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 & 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. doi: 10.1007/s00030-010-0075-0. Google Scholar

[2]

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

[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. doi: 10.1016/j.na.2009.11.002. Google Scholar

[4]

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

[5]

V. V. Chepyzhov and M. I. Vishik, "Attractors for Equations of Mathematical Physics,", American Mathematical Society Colloquium Publications, 49 (2002). Google Scholar

[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. doi: 10.1002/mana.200410431. Google Scholar

[7]

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

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[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. Google Scholar

[16]

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

[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. Google Scholar

[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. doi: 10.3934/cpaa.2008.7.819. Google Scholar

[19]

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

[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., (). Google Scholar

[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. doi: 10.1016/j.nonrwa.2008.02.013. Google Scholar

[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. doi: 10.3934/dcdss.2009.2.113. Google Scholar

[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. doi: 10.3934/dcds.2008.22.1041. Google Scholar

[24]

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

[25]

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

[26]

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

[27]

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

[28]

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

[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. doi: 10.1142/S0218202505000327. Google Scholar

[30]

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

[31]

S. Gatti, M. Grasselli, A. Miranville and V. Pata, Memory relaxation of the one-dimensional Cahn-Hilliard equation,, in, 71 (2006), 101. doi: 10.1142/9789812774293_0006. Google Scholar

[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. doi: 10.1017/S0308210509000365. Google Scholar

[33]

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

[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. doi: 10.3934/cpaa.2009.8.881. Google Scholar

[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. doi: 10.1016/j.physd.2010.12.007. Google Scholar

[36]

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

[37]

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

[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. doi: 10.1007/s00028-009-0017-7. Google Scholar

[39]

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

[40]

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

[41]

M. Grasselli and V. Pata, Uniform attractors of nonautonomous dynamical systems with memory,, in, 50 (2002), 155. Google Scholar

[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. doi: 10.1016/j.jde.2007.05.003. Google Scholar

[43]

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

[44]

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

[45]

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

[46]

T. Kato, "Perturbation Theory for Linear Operators,", Reprint of the 1980 edition, (1980). Google Scholar

[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. doi: 10.1016/S0010-4655(00)00159-4. Google Scholar

[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. Google Scholar

[49]

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

[50]

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

[51]

A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains,, in, (2008), 103. doi: 10.1016/S1874-5717(08)00003-0. Google Scholar

[52]

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

[53]

A. Novick-Cohen, On the viscous Cahn-Hilliard equation,, in, (1988), 1985. Google Scholar

[54]

A. Novick-Cohen, The Cahn-Hilliard equation,, in, (2008), 201. doi: 10.1016/S1874-5717(08)00004-2. Google Scholar

[55]

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

[56]

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

[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. doi: 10.1007/s10231-005-0175-3. Google Scholar

[58]

T. Qian, X.-P. Wang and P. Sheng, Molecular hydrodynamics of the moving contact line in two-phase immiscible flows,, Comm. Comp. Phys., 1 (2006), 1. Google Scholar

[59]

R. Racke, The Cahn-Hilliard equation with dynamic boundary conditions,, in, 20 (2004), 266. Google Scholar

[60]

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

[61]

P. Rybka and K.-H. Hoffmann, Convergence of solutions to Cahn-Hilliard equation,, Comm. Partial Differential Equations, 24 (1999), 1055. doi: 10.1080/03605309908821458. Google Scholar

[62]

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. doi: 10.1142/S0218202507001978. Google Scholar

[63]

R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics,", Second edition, 68 (1997). Google Scholar

[64]

V. Vergara, A conserved phase field system with memory and relaxed chemical potential,, J. Math. Anal. Appl., 328 (2007), 789. doi: 10.1016/j.jmaa.2006.05.075. Google Scholar

[65]

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

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. doi: 10.1007/s00030-010-0075-0. Google Scholar

[2]

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

[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. doi: 10.1016/j.na.2009.11.002. Google Scholar

[4]

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

[5]

V. V. Chepyzhov and M. I. Vishik, "Attractors for Equations of Mathematical Physics,", American Mathematical Society Colloquium Publications, 49 (2002). Google Scholar

[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. doi: 10.1002/mana.200410431. Google Scholar

[7]

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

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[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. Google Scholar

[16]

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

[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. Google Scholar

[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. doi: 10.3934/cpaa.2008.7.819. Google Scholar

[19]

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

[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., (). Google Scholar

[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. doi: 10.1016/j.nonrwa.2008.02.013. Google Scholar

[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. doi: 10.3934/dcdss.2009.2.113. Google Scholar

[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. doi: 10.3934/dcds.2008.22.1041. Google Scholar

[24]

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

[25]

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

[26]

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

[27]

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

[28]

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

[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. doi: 10.1142/S0218202505000327. Google Scholar

[30]

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

[31]

S. Gatti, M. Grasselli, A. Miranville and V. Pata, Memory relaxation of the one-dimensional Cahn-Hilliard equation,, in, 71 (2006), 101. doi: 10.1142/9789812774293_0006. Google Scholar

[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. doi: 10.1017/S0308210509000365. Google Scholar

[33]

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

[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. doi: 10.3934/cpaa.2009.8.881. Google Scholar

[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. doi: 10.1016/j.physd.2010.12.007. Google Scholar

[36]

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

[37]

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

[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. doi: 10.1007/s00028-009-0017-7. Google Scholar

[39]

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

[40]

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

[41]

M. Grasselli and V. Pata, Uniform attractors of nonautonomous dynamical systems with memory,, in, 50 (2002), 155. Google Scholar

[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. doi: 10.1016/j.jde.2007.05.003. Google Scholar

[43]

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

[44]

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

[45]

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

[46]

T. Kato, "Perturbation Theory for Linear Operators,", Reprint of the 1980 edition, (1980). Google Scholar

[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. doi: 10.1016/S0010-4655(00)00159-4. Google Scholar

[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. Google Scholar

[49]

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

[50]

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

[51]

A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains,, in, (2008), 103. doi: 10.1016/S1874-5717(08)00003-0. Google Scholar

[52]

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

[53]

A. Novick-Cohen, On the viscous Cahn-Hilliard equation,, in, (1988), 1985. Google Scholar

[54]

A. Novick-Cohen, The Cahn-Hilliard equation,, in, (2008), 201. doi: 10.1016/S1874-5717(08)00004-2. Google Scholar

[55]

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

[56]

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

[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. doi: 10.1007/s10231-005-0175-3. Google Scholar

[58]

T. Qian, X.-P. Wang and P. Sheng, Molecular hydrodynamics of the moving contact line in two-phase immiscible flows,, Comm. Comp. Phys., 1 (2006), 1. Google Scholar

[59]

R. Racke, The Cahn-Hilliard equation with dynamic boundary conditions,, in, 20 (2004), 266. Google Scholar

[60]

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

[61]

P. Rybka and K.-H. Hoffmann, Convergence of solutions to Cahn-Hilliard equation,, Comm. Partial Differential Equations, 24 (1999), 1055. doi: 10.1080/03605309908821458. Google Scholar

[62]

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. doi: 10.1142/S0218202507001978. Google Scholar

[63]

R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics,", Second edition, 68 (1997). Google Scholar

[64]

V. Vergara, A conserved phase field system with memory and relaxed chemical potential,, J. Math. Anal. Appl., 328 (2007), 789. doi: 10.1016/j.jmaa.2006.05.075. Google Scholar

[65]

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

[1]

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