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

A doubly nonlinear parabolic equation with a singular potential

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

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.
Citation: Laurence Cherfils, Stefania Gatti, Alain Miranville. A doubly nonlinear parabolic equation with a singular potential. Discrete and 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, Amsterdam, 1992.

[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), 557-566. doi: doi:10.1016/j.jmaa.2008.01.077.

[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), 1029-1030. doi: doi:10.1016/j.jmaa.2008.07.058.

[4]

A. Eden, C. Foias, B. Nicolaenko and R.Temam, "Exponential Attractors for Dissipative Evolution Equations," in "Research in Applied Mathematics," Vol. 37, John-Wiley, New York, 1994.

[5]

A. Eden, B. Michaux and J.-M. Rakotoson, Doubly nonlinear parabolic-type equations as dynamical systems, J. Dyn. Diff. Eqns., 3 (1991), 87-131. doi: doi:10.1007/BF01049490.

[6]

A. Eden and J.-M. Rakotoson, Exponential attractors for a doubly nonlinear equation, J. Math. Anal. Appl., 185 (1994), 321-339. doi: doi:10.1006/jmaa.1994.1251.

[7]

M. Gurtin, Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance, Physica D, 92 (1996), 178-192. doi: doi:10.1016/0167-2789(95)00173-5.

[8]

O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type," in "Translations of Mathematical Monographs," Vol. 23, American Mathematical Society, Providence, R.I., 1967.

[9]

J. Málek and D. Prážak, Large time behavior via the method of $l$-trajectories, J. Diff. Eqns., 181 (2002), 243-279. doi: doi:10.1006/jdeq.2001.4087.

[10]

A. Miranville, Finite dimensional global attractor for a class of doubly nonlinear parabolic equations, Cent. Eur. J. Math., 4 (2006), 163-182. doi: doi:10.1007/s11533-005-0010-5.

[11]

A. Miranville and S. Zelik, Finite-dimensionality of attractors for degenerate equations of elliptic-parabolic type, Nonlinearity, 20 (2007), 1773-1797. doi: doi:10.1088/0951-7715/20/8/001.

[12]

A. Rougirel, Convergence to steady state and attractors for doubly nonlinear equations, J. Math. Anal. Appl., 339 (2008), 281-294. doi: doi:10.1016/j.jmaa.2007.06.028.

[13]

R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics," Springer, New York, 1988.

show all references

References:
[1]

A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations," North-Holland, Amsterdam, 1992.

[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), 557-566. doi: doi:10.1016/j.jmaa.2008.01.077.

[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), 1029-1030. doi: doi:10.1016/j.jmaa.2008.07.058.

[4]

A. Eden, C. Foias, B. Nicolaenko and R.Temam, "Exponential Attractors for Dissipative Evolution Equations," in "Research in Applied Mathematics," Vol. 37, John-Wiley, New York, 1994.

[5]

A. Eden, B. Michaux and J.-M. Rakotoson, Doubly nonlinear parabolic-type equations as dynamical systems, J. Dyn. Diff. Eqns., 3 (1991), 87-131. doi: doi:10.1007/BF01049490.

[6]

A. Eden and J.-M. Rakotoson, Exponential attractors for a doubly nonlinear equation, J. Math. Anal. Appl., 185 (1994), 321-339. doi: doi:10.1006/jmaa.1994.1251.

[7]

M. Gurtin, Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance, Physica D, 92 (1996), 178-192. doi: doi:10.1016/0167-2789(95)00173-5.

[8]

O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type," in "Translations of Mathematical Monographs," Vol. 23, American Mathematical Society, Providence, R.I., 1967.

[9]

J. Málek and D. Prážak, Large time behavior via the method of $l$-trajectories, J. Diff. Eqns., 181 (2002), 243-279. doi: doi:10.1006/jdeq.2001.4087.

[10]

A. Miranville, Finite dimensional global attractor for a class of doubly nonlinear parabolic equations, Cent. Eur. J. Math., 4 (2006), 163-182. doi: doi:10.1007/s11533-005-0010-5.

[11]

A. Miranville and S. Zelik, Finite-dimensionality of attractors for degenerate equations of elliptic-parabolic type, Nonlinearity, 20 (2007), 1773-1797. doi: doi:10.1088/0951-7715/20/8/001.

[12]

A. Rougirel, Convergence to steady state and attractors for doubly nonlinear equations, J. Math. Anal. Appl., 339 (2008), 281-294. doi: doi:10.1016/j.jmaa.2007.06.028.

[13]

R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics," Springer, New York, 1988.

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