C.G., Exponential attractors for a cahn – hilliard model in bounded domains with permeable walls, Electron. J. differential equations, 143, 23 pp, (2006), (electronic) · Zbl 1113.35031
[23] Gal, C.G., Robust exponential attractors for a conserved cahn – hillard model with singularly perturbed boundary conditions, Commun. pure appl. anal., 7, 819-836, (2008) · Zbl 1165.35327
[24] Gal, C.G.; Wu, H., Asymptotic behavior of a cahn – hilliard equation with Wentzell boundary conditions and mass conservation, Discrete contin. dyn. syst., 22, 1041-1063, (2008) · Zbl 1158.35052
[25] Schimperna, G., Abstract approach to evolution equations of phase field type and applications, J. differential equations, 164, 395-430, (2000) · Zbl 0978.35075
[26] Fasano, A.; Primicerio, M.; Rubinstein, L., A model problem for heat conduction with a free boundary in a concentrated capacity, J. inst. math. appl., 26, 327-347, (1980) · Zbl 0456.35093
[27] Magenes, E., Some new results on a Stefan problem in a concentrated capacity, Rend. mat. accad. lincei, IX, 3, 23-34, (1992) · Zbl 0767.35110
[28] Savaré, G.; Visintin, A., Variational convergence of nonlinear diffusion equations: applications to concentrated capacity problems with change of phase, Atti accad. naz. lincei cl. sci. fis. mat. natur. rend. lincei (9) mat. appl., 8, 49-89, (1997) · Zbl 0888.35139
[29] Schimperna, G., Weak solution to a phase-field transmission problem in a concentrated capacity, Math. methods appl. sci., 22, 1235-1254, (1999) · Zbl 0933.35198
[30] Miranville, A.; Zelik, S., Robust exponential attractors for cahn – hilliard type equations with singular potentials, Math. methods appl. sci., 27, 545-582, (2004) · Zbl 1050.35113
[31] Brézis, H., Intégrales convexes dans LES espaces de Sobolev, Israel J. math., 13, 9-23, (1972), (in French)
[32] Grun-Rehomme, M., Caractérisation du sous-différentiel d’intégrandes convexes dans LES espaces de Sobolev, J. math. pures appl. (9), 56, 149-156, (1977), (in French) · Zbl 0314.35001
[33] Colli, P.; Gilardi, G.; Grasselli, M.; Schimperna, G., The conserved phase-field system with memory, Adv. math. sci. appl., 11, 265-291, (2001) · Zbl 0982.80006
[34] Rocca, E.; Schimperna, G., Universal attractor for some singular phase transition systems, Physica D, 192, 279-307, (2004) · Zbl 1062.82015
[35] Pata, V.; Zelik, S., A result on the existence of global attractors for semigroups of closed operators, Commun. pure appl. anal., 6, 481-486, (2007) · Zbl 1152.47046
[36] Barbu, V., Nonlinear semigroups and differential equations in Banach spaces, (1976), Noordhoff Leyden
[37] Temam, R., Infinite-dimensional dynamical systems in mechanics and physics, (1988), Springer-Verlag New York · Zbl 0662.35001
[38] Geymonat, G., Trace theorems for Sobolev spaces on Lipschitz domains. necessary conditions, Ann. math. blaise Pascal, 14, 187-197, (2007) · Zbl 1161.46019
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