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January  2020, 25(1): 301-319. doi: 10.3934/dcdsb.2019183

Pullback exponential attractors for differential equations with variable delays

Mohamed Ali Hammami a, , Lassaad Mchiri a,b, , Sana Netchaoui a, and  Stefanie Sonner c,,

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

* Corresponding author: Stefanie Sonner

Received  February 2019 Published  January 2020 Early access  July 2019

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.

Keywords: Non-autonomous delay differential equations, variable delays, pullback exponential attractors, fractal dimension, non-autonomous dynamical systems.
Mathematics Subject Classification: Primary: 37L30, 37B55; Secondary: 34D45, 37L25.
Citation: Mohamed Ali Hammami, Lassaad Mchiri, Sana Netchaoui, Stefanie Sonner. Pullback exponential attractors for differential equations with variable delays. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 301-319. doi: 10.3934/dcdsb.2019183
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential Attractors for Dissipative Evolution Equations, John Wiley and Sons Ltd., Chichester, 1994.  Google Scholar

[8] D. E. Edmunds and H. Triebel, Function Spaces, Entropy Numbers and Differential Operators, Cambridge University Press, New York, 1996.  doi: 10.1017/CBO9780511662201.  Google Scholar
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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.  Google Scholar

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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.  Google Scholar

[11]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs, 25, American Mathematical Society, Providence, RI, 1988.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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S. Sonner, Systems of Quasi-Linear PDEs Arising in the Modelling of Biofilms and Related Dynamical Questions, PhD thesis, Technische Universität München, Germany (2012). Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential Attractors for Dissipative Evolution Equations, John Wiley and Sons Ltd., Chichester, 1994.  Google Scholar

[8] D. E. Edmunds and H. Triebel, Function Spaces, Entropy Numbers and Differential Operators, Cambridge University Press, New York, 1996.  doi: 10.1017/CBO9780511662201.  Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[11]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs, 25, American Mathematical Society, Providence, RI, 1988.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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). Google Scholar

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