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Advances in the LaSalle-type theorems for stochastic functional differential equations with infinite delay
Pullback exponential attractors for differential equations with variable delays
| a. | Department of Mathematics, University of Sfax, Route de la Soukra km 4, Sfax 3038, Tunisia |
| b. | Department of Statistics and Operations Research, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia |
| c. | IMAPP Mathematics, Radboud Universiteit Nijmegen, PO Box 9010, 6500GL Nijmegen, The Netherlands |
We show how recent existence results for pullback exponential attractors can be applied to non-autonomous delay differential equations with time-varying delays. Moreover, we derive explicit estimates for the fractal dimension of the attractors.
As a special case, autonomous delay differential equations are also discussed, where our results improve previously obtained bounds for the fractal dimension of exponential attractors.
References:
| [1] |
T. Caraballo, P. Marin-Rubio and J. Valero,
Autonomous and non-autonomous attractors for differential equations with delays, J. Differential Equations, 208 (2005), 9-41.
doi: 10.1016/j.jde.2003.09.008. |
| [2] |
T. Caraballo, J. A. Langa and J. C. Robinson,
Attractors for differential equations with variable delays, J. Math. Anal. Appl., 260 (2001), 421-438.
doi: 10.1006/jmaa.2000.7464. |
| [3] |
A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, Applied Mathematical Sciences, 182. Springer, New York, 2013.
doi: 10.1007/978-1-4614-4581-4. |
| [4] |
A. N. Carvalho and S. Sonner,
Pullback exponential attractors for evolution processes in Banach spaces: theoretical results, Commun. Pure Appl. Anal., 12 (2013), 3047-3071.
doi: 10.3934/cpaa.2013.12.3047. |
| [5] |
A. N. Carvalho and S. Sonner,
Pullback exponential attractors for evolution processes in Banach spaces: properties and applications, Commun. Pure Appl. Anal., 13 (2014), 1141-1165.
doi: 10.3934/cpaa.2014.13.1141. |
| [6] |
R. Czaja and M. A. Efendiev,
Pullback exponential attractors for nonautonomous equations part I: Semilinear parabolic equations, J. Math. Anal. Appl., 381 (2011), 748-765.
doi: 10.1016/j.jmaa.2011.03.053. |
| [7] |
A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential Attractors for Dissipative Evolution Equations, John Wiley and Sons Ltd., Chichester, 1994. |
| [8] |
D. E. Edmunds and H. Triebel, Function Spaces, Entropy Numbers and Differential Operators, Cambridge University Press, New York, 1996.
doi: 10.1017/CBO9780511662201.
|
| [9] |
M. A. 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. |
| [10] |
S. Habibi,
Estimates on the dimension of an exponential attractor for a delay differential equation, Math. Slovaca, 64 (2014), 1237-1248.
doi: 10.2478/s12175-014-0272-0. |
| [11] |
J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs, 25, American Mathematical Society, Providence, RI, 1988. |
| [12] |
J. A. Langa, A. Miranville and J. Real,
Pullback exponential attractors, Discrete Contin. Dyn. Syst., 26 (2010), 1329-1357.
doi: 10.3934/dcds.2010.26.1329. |
| [13] |
D. Pražák,
On the dynamics of equations with infinite delay, Cent. Eur. J. Math., 4 (2006), 635-647.
doi: 10.2478/s11533-006-0024-7. |
| [14] |
H. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences, Texts in Applied Mathematics, 57, Springer, New York, 2011.
doi: 10.1007/978-1-4419-7646-8. |
| [15] |
S. Sonner, Systems of Quasi-Linear PDEs Arising in the Modelling of Biofilms and Related Dynamical Questions, PhD thesis, Technische Universität München, Germany (2012). |
show all references
References:
| [1] |
T. Caraballo, P. Marin-Rubio and J. Valero,
Autonomous and non-autonomous attractors for differential equations with delays, J. Differential Equations, 208 (2005), 9-41.
doi: 10.1016/j.jde.2003.09.008. |
| [2] |
T. Caraballo, J. A. Langa and J. C. Robinson,
Attractors for differential equations with variable delays, J. Math. Anal. Appl., 260 (2001), 421-438.
doi: 10.1006/jmaa.2000.7464. |
| [3] |
A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, Applied Mathematical Sciences, 182. Springer, New York, 2013.
doi: 10.1007/978-1-4614-4581-4. |
| [4] |
A. N. Carvalho and S. Sonner,
Pullback exponential attractors for evolution processes in Banach spaces: theoretical results, Commun. Pure Appl. Anal., 12 (2013), 3047-3071.
doi: 10.3934/cpaa.2013.12.3047. |
| [5] |
A. N. Carvalho and S. Sonner,
Pullback exponential attractors for evolution processes in Banach spaces: properties and applications, Commun. Pure Appl. Anal., 13 (2014), 1141-1165.
doi: 10.3934/cpaa.2014.13.1141. |
| [6] |
R. Czaja and M. A. Efendiev,
Pullback exponential attractors for nonautonomous equations part I: Semilinear parabolic equations, J. Math. Anal. Appl., 381 (2011), 748-765.
doi: 10.1016/j.jmaa.2011.03.053. |
| [7] |
A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential Attractors for Dissipative Evolution Equations, John Wiley and Sons Ltd., Chichester, 1994. |
| [8] |
D. E. Edmunds and H. Triebel, Function Spaces, Entropy Numbers and Differential Operators, Cambridge University Press, New York, 1996.
doi: 10.1017/CBO9780511662201.
|
| [9] |
M. A. 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. |
| [10] |
S. Habibi,
Estimates on the dimension of an exponential attractor for a delay differential equation, Math. Slovaca, 64 (2014), 1237-1248.
doi: 10.2478/s12175-014-0272-0. |
| [11] |
J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs, 25, American Mathematical Society, Providence, RI, 1988. |
| [12] |
J. A. Langa, A. Miranville and J. Real,
Pullback exponential attractors, Discrete Contin. Dyn. Syst., 26 (2010), 1329-1357.
doi: 10.3934/dcds.2010.26.1329. |
| [13] |
D. Pražák,
On the dynamics of equations with infinite delay, Cent. Eur. J. Math., 4 (2006), 635-647.
doi: 10.2478/s11533-006-0024-7. |
| [14] |
H. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences, Texts in Applied Mathematics, 57, Springer, New York, 2011.
doi: 10.1007/978-1-4419-7646-8. |
| [15] |
S. Sonner, Systems of Quasi-Linear PDEs Arising in the Modelling of Biofilms and Related Dynamical Questions, PhD thesis, Technische Universität München, Germany (2012). |
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