American Institute of Mathematical Sciences

September  2014, 13(5): 1855-1890. doi: 10.3934/cpaa.2014.13.1855

Non-isothermal viscous Cahn-Hilliard equation with inertial term and dynamic boundary conditions

 1 Dipartimento di Matematica “F. Enriques”, Università degli Studi di Milano, Via C. Saldini, 50, I-20133 Milano 2 Dipartimento di Matematica, Politecnico di Milano, 20133 Milano 3 School of Mathematical Sciences, Fudan University, Han Dan Road 220, 200433 Shanghai

Received  October 2013 Revised  February 2014 Published  June 2014

We consider a non-isothermal modified viscous Cahn-Hilliard equation which was previously analyzed by M. Grasselli et al. Such an equation is characterized by an inertial term and it is coupled with a hyperbolic heat equation from the Maxwell-Cattaneo's law. We analyze the case in which the order parameter is subject to a dynamic boundary condition. This assumption requires a more refined strategy to extend the previous results to the present case. More precisely, we first prove the well-posedness for solutions with finite energy as well as for weak solutions. Then we establish the existence of a global attractor. Finally, we prove the convergence of any given weak solution to a single equilibrium by using a suitable Łojasiewicz-Simon inequality.
Citation: Cecilia Cavaterra, Maurizio Grasselli, Hao Wu. Non-isothermal viscous Cahn-Hilliard equation with inertial term and dynamic boundary conditions. Communications on Pure and Applied Analysis, 2014, 13 (5) : 1855-1890. doi: 10.3934/cpaa.2014.13.1855
References:
 [1] A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, Studies Math. Appl., 25, North-Holland Publishing Co., Amsterdam, 1992. [2] A. Bonfoh, Existence and continuity of uniform exponential attractors for a singular perturbation of a generalized Cahn-Hilliard equation, Asymptot. Anal., 43 (2005), 233-247. [3] A. Bonfoh, M. Grasselli and A. Miranville, Long time behavior of a singular perturbation of the viscous Cahn-Hilliard-Gurtin equation, Math. Methods Appl. Sci., 31 (2008), 695-734. doi: 10.1002/mma.938. [4] A. Bonfoh, M. Grasselli and A. Miranville, Singularly perturbed 1D Cahn-Hilliard equation revisited, NoDEA Nonlinear Differential Equations Appl., 17 (2010), 663-695. doi: 10.1007/s00030-010-0075-0. [5] A. Bonfoh, M. Grasselli and A. Miranville, Inertial manifolds for a singular perturbation of the viscous Cahn-Hilliard-Gurtin equation, Topol. Methods Nonlinear Anal., 35 (2010), 155-185. [6] C. Cavaterra, C. G. Gal and M. Grasselli, Cahn-Hilliard equations with memory and dynamic boundary conditions, Asymptot. Anal., 71 (2011), 123-162. [7] L. Cherfils, A. Miranville and S. Zelik, The Cahn-Hilliard equation with logarithmic potentials, Milan J. Math., 79 (2011), 561-596. doi: 10.1007/s00032-011-0165-4. [8] R. Chill, E. Fašangová and J. Prüss, Convergence to steady states of solutions of the Cahn-Hilliard equation with dynamic boundary conditions, Math. Nachr., 279 (2006), 1448-1462. doi: 10.1002/mana.200410431. [9] E. Feireisl and F. Simondon, Convergence for semilinear degenerate parabolic equations in several space dimensions, J. Dynam. Differential Equations, 12 (2000), 647-673. doi: 10.1023/A:1026467729263. [10] H. P. Fischer, P. Maass and W. Dieterich, Novel surface modes in spinodal decomposition, Phys. Rev. Lett., 79 (1997), 893-896. [11] H. P. Fischer, P. Maass and W. Dieterich, Diverging time and length scales of spinodal decomposition modes in thin flows, Europhys. Lett., 42 (1998), 49-54. [12] C. G. Gal, Global well-posedness for the non-isothermal Cahn-Hilliard equation with dynamic boundary conditions, Adv. Differential Equations, 12 (2007), 1241-1274. [13] C. G. Gal, Well-posedness and long time behavior of the non-isothermal viscous Cahn-Hilliard equation with dynamic boundary conditions, Dyn. Partial Differ. Equ., 5 (2008), 39-67. doi: 10.4310/DPDE.2008.v5.n1.a2. [14] C. G. Gal and M. Grasselli, Singular limit of viscous Cahn-Hilliard equations with memory and dynamic boundary conditions, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 1581-1610. doi: 10.3934/dcdsb.2013.18.1581. [15] C. G. Gal and A. Miranville, Uniform global attractors for non-isothermal viscous and non-viscous Cahn-Hilliard equations with dynamic boundary conditions, Nonlinear Anal. Real World Appl., 10 (2009), 1738-1766. doi: 10.1016/j.nonrwa.2008.02.013. [16] C. G. Gal and A. Miranville, Robust exponential attractors and convergence to equilibria for non-isothermal Cahn-Hilliard equations with dynamic boundary conditions, Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 113-147. doi: 10.3934/dcdss.2009.2.113. [17] C. G. Gal and H. Wu, Asymptotic behavior of a Cahn-Hilliard equation with Wentzell boundary conditions and mass conservation, Discrete Contin. Dyn. Syst. Ser. A, 22 (2008), 1041-1063. doi: 10.3934/dcds.2008.22.1041. [18] P. Galenko, Phase-field model with relaxation of the diffusion flux in nonequilibrium solidification of a binary system, Phys. Lett. A, 287 (2001), 190-197. [19] P. Galenko and D. Jou, Diffuse-interface model for rapid phase transformations in nonequilibrium systems, Phys. Rev. E, 71 (2005), 046125 (13 pp.). [20] P. Galenko and D. Jou, Kinetic contribution to the fast spinodal decomposition controlled by diffusion, Phys. A, 388 (2009), 3113-3123. doi: 10.1016/j.physa.2009.04.003. [21] S. Gatti, M. Grasselli, A. Miranville and V. Pata, On the hyperbolic relaxation of the one-dimensional Cahn-Hilliard equation, J. Math. Anal. Appl., 312 (2005), 230-247. doi: 10.1016/j.jmaa.2005.03.029. [22] S. Gatti, M. Grasselli, A. Miranville and V. Pata, Hyperbolic relaxation of the viscous Cahn-Hilliard equation in 3D, Math. Models Methods Appl. Sci., 15 (2005), 165-198. doi: 10.1142/S0218202505000327. [23] G. Gilardi, A. Miranville and G. Schimperna, On the Cahn-Hilliard equation with irregular potentials and dynamic boundary conditions, Comm. Pure Appl. Anal., 8 (2009), 881-912. doi: 10.3934/cpaa.2009.8.881. [24] G. Gilardi, A. Miranville and G. Schimperna, Long time behavior of the Cahn-Hilliard equation with irregular potentials and dynamic boundary conditions, Chin. Ann. Math. Ser. B, 31 (2010), 679-712. doi: 10.1007/s11401-010-0602-7. [25] M. Grasselli and V. Pata, Asymptotic behavior of a parabolic-hyperbolic system, Comm. Pure Appl. Anal., 4 (2004), 849-881. doi: 10.3934/cpaa.2004.3.849. [26] M. Grasselli, H. Petzeltová and G. Schimperna, Convergence to stationary solutions for a parabolic-hyperbolic phase-field system, Commun. Pure Appl. Anal., 5 (2006), 827-838. doi: 10.3934/cpaa.2006.5.827. [27] M. Grasselli, H. Petzeltová and G. Schimperna, Asymptotic behavior of a nonisothermal viscous Cahn-Hilliard equation with inertial term, J. Differential Equations, 239 (2007), 38-60. doi: 10.1016/j.jde.2007.05.003. [28] M. Grasselli, G. Schimperna and S. Zelik, On the 2D Cahn-Hilliard equation with inertial term, Comm. Partial Differential Equations, 34 (2009), 137-170. doi: 10.1080/03605300802608247. [29] M. Grasselli, G. Schimperna and S. Zelik, Trajectory and smooth attractors for Cahn-Hilliard equations with inertial term, Nonlinearity, 23 (2010), 707-737. doi: 10.1088/0951-7715/23/3/016. [30] M. Grasselli, G. Schimperna, A. Segatti and S. Zelik, On the 3D Cahn-Hilliard equation with inertial term, J. Evol. Equ., 9 (2009), 371-404. doi: 10.1007/s00028-009-0017-7. [31] A. Haraux and M. A. Jendoubi, Convergence of bounded weak solutions of the wave equation with dissipation and analytic nonlinearity, Calc. Var. Partial Differential Equations, 9 (1999), 95-124. doi: 10.1007/s005260050133. [32] A. Haraux and M. A. Jendoubi, Decay estimates to equilibrium for some evolution equations with an analytic nonlinearity, Asymptot. Anal., 26 (2001), 21-36. [33] S.-Z. Huang and P. Takáč, Convergence in gradient-like systems which are asymptotically autonomous and analytic, Nonlinear Anal., 46 (2001), 675-698. doi: 10.1016/S0362-546X(00)00145-0. [34] M. B. Kania, Global attractor for the perturbed viscous Cahn-Hilliard equation, Colloq. Math., 109 (2007), 217-229. doi: 10.4064/cm109-2-4. [35] R. Kenzler, F. Eurich, P. Maass, B. Rinn, J. Schropp, E. Bohl and W. Dieterich, Phase separation in confined geometries: Solving the Cahn-Hilliard equation with generic boundary conditions, Comput. Phys. Commun., 133 (2001), 139-157. doi: 10.1016/S0010-4655(00)00159-4. [36] A. Miranville amd S. Zelik, Exponential attractors for the Cahn-Hilliard equation with dynamic boundary conditions, Math. Methods Appl. Sci., 28 (2005), 709-735. doi: 10.1002/mma.590. [37] A. Miranville and S. Zelik, The Cahn-Hilliard equation with singular potentials and dynamic boundary conditions, Discrete Contin. Dyn. Syst. Ser. A, 28 (2010), 275-310. doi: 10.3934/dcds.2010.28.275. [38] P. Monk, Finite Element Methods for Maxwell's Equations, Clarendon Press, Oxford, 2003. doi: 10.1093/acprof:oso/9780198508885.001.0001. [39] A. Novick-Cohen, On the viscous Cahn-Hilliard equation in Material instabilities in continuum mechanics (Edinburgh, 1985-1986), [40] A. Novick-Cohen, The Cahn-Hilliard equation: mathematical and modeling perspectives, Adv. Math. Sci. Appl., 8 (1998), 965-985. [41] A. Novick-Cohen, The Cahn-Hilliard equation, in Handbook of Differential Equations: Evolutionary Equations, Vol. IV, Elsevier/North-Holland, Amsterdam, (2008), 201-228. doi: 10.1016/S1874-5717(08)00004-2. [42] V. Pata and S. Zelik, A result on the existence of global attractors for semigroups of closed operators, Commun. Pure Appl. Anal., 6 (2007), 481-486. doi: 10.3934/cpaa.2007.6.481. [43] J. Prüss, R. Racke and S. Zheng, Maximal regularity and asymptotic behavior of solutions for the Cahn-Hilliard equation with dynamic boundary conditions, Ann. Mat. Pura Appl., 185 (2006), 627-648. doi: 10.1007/s10231-005-0175-3. [44] R. Racke and S. Zheng, The Cahn-Hilliard equation with dynamical boundary conditions, Adv. Differential Equations, 8 (2003), 83-110. [45] A. Segatti, On the hyperbolic relaxation of the Cahn-Hilliard equation in 3D: approximation and long time behaviour, Math. Models Methods Appl. Sci., 17 (2007), 411-437. doi: 10.1142/S0218202507001978. [46] J. Sprekels and H. Wu, A note on parabolic equation with nonlinear dynamical boundary condition, Nonlinear Anal. T.M.A., 72 (2010), 3028-3048. doi: 10.1016/j.na.2009.11.043. [47] B. Straughan, Heat Waves, Appl. Math. Sci., 177, Springer, New York, 2011. [48] R. Temam, Infinite Dimensional Dynamical System in Mechanics and Physics, Appl. Math. Sci. 68, Springer, New York, 1988. doi: 10.1007/978-1-4684-0313-8. [49] H. Wu, Convergence to equilibrium for a Cahn-Hilliard model with the Wentzell boundary condition, Asymptotic Analysis, 54 (2007), 71-92. [50] H. Wu, M. Grasselli and S. Zheng, Convergence to equilibrium for a parabolic-hyperbolic phase-field system with Neumann boundary conditions, Math. Models Methods Appl. Sci., 17 (2007), 125-153. doi: 10.1142/S0218202507001851. [51] H. Wu, M. Grasselli and S. Zheng, Convergence to equilibrium for a nonlinear parabolic-hyperbolic phase-field system with dynamic boundary condition, J. Math. Anal. Appl., 329 (2007), 948-976. doi: 10.1016/j.jmaa.2006.07.011. [52] H. Wu and S. Zheng, Convergence to equilibrium for the Cahn-Hilliard equation with dynamic boundary condition, J. Differential Equations, 204 (2004), 511-531. doi: 10.1016/j.jde.2004.05.004. [53] S. Zheng and A. Milani, Exponential attractors and inertial manifolds for singular perturbations of the Cahn-Hilliard equations, Nonlinear Anal., 57 (2004), 843-877. doi: 10.1016/j.na.2004.03.023. [54] S. Zheng and A. Milani, Global attractors for singular perturbations of the Cahn-Hilliard equations, J. Differential Equations, 209 (2005), 101-139. doi: 10.1016/j.jde.2004.08.026.

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References:
 [1] A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, Studies Math. Appl., 25, North-Holland Publishing Co., Amsterdam, 1992. [2] A. Bonfoh, Existence and continuity of uniform exponential attractors for a singular perturbation of a generalized Cahn-Hilliard equation, Asymptot. Anal., 43 (2005), 233-247. [3] A. Bonfoh, M. Grasselli and A. Miranville, Long time behavior of a singular perturbation of the viscous Cahn-Hilliard-Gurtin equation, Math. Methods Appl. Sci., 31 (2008), 695-734. doi: 10.1002/mma.938. [4] A. Bonfoh, M. Grasselli and A. Miranville, Singularly perturbed 1D Cahn-Hilliard equation revisited, NoDEA Nonlinear Differential Equations Appl., 17 (2010), 663-695. doi: 10.1007/s00030-010-0075-0. [5] A. Bonfoh, M. Grasselli and A. Miranville, Inertial manifolds for a singular perturbation of the viscous Cahn-Hilliard-Gurtin equation, Topol. Methods Nonlinear Anal., 35 (2010), 155-185. [6] C. Cavaterra, C. G. Gal and M. Grasselli, Cahn-Hilliard equations with memory and dynamic boundary conditions, Asymptot. Anal., 71 (2011), 123-162. [7] L. Cherfils, A. Miranville and S. Zelik, The Cahn-Hilliard equation with logarithmic potentials, Milan J. Math., 79 (2011), 561-596. doi: 10.1007/s00032-011-0165-4. [8] R. Chill, E. Fašangová and J. Prüss, Convergence to steady states of solutions of the Cahn-Hilliard equation with dynamic boundary conditions, Math. Nachr., 279 (2006), 1448-1462. doi: 10.1002/mana.200410431. [9] E. Feireisl and F. Simondon, Convergence for semilinear degenerate parabolic equations in several space dimensions, J. Dynam. Differential Equations, 12 (2000), 647-673. doi: 10.1023/A:1026467729263. [10] H. P. Fischer, P. Maass and W. Dieterich, Novel surface modes in spinodal decomposition, Phys. Rev. Lett., 79 (1997), 893-896. [11] H. P. Fischer, P. Maass and W. Dieterich, Diverging time and length scales of spinodal decomposition modes in thin flows, Europhys. Lett., 42 (1998), 49-54. [12] C. G. Gal, Global well-posedness for the non-isothermal Cahn-Hilliard equation with dynamic boundary conditions, Adv. Differential Equations, 12 (2007), 1241-1274. [13] C. G. Gal, Well-posedness and long time behavior of the non-isothermal viscous Cahn-Hilliard equation with dynamic boundary conditions, Dyn. Partial Differ. Equ., 5 (2008), 39-67. doi: 10.4310/DPDE.2008.v5.n1.a2. [14] C. G. Gal and M. Grasselli, Singular limit of viscous Cahn-Hilliard equations with memory and dynamic boundary conditions, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 1581-1610. doi: 10.3934/dcdsb.2013.18.1581. [15] C. G. Gal and A. Miranville, Uniform global attractors for non-isothermal viscous and non-viscous Cahn-Hilliard equations with dynamic boundary conditions, Nonlinear Anal. Real World Appl., 10 (2009), 1738-1766. doi: 10.1016/j.nonrwa.2008.02.013. [16] C. G. Gal and A. Miranville, Robust exponential attractors and convergence to equilibria for non-isothermal Cahn-Hilliard equations with dynamic boundary conditions, Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 113-147. doi: 10.3934/dcdss.2009.2.113. [17] C. G. Gal and H. Wu, Asymptotic behavior of a Cahn-Hilliard equation with Wentzell boundary conditions and mass conservation, Discrete Contin. Dyn. Syst. Ser. A, 22 (2008), 1041-1063. doi: 10.3934/dcds.2008.22.1041. [18] P. Galenko, Phase-field model with relaxation of the diffusion flux in nonequilibrium solidification of a binary system, Phys. Lett. A, 287 (2001), 190-197. [19] P. Galenko and D. Jou, Diffuse-interface model for rapid phase transformations in nonequilibrium systems, Phys. Rev. E, 71 (2005), 046125 (13 pp.). [20] P. Galenko and D. Jou, Kinetic contribution to the fast spinodal decomposition controlled by diffusion, Phys. A, 388 (2009), 3113-3123. doi: 10.1016/j.physa.2009.04.003. [21] S. Gatti, M. Grasselli, A. Miranville and V. Pata, On the hyperbolic relaxation of the one-dimensional Cahn-Hilliard equation, J. Math. Anal. Appl., 312 (2005), 230-247. doi: 10.1016/j.jmaa.2005.03.029. [22] S. Gatti, M. Grasselli, A. Miranville and V. Pata, Hyperbolic relaxation of the viscous Cahn-Hilliard equation in 3D, Math. Models Methods Appl. Sci., 15 (2005), 165-198. doi: 10.1142/S0218202505000327. [23] G. Gilardi, A. Miranville and G. Schimperna, On the Cahn-Hilliard equation with irregular potentials and dynamic boundary conditions, Comm. Pure Appl. Anal., 8 (2009), 881-912. doi: 10.3934/cpaa.2009.8.881. [24] G. Gilardi, A. Miranville and G. Schimperna, Long time behavior of the Cahn-Hilliard equation with irregular potentials and dynamic boundary conditions, Chin. Ann. Math. Ser. B, 31 (2010), 679-712. doi: 10.1007/s11401-010-0602-7. [25] M. Grasselli and V. Pata, Asymptotic behavior of a parabolic-hyperbolic system, Comm. Pure Appl. Anal., 4 (2004), 849-881. doi: 10.3934/cpaa.2004.3.849. [26] M. Grasselli, H. Petzeltová and G. Schimperna, Convergence to stationary solutions for a parabolic-hyperbolic phase-field system, Commun. Pure Appl. Anal., 5 (2006), 827-838. doi: 10.3934/cpaa.2006.5.827. [27] M. Grasselli, H. Petzeltová and G. Schimperna, Asymptotic behavior of a nonisothermal viscous Cahn-Hilliard equation with inertial term, J. Differential Equations, 239 (2007), 38-60. doi: 10.1016/j.jde.2007.05.003. [28] M. Grasselli, G. Schimperna and S. Zelik, On the 2D Cahn-Hilliard equation with inertial term, Comm. Partial Differential Equations, 34 (2009), 137-170. doi: 10.1080/03605300802608247. [29] M. Grasselli, G. Schimperna and S. Zelik, Trajectory and smooth attractors for Cahn-Hilliard equations with inertial term, Nonlinearity, 23 (2010), 707-737. doi: 10.1088/0951-7715/23/3/016. [30] M. Grasselli, G. Schimperna, A. Segatti and S. Zelik, On the 3D Cahn-Hilliard equation with inertial term, J. Evol. Equ., 9 (2009), 371-404. doi: 10.1007/s00028-009-0017-7. [31] A. Haraux and M. A. Jendoubi, Convergence of bounded weak solutions of the wave equation with dissipation and analytic nonlinearity, Calc. Var. Partial Differential Equations, 9 (1999), 95-124. doi: 10.1007/s005260050133. [32] A. Haraux and M. A. Jendoubi, Decay estimates to equilibrium for some evolution equations with an analytic nonlinearity, Asymptot. Anal., 26 (2001), 21-36. [33] S.-Z. Huang and P. Takáč, Convergence in gradient-like systems which are asymptotically autonomous and analytic, Nonlinear Anal., 46 (2001), 675-698. doi: 10.1016/S0362-546X(00)00145-0. [34] M. B. Kania, Global attractor for the perturbed viscous Cahn-Hilliard equation, Colloq. Math., 109 (2007), 217-229. doi: 10.4064/cm109-2-4. [35] R. Kenzler, F. Eurich, P. Maass, B. Rinn, J. Schropp, E. Bohl and W. Dieterich, Phase separation in confined geometries: Solving the Cahn-Hilliard equation with generic boundary conditions, Comput. Phys. Commun., 133 (2001), 139-157. doi: 10.1016/S0010-4655(00)00159-4. [36] A. Miranville amd S. Zelik, Exponential attractors for the Cahn-Hilliard equation with dynamic boundary conditions, Math. Methods Appl. Sci., 28 (2005), 709-735. doi: 10.1002/mma.590. [37] A. Miranville and S. Zelik, The Cahn-Hilliard equation with singular potentials and dynamic boundary conditions, Discrete Contin. Dyn. Syst. Ser. A, 28 (2010), 275-310. doi: 10.3934/dcds.2010.28.275. [38] P. Monk, Finite Element Methods for Maxwell's Equations, Clarendon Press, Oxford, 2003. doi: 10.1093/acprof:oso/9780198508885.001.0001. [39] A. Novick-Cohen, On the viscous Cahn-Hilliard equation in Material instabilities in continuum mechanics (Edinburgh, 1985-1986), [40] A. Novick-Cohen, The Cahn-Hilliard equation: mathematical and modeling perspectives, Adv. Math. Sci. Appl., 8 (1998), 965-985. [41] A. Novick-Cohen, The Cahn-Hilliard equation, in Handbook of Differential Equations: Evolutionary Equations, Vol. IV, Elsevier/North-Holland, Amsterdam, (2008), 201-228. doi: 10.1016/S1874-5717(08)00004-2. [42] V. Pata and S. Zelik, A result on the existence of global attractors for semigroups of closed operators, Commun. Pure Appl. Anal., 6 (2007), 481-486. doi: 10.3934/cpaa.2007.6.481. [43] J. Prüss, R. Racke and S. Zheng, Maximal regularity and asymptotic behavior of solutions for the Cahn-Hilliard equation with dynamic boundary conditions, Ann. Mat. Pura Appl., 185 (2006), 627-648. doi: 10.1007/s10231-005-0175-3. [44] R. Racke and S. Zheng, The Cahn-Hilliard equation with dynamical boundary conditions, Adv. Differential Equations, 8 (2003), 83-110. [45] A. Segatti, On the hyperbolic relaxation of the Cahn-Hilliard equation in 3D: approximation and long time behaviour, Math. Models Methods Appl. Sci., 17 (2007), 411-437. doi: 10.1142/S0218202507001978. [46] J. Sprekels and H. Wu, A note on parabolic equation with nonlinear dynamical boundary condition, Nonlinear Anal. T.M.A., 72 (2010), 3028-3048. doi: 10.1016/j.na.2009.11.043. [47] B. Straughan, Heat Waves, Appl. Math. Sci., 177, Springer, New York, 2011. [48] R. Temam, Infinite Dimensional Dynamical System in Mechanics and Physics, Appl. Math. Sci. 68, Springer, New York, 1988. doi: 10.1007/978-1-4684-0313-8. [49] H. Wu, Convergence to equilibrium for a Cahn-Hilliard model with the Wentzell boundary condition, Asymptotic Analysis, 54 (2007), 71-92. [50] H. Wu, M. Grasselli and S. Zheng, Convergence to equilibrium for a parabolic-hyperbolic phase-field system with Neumann boundary conditions, Math. Models Methods Appl. Sci., 17 (2007), 125-153. doi: 10.1142/S0218202507001851. [51] H. Wu, M. Grasselli and S. Zheng, Convergence to equilibrium for a nonlinear parabolic-hyperbolic phase-field system with dynamic boundary condition, J. Math. Anal. Appl., 329 (2007), 948-976. doi: 10.1016/j.jmaa.2006.07.011. [52] H. Wu and S. Zheng, Convergence to equilibrium for the Cahn-Hilliard equation with dynamic boundary condition, J. Differential Equations, 204 (2004), 511-531. doi: 10.1016/j.jde.2004.05.004. [53] S. Zheng and A. Milani, Exponential attractors and inertial manifolds for singular perturbations of the Cahn-Hilliard equations, Nonlinear Anal., 57 (2004), 843-877. doi: 10.1016/j.na.2004.03.023. [54] S. Zheng and A. Milani, Global attractors for singular perturbations of the Cahn-Hilliard equations, J. Differential Equations, 209 (2005), 101-139. doi: 10.1016/j.jde.2004.08.026.
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