Figure 1.
Trajectories of the semigroup S(t)
- Previous Article
- EECT Home
- This Issue
-
Next Article
Waves and diffusion on metric graphs with general vertex conditions
September
2019, 8(3): 663-668.
doi: 10.3934/eect.2019031
Two questions arising in the theory of attractors
Politecnico di Milano - Dipartimento di Matematica, Via Bonardi 9, 20133 Milano, Italy |
In this note, we dwell on the notions of global and exponential attractors for strongly continuous semigroups acting on a complete metric space. Two natural questions arising in the theory are addressed.
Mathematics Subject Classification:
Primary: 34D45; Secondary: 47H20.
Citation:
Vittorino Pata. Two questions arising in the theory of attractors. Evolution Equations and Control Theory,
2019, 8
(3)
: 663-668.
doi: 10.3934/eect.2019031
![]()
References:
| [1] |
A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North-HollandAmsterdam, 1992. |
| [2] |
V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, Amer. Math. Soc. Providence, 2002. |
| [3] |
A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential Attractors for Dissipative Evolution Equations, MassonParis, 1994. |
| [4] |
M. Efendiev, A. Miranville and S. Zelik,
Exponential attractors for a nonlinear reaction-diffusion system in ${\mathbb R}^3$, C.R. Acad. Sci. Paris Sér. I Math., 330 (2000), 713-718.
doi: 10.1016/S0764-4442(00)00259-7. |
| [5] |
J. K. Hale, Asymptotic Behavior of Dissipative Systems, Amer. Math. Soc. Providence, 1988. |
| [6] |
A. Haraux, Systèmes Dynamiques Dissipatifs Et Applications, MassonParis, 1991. |
| [7] |
A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains, Handbook of Differential Equations: Evolutionary Equations, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 4 (2008), 103–200.
doi: 10.1016/S1874-5717(08)00003-0. |
| [8] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983.
doi: 10.1007/978-1-4612-5561-1. |
| [9] |
R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Springer, New York, 1997.
doi: 10.1007/978-1-4612-0645-3. |
show all references
References:
| [1] |
A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North-HollandAmsterdam, 1992. |
| [2] |
V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, Amer. Math. Soc. Providence, 2002. |
| [3] |
A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential Attractors for Dissipative Evolution Equations, MassonParis, 1994. |
| [4] |
M. Efendiev, A. Miranville and S. Zelik,
Exponential attractors for a nonlinear reaction-diffusion system in ${\mathbb R}^3$, C.R. Acad. Sci. Paris Sér. I Math., 330 (2000), 713-718.
doi: 10.1016/S0764-4442(00)00259-7. |
| [5] |
J. K. Hale, Asymptotic Behavior of Dissipative Systems, Amer. Math. Soc. Providence, 1988. |
| [6] |
A. Haraux, Systèmes Dynamiques Dissipatifs Et Applications, MassonParis, 1991. |
| [7] |
A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains, Handbook of Differential Equations: Evolutionary Equations, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 4 (2008), 103–200.
doi: 10.1016/S1874-5717(08)00003-0. |
| [8] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983.
doi: 10.1007/978-1-4612-5561-1. |
| [9] |
R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Springer, New York, 1997.
doi: 10.1007/978-1-4612-0645-3. |
Figure 2.
Portrait of the exponential attractor ε
| [1] |
V. Pata, Sergey Zelik. A result on the existence of global attractors for semigroups of closed operators. Communications on Pure and Applied Analysis, 2007, 6 (2) : 481-486. doi: 10.3934/cpaa.2007.6.481 |
| [2] |
Samir EL Mourchid. On a hypercylicity criterion for strongly continuous semigroups. Discrete and Continuous Dynamical Systems, 2005, 13 (2) : 271-275. doi: 10.3934/dcds.2005.13.271 |
| [3] |
Michele Coti Zelati. Global and exponential attractors for the singularly perturbed extensible beam. Discrete and Continuous Dynamical Systems, 2009, 25 (3) : 1041-1060. doi: 10.3934/dcds.2009.25.1041 |
| [4] |
Filippo Dell'Oro. Global attractors for strongly damped wave equations with subcritical-critical nonlinearities. Communications on Pure and Applied Analysis, 2013, 12 (2) : 1015-1027. doi: 10.3934/cpaa.2013.12.1015 |
| [5] |
Angela A. Albanese, Xavier Barrachina, Elisabetta M. Mangino, Alfredo Peris. Distributional chaos for strongly continuous semigroups of operators. Communications on Pure and Applied Analysis, 2013, 12 (5) : 2069-2082. doi: 10.3934/cpaa.2013.12.2069 |
| [6] |
Manoel J. Dos Santos, Baowei Feng, Dilberto S. Almeida Júnior, Mauro L. Santos. Global and exponential attractors for a nonlinear porous elastic system with delay term. Discrete and Continuous Dynamical Systems - B, 2021, 26 (5) : 2805-2828. doi: 10.3934/dcdsb.2020206 |
| [7] |
Alessia Berti, Valeria Berti, Ivana Bochicchio. Global and exponential attractors for a Ginzburg-Landau model of superfluidity. Discrete and Continuous Dynamical Systems - S, 2011, 4 (2) : 247-271. doi: 10.3934/dcdss.2011.4.247 |
| [8] |
José A. Langa, Alain Miranville, José Real. Pullback exponential attractors. Discrete and Continuous Dynamical Systems, 2010, 26 (4) : 1329-1357. doi: 10.3934/dcds.2010.26.1329 |
| [9] |
Manil T. Mohan. Global attractors, exponential attractors and determining modes for the three dimensional Kelvin-Voigt fluids with "fading memory". Evolution Equations and Control Theory, 2022, 11 (1) : 125-167. doi: 10.3934/eect.2020105 |
| [10] |
Jin Zhang, Peter E. Kloeden, Meihua Yang, Chengkui Zhong. Global exponential κ-dissipative semigroups and exponential attraction. Discrete and Continuous Dynamical Systems, 2017, 37 (6) : 3487-3502. doi: 10.3934/dcds.2017148 |
| [11] |
A. Kh. Khanmamedov. Global attractors for strongly damped wave equations with displacement dependent damping and nonlinear source term of critical exponent. Discrete and Continuous Dynamical Systems, 2011, 31 (1) : 119-138. doi: 10.3934/dcds.2011.31.119 |
| [12] |
Tomáš Gedeon. Attractors in continuous –time switching networks. Communications on Pure and Applied Analysis, 2003, 2 (2) : 187-209. doi: 10.3934/cpaa.2003.2.187 |
| [13] |
Kei Matsuura, Mitsuharu Otani. Exponential attractors for a quasilinear parabolic equation. Conference Publications, 2007, 2007 (Special) : 713-720. doi: 10.3934/proc.2007.2007.713 |
| [14] |
Lin Yang, Yejuan Wang, Tomás Caraballo. Regularity of global attractors and exponential attractors for $ 2 $D quasi-geostrophic equations with fractional dissipation. Discrete and Continuous Dynamical Systems - B, 2022, 27 (3) : 1345-1377. doi: 10.3934/dcdsb.2021093 |
| [15] |
Monica Conti, Vittorino Pata. On the regularity of global attractors. Discrete and Continuous Dynamical Systems, 2009, 25 (4) : 1209-1217. doi: 10.3934/dcds.2009.25.1209 |
| [16] |
Manil T. Mohan. Global and exponential attractors for the 3D Kelvin-Voigt-Brinkman-Forchheimer equations. Discrete and Continuous Dynamical Systems - B, 2020, 25 (9) : 3393-3436. doi: 10.3934/dcdsb.2020067 |
| [17] |
Etsushi Nakaguchi, Koichi Osaki. Global solutions and exponential attractors of a parabolic-parabolic system for chemotaxis with subquadratic degradation. Discrete and Continuous Dynamical Systems - B, 2013, 18 (10) : 2627-2646. doi: 10.3934/dcdsb.2013.18.2627 |
| [18] |
Pengyan Ding, Zhijian Yang. Attractors of the strongly damped Kirchhoff wave equation on $\mathbb{R}^{N}$. Communications on Pure and Applied Analysis, 2019, 18 (2) : 825-843. doi: 10.3934/cpaa.2019040 |
| [19] |
Veronica Belleri, Vittorino Pata. Attractors for semilinear strongly damped wave equations on $\mathbb R^3$. Discrete and Continuous Dynamical Systems, 2001, 7 (4) : 719-735. doi: 10.3934/dcds.2001.7.719 |
| [20] |
Vladimir V. Chepyzhov, Monica Conti, Vittorino Pata. A minimal approach to the theory of global attractors. Discrete and Continuous Dynamical Systems, 2012, 32 (6) : 2079-2088. doi: 10.3934/dcds.2012.32.2079 |
2021 Impact Factor: 1.169
Tools
Metrics
Other articles
by authors
[Back to Top]