# American Institute of Mathematical Sciences

November  2013, 12(6): 2811-2827. doi: 10.3934/cpaa.2013.12.2811

## 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

Received  August 2011 Revised  January 2012 Published  May 2013

The aim of this article is to study the existence and uniqueness of solutions for an equation of Allen-Cahn type and to prove the existence of the finite-dimensional global attractor as well as the existence of exponential attractors.
Citation: Haydi Israel. Well-posedness and long time behavior of an Allen-Cahn type equation. Communications on Pure and Applied Analysis, 2013, 12 (6) : 2811-2827. doi: 10.3934/cpaa.2013.12.2811
##### 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.

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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