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 & Applied Analysis, 2014, 13 (5) : 1855-1890. doi: 10.3934/cpaa.2014.13.1855
References:
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A. V. Babin and M. I. Vishik, Attractors of Evolution Equations,, Studies Math. Appl., 25 (1992).   Google Scholar

[2]

A. Bonfoh, Existence and continuity of uniform exponential attractors for a singular perturbation of a generalized Cahn-Hilliard equation,, \emph{Asymptot. Anal.}, 43 (2005), 233.   Google Scholar

[3]

A. Bonfoh, M. Grasselli and A. Miranville, Long time behavior of a singular perturbation of the viscous Cahn-Hilliard-Gurtin equation,, \emph{Math. Methods Appl. Sci.}, 31 (2008), 695.  doi: 10.1002/mma.938.  Google Scholar

[4]

A. Bonfoh, M. Grasselli and A. Miranville, Singularly perturbed 1D Cahn-Hilliard equation revisited,, \emph{NoDEA Nonlinear Differential Equations Appl.}, 17 (2010), 663.  doi: 10.1007/s00030-010-0075-0.  Google Scholar

[5]

A. Bonfoh, M. Grasselli and A. Miranville, Inertial manifolds for a singular perturbation of the viscous Cahn-Hilliard-Gurtin equation,, \emph{Topol. Methods Nonlinear Anal.}, 35 (2010), 155.   Google Scholar

[6]

C. Cavaterra, C. G. Gal and M. Grasselli, Cahn-Hilliard equations with memory and dynamic boundary conditions,, \emph{Asymptot. Anal.}, 71 (2011), 123.   Google Scholar

[7]

L. Cherfils, A. Miranville and S. Zelik, The Cahn-Hilliard equation with logarithmic potentials,, \emph{Milan J. Math.}, 79 (2011), 561.  doi: 10.1007/s00032-011-0165-4.  Google Scholar

[8]

R. Chill, E. Fašangová and J. Prüss, Convergence to steady states of solutions of the Cahn-Hilliard equation with dynamic boundary conditions,, \emph{Math. Nachr.}, 279 (2006), 1448.  doi: 10.1002/mana.200410431.  Google Scholar

[9]

E. Feireisl and F. Simondon, Convergence for semilinear degenerate parabolic equations in several space dimensions,, \emph{J. Dynam. Differential Equations}, 12 (2000), 647.  doi: 10.1023/A:1026467729263.  Google Scholar

[10]

H. P. Fischer, P. Maass and W. Dieterich, Novel surface modes in spinodal decomposition,, \emph{Phys. Rev. Lett.}, 79 (1997), 893.   Google Scholar

[11]

H. P. Fischer, P. Maass and W. Dieterich, Diverging time and length scales of spinodal decomposition modes in thin flows,, \emph{Europhys. Lett.}, 42 (1998), 49.   Google Scholar

[12]

C. G. Gal, Global well-posedness for the non-isothermal Cahn-Hilliard equation with dynamic boundary conditions,, \emph{Adv. Differential Equations}, 12 (2007), 1241.   Google Scholar

[13]

C. G. Gal, Well-posedness and long time behavior of the non-isothermal viscous Cahn-Hilliard equation with dynamic boundary conditions,, \emph{Dyn. Partial Differ. Equ.}, 5 (2008), 39.  doi: 10.4310/DPDE.2008.v5.n1.a2.  Google Scholar

[14]

C. G. Gal and M. Grasselli, Singular limit of viscous Cahn-Hilliard equations with memory and dynamic boundary conditions,, \emph{Discrete Contin. Dyn. Syst. Ser. B}, 18 (2013), 1581.  doi: 10.3934/dcdsb.2013.18.1581.  Google Scholar

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C. G. Gal and A. Miranville, Uniform global attractors for non-isothermal viscous and non-viscous Cahn-Hilliard equations with dynamic boundary conditions,, \emph{Nonlinear Anal. Real World Appl.}, 10 (2009), 1738.  doi: 10.1016/j.nonrwa.2008.02.013.  Google Scholar

[16]

C. G. Gal and A. Miranville, Robust exponential attractors and convergence to equilibria for non-isothermal Cahn-Hilliard equations with dynamic boundary conditions,, \emph{Discrete Contin. Dyn. Syst. Ser. S}, 2 (2009), 113.  doi: 10.3934/dcdss.2009.2.113.  Google Scholar

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C. G. Gal and H. Wu, Asymptotic behavior of a Cahn-Hilliard equation with Wentzell boundary conditions and mass conservation,, \emph{Discrete Contin. Dyn. Syst. Ser. A}, 22 (2008), 1041.  doi: 10.3934/dcds.2008.22.1041.  Google Scholar

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P. Galenko, Phase-field model with relaxation of the diffusion flux in nonequilibrium solidification of a binary system,, \emph{Phys. Lett. A}, 287 (2001), 190.   Google Scholar

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P. Galenko and D. Jou, Diffuse-interface model for rapid phase transformations in nonequilibrium systems,, \emph{Phys. Rev. E}, 71 (2005).   Google Scholar

[20]

P. Galenko and D. Jou, Kinetic contribution to the fast spinodal decomposition controlled by diffusion,, \emph{Phys. A}, 388 (2009), 3113.  doi: 10.1016/j.physa.2009.04.003.  Google Scholar

[21]

S. Gatti, M. Grasselli, A. Miranville and V. Pata, On the hyperbolic relaxation of the one-dimensional Cahn-Hilliard equation,, \emph{J. Math. Anal. Appl.}, 312 (2005), 230.  doi: 10.1016/j.jmaa.2005.03.029.  Google Scholar

[22]

S. Gatti, M. Grasselli, A. Miranville and V. Pata, Hyperbolic relaxation of the viscous Cahn-Hilliard equation in 3D,, \emph{Math. Models Methods Appl. Sci.}, 15 (2005), 165.  doi: 10.1142/S0218202505000327.  Google Scholar

[23]

G. Gilardi, A. Miranville and G. Schimperna, On the Cahn-Hilliard equation with irregular potentials and dynamic boundary conditions,, \emph{Comm. Pure Appl. Anal.}, 8 (2009), 881.  doi: 10.3934/cpaa.2009.8.881.  Google Scholar

[24]

G. Gilardi, A. Miranville and G. Schimperna, Long time behavior of the Cahn-Hilliard equation with irregular potentials and dynamic boundary conditions,, \emph{Chin. Ann. Math. Ser. B}, 31 (2010), 679.  doi: 10.1007/s11401-010-0602-7.  Google Scholar

[25]

M. Grasselli and V. Pata, Asymptotic behavior of a parabolic-hyperbolic system,, \emph{Comm. Pure Appl. Anal.}, 4 (2004), 849.  doi: 10.3934/cpaa.2004.3.849.  Google Scholar

[26]

M. Grasselli, H. Petzeltová and G. Schimperna, Convergence to stationary solutions for a parabolic-hyperbolic phase-field system,, \emph{Commun. Pure Appl. Anal.}, 5 (2006), 827.  doi: 10.3934/cpaa.2006.5.827.  Google Scholar

[27]

M. Grasselli, H. Petzeltová and G. Schimperna, Asymptotic behavior of a nonisothermal viscous Cahn-Hilliard equation with inertial term,, \emph{J. Differential Equations}, 239 (2007), 38.  doi: 10.1016/j.jde.2007.05.003.  Google Scholar

[28]

M. Grasselli, G. Schimperna and S. Zelik, On the 2D Cahn-Hilliard equation with inertial term,, \emph{Comm. Partial Differential Equations}, 34 (2009), 137.  doi: 10.1080/03605300802608247.  Google Scholar

[29]

M. Grasselli, G. Schimperna and S. Zelik, Trajectory and smooth attractors for Cahn-Hilliard equations with inertial term,, \emph{Nonlinearity}, 23 (2010), 707.  doi: 10.1088/0951-7715/23/3/016.  Google Scholar

[30]

M. Grasselli, G. Schimperna, A. Segatti and S. Zelik, On the 3D Cahn-Hilliard equation with inertial term,, \emph{J. Evol. Equ.}, 9 (2009), 371.  doi: 10.1007/s00028-009-0017-7.  Google Scholar

[31]

A. Haraux and M. A. Jendoubi, Convergence of bounded weak solutions of the wave equation with dissipation and analytic nonlinearity,, \emph{Calc. Var. Partial Differential Equations}, 9 (1999), 95.  doi: 10.1007/s005260050133.  Google Scholar

[32]

A. Haraux and M. A. Jendoubi, Decay estimates to equilibrium for some evolution equations with an analytic nonlinearity,, \emph{Asymptot. Anal.}, 26 (2001), 21.   Google Scholar

[33]

S.-Z. Huang and P. Takáč, Convergence in gradient-like systems which are asymptotically autonomous and analytic,, \emph{Nonlinear Anal.}, 46 (2001), 675.  doi: 10.1016/S0362-546X(00)00145-0.  Google Scholar

[34]

M. B. Kania, Global attractor for the perturbed viscous Cahn-Hilliard equation,, \emph{Colloq. Math.}, 109 (2007), 217.  doi: 10.4064/cm109-2-4.  Google Scholar

[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,, \emph{Comput. Phys. Commun.}, 133 (2001), 139.  doi: 10.1016/S0010-4655(00)00159-4.  Google Scholar

[36]

A. Miranville amd S. Zelik, Exponential attractors for the Cahn-Hilliard equation with dynamic boundary conditions,, \emph{Math. Methods Appl. Sci.}, 28 (2005), 709.  doi: 10.1002/mma.590.  Google Scholar

[37]

A. Miranville and S. Zelik, The Cahn-Hilliard equation with singular potentials and dynamic boundary conditions,, \emph{Discrete Contin. Dyn. Syst. Ser. A}, 28 (2010), 275.  doi: 10.3934/dcds.2010.28.275.  Google Scholar

[38]

P. Monk, Finite Element Methods for Maxwell's Equations,, Clarendon Press, (2003).  doi: 10.1093/acprof:oso/9780198508885.001.0001.  Google Scholar

[39]

A. Novick-Cohen, On the viscous Cahn-Hilliard equation, in \emph{Material instabilities in continuum mechanics (Edinburgh, (): 1985.   Google Scholar

[40]

A. Novick-Cohen, The Cahn-Hilliard equation: mathematical and modeling perspectives,, \emph{Adv. Math. Sci. Appl.}, 8 (1998), 965.   Google Scholar

[41]

A. Novick-Cohen, The Cahn-Hilliard equation, in Handbook of Differential Equations: Evolutionary Equations, Vol. IV,, Elsevier/North-Holland, (2008), 201.  doi: 10.1016/S1874-5717(08)00004-2.  Google Scholar

[42]

V. Pata and S. Zelik, A result on the existence of global attractors for semigroups of closed operators,, \emph{Commun. Pure Appl. Anal.}, 6 (2007), 481.  doi: 10.3934/cpaa.2007.6.481.  Google Scholar

[43]

J. Prüss, R. Racke and S. Zheng, Maximal regularity and asymptotic behavior of solutions for the Cahn-Hilliard equation with dynamic boundary conditions,, \emph{Ann. Mat. Pura Appl.}, 185 (2006), 627.  doi: 10.1007/s10231-005-0175-3.  Google Scholar

[44]

R. Racke and S. Zheng, The Cahn-Hilliard equation with dynamical boundary conditions,, \emph{Adv. Differential Equations}, 8 (2003), 83.   Google Scholar

[45]

A. Segatti, On the hyperbolic relaxation of the Cahn-Hilliard equation in 3D: approximation and long time behaviour,, \emph{Math. Models Methods Appl. Sci.}, 17 (2007), 411.  doi: 10.1142/S0218202507001978.  Google Scholar

[46]

J. Sprekels and H. Wu, A note on parabolic equation with nonlinear dynamical boundary condition,, \emph{Nonlinear Anal. T.M.A.}, 72 (2010), 3028.  doi: 10.1016/j.na.2009.11.043.  Google Scholar

[47]

B. Straughan, Heat Waves,, Appl. Math. Sci., 177 (2011).   Google Scholar

[48]

R. Temam, Infinite Dimensional Dynamical System in Mechanics and Physics,, Appl. Math. Sci. \textbf{68}, 68 (1988).  doi: 10.1007/978-1-4684-0313-8.  Google Scholar

[49]

H. Wu, Convergence to equilibrium for a Cahn-Hilliard model with the Wentzell boundary condition,, \emph{Asymptotic Analysis}, 54 (2007), 71.   Google Scholar

[50]

H. Wu, M. Grasselli and S. Zheng, Convergence to equilibrium for a parabolic-hyperbolic phase-field system with Neumann boundary conditions,, \emph{Math. Models Methods Appl. Sci.}, 17 (2007), 125.  doi: 10.1142/S0218202507001851.  Google Scholar

[51]

H. Wu, M. Grasselli and S. Zheng, Convergence to equilibrium for a nonlinear parabolic-hyperbolic phase-field system with dynamic boundary condition,, \emph{J. Math. Anal. Appl.}, 329 (2007), 948.  doi: 10.1016/j.jmaa.2006.07.011.  Google Scholar

[52]

H. Wu and S. Zheng, Convergence to equilibrium for the Cahn-Hilliard equation with dynamic boundary condition,, \emph{J. Differential Equations}, 204 (2004), 511.  doi: 10.1016/j.jde.2004.05.004.  Google Scholar

[53]

S. Zheng and A. Milani, Exponential attractors and inertial manifolds for singular perturbations of the Cahn-Hilliard equations,, \emph{Nonlinear Anal.}, 57 (2004), 843.  doi: 10.1016/j.na.2004.03.023.  Google Scholar

[54]

S. Zheng and A. Milani, Global attractors for singular perturbations of the Cahn-Hilliard equations,, \emph{J. Differential Equations}, 209 (2005), 101.  doi: 10.1016/j.jde.2004.08.026.  Google Scholar

show all references

References:
[1]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations,, Studies Math. Appl., 25 (1992).   Google Scholar

[2]

A. Bonfoh, Existence and continuity of uniform exponential attractors for a singular perturbation of a generalized Cahn-Hilliard equation,, \emph{Asymptot. Anal.}, 43 (2005), 233.   Google Scholar

[3]

A. Bonfoh, M. Grasselli and A. Miranville, Long time behavior of a singular perturbation of the viscous Cahn-Hilliard-Gurtin equation,, \emph{Math. Methods Appl. Sci.}, 31 (2008), 695.  doi: 10.1002/mma.938.  Google Scholar

[4]

A. Bonfoh, M. Grasselli and A. Miranville, Singularly perturbed 1D Cahn-Hilliard equation revisited,, \emph{NoDEA Nonlinear Differential Equations Appl.}, 17 (2010), 663.  doi: 10.1007/s00030-010-0075-0.  Google Scholar

[5]

A. Bonfoh, M. Grasselli and A. Miranville, Inertial manifolds for a singular perturbation of the viscous Cahn-Hilliard-Gurtin equation,, \emph{Topol. Methods Nonlinear Anal.}, 35 (2010), 155.   Google Scholar

[6]

C. Cavaterra, C. G. Gal and M. Grasselli, Cahn-Hilliard equations with memory and dynamic boundary conditions,, \emph{Asymptot. Anal.}, 71 (2011), 123.   Google Scholar

[7]

L. Cherfils, A. Miranville and S. Zelik, The Cahn-Hilliard equation with logarithmic potentials,, \emph{Milan J. Math.}, 79 (2011), 561.  doi: 10.1007/s00032-011-0165-4.  Google Scholar

[8]

R. Chill, E. Fašangová and J. Prüss, Convergence to steady states of solutions of the Cahn-Hilliard equation with dynamic boundary conditions,, \emph{Math. Nachr.}, 279 (2006), 1448.  doi: 10.1002/mana.200410431.  Google Scholar

[9]

E. Feireisl and F. Simondon, Convergence for semilinear degenerate parabolic equations in several space dimensions,, \emph{J. Dynam. Differential Equations}, 12 (2000), 647.  doi: 10.1023/A:1026467729263.  Google Scholar

[10]

H. P. Fischer, P. Maass and W. Dieterich, Novel surface modes in spinodal decomposition,, \emph{Phys. Rev. Lett.}, 79 (1997), 893.   Google Scholar

[11]

H. P. Fischer, P. Maass and W. Dieterich, Diverging time and length scales of spinodal decomposition modes in thin flows,, \emph{Europhys. Lett.}, 42 (1998), 49.   Google Scholar

[12]

C. G. Gal, Global well-posedness for the non-isothermal Cahn-Hilliard equation with dynamic boundary conditions,, \emph{Adv. Differential Equations}, 12 (2007), 1241.   Google Scholar

[13]

C. G. Gal, Well-posedness and long time behavior of the non-isothermal viscous Cahn-Hilliard equation with dynamic boundary conditions,, \emph{Dyn. Partial Differ. Equ.}, 5 (2008), 39.  doi: 10.4310/DPDE.2008.v5.n1.a2.  Google Scholar

[14]

C. G. Gal and M. Grasselli, Singular limit of viscous Cahn-Hilliard equations with memory and dynamic boundary conditions,, \emph{Discrete Contin. Dyn. Syst. Ser. B}, 18 (2013), 1581.  doi: 10.3934/dcdsb.2013.18.1581.  Google Scholar

[15]

C. G. Gal and A. Miranville, Uniform global attractors for non-isothermal viscous and non-viscous Cahn-Hilliard equations with dynamic boundary conditions,, \emph{Nonlinear Anal. Real World Appl.}, 10 (2009), 1738.  doi: 10.1016/j.nonrwa.2008.02.013.  Google Scholar

[16]

C. G. Gal and A. Miranville, Robust exponential attractors and convergence to equilibria for non-isothermal Cahn-Hilliard equations with dynamic boundary conditions,, \emph{Discrete Contin. Dyn. Syst. Ser. S}, 2 (2009), 113.  doi: 10.3934/dcdss.2009.2.113.  Google Scholar

[17]

C. G. Gal and H. Wu, Asymptotic behavior of a Cahn-Hilliard equation with Wentzell boundary conditions and mass conservation,, \emph{Discrete Contin. Dyn. Syst. Ser. A}, 22 (2008), 1041.  doi: 10.3934/dcds.2008.22.1041.  Google Scholar

[18]

P. Galenko, Phase-field model with relaxation of the diffusion flux in nonequilibrium solidification of a binary system,, \emph{Phys. Lett. A}, 287 (2001), 190.   Google Scholar

[19]

P. Galenko and D. Jou, Diffuse-interface model for rapid phase transformations in nonequilibrium systems,, \emph{Phys. Rev. E}, 71 (2005).   Google Scholar

[20]

P. Galenko and D. Jou, Kinetic contribution to the fast spinodal decomposition controlled by diffusion,, \emph{Phys. A}, 388 (2009), 3113.  doi: 10.1016/j.physa.2009.04.003.  Google Scholar

[21]

S. Gatti, M. Grasselli, A. Miranville and V. Pata, On the hyperbolic relaxation of the one-dimensional Cahn-Hilliard equation,, \emph{J. Math. Anal. Appl.}, 312 (2005), 230.  doi: 10.1016/j.jmaa.2005.03.029.  Google Scholar

[22]

S. Gatti, M. Grasselli, A. Miranville and V. Pata, Hyperbolic relaxation of the viscous Cahn-Hilliard equation in 3D,, \emph{Math. Models Methods Appl. Sci.}, 15 (2005), 165.  doi: 10.1142/S0218202505000327.  Google Scholar

[23]

G. Gilardi, A. Miranville and G. Schimperna, On the Cahn-Hilliard equation with irregular potentials and dynamic boundary conditions,, \emph{Comm. Pure Appl. Anal.}, 8 (2009), 881.  doi: 10.3934/cpaa.2009.8.881.  Google Scholar

[24]

G. Gilardi, A. Miranville and G. Schimperna, Long time behavior of the Cahn-Hilliard equation with irregular potentials and dynamic boundary conditions,, \emph{Chin. Ann. Math. Ser. B}, 31 (2010), 679.  doi: 10.1007/s11401-010-0602-7.  Google Scholar

[25]

M. Grasselli and V. Pata, Asymptotic behavior of a parabolic-hyperbolic system,, \emph{Comm. Pure Appl. Anal.}, 4 (2004), 849.  doi: 10.3934/cpaa.2004.3.849.  Google Scholar

[26]

M. Grasselli, H. Petzeltová and G. Schimperna, Convergence to stationary solutions for a parabolic-hyperbolic phase-field system,, \emph{Commun. Pure Appl. Anal.}, 5 (2006), 827.  doi: 10.3934/cpaa.2006.5.827.  Google Scholar

[27]

M. Grasselli, H. Petzeltová and G. Schimperna, Asymptotic behavior of a nonisothermal viscous Cahn-Hilliard equation with inertial term,, \emph{J. Differential Equations}, 239 (2007), 38.  doi: 10.1016/j.jde.2007.05.003.  Google Scholar

[28]

M. Grasselli, G. Schimperna and S. Zelik, On the 2D Cahn-Hilliard equation with inertial term,, \emph{Comm. Partial Differential Equations}, 34 (2009), 137.  doi: 10.1080/03605300802608247.  Google Scholar

[29]

M. Grasselli, G. Schimperna and S. Zelik, Trajectory and smooth attractors for Cahn-Hilliard equations with inertial term,, \emph{Nonlinearity}, 23 (2010), 707.  doi: 10.1088/0951-7715/23/3/016.  Google Scholar

[30]

M. Grasselli, G. Schimperna, A. Segatti and S. Zelik, On the 3D Cahn-Hilliard equation with inertial term,, \emph{J. Evol. Equ.}, 9 (2009), 371.  doi: 10.1007/s00028-009-0017-7.  Google Scholar

[31]

A. Haraux and M. A. Jendoubi, Convergence of bounded weak solutions of the wave equation with dissipation and analytic nonlinearity,, \emph{Calc. Var. Partial Differential Equations}, 9 (1999), 95.  doi: 10.1007/s005260050133.  Google Scholar

[32]

A. Haraux and M. A. Jendoubi, Decay estimates to equilibrium for some evolution equations with an analytic nonlinearity,, \emph{Asymptot. Anal.}, 26 (2001), 21.   Google Scholar

[33]

S.-Z. Huang and P. Takáč, Convergence in gradient-like systems which are asymptotically autonomous and analytic,, \emph{Nonlinear Anal.}, 46 (2001), 675.  doi: 10.1016/S0362-546X(00)00145-0.  Google Scholar

[34]

M. B. Kania, Global attractor for the perturbed viscous Cahn-Hilliard equation,, \emph{Colloq. Math.}, 109 (2007), 217.  doi: 10.4064/cm109-2-4.  Google Scholar

[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,, \emph{Comput. Phys. Commun.}, 133 (2001), 139.  doi: 10.1016/S0010-4655(00)00159-4.  Google Scholar

[36]

A. Miranville amd S. Zelik, Exponential attractors for the Cahn-Hilliard equation with dynamic boundary conditions,, \emph{Math. Methods Appl. Sci.}, 28 (2005), 709.  doi: 10.1002/mma.590.  Google Scholar

[37]

A. Miranville and S. Zelik, The Cahn-Hilliard equation with singular potentials and dynamic boundary conditions,, \emph{Discrete Contin. Dyn. Syst. Ser. A}, 28 (2010), 275.  doi: 10.3934/dcds.2010.28.275.  Google Scholar

[38]

P. Monk, Finite Element Methods for Maxwell's Equations,, Clarendon Press, (2003).  doi: 10.1093/acprof:oso/9780198508885.001.0001.  Google Scholar

[39]

A. Novick-Cohen, On the viscous Cahn-Hilliard equation, in \emph{Material instabilities in continuum mechanics (Edinburgh, (): 1985.   Google Scholar

[40]

A. Novick-Cohen, The Cahn-Hilliard equation: mathematical and modeling perspectives,, \emph{Adv. Math. Sci. Appl.}, 8 (1998), 965.   Google Scholar

[41]

A. Novick-Cohen, The Cahn-Hilliard equation, in Handbook of Differential Equations: Evolutionary Equations, Vol. IV,, Elsevier/North-Holland, (2008), 201.  doi: 10.1016/S1874-5717(08)00004-2.  Google Scholar

[42]

V. Pata and S. Zelik, A result on the existence of global attractors for semigroups of closed operators,, \emph{Commun. Pure Appl. Anal.}, 6 (2007), 481.  doi: 10.3934/cpaa.2007.6.481.  Google Scholar

[43]

J. Prüss, R. Racke and S. Zheng, Maximal regularity and asymptotic behavior of solutions for the Cahn-Hilliard equation with dynamic boundary conditions,, \emph{Ann. Mat. Pura Appl.}, 185 (2006), 627.  doi: 10.1007/s10231-005-0175-3.  Google Scholar

[44]

R. Racke and S. Zheng, The Cahn-Hilliard equation with dynamical boundary conditions,, \emph{Adv. Differential Equations}, 8 (2003), 83.   Google Scholar

[45]

A. Segatti, On the hyperbolic relaxation of the Cahn-Hilliard equation in 3D: approximation and long time behaviour,, \emph{Math. Models Methods Appl. Sci.}, 17 (2007), 411.  doi: 10.1142/S0218202507001978.  Google Scholar

[46]

J. Sprekels and H. Wu, A note on parabolic equation with nonlinear dynamical boundary condition,, \emph{Nonlinear Anal. T.M.A.}, 72 (2010), 3028.  doi: 10.1016/j.na.2009.11.043.  Google Scholar

[47]

B. Straughan, Heat Waves,, Appl. Math. Sci., 177 (2011).   Google Scholar

[48]

R. Temam, Infinite Dimensional Dynamical System in Mechanics and Physics,, Appl. Math. Sci. \textbf{68}, 68 (1988).  doi: 10.1007/978-1-4684-0313-8.  Google Scholar

[49]

H. Wu, Convergence to equilibrium for a Cahn-Hilliard model with the Wentzell boundary condition,, \emph{Asymptotic Analysis}, 54 (2007), 71.   Google Scholar

[50]

H. Wu, M. Grasselli and S. Zheng, Convergence to equilibrium for a parabolic-hyperbolic phase-field system with Neumann boundary conditions,, \emph{Math. Models Methods Appl. Sci.}, 17 (2007), 125.  doi: 10.1142/S0218202507001851.  Google Scholar

[51]

H. Wu, M. Grasselli and S. Zheng, Convergence to equilibrium for a nonlinear parabolic-hyperbolic phase-field system with dynamic boundary condition,, \emph{J. Math. Anal. Appl.}, 329 (2007), 948.  doi: 10.1016/j.jmaa.2006.07.011.  Google Scholar

[52]

H. Wu and S. Zheng, Convergence to equilibrium for the Cahn-Hilliard equation with dynamic boundary condition,, \emph{J. Differential Equations}, 204 (2004), 511.  doi: 10.1016/j.jde.2004.05.004.  Google Scholar

[53]

S. Zheng and A. Milani, Exponential attractors and inertial manifolds for singular perturbations of the Cahn-Hilliard equations,, \emph{Nonlinear Anal.}, 57 (2004), 843.  doi: 10.1016/j.na.2004.03.023.  Google Scholar

[54]

S. Zheng and A. Milani, Global attractors for singular perturbations of the Cahn-Hilliard equations,, \emph{J. Differential Equations}, 209 (2005), 101.  doi: 10.1016/j.jde.2004.08.026.  Google Scholar

[1]

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