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April  2021, 14(4): 1345-1358. doi: 10.3934/dcdss.2020367

Pullback exponential attractors for differential equations with delay

Sana Netchaoui 1, , Mohamed Ali Hammami 1, and  Tomás Caraballo 2,

1. 

Department of Mathematics, Faculty of Sciences of Sfax, University of Sfax, Route de la Soukra km 4 Sfax 3038, Tunisia
2. 

Departamento de Ecuaciones Diferenciales y Análisis Numérico, Facultad de Matemáticas, Universidad de Sevilla, c/ Tarfia s/n, 41012-Sevilla, Spain

Received  August 2019 Revised  January 2020 Published  April 2021 Early access  May 2020

Fund Project: This work has been partially supported by FEDER and the Spanish Ministerio de Ciencia, Innovación y Universidades under project PGC2018-096540-B-I00, and Proyecto I+D+i Programa Operativo FEDER Andalucía US-1254251

We show the existence of an exponential attractor for non-autono-mous dynamical system with bounded delay. We considered the case of strong dissipativity then prove that the result remains for the weak dissipativity. We conclude then the existence of the global attractor and ensure the boundedness of its fractal dimension.

Keywords: non-autonomous delay differential equations, variable delays, distributed delay, integro-differential equation, pullback exponential attractors, fractal dimension, non-autonomous dynamical systems.
Mathematics Subject Classification: Primary 37B55; Secondary 34D45, 37L25.
Citation: Sana Netchaoui, Mohamed Ali Hammami, Tomás Caraballo. Pullback exponential attractors for differential equations with delay. Discrete and Continuous Dynamical Systems - S, 2021, 14 (4) : 1345-1358. doi: 10.3934/dcdss.2020367
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.

[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 problems, J. Math. Anal. Appl., 381 (2011), 748-765.  doi: 10.1016/j.jmaa.2011.03.053.

[7]

J. M. Cushing, Integrodifferential Equations and Delay Models in Population Dynamics, Lecture Notes in Biomathematics, vol. 20, Springer-Verlag, Heidelberg, 1977.

[8]

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

[9]

D. E. Edmunds and H. Triebel, Function Spaces, Entropy Numbers, Differential Operators, Cambridge Tracts in Mathematics, 120. Cambridge University Press, Cambridge, 1996. doi: 10.1017/CBO9780511662201.

[10]

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.

[11]

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.

[12]

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

[13]

M. A.Hammami, L. Mchiri, S. Netchaoui and S. Sonner, Pullback exponential attractors for differential equations with variable delays, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 301-319.  doi: 10.3934/dcdsb.2019183.

[14]

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.

[15]

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.

[16]

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.

[17]

S. Sonner, Systems of Quasi-Linear PDEs Arising in the Modelling of Biofilms and Related Dynamical Questions, Ph.D. 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.

[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 problems, J. Math. Anal. Appl., 381 (2011), 748-765.  doi: 10.1016/j.jmaa.2011.03.053.

[7]

J. M. Cushing, Integrodifferential Equations and Delay Models in Population Dynamics, Lecture Notes in Biomathematics, vol. 20, Springer-Verlag, Heidelberg, 1977.

[8]

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

[9]

D. E. Edmunds and H. Triebel, Function Spaces, Entropy Numbers, Differential Operators, Cambridge Tracts in Mathematics, 120. Cambridge University Press, Cambridge, 1996. doi: 10.1017/CBO9780511662201.

[10]

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.

[11]

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.

[12]

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

[13]

M. A.Hammami, L. Mchiri, S. Netchaoui and S. Sonner, Pullback exponential attractors for differential equations with variable delays, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 301-319.  doi: 10.3934/dcdsb.2019183.

[14]

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.

[15]

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.

[16]

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.

[17]

S. Sonner, Systems of Quasi-Linear PDEs Arising in the Modelling of Biofilms and Related Dynamical Questions, Ph.D. thesis, Technische Universität München, Germany, 2012.

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