American Institute of Mathematical Sciences

Advanced
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, (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

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

[1]

Antonio Segatti. Global attractor for a class of doubly nonlinear abstract evolution equations. Discrete & Continuous Dynamical Systems - A, 2006, 14 (4) : 801-820. doi: 10.3934/dcds.2006.14.801
[2]

Dalibor Pražák. Exponential attractor for the delayed logistic equation with a nonlinear diffusion. Conference Publications, 2003, 2003 (Special) : 717-726. doi: 10.3934/proc.2003.2003.717
[3]

Wided Kechiche. Regularity of the global attractor for a nonlinear Schrödinger equation with a point defect. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1233-1252. doi: 10.3934/cpaa.2017060
[4]

S.V. Zelik. The attractor for a nonlinear hyperbolic equation in the unbounded domain. Discrete & Continuous Dynamical Systems - A, 2001, 7 (3) : 593-641. doi: 10.3934/dcds.2001.7.593
[5]

Dalibor Pražák. On the dimension of the attractor for the wave equation with nonlinear damping. Communications on Pure & Applied Analysis, 2005, 4 (1) : 165-174. doi: 10.3934/cpaa.2005.4.165
[6]

Rolci Cipolatti, Otared Kavian. On a nonlinear Schrödinger equation modelling ultra-short laser pulses with a large noncompact global attractor. Discrete & Continuous Dynamical Systems - A, 2007, 17 (1) : 121-132. doi: 10.3934/dcds.2007.17.121
[7]

Oleksiy V. Kapustyan, Pavlo O. Kasyanov, José Valero. Structure and regularity of the global attractor of a reaction-diffusion equation with non-smooth nonlinear term. Discrete & Continuous Dynamical Systems - A, 2014, 34 (10) : 4155-4182. doi: 10.3934/dcds.2014.34.4155
[8]

Zhiming Liu, Zhijian Yang. Global attractor of multi-valued operators with applications to a strongly damped nonlinear wave equation without uniqueness. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 223-240. doi: 10.3934/dcdsb.2019179
[9]

Francesca Bucci, Igor Chueshov, Irena Lasiecka. Global attractor for a composite system of nonlinear wave and plate equations. Communications on Pure & Applied Analysis, 2007, 6 (1) : 113-140. doi: 10.3934/cpaa.2007.6.113
[10]

Wenxian Shen. Global attractor and rotation number of a class of nonlinear noisy oscillators. Discrete & Continuous Dynamical Systems - A, 2007, 18 (2&3) : 597-611. doi: 10.3934/dcds.2007.18.597
[11]

Moncef Aouadi, Alain Miranville. Quasi-stability and global attractor in nonlinear thermoelastic diffusion plate with memory. Evolution Equations & Control Theory, 2015, 4 (3) : 241-263. doi: 10.3934/eect.2015.4.241
[12]

Milena Stanislavova. On the global attractor for the damped Benjamin-Bona-Mahony equation. Conference Publications, 2005, 2005 (Special) : 824-832. doi: 10.3934/proc.2005.2005.824
[13]

Zhijian Yang, Zhiming Liu. Global attractor for a strongly damped wave equation with fully supercritical nonlinearities. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 2181-2205. doi: 10.3934/dcds.2017094
[14]

D. Hilhorst, L. A. Peletier, A. I. Rotariu, G. Sivashinsky. Global attractor and inertial sets for a nonlocal Kuramoto-Sivashinsky equation. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 557-580. doi: 10.3934/dcds.2004.10.557
[15]

Azer Khanmamedov, Sema Simsek. Existence of the global attractor for the plate equation with nonlocal nonlinearity in $ \mathbb{R} ^{n}$. Discrete & Continuous Dynamical Systems - B, 2016, 21 (1) : 151-172. doi: 10.3934/dcdsb.2016.21.151
[16]

Tomás Caraballo, Marta Herrera-Cobos, Pedro Marín-Rubio. Global attractor for a nonlocal p-Laplacian equation without uniqueness of solution. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1801-1816. doi: 10.3934/dcdsb.2017107
[17]

Zhaojuan Wang, Shengfan Zhou. Random attractor and random exponential attractor for stochastic non-autonomous damped cubic wave equation with linear multiplicative white noise. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4767-4817. doi: 10.3934/dcds.2018210
[18]

Olivier Goubet, Wided Kechiche. Uniform attractor for non-autonomous nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2011, 10 (2) : 639-651. doi: 10.3934/cpaa.2011.10.639
[19]

Messoud Efendiev, Anna Zhigun. On an exponential attractor for a class of PDEs with degenerate diffusion and chemotaxis. Discrete & Continuous Dynamical Systems - A, 2018, 38 (2) : 651-673. doi: 10.3934/dcds.2018028
[20]

Xingni Tan, Fuqi Yin, Guihong Fan. Random exponential attractor for stochastic discrete long wave-short wave resonance equation with multiplicative white noise. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 0-0. doi: 10.3934/dcdsb.2020055

2018 Impact Factor: 0.545

Tools

Metrics

  • PDF downloads (26)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences