# American Institute of Mathematical Sciences

June  2012, 5(3): 485-505. doi: 10.3934/dcdss.2012.5.485

## Asymptotic behavior of the Caginalp phase-field system with coupled dynamic boundary conditions

 1 Politecnico di Milano - Dipartimento di Matematica "F. Brioschi", Via Bonardi 9, 20133 Milano 2 Dipartimento di Matematica, Università di Modena e Reggio Emilia, Via Campi 213/B, I-41125 Modena, Italy 3 Université de Poitiers, Laboratoire de Mathématiques et Applications, UMR CNRS 6086 - SP2MI, Boulevard Marie et Pierre Curie - Téléport 2, F-86962 Chasseneuil Futuroscope Cedex

Received  May 2011 Revised  July 2011 Published  October 2011

This paper deals with the longtime behavior of the Caginalp phase-field system with coupled dynamic boundary conditions on both state variables. We prove that the system generates a dissipative semigroup in a suitable phase-space and possesses the finite-dimensional smooth global attractor and an exponential attractor.
Citation: Monica Conti, Stefania Gatti, Alain Miranville. Asymptotic behavior of the Caginalp phase-field system with coupled dynamic boundary conditions. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 485-505. doi: 10.3934/dcdss.2012.5.485
##### References:
 [1] G. Caginalp, An analysis of a phase field model of a free boundary, Arch. Rational Mech. Anal., 92 (1986), 205-245. doi: 10.1007/BF00254827.  Google Scholar [2] P. Fabrie, C. Galusinski, A. Miranville and S. Zelik, Uniform exponential attractors for a singularly perturbed damped wave equation. Partial differential equations and applications, Discrete Contin. Dynam. Systems, 10 (2004), 211-238. doi: 10.3934/dcds.2004.10.211.  Google Scholar [3] H. P. Fischer, P. Maass and W. Dieterich, Novel surface modes in spinodal decomposition, Phys. Rev. Letters, 79 (1997), 893-896. doi: 10.1103/PhysRevLett.79.893.  Google Scholar [4] 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. doi: 10.1209/epl/i1998-00550-y.  Google Scholar [5] H. P. Fischer, J. Reinhard, W. Dieterich, J.-F. Gouyet, P. Maass, A. Majhofer and D. Reinel, Time-dependent density functional theory and the kinetics of lattice gas systems in contact with a wall, J. Chem. Phys., 108 (1998), 3028-3037. doi: 10.1063/1.475690.  Google Scholar [6] C. G. Gal, M. Grasselli and A. Miranville, Robust exponential attractors for singularly perturbed phase-field equations with dynamic boundary conditions, NoDEA Nonlinear Differential Equations Appl., 15 (2008), 535–-556. doi: 10.1007/s00030-008-7029-9.  Google Scholar [7] C. G. Gal, M. Grasselli and A. Miranville, Nonisothermal Allen-Cahn equations with coupled dynamic boundary conditions, in "Nonlinear Phenomena with Energy Dissipation," GAKUTO Internat. Ser. Math. Sci. Appl., 29, Gakkotōsho, Tokyo, 2008, 117–-139.  Google Scholar [8] S. Gatti and A. Miranville, Asymptotic behavior of a phase-field system with dynamic boundary conditions, in "Differential Equations: Inverse and Direct Problems," Lecture Notes in Pure and Applied Mathematics, 251, Chapman & Hall/CRC, Boca Raton, FL, (2006), 149-170.  Google Scholar [9] M. Grasselli, A. Miranville and G. Schimperna, The Caginalp phase-field system with coupled dynamic boundary conditions and singular potentials, Discrete Contin. Dynam. Systems, 28 (2010), 67-98. doi: 10.3934/dcds.2010.28.67.  Google Scholar [10] O. A. Ladyžhenskaja, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type," Translations of Mathematical Monographs, Vol. 23, AMS, Providence, RI, 1967.  Google Scholar [11] A. Miranville and S. Zelik, Robust exponential attractors for singularly perturbed phase-field type equations,, Electron. J. Differential Equations, 2002 ().   Google Scholar [12] A. Miranville and S. Zelik, Exponential attractors for the Cahn-Hilliard equation with dynamic boundary conditions, Math. Meth. Appl. Sci., 28 (2005), 709-735. doi: 10.1002/mma.590.  Google Scholar [13] A. Miranville and S. Zelik, The Cahn-Hilliard equation with singular potentials and dynamic boundary conditions, Discrete Contin. Dynam. Systems, 28 (2010), 275-310. doi: 10.3934/dcds.2010.28.275.  Google Scholar [14] R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics," 2nd edition, Applied Mathematical Sciences, 68, Springer-Verlag, New York, 1997.  Google Scholar [15] E. Zeidler, "Nonlinear Functional Analysis and its Applications. Part I. Fixed-Point Theorems," Springer-Verlag, New York, 1986.  Google Scholar

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##### References:
 [1] G. Caginalp, An analysis of a phase field model of a free boundary, Arch. Rational Mech. Anal., 92 (1986), 205-245. doi: 10.1007/BF00254827.  Google Scholar [2] P. Fabrie, C. Galusinski, A. Miranville and S. Zelik, Uniform exponential attractors for a singularly perturbed damped wave equation. Partial differential equations and applications, Discrete Contin. Dynam. Systems, 10 (2004), 211-238. doi: 10.3934/dcds.2004.10.211.  Google Scholar [3] H. P. Fischer, P. Maass and W. Dieterich, Novel surface modes in spinodal decomposition, Phys. Rev. Letters, 79 (1997), 893-896. doi: 10.1103/PhysRevLett.79.893.  Google Scholar [4] 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. doi: 10.1209/epl/i1998-00550-y.  Google Scholar [5] H. P. Fischer, J. Reinhard, W. Dieterich, J.-F. Gouyet, P. Maass, A. Majhofer and D. Reinel, Time-dependent density functional theory and the kinetics of lattice gas systems in contact with a wall, J. Chem. Phys., 108 (1998), 3028-3037. doi: 10.1063/1.475690.  Google Scholar [6] C. G. Gal, M. Grasselli and A. Miranville, Robust exponential attractors for singularly perturbed phase-field equations with dynamic boundary conditions, NoDEA Nonlinear Differential Equations Appl., 15 (2008), 535–-556. doi: 10.1007/s00030-008-7029-9.  Google Scholar [7] C. G. Gal, M. Grasselli and A. Miranville, Nonisothermal Allen-Cahn equations with coupled dynamic boundary conditions, in "Nonlinear Phenomena with Energy Dissipation," GAKUTO Internat. Ser. Math. Sci. Appl., 29, Gakkotōsho, Tokyo, 2008, 117–-139.  Google Scholar [8] S. Gatti and A. Miranville, Asymptotic behavior of a phase-field system with dynamic boundary conditions, in "Differential Equations: Inverse and Direct Problems," Lecture Notes in Pure and Applied Mathematics, 251, Chapman & Hall/CRC, Boca Raton, FL, (2006), 149-170.  Google Scholar [9] M. Grasselli, A. Miranville and G. Schimperna, The Caginalp phase-field system with coupled dynamic boundary conditions and singular potentials, Discrete Contin. Dynam. Systems, 28 (2010), 67-98. doi: 10.3934/dcds.2010.28.67.  Google Scholar [10] O. A. Ladyžhenskaja, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type," Translations of Mathematical Monographs, Vol. 23, AMS, Providence, RI, 1967.  Google Scholar [11] A. Miranville and S. Zelik, Robust exponential attractors for singularly perturbed phase-field type equations,, Electron. J. Differential Equations, 2002 ().   Google Scholar [12] A. Miranville and S. Zelik, Exponential attractors for the Cahn-Hilliard equation with dynamic boundary conditions, Math. Meth. Appl. Sci., 28 (2005), 709-735. doi: 10.1002/mma.590.  Google Scholar [13] A. Miranville and S. Zelik, The Cahn-Hilliard equation with singular potentials and dynamic boundary conditions, Discrete Contin. Dynam. Systems, 28 (2010), 275-310. doi: 10.3934/dcds.2010.28.275.  Google Scholar [14] R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics," 2nd edition, Applied Mathematical Sciences, 68, Springer-Verlag, New York, 1997.  Google Scholar [15] E. Zeidler, "Nonlinear Functional Analysis and its Applications. Part I. Fixed-Point Theorems," Springer-Verlag, New York, 1986.  Google Scholar
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