A doubly nonlinear parabolic equation with a singular potential

Laurence Cherfils; Stefania Gatti; Alain Miranville

doi:10.3934/dcdss.2011.4.51

Article Contents

2011, Volume 4, Issue 1: 51-66. Doi: 10.3934/dcdss.2011.4.51

This issue Previous Article Large-time asymptotics of the generalized Benjamin-Ono-Burgers equation Next Article Some remarks on the qualitative properties of solutions to a predator-prey model posed on non coincident spatial domains

A doubly nonlinear parabolic equation with a singular potential

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

Received: June 30, 2009

Revised: August 31, 2009

Published: January 2011

Abstract Related Papers Cited by

Abstract

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:

Mathematics Subject Classification: 35B41, 35B45, 35K65.

Citation:

Related Papers

Cited by

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.