# American Institute of Mathematical Sciences

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## Pullback exponential attractors for differential equations with delay

 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  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.

Citation: Sana Netchaoui, Mohamed Ali Hammami, Tomás Caraballo. Pullback exponential attractors for differential equations with delay. Discrete & Continuous Dynamical Systems - S, 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.  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.  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 problems, J. Math. Anal. Appl., 381 (2011), 748-765.  doi: 10.1016/j.jmaa.2011.03.053.  Google Scholar [7] J. M. Cushing, Integrodifferential Equations and Delay Models in Population Dynamics, Lecture Notes in Biomathematics, vol. 20, Springer-Verlag, Heidelberg, 1977.  Google Scholar [8] A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential Attractors for Dissipative Evolution Equations, John Wiley and Sons Ltd., Chichester, 1994.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [12] J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs, 25, American Mathematical Society, Providence, RI., 1988.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar

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##### 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.  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 problems, J. Math. Anal. Appl., 381 (2011), 748-765.  doi: 10.1016/j.jmaa.2011.03.053.  Google Scholar [7] J. M. Cushing, Integrodifferential Equations and Delay Models in Population Dynamics, Lecture Notes in Biomathematics, vol. 20, Springer-Verlag, Heidelberg, 1977.  Google Scholar [8] A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential Attractors for Dissipative Evolution Equations, John Wiley and Sons Ltd., Chichester, 1994.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [12] J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs, 25, American Mathematical Society, Providence, RI., 1988.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar
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