A doubly nonlinear parabolic equation with a singular potential

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February 2011, 4(1): 51-66. doi: 10.3934/dcdss.2011.4.51

A doubly nonlinear parabolic equation with a singular potential

Laurence Cherfils ^1, , Stefania Gatti ^2, and Alain Miranville ^3,

1.	Université de La Rochelle, Laboratoire MIA, Avenue Michel Crépeau, F-17042 La Rochelle Cedex, France
2.	Dipartimento di Matematica, Università di Modena e Reggio Emilia, Via Campi 213/B, I-41100 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, France

Received July 2009 Revised September 2009 Published October 2010

Abstract
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Our aim in this paper is to study the long time behavior, in terms of finite dimensional attractors, of doubly nonlinear Allen-Cahn type equations with singular potentials.

Keywords: exponential attractor., Doubly nonlinear equation, global attractor, singular potential.

Mathematics Subject Classification: 35B41, 35B45, 35K6.

Citation: Laurence Cherfils, Stefania Gatti, Alain Miranville. A doubly nonlinear parabolic equation with a singular potential. Discrete & Continuous Dynamical Systems - S, 2011, 4 (1) : 51-66. doi: 10.3934/dcdss.2011.4.51

References:

[1]	A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations,", North-Holland, (1992). 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 potential,, J. Math. Anal. Appl., 343 (2008), 557. doi: doi:10.1016/j.jmaa.2008.01.077. Google Scholar
[3]	L. Cherfils, S. Gatti and A. Miranville, Corrigendum to "Existence of global solutions to the Caginalp phase-field system with dynamic boundary conditions and singular potential,", J. Math. Anal. Appl., 348 (2008), 1029. doi: doi:10.1016/j.jmaa.2008.07.058. Google Scholar
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[5]	A. Eden, B. Michaux and J.-M. Rakotoson, Doubly nonlinear parabolic-type equations as dynamical systems,, J. Dyn. Diff. Eqns., 3 (1991), 87. doi: doi:10.1007/BF01049490. Google Scholar
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[7]	M. Gurtin, Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance,, Physica D, 92 (1996), 178. doi: doi:10.1016/0167-2789(95)00173-5. Google Scholar
[8]	O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type,", in, 23 (1967). Google Scholar
[9]	J. Málek and D. Prážak, Large time behavior via the method of $l$-trajectories,, J. Diff. Eqns., 181 (2002), 243. doi: doi:10.1006/jdeq.2001.4087. Google Scholar
[10]	A. Miranville, Finite dimensional global attractor for a class of doubly nonlinear parabolic equations,, Cent. Eur. J. Math., 4 (2006), 163. doi: doi:10.1007/s11533-005-0010-5. Google Scholar
[11]	A. Miranville and S. Zelik, Finite-dimensionality of attractors for degenerate equations of elliptic-parabolic type,, Nonlinearity, 20 (2007), 1773. doi: doi:10.1088/0951-7715/20/8/001. Google Scholar
[12]	A. Rougirel, Convergence to steady state and attractors for doubly nonlinear equations,, J. Math. Anal. Appl., 339 (2008), 281. doi: doi:10.1016/j.jmaa.2007.06.028. Google Scholar
[13]	R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics,", Springer, (1988). Google Scholar

show all references

References:

[1]	A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations,", North-Holland, (1992). 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 potential,, J. Math. Anal. Appl., 343 (2008), 557. doi: doi:10.1016/j.jmaa.2008.01.077. Google Scholar
[3]	L. Cherfils, S. Gatti and A. Miranville, Corrigendum to "Existence of global solutions to the Caginalp phase-field system with dynamic boundary conditions and singular potential,", J. Math. Anal. Appl., 348 (2008), 1029. doi: doi:10.1016/j.jmaa.2008.07.058. Google Scholar
[4]	A. Eden, C. Foias, B. Nicolaenko and R.Temam, "Exponential Attractors for Dissipative Evolution Equations,", in, 37 (1994). Google Scholar
[5]	A. Eden, B. Michaux and J.-M. Rakotoson, Doubly nonlinear parabolic-type equations as dynamical systems,, J. Dyn. Diff. Eqns., 3 (1991), 87. doi: doi:10.1007/BF01049490. Google Scholar
[6]	A. Eden and J.-M. Rakotoson, Exponential attractors for a doubly nonlinear equation,, J. Math. Anal. Appl., 185 (1994), 321. doi: doi:10.1006/jmaa.1994.1251. Google Scholar
[7]	M. Gurtin, Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance,, Physica D, 92 (1996), 178. doi: doi:10.1016/0167-2789(95)00173-5. Google Scholar
[8]	O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type,", in, 23 (1967). Google Scholar
[9]	J. Málek and D. Prážak, Large time behavior via the method of $l$-trajectories,, J. Diff. Eqns., 181 (2002), 243. doi: doi:10.1006/jdeq.2001.4087. Google Scholar
[10]	A. Miranville, Finite dimensional global attractor for a class of doubly nonlinear parabolic equations,, Cent. Eur. J. Math., 4 (2006), 163. doi: doi:10.1007/s11533-005-0010-5. Google Scholar
[11]	A. Miranville and S. Zelik, Finite-dimensionality of attractors for degenerate equations of elliptic-parabolic type,, Nonlinearity, 20 (2007), 1773. doi: doi:10.1088/0951-7715/20/8/001. Google Scholar
[12]	A. Rougirel, Convergence to steady state and attractors for doubly nonlinear equations,, J. Math. Anal. Appl., 339 (2008), 281. doi: doi:10.1016/j.jmaa.2007.06.028. Google Scholar
[13]	R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics,", Springer, (1988). Google Scholar

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