-
Previous Article
Regular solutions and global attractors for reaction-diffusion systems without uniqueness
- CPAA Home
- This Issue
-
Next Article
Stabilization of the simplest normal parabolic equation
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 |
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 \emph{Material instabilities in continuum mechanics (Edinburgh, (): 1985.
|
[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. |
show all references
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 \emph{Material instabilities in continuum mechanics (Edinburgh, (): 1985.
|
[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. |
[1] |
Matthieu Brachet, Philippe Parnaudeau, Morgan Pierre. Convergence to equilibrium for time and space discretizations of the Cahn-Hilliard equation. Discrete and Continuous Dynamical Systems - S, 2022, 15 (8) : 1987-2031. doi: 10.3934/dcdss.2022110 |
[2] |
Desheng Li, Xuewei Ju. On dynamical behavior of viscous Cahn-Hilliard equation. Discrete and Continuous Dynamical Systems, 2012, 32 (6) : 2207-2221. doi: 10.3934/dcds.2012.32.2207 |
[3] |
Aibo Liu, Changchun Liu. Cauchy problem for a sixth order Cahn-Hilliard type equation with inertial term. Evolution Equations and Control Theory, 2015, 4 (3) : 315-324. doi: 10.3934/eect.2015.4.315 |
[4] |
Maurizio Grasselli, Nicolas Lecoq, Morgan Pierre. A long-time stable fully discrete approximation of the Cahn-Hilliard equation with inertial term. Conference Publications, 2011, 2011 (Special) : 543-552. doi: 10.3934/proc.2011.2011.543 |
[5] |
Georgia Karali, Yuko Nagase. On the existence of solution for a Cahn-Hilliard/Allen-Cahn equation. Discrete and Continuous Dynamical Systems - S, 2014, 7 (1) : 127-137. doi: 10.3934/dcdss.2014.7.127 |
[6] |
Dimitra Antonopoulou, Georgia Karali. Existence of solution for a generalized stochastic Cahn-Hilliard equation on convex domains. Discrete and Continuous Dynamical Systems - B, 2011, 16 (1) : 31-55. doi: 10.3934/dcdsb.2011.16.31 |
[7] |
Tohru Nakamura, Shuichi Kawashima. Viscous shock profile and singular limit for hyperbolic systems with Cattaneo's law. Kinetic and Related Models, 2018, 11 (4) : 795-819. doi: 10.3934/krm.2018032 |
[8] |
Ciprian G. Gal, Alain Miranville. Robust exponential attractors and convergence to equilibria for non-isothermal Cahn-Hilliard equations with dynamic boundary conditions. Discrete and Continuous Dynamical Systems - S, 2009, 2 (1) : 113-147. doi: 10.3934/dcdss.2009.2.113 |
[9] |
Alain Miranville. Existence of solutions for Cahn-Hilliard type equations. Conference Publications, 2003, 2003 (Special) : 630-637. doi: 10.3934/proc.2003.2003.630 |
[10] |
Changchun Liu, Hui Tang. Existence of periodic solution for a Cahn-Hilliard/Allen-Cahn equation in two space dimensions. Evolution Equations and Control Theory, 2017, 6 (2) : 219-237. doi: 10.3934/eect.2017012 |
[11] |
Erica Ipocoana, Andrea Zafferi. Further regularity and uniqueness results for a non-isothermal Cahn-Hilliard equation. Communications on Pure and Applied Analysis, 2021, 20 (2) : 763-782. doi: 10.3934/cpaa.2020289 |
[12] |
Irena Pawłow, Wojciech M. Zajączkowski. On a class of sixth order viscous Cahn-Hilliard type equations. Discrete and Continuous Dynamical Systems - S, 2013, 6 (2) : 517-546. doi: 10.3934/dcdss.2013.6.517 |
[13] |
Riccarda Rossi. On two classes of generalized viscous Cahn-Hilliard equations. Communications on Pure and Applied Analysis, 2005, 4 (2) : 405-430. doi: 10.3934/cpaa.2005.4.405 |
[14] |
Laurence Cherfils, Alain Miranville, Sergey Zelik. On a generalized Cahn-Hilliard equation with biological applications. Discrete and Continuous Dynamical Systems - B, 2014, 19 (7) : 2013-2026. doi: 10.3934/dcdsb.2014.19.2013 |
[15] |
Álvaro Hernández, Michał Kowalczyk. Rotationally symmetric solutions to the Cahn-Hilliard equation. Discrete and Continuous Dynamical Systems, 2017, 37 (2) : 801-827. doi: 10.3934/dcds.2017033 |
[16] |
Nguyen Huy Tuan. Existence and limit problem for fractional fourth order subdiffusion equation and Cahn-Hilliard equation. Discrete and Continuous Dynamical Systems - S, 2021, 14 (12) : 4551-4574. doi: 10.3934/dcdss.2021113 |
[17] |
Sergey Zelik, Jon Pennant. Global well-posedness in uniformly local spaces for the Cahn-Hilliard equation in $\mathbb{R}^3$. Communications on Pure and Applied Analysis, 2013, 12 (1) : 461-480. doi: 10.3934/cpaa.2013.12.461 |
[18] |
Kelong Cheng, Cheng Wang, Steven M. Wise, Zixia Yuan. Global-in-time Gevrey regularity solutions for the functionalized Cahn-Hilliard equation. Discrete and Continuous Dynamical Systems - S, 2020, 13 (8) : 2211-2229. doi: 10.3934/dcdss.2020186 |
[19] |
Irena Pawłow, Wojciech M. Zajączkowski. The global solvability of a sixth order Cahn-Hilliard type equation via the Bäcklund transformation. Communications on Pure and Applied Analysis, 2014, 13 (2) : 859-880. doi: 10.3934/cpaa.2014.13.859 |
[20] |
Anna Kostianko, Sergey Zelik. Inertial manifolds for the 3D Cahn-Hilliard equations with periodic boundary conditions. Communications on Pure and Applied Analysis, 2015, 14 (5) : 2069-2094. doi: 10.3934/cpaa.2015.14.2069 |
2020 Impact Factor: 1.916
Tools
Metrics
Other articles
by authors
[Back to Top]