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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

Haydi Israel 1,

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.
Keywords: global attractor, exponential attractor., Allen-Cahn equation, continuity.
Mathematics Subject Classification: 37L30, 35B45, 35A05, 35B4.
Citation: Haydi Israel. Well-posedness and long time behavior of an Allen-Cahn type equation. Communications on Pure & 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.  Google Scholar

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[3]

A. Eden, C. Foias, B. Nicolaenko and R. and Temam, "Exponential Attractors for Dissipative Evolution Equations," Masson, Paris, 1994.  Google Scholar

[4]

M.Efendiev, A. Miranville and S. Zelik, Exponential attractors for a singularly perturbed Cahn-Hilliard system, Math. Nachr., 272 (2004), 11-31.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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[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.  Google Scholar

[10]

A. Miranville, Some generalizations of the Cahn-Hilliard equation, Asymptot. Anal., 22 (2000), 235-259.  Google Scholar

[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.  Google Scholar

[12]

C. Robinson, "Infinite-dimensional Dynamical Systems,'' Cambridge Universtity Press, Cambridge, 2001.  Google Scholar

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R. Temam, "Infinite-dimensional Dynamical Systems in Mechanics and Physics,'' Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4684-0313-8.  Google Scholar

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.  Google Scholar

[2]

M. Carrive, A. Miranville, A. Piétrus and J. M. Rakotoson, Weakly coupled dynamical systems and applications, Asymptotic Analysis, 30 (2002), 161-185.  Google Scholar

[3]

A. Eden, C. Foias, B. Nicolaenko and R. and Temam, "Exponential Attractors for Dissipative Evolution Equations," Masson, Paris, 1994.  Google Scholar

[4]

M.Efendiev, A. Miranville and S. Zelik, Exponential attractors for a singularly perturbed Cahn-Hilliard system, Math. Nachr., 272 (2004), 11-31.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

A. Miranville, Some generalizations of the Cahn-Hilliard equation, Asymptot. Anal., 22 (2000), 235-259.  Google Scholar

[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.  Google Scholar

[12]

C. Robinson, "Infinite-dimensional Dynamical Systems,'' Cambridge Universtity Press, Cambridge, 2001.  Google Scholar

[13]

R. Temam, "Infinite-dimensional Dynamical Systems in Mechanics and Physics,'' Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4684-0313-8.  Google Scholar

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