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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

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, Amsterdam, 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), 557-566. 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), 1029-1030. 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 "Research in Applied Mathematics," Vol. 37, John-Wiley, New York, 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), 87-131. 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), 321-339. 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), 178-192. 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 "Translations of Mathematical Monographs," Vol. 23, American Mathematical Society, Providence, R.I., 1967.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

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

show all references

References:
[1]

A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations," North-Holland, Amsterdam, 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), 557-566. 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), 1029-1030. 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 "Research in Applied Mathematics," Vol. 37, John-Wiley, New York, 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), 87-131. 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), 321-339. 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), 178-192. 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 "Translations of Mathematical Monographs," Vol. 23, American Mathematical Society, Providence, R.I., 1967.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

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

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