August  2013, 18(6): 1581-1610. doi: 10.3934/dcdsb.2013.18.1581

Singular limit of viscous Cahn-Hilliard equations with memory and dynamic boundary conditions

1. 

Department of Mathematics, Florida International University, Miami, FL, 33199, United States

2. 

Dipartimento di Matematica, Politecnico di Milano, 20133 Milano

Received  October 2011 Revised  December 2011 Published  March 2013

We consider a modified Cahn-Hiliard equation where the velocity of the order parameter $u$ depends on the past history of $\Delta \mu $, $\mu $ being the chemical potential with an additional viscous term $ \alpha u_{t},$ $\alpha >0.$ In addition, the usual no-flux boundary condition for $u$ is replaced by a nonlinear dynamic boundary condition which accounts for possible interactions with the boundary. The aim of this work is to analyze the passage to the singular limit when the memory kernel collapses into a Dirac mass. In particular, we discuss the convergence of solutions on finite time-intervals and we also establish stability results for global and exponential attractors.
Citation: Ciprian G. Gal, Maurizio Grasselli. Singular limit of viscous Cahn-Hilliard equations with memory and dynamic boundary conditions. Discrete and Continuous Dynamical Systems - B, 2013, 18 (6) : 1581-1610. doi: 10.3934/dcdsb.2013.18.1581
References:
[1]

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.

[2]

J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. I. Interfacial energy, J. Chem. Phys., 28 (1958), 258-267.

[3]

C. Cavaterra, C. G. Gal, M. Grasselli and A. Miranville, Phase-field systems with nonlinear coupling and dynamic boundary conditions, Nonlinear Anal., 72 (2010), 2375-2399. doi: 10.1016/j.na.2009.11.002.

[4]

C. Cavaterra, C. G. Gal and M. Grasselli, Cahn-Hilliard equations with memory and dynamic boundary conditions, Asymptot. Anal., 71 (2011), 123-162.

[5]

V. V. Chepyzhov and M. I. Vishik, "Attractors for Equations of Mathematical Physics," American Mathematical Society Colloquium Publications, 49, American Mathematical Society, Providence, RI, 2002.

[6]

R. Chill, E. Fašangová and J. Prüss, Convergence to steady states of solutions of the Cahn-Hilliard and Caginalp equations with dynamic boundary conditions, Math. Nachr., 13 (2006), 1448-1462. doi: 10.1002/mana.200410431.

[7]

M. Conti and M. Coti Zelati, Attractors for the non-viscous Cahn-Hilliard equation with memory in 2D, Nonlinear Anal., 72 (2010), 1668-1682. doi: 10.1016/j.na.2009.09.006.

[8]

M. Conti, S. Gatti, M. Grasselli and V. Pata, Two-dimensional reaction-diffusion equations with memory, Quart. Appl. Math., 68 (2010), 607-643.

[9]

M. Conti and G. Mola, 3-D viscous Cahn-Hilliard equation with memory, Math. Meth. Appl. Sci., 32 (2009), 1370-1395. doi: 10.1002/mma.1091.

[10]

M. Conti, V. Pata and M. Squassina, Singular limit of differential system with memory, Indiana Univ. Math. J., 55 (2006), 169-215. doi: 10.1512/iumj.2006.55.2661.

[11]

C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rational Mech. Anal., 37 (1970), 297-308.

[12]

E. B. Dussan, On the spreading of liquids on solid surfaces: Static and dynamic contact lines, Ann. Rev. Fluid Mech., 11 (1979), 371-400.

[13]

M. Efendiev, A. Miranville and S. Zelik, Exponential attractors for a singularly perturbed Cahn-Hilliard system, Math. Nachr., 272 (2004), 11-31. doi: 10.1002/mana.200310186.

[14]

H. P. Fischer, Ph. Maass and W. Dieterich, Novel surface modes of spinodal decomposition, Phys. Rev. Letters, 79 (1997), 893-896.

[15]

H. P. Fischer, Ph. Maass and W. Dieterich, Diverging time and length scales of spinodal decomposition modes in thin flows, Europhys. Letters, 42 (1998), 49-54.

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C. G. Gal, A Cahn-Hilliard model in bounded domains with permeable walls, Math. Methods Appl. Sci., 29 (2006), 2009-2036. doi: 10.1002/mma.757.

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C. G. Gal, Well-posedness and long time behavior of the non-isothermal viscous Cahn-Hilliard model with dynamic boundary conditions, Dyn. Partial Differ. Equ., 5 (2008), 39-67.

[18]

C. G. Gal, Robust exponential attractors for a conserved Cahn-Hilliard model with singularly perturbed boundary conditions, Commun. Pure Appl. Anal., 7 (2008), 819-836. doi: 10.3934/cpaa.2008.7.819.

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C. G. Gal, Exponential attractors for a Cahn-Hilliard model in bounded domains with permeable walls, Electron. J. Differential Equations, 2006, 23 pp. (electronic).

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C. G. Gal, G. R. Goldstein, J. A. Goldstein, S. Romanelli and M. Warma, Fredholm alternative, semilinear elliptic problems, and Wentzell boundary conditions, submitted.

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C. G. Gal and A. Miranville, Uniform global attractors for non-isothermal Cahn-Hilliard equations with dynamic boundary conditions, Nonlinear Anal. Real World Appl., 10 (2009), 1738-1766. doi: 10.1016/j.nonrwa.2008.02.013.

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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.

[23]

C. G. Gal and H. Wu, Asymptotic behavior of a Cahn-Hilliard equation with Wentzell boundary conditions and mass conservation, Discrete Contin. Dyn. Syst., 22 (2008), 1041-1063. doi: 10.3934/dcds.2008.22.1041.

[24]

P. Galenko and D. Jou, Diffuse-interface model for rapid phase transformations in nonequilibrium systems, Phys. Rev. E, 71 (2005), 046125, 13 pp.

[25]

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.

[26]

P. Galenko and V. Lebedev, Analysis of the dispersion relation in spinodal decomposition of a binary system, Philos. Mag. Lett., 87 (2007), 821-827.

[27]

P. Galenko and V. Lebedev, Local nonequilibrium effect on spinodal decomposition in a binary system, Int. J. Thermodyn., 11 (2008), 21-28.

[28]

P. Galenko and V. Lebedev, Nonequilibrium effects in spinodal decomposition of a binary system, Phys. Lett. A, 372 (2008), 985-989.

[29]

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.

[30]

S. Gatti, M. Grasselli, A. Miranville, V. Pata, Memory relaxation of first order evolution equations, Nonlinearity, 18 (2005), 1859-1883. doi: 10.1088/0951-7715/18/4/023.

[31]

S. Gatti, M. Grasselli, A. Miranville and V. Pata, Memory relaxation of the one-dimensional Cahn-Hilliard equation, in "Dissipative Phase Transitions," Ser. Adv. Math. Appl. Sci., 71, World Sci. Publ., Hackensack, NJ, (2006), 101-114. doi: 10.1142/9789812774293_0006.

[32]

S. Gatti, A. Miranville, V. Pata and S. Zelik, Continuous families of exponential attractors for singularly perturbed equations with memory, Proc. Royal Soc. Edinburgh Sect. A, 140 (2010), 329-366. doi: 10.1017/S0308210509000365.

[33]

G. Gilardi, On a conserved phase field model with irregular potentials and dynamic boundary conditions, Rend. Cl. Sci. Mat. Nat., 141 (2007), 129-161.

[34]

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.

[35]

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. doi: 10.1016/j.physd.2010.12.007.

[36]

M. Grasselli, On the large time behavior of a phase-field system with memory, Asymptot. Anal., 56 (2008), 229-249.

[37]

M. Grasselli, V. Pata and F. M. Vegni, Longterm dynamics of a conserved phase-field system with memory, Asymptot. Anal., 33 (2003), 261-320.

[38]

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.

[39]

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.

[40]

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.

[41]

M. Grasselli and V. Pata, Uniform attractors of nonautonomous dynamical systems with memory, in "Evolution Equations, Semigroups and Functional Analysis" (eds. A. Lorenzi and B. Ruf) (Milano, 2000), Progr. Nonlinear Differential Equations Appl., 50, Birkhäuser, Basel, (2002), 155-178.

[42]

M. Grasselli, H. Petzeltová and G. Schimperna, Asymptotic behaviour 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.

[43]

M. E. Gurtin, Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance, Phys. D, 92 (1996), 178-192. doi: 10.1016/0167-2789(95)00173-5.

[44]

M. B. Kania, Global attractor for the perturbed viscous Cahn-Hilliard equation, Colloq. Math., 109 (2007), 217-229. doi: 10.4064/cm109-2-4.

[45]

M. B. Kania, Upper semicontinuity of global attractors for the perturbed viscous Cahn-Hilliard equations, Topol. Methods Nonlinear Anal., 32 (2008), 327-345.

[46]

T. Kato, "Perturbation Theory for Linear Operators," Reprint of the 1980 edition, Classics in Mathematics, Springer-Verlag, Berlin, 1995.

[47]

R. Kenzler, F. Eurich, Ph. Maass, B. Rinn, J. Schropp, E. Bohl and W. Dieterich, Phase separation in confined geometries: Solving the Cahn-Hilliard equation with generic boundary conditions, Computer Phys. Comm., 133 (2001), 139-157. doi: 10.1016/S0010-4655(00)00159-4.

[48]

N. Lecoq, H. Zapolsky and P. Galenko, Evolution of the structure factor in a hyperbolic model of spinodal decomposition, Eur. Phys. J. Special Topics, 177 (2009), 165-175.

[49]

A. Lorenzi and E. Rocca, Weak solutions for the fully hyperbolic phase-field system of conserved type, J. Evol. Equ., 7 (2007), 59-78. doi: 10.1007/s00028-006-0235-1.

[50]

A. Miranville and S. Zelik, Exponential attractors for the Cahn-Hilliard equation with dynamic boundary conditions, Math. Models Appl. Sci., 28 (2005), 709-735. doi: 10.1002/mma.590.

[51]

A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains, in "Handbook of Differential Equations: Evolutionary Equations," Vol. IV, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, (2008), 103-200. doi: 10.1016/S1874-5717(08)00003-0.

[52]

A. Miranville and S. Zelik, The Cahn-Hilliard equation with singular potentials and dynamic boundary conditions, Discrete Contin. Dyn. Syst., 28 (2010), 275-310. doi: 10.3934/dcds.2010.28.275.

[53]

A. Novick-Cohen, On the viscous Cahn-Hilliard equation, in "Material Instabilities in Continuum Mechanics" (Edinburgh, 1985-1986), Oxford Sci. Publ., Oxford Univ. Press, New York, (1988), 329-342.

[54]

A. Novick-Cohen, The Cahn-Hilliard equation, in "Handbook of Differential Equations: Evolutionary Equations," Vol. IV, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, (2008), 201-228. doi: 10.1016/S1874-5717(08)00004-2.

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V. Pata and S. Zelik, A remark on the damped wave equation, Commun. Pure Appl. Anal., 5 (2006), 609-614.

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V. Pata and A. Zucchi, Attractors for a damped hyperbolic equation with linear memory, Adv. Math. Sci. Appl., 11 (2001), 505-529.

[57]

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. (4), 185 (2006), 627-648. doi: 10.1007/s10231-005-0175-3.

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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.

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show all references

References:
[1]

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.

[2]

J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. I. Interfacial energy, J. Chem. Phys., 28 (1958), 258-267.

[3]

C. Cavaterra, C. G. Gal, M. Grasselli and A. Miranville, Phase-field systems with nonlinear coupling and dynamic boundary conditions, Nonlinear Anal., 72 (2010), 2375-2399. doi: 10.1016/j.na.2009.11.002.

[4]

C. Cavaterra, C. G. Gal and M. Grasselli, Cahn-Hilliard equations with memory and dynamic boundary conditions, Asymptot. Anal., 71 (2011), 123-162.

[5]

V. V. Chepyzhov and M. I. Vishik, "Attractors for Equations of Mathematical Physics," American Mathematical Society Colloquium Publications, 49, American Mathematical Society, Providence, RI, 2002.

[6]

R. Chill, E. Fašangová and J. Prüss, Convergence to steady states of solutions of the Cahn-Hilliard and Caginalp equations with dynamic boundary conditions, Math. Nachr., 13 (2006), 1448-1462. doi: 10.1002/mana.200410431.

[7]

M. Conti and M. Coti Zelati, Attractors for the non-viscous Cahn-Hilliard equation with memory in 2D, Nonlinear Anal., 72 (2010), 1668-1682. doi: 10.1016/j.na.2009.09.006.

[8]

M. Conti, S. Gatti, M. Grasselli and V. Pata, Two-dimensional reaction-diffusion equations with memory, Quart. Appl. Math., 68 (2010), 607-643.

[9]

M. Conti and G. Mola, 3-D viscous Cahn-Hilliard equation with memory, Math. Meth. Appl. Sci., 32 (2009), 1370-1395. doi: 10.1002/mma.1091.

[10]

M. Conti, V. Pata and M. Squassina, Singular limit of differential system with memory, Indiana Univ. Math. J., 55 (2006), 169-215. doi: 10.1512/iumj.2006.55.2661.

[11]

C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rational Mech. Anal., 37 (1970), 297-308.

[12]

E. B. Dussan, On the spreading of liquids on solid surfaces: Static and dynamic contact lines, Ann. Rev. Fluid Mech., 11 (1979), 371-400.

[13]

M. Efendiev, A. Miranville and S. Zelik, Exponential attractors for a singularly perturbed Cahn-Hilliard system, Math. Nachr., 272 (2004), 11-31. doi: 10.1002/mana.200310186.

[14]

H. P. Fischer, Ph. Maass and W. Dieterich, Novel surface modes of spinodal decomposition, Phys. Rev. Letters, 79 (1997), 893-896.

[15]

H. P. Fischer, Ph. Maass and W. Dieterich, Diverging time and length scales of spinodal decomposition modes in thin flows, Europhys. Letters, 42 (1998), 49-54.

[16]

C. G. Gal, A Cahn-Hilliard model in bounded domains with permeable walls, Math. Methods Appl. Sci., 29 (2006), 2009-2036. doi: 10.1002/mma.757.

[17]

C. G. Gal, Well-posedness and long time behavior of the non-isothermal viscous Cahn-Hilliard model with dynamic boundary conditions, Dyn. Partial Differ. Equ., 5 (2008), 39-67.

[18]

C. G. Gal, Robust exponential attractors for a conserved Cahn-Hilliard model with singularly perturbed boundary conditions, Commun. Pure Appl. Anal., 7 (2008), 819-836. doi: 10.3934/cpaa.2008.7.819.

[19]

C. G. Gal, Exponential attractors for a Cahn-Hilliard model in bounded domains with permeable walls, Electron. J. Differential Equations, 2006, 23 pp. (electronic).

[20]

C. G. Gal, G. R. Goldstein, J. A. Goldstein, S. Romanelli and M. Warma, Fredholm alternative, semilinear elliptic problems, and Wentzell boundary conditions, submitted.

[21]

C. G. Gal and A. Miranville, Uniform global attractors for non-isothermal Cahn-Hilliard equations with dynamic boundary conditions, Nonlinear Anal. Real World Appl., 10 (2009), 1738-1766. doi: 10.1016/j.nonrwa.2008.02.013.

[22]

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.

[23]

C. G. Gal and H. Wu, Asymptotic behavior of a Cahn-Hilliard equation with Wentzell boundary conditions and mass conservation, Discrete Contin. Dyn. Syst., 22 (2008), 1041-1063. doi: 10.3934/dcds.2008.22.1041.

[24]

P. Galenko and D. Jou, Diffuse-interface model for rapid phase transformations in nonequilibrium systems, Phys. Rev. E, 71 (2005), 046125, 13 pp.

[25]

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.

[26]

P. Galenko and V. Lebedev, Analysis of the dispersion relation in spinodal decomposition of a binary system, Philos. Mag. Lett., 87 (2007), 821-827.

[27]

P. Galenko and V. Lebedev, Local nonequilibrium effect on spinodal decomposition in a binary system, Int. J. Thermodyn., 11 (2008), 21-28.

[28]

P. Galenko and V. Lebedev, Nonequilibrium effects in spinodal decomposition of a binary system, Phys. Lett. A, 372 (2008), 985-989.

[29]

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.

[30]

S. Gatti, M. Grasselli, A. Miranville, V. Pata, Memory relaxation of first order evolution equations, Nonlinearity, 18 (2005), 1859-1883. doi: 10.1088/0951-7715/18/4/023.

[31]

S. Gatti, M. Grasselli, A. Miranville and V. Pata, Memory relaxation of the one-dimensional Cahn-Hilliard equation, in "Dissipative Phase Transitions," Ser. Adv. Math. Appl. Sci., 71, World Sci. Publ., Hackensack, NJ, (2006), 101-114. doi: 10.1142/9789812774293_0006.

[32]

S. Gatti, A. Miranville, V. Pata and S. Zelik, Continuous families of exponential attractors for singularly perturbed equations with memory, Proc. Royal Soc. Edinburgh Sect. A, 140 (2010), 329-366. doi: 10.1017/S0308210509000365.

[33]

G. Gilardi, On a conserved phase field model with irregular potentials and dynamic boundary conditions, Rend. Cl. Sci. Mat. Nat., 141 (2007), 129-161.

[34]

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.

[35]

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. doi: 10.1016/j.physd.2010.12.007.

[36]

M. Grasselli, On the large time behavior of a phase-field system with memory, Asymptot. Anal., 56 (2008), 229-249.

[37]

M. Grasselli, V. Pata and F. M. Vegni, Longterm dynamics of a conserved phase-field system with memory, Asymptot. Anal., 33 (2003), 261-320.

[38]

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.

[39]

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.

[40]

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.

[41]

M. Grasselli and V. Pata, Uniform attractors of nonautonomous dynamical systems with memory, in "Evolution Equations, Semigroups and Functional Analysis" (eds. A. Lorenzi and B. Ruf) (Milano, 2000), Progr. Nonlinear Differential Equations Appl., 50, Birkhäuser, Basel, (2002), 155-178.

[42]

M. Grasselli, H. Petzeltová and G. Schimperna, Asymptotic behaviour 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.

[43]

M. E. Gurtin, Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance, Phys. D, 92 (1996), 178-192. doi: 10.1016/0167-2789(95)00173-5.

[44]

M. B. Kania, Global attractor for the perturbed viscous Cahn-Hilliard equation, Colloq. Math., 109 (2007), 217-229. doi: 10.4064/cm109-2-4.

[45]

M. B. Kania, Upper semicontinuity of global attractors for the perturbed viscous Cahn-Hilliard equations, Topol. Methods Nonlinear Anal., 32 (2008), 327-345.

[46]

T. Kato, "Perturbation Theory for Linear Operators," Reprint of the 1980 edition, Classics in Mathematics, Springer-Verlag, Berlin, 1995.

[47]

R. Kenzler, F. Eurich, Ph. Maass, B. Rinn, J. Schropp, E. Bohl and W. Dieterich, Phase separation in confined geometries: Solving the Cahn-Hilliard equation with generic boundary conditions, Computer Phys. Comm., 133 (2001), 139-157. doi: 10.1016/S0010-4655(00)00159-4.

[48]

N. Lecoq, H. Zapolsky and P. Galenko, Evolution of the structure factor in a hyperbolic model of spinodal decomposition, Eur. Phys. J. Special Topics, 177 (2009), 165-175.

[49]

A. Lorenzi and E. Rocca, Weak solutions for the fully hyperbolic phase-field system of conserved type, J. Evol. Equ., 7 (2007), 59-78. doi: 10.1007/s00028-006-0235-1.

[50]

A. Miranville and S. Zelik, Exponential attractors for the Cahn-Hilliard equation with dynamic boundary conditions, Math. Models Appl. Sci., 28 (2005), 709-735. doi: 10.1002/mma.590.

[51]

A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains, in "Handbook of Differential Equations: Evolutionary Equations," Vol. IV, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, (2008), 103-200. doi: 10.1016/S1874-5717(08)00003-0.

[52]

A. Miranville and S. Zelik, The Cahn-Hilliard equation with singular potentials and dynamic boundary conditions, Discrete Contin. Dyn. Syst., 28 (2010), 275-310. doi: 10.3934/dcds.2010.28.275.

[53]

A. Novick-Cohen, On the viscous Cahn-Hilliard equation, in "Material Instabilities in Continuum Mechanics" (Edinburgh, 1985-1986), Oxford Sci. Publ., Oxford Univ. Press, New York, (1988), 329-342.

[54]

A. Novick-Cohen, The Cahn-Hilliard equation, in "Handbook of Differential Equations: Evolutionary Equations," Vol. IV, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, (2008), 201-228. doi: 10.1016/S1874-5717(08)00004-2.

[55]

V. Pata and S. Zelik, A remark on the damped wave equation, Commun. Pure Appl. Anal., 5 (2006), 609-614.

[56]

V. Pata and A. Zucchi, Attractors for a damped hyperbolic equation with linear memory, Adv. Math. Sci. Appl., 11 (2001), 505-529.

[57]

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. (4), 185 (2006), 627-648. doi: 10.1007/s10231-005-0175-3.

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