# American Institute of Mathematical Sciences

November  2012, 11(6): 2261-2290. doi: 10.3934/cpaa.2012.11.2261

## Long time behavior of the Caginalp system with singular potentials and dynamic boundary conditions

 1 Université de La Rochelle, Laboratoire de Mathématiques Images et Applications EA 3165, Avenue Michel Crépeau, 17042 La Rochelle Cedex 1 2 Dipartimento di Matematica, Università di Modena e Reggio Emilia, Via Campi 213/B, I-41125 Modena 3 Université de Poitiers, Mathématiques SP2MI, 86962 Chasseneuil Futuroscope Cedex

Received  December 2010 Revised  December 2010 Published  April 2012

This paper is devoted to the study of the well-posedness and the long time behavior of the Caginalp phase-field model with singular potentials and dynamic boundary conditions. Thanks to a suitable definition of solutions, coinciding with the strong ones under proper assumptions on the bulk and surface potentials, we are able to get dissipative estimates, leading to the existence of the global attractor with finite fractal dimension, as well as of an exponential attractor.
Citation: Laurence Cherfils, Stefania Gatti, Alain Miranville. Long time behavior of the Caginalp system with singular potentials and dynamic boundary conditions. Communications on Pure & Applied Analysis, 2012, 11 (6) : 2261-2290. doi: 10.3934/cpaa.2012.11.2261
##### 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] L. Cherfils, S. Gatti and A. Miranville, Existence of global solutions to the Caginalp phase-field system with dynamic boundary conditions and singular potentials (Corrigendum, J. Math. Anal. Appl. 348 (2008), 1029-1030), J. Math. Anal. Appl., 343 (2008), 557-566. doi: 10.1016/j.jmaa.2008.07.058.  Google Scholar [3] L. Cherfils, S. Gatti and A. Miranville, Finite dimensional attractors for the Caginalp system with singular potentials and dynamic boundary conditions, Bull. Transilvania University Bra\csov-Series III: Mathematics, Informatics, Physics, 2 (2009), 25-34.  Google Scholar [4] L. Cherfils and A. Miranville, On the Caginalp system with dynamic boundary conditions and singular potentials, Appl. Math., 54 (2009), 89-115. doi: 10.1007/s10492-009-0008-6.  Google Scholar [5] 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.  Google Scholar [6] P. Fabrie, C. Galusinski, A. Miranville and S. Zelik, Uniform exponential attractors for a singularly perturbed damped wave equation, Discrete Contin. Dynam. Systems, 10 (2004), 211-238. doi: 10.3934/dcds.2004.10.211.  Google Scholar [7] H. P. Fischer, P. Maass and W. Dieterich, Novel surface modes in spinodal decomposition, Phys. Rev. Lett., 79 (1997), 893-896. doi: 10.1103/PhysRevLett.79.893.  Google Scholar [8] 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 [9] 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 [10] 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," Taylor and Francis, (2006), 149-170. doi: 10.1201/9781420011135.ch9.  Google Scholar [11] G. Gilardi, A. Miranville and G. Schimperna, On the Cahn-Hilliard equation with irregular potentials and dynamic boundary conditions, Commun. Pure Appl. Anal., 8 (2009), 881-912. doi: 10.3934/cpaa.2009.8.881.  Google Scholar [12] G. Gilardi, A. Miranville and G. Schimperna, Long time behavior of the Cahn-Hilliard equation with irregular potentials and dynamic boundary conditions, Chinese Ann. Math., Ser. B, 31 (2010), 679-712. doi: 10.1007/s11401-010-0602-7.  Google Scholar [13] 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 [14] J. Málek and D. Prážak, Large time behavior via the method of $l$-trajectories, J. Diff. Eqns., 181 (2002), 243-279. doi: 10.1006/jdeq.2001.4087.  Google Scholar [15] A. Miranville and S. Zelik, Robust exponential attractors for Cahn-Hilliard type equations with singular potentials, Math. Methods Appl. Sci., 27 (2004), 545-582. doi: 10.1002/mma.464.  Google Scholar [16] A. Miranville and 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.  Google Scholar [17] 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" (eds. C.M. Dafermos and M. Pokorny), Elsevier/North-Holland, (2008), 103-200. doi: 10.1016/S1874-5717(08)00003-0.  Google Scholar [18] 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 [19] G. Ruiz Goldstein, A. Miranville and G. Schimperna, A Cahn-Hilliard equation in a domain with non-permeable walls, Phys. D, 240 (2011), 754-766. doi: 10.1016/j.physd.2010.12.007.  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] L. Cherfils, S. Gatti and A. Miranville, Existence of global solutions to the Caginalp phase-field system with dynamic boundary conditions and singular potentials (Corrigendum, J. Math. Anal. Appl. 348 (2008), 1029-1030), J. Math. Anal. Appl., 343 (2008), 557-566. doi: 10.1016/j.jmaa.2008.07.058.  Google Scholar [3] L. Cherfils, S. Gatti and A. Miranville, Finite dimensional attractors for the Caginalp system with singular potentials and dynamic boundary conditions, Bull. Transilvania University Bra\csov-Series III: Mathematics, Informatics, Physics, 2 (2009), 25-34.  Google Scholar [4] L. Cherfils and A. Miranville, On the Caginalp system with dynamic boundary conditions and singular potentials, Appl. Math., 54 (2009), 89-115. doi: 10.1007/s10492-009-0008-6.  Google Scholar [5] 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.  Google Scholar [6] P. Fabrie, C. Galusinski, A. Miranville and S. Zelik, Uniform exponential attractors for a singularly perturbed damped wave equation, Discrete Contin. Dynam. Systems, 10 (2004), 211-238. doi: 10.3934/dcds.2004.10.211.  Google Scholar [7] H. P. Fischer, P. Maass and W. Dieterich, Novel surface modes in spinodal decomposition, Phys. Rev. Lett., 79 (1997), 893-896. doi: 10.1103/PhysRevLett.79.893.  Google Scholar [8] 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 [9] 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 [10] 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," Taylor and Francis, (2006), 149-170. doi: 10.1201/9781420011135.ch9.  Google Scholar [11] G. Gilardi, A. Miranville and G. Schimperna, On the Cahn-Hilliard equation with irregular potentials and dynamic boundary conditions, Commun. Pure Appl. Anal., 8 (2009), 881-912. doi: 10.3934/cpaa.2009.8.881.  Google Scholar [12] G. Gilardi, A. Miranville and G. Schimperna, Long time behavior of the Cahn-Hilliard equation with irregular potentials and dynamic boundary conditions, Chinese Ann. Math., Ser. B, 31 (2010), 679-712. doi: 10.1007/s11401-010-0602-7.  Google Scholar [13] 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 [14] J. Málek and D. Prážak, Large time behavior via the method of $l$-trajectories, J. Diff. Eqns., 181 (2002), 243-279. doi: 10.1006/jdeq.2001.4087.  Google Scholar [15] A. Miranville and S. Zelik, Robust exponential attractors for Cahn-Hilliard type equations with singular potentials, Math. Methods Appl. Sci., 27 (2004), 545-582. doi: 10.1002/mma.464.  Google Scholar [16] A. Miranville and 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.  Google Scholar [17] 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" (eds. C.M. Dafermos and M. Pokorny), Elsevier/North-Holland, (2008), 103-200. doi: 10.1016/S1874-5717(08)00003-0.  Google Scholar [18] 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 [19] G. Ruiz Goldstein, A. Miranville and G. Schimperna, A Cahn-Hilliard equation in a domain with non-permeable walls, Phys. D, 240 (2011), 754-766. doi: 10.1016/j.physd.2010.12.007.  Google Scholar
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