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Well-posedness and long time behavior of an Allen-Cahn type equation
| 1. | UMR 6086 CNRS. Laboratoire de Mathématiques et Applications - Université de Poitiers, SP2MI - Boulevard Marie et Pierre Curie - Téléport 2, BP30179 - 86962 Futuroscope Chasseneuil Cedex, France |
References:
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A. Bonfoh and A. Miranville, On Cahn-Hilliard-Gurtin equations, in"Proceedings of the Third World Congress of Nonlinear Analysts, Part 5 (Catania, 2000)'', 47 (2001), 3455-3466.
doi: 10.1016/S0362-546X(01)00463-1. |
| [2] |
M. Carrive, A. Miranville, A. Piétrus and J. M. Rakotoson, Weakly coupled dynamical systems and applications, Asymptotic Analysis, 30 (2002), 161-185. |
| [3] |
A. Eden, C. Foias, B. Nicolaenko and R. and Temam, "Exponential Attractors for Dissipative Evolution Equations," Masson, Paris, 1994. |
| [4] |
M.Efendiev, A. Miranville and S. Zelik, Exponential attractors for a singularly perturbed Cahn-Hilliard system, Math. Nachr., 272 (2004), 11-31. |
| [5] |
G. Karali, and A. Katsoulakis, The role of multiple microscopic mechanisms in cluster interface evolution, J. Differential Equations, 235 (2007), 418-438.
doi: 10.1016/j.jde.2006.12.021. |
| [6] |
G. Karali and T. Ricciardi, On the convergence of a fourth order evolution equation to the Allen-Cahn equation, Nonlinear Anal., 72 (2010), 4271-4281.
doi: 10.1016/j.na.2010.02.003. |
| [7] |
A. Katsoulakis and G. Vlachos, From microscopic interactions to macroscopic laws of cluster evolution, Phys. Rev. Lett., 84 (2000), 1511-1514.
doi: 10.1103/PhysRevLett.84.1511. |
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S. Mikhailov, M. Hildebrand and G. Ertl, Nonequilibrium nanostructures in condensed reactive systems, in "Coherent Structures in Complex Systems (Sitges, 2000),'' 567 (2001), 252-269.
doi: 10.1007/3-540-44698-2_16. |
| [9] |
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, (2008), 103-200.
doi: 10.1016/S1874-5717(08)00003-0. |
| [10] |
A. Miranville, Some generalizations of the Cahn-Hilliard equation, Asymptot. Anal., 22 (2000), 235-259. |
| [11] |
A. Miranville, Long-time behavior of some models of Cahn-Hilliard equations in deformable continua, Nonlinear Anal. Real World Appl., 2 (2001), 273-304.
doi: 10.1016/S0362-546X(00)00104-8. |
| [12] |
C. Robinson, "Infinite-dimensional Dynamical Systems,'' Cambridge Universtity Press, Cambridge, 2001. |
| [13] |
R. Temam, "Infinite-dimensional Dynamical Systems in Mechanics and Physics,'' Springer-Verlag, New York, 1988.
doi: 10.1007/978-1-4684-0313-8. |
show all references
References:
| [1] |
A. Bonfoh and A. Miranville, On Cahn-Hilliard-Gurtin equations, in"Proceedings of the Third World Congress of Nonlinear Analysts, Part 5 (Catania, 2000)'', 47 (2001), 3455-3466.
doi: 10.1016/S0362-546X(01)00463-1. |
| [2] |
M. Carrive, A. Miranville, A. Piétrus and J. M. Rakotoson, Weakly coupled dynamical systems and applications, Asymptotic Analysis, 30 (2002), 161-185. |
| [3] |
A. Eden, C. Foias, B. Nicolaenko and R. and Temam, "Exponential Attractors for Dissipative Evolution Equations," Masson, Paris, 1994. |
| [4] |
M.Efendiev, A. Miranville and S. Zelik, Exponential attractors for a singularly perturbed Cahn-Hilliard system, Math. Nachr., 272 (2004), 11-31. |
| [5] |
G. Karali, and A. Katsoulakis, The role of multiple microscopic mechanisms in cluster interface evolution, J. Differential Equations, 235 (2007), 418-438.
doi: 10.1016/j.jde.2006.12.021. |
| [6] |
G. Karali and T. Ricciardi, On the convergence of a fourth order evolution equation to the Allen-Cahn equation, Nonlinear Anal., 72 (2010), 4271-4281.
doi: 10.1016/j.na.2010.02.003. |
| [7] |
A. Katsoulakis and G. Vlachos, From microscopic interactions to macroscopic laws of cluster evolution, Phys. Rev. Lett., 84 (2000), 1511-1514.
doi: 10.1103/PhysRevLett.84.1511. |
| [8] |
S. Mikhailov, M. Hildebrand and G. Ertl, Nonequilibrium nanostructures in condensed reactive systems, in "Coherent Structures in Complex Systems (Sitges, 2000),'' 567 (2001), 252-269.
doi: 10.1007/3-540-44698-2_16. |
| [9] |
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, (2008), 103-200.
doi: 10.1016/S1874-5717(08)00003-0. |
| [10] |
A. Miranville, Some generalizations of the Cahn-Hilliard equation, Asymptot. Anal., 22 (2000), 235-259. |
| [11] |
A. Miranville, Long-time behavior of some models of Cahn-Hilliard equations in deformable continua, Nonlinear Anal. Real World Appl., 2 (2001), 273-304.
doi: 10.1016/S0362-546X(00)00104-8. |
| [12] |
C. Robinson, "Infinite-dimensional Dynamical Systems,'' Cambridge Universtity Press, Cambridge, 2001. |
| [13] |
R. Temam, "Infinite-dimensional Dynamical Systems in Mechanics and Physics,'' Springer-Verlag, New York, 1988.
doi: 10.1007/978-1-4684-0313-8. |
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