Pullback attractors via quasi-stability for non-autonomous lattice dynamical systems

Radosław Czaja

doi:10.3934/dcdsb.2021276

Article Contents

2022, Volume 27, Issue 9: 5317-5342. Doi: 10.3934/dcdsb.2021276

This issue Previous Article Tempered fractional order compartment models and applications in biology Next Article Dynamic transitions for the S-K-T competition system

Pullback attractors via quasi-stability for non-autonomous lattice dynamical systems

Radosław Czaja

Institute of Mathematics, University of Silesia in Katowice, Bankowa 14, 40-007 Katowice, Poland

In memory of María José Garrido Atienza

Received: February 2021

Revised: October 2021

Early access: November 2021

Published: September 2022

Abstract Full Text(HTML) Related Papers Cited by

Abstract

In this paper we study long-time behavior of first-order non-autono-mous lattice dynamical systems in square summable space of double-sided sequences using the cooperation between the discretized diffusion operator and the discretized reaction term. We obtain existence of a pullback global attractor and construct pullback exponential attractor applying the introduced notion of quasi-stability of the corresponding evolution process.

Keywords:

Mathematics Subject Classification: Primary: 37L30; Secondary: 37L60, 34D45.

Citation:

Full Text(HTML)

Related Papers

Cited by

References

[1]	A. Y. Abdallah, Exponential attractors for first-order lattice dynamical systems, J. Math. Anal. Appl., 339 (2008), 217-224. doi: 10.1016/j.jmaa.2007.06.054.
[2]	A. Y. Abdallah, Uniform exponential attractors for first order non-autonomous lattice dynamical systems, J. Differential Equations, 251 (2011), 1489-1504. doi: 10.1016/j.jde.2011.05.030.
[3]	A. Y. Abdallah, Attractors for first order lattice systems with almost periodic nonlinear part, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 1241-1255. doi: 10.3934/dcdsb.2019218.
[4]	P. W. Bates, K. Lu and B. Wang, Attractors for lattice dynamical systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 11 (2001), 143-153. doi: 10.1142/S0218127401002031.
[5]	T. Caraballo, G. Łukaszewicz and J. Real, Pullback attractors for asymptotically compact non-autonomous dynamical systems, Nonlinear Anal., 64 (2006), 484-498. doi: 10.1016/j.na.2005.03.111.
[6]	T. Caraballo, F. Morillas and J. Valero, Asymptotic behaviour of a logistic lattice system, Discrete Contin. Dyn. Syst., 34 (2014), 4019-4037. doi: 10.3934/dcds.2014.34.4019.
[7]	T. Caraballo and S. Sonner, Random pullback exponential attractors: General existence results for random dynamical systems in Banach spaces, Discrete Contin. Dyn. Syst., 37 (2017), 6383-6403. doi: 10.3934/dcds.2017277.
[8]	A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, Springer, New York, 2013. doi: 10.1007/978-1-4614-4581-4.
[9]	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.
[10]	J. W. Cholewa and R. Czaja, Lattice dynamical systems: Dissipative mechanism and fractal dimension of global and exponential attractors, J. Evol. Equ., 20 (2020), 485-515. doi: 10.1007/s00028-019-00535-3.
[11]	J. W. Cholewa and R. Czaja, On fractal dimension of global and exponential attractors for dissipative higher order parabolic problems in $\mathbb{R}^{N}$ with general potential, Contemporary Approaches and Methods in Fundamental Mathematics and Mechanics, (2021), 293–314. doi: 10.1007/978-3-030-50302-4_13.
[12]	I. Chueshov, Dynamics of Quasi-Stable Dissipative Systems, Universitext. Springer, Cham, 2015. doi: 10.1007/978-3-319-22903-4.
[13]	I. Chueshov and I. Lasiecka, Long-Time Behavior of Second Order Evolution Equations with Nonlinear Damping, Mem. Amer. Math. Soc. 195 (2008). doi: 10.1090/memo/0912.
[14]	R. Czaja, Bi-space pullback attractors for closed processes, São Paulo J. Math. Sci., 6 (2012), 227-246. doi: 10.11606/issn.2316-9028.v6i2p227-246.
[15]	R. Czaja, Pullback exponential attractors with admissible exponential growth in the past, Nonlinear Anal., 104 (2014), 90-108. doi: 10.1016/j.na.2014.03.020.
[16]	R. Czaja and M. Efendiev, Pullback exponential attractors for nonautonomous equations, Part I: Semilinear parabolic problems, J. Math. Anal. Appl., 381 (2011), 744-765. doi: 10.1016/j.jmaa.2011.03.053.
[17]	X. Han, Exponential attractors for lattice dynamical systems in weighted spaces, Discrete Contin. Dyn. Syst., 31 (2011), 445-467. doi: 10.3934/dcds.2011.31.445.
[18]	X. Han and P. E. Kloeden, Non-autonomous lattice systems with switching effects and delayed recovery, J. Differential Equations, 261 (2016), 2986-3009. doi: 10.1016/j.jde.2016.05.015.
[19]	X. Li, K. Wei and H. Zhang, Exponential attractors for lattice dynamical systems in weighted spaces, Acta Appl. Math., 114 (2011), 157-172. doi: 10.1007/s10440-011-9606-x.
[20]	X. Li and C. Zhong, Attractors for partly dissipative lattice dynamic systems in $\ell^2\times\ell^2$, J. Comput. Appl. Math., 177 (2005), 159-174. doi: 10.1016/j.cam.2004.09.014.
[21]	T. F. Ma, R. N. Monteiro and A. C. Pereira, Pullback dynamics of non-autonomous Timoshenko systems, Appl. Math. Optim., 80 (2019), 391-413. doi: 10.1007/s00245-017-9469-2.
[22]	P. Marín-Rubio and J. Real, On the relation between two different concepts of pullback attractors for non-autonomous dynamical systems, Nonlinear Anal., 71 (2009), 3956-3963. doi: 10.1016/j.na.2009.02.065.
[23]	B. Wang, Attractors for reaction-diffusion equations in unbounded domains, Phys. D, 128 (1999), 41-52. doi: 10.1016/S0167-2789(98)00304-2.
[24]	B. Wang, Asymptotic behavior of non-autonomous lattice systems, J. Math. Anal. Appl., 331 (2007), 121-136. doi: 10.1016/j.jmaa.2006.08.070.
[25]	S. Zhou, Attractors for first order dissipative lattice dynamical systems, Phys. D, 178 (2003), 51-61. doi: 10.1016/S0167-2789(02)00807-2.
[26]	S. Zhou, Attractors and approximations for lattice dynamical systems, J. Differential Equations, 200 (2004), 342-368. doi: 10.1016/j.jde.2004.02.005.
[27]	S. Zhou and X. Han, Pullback exponential attractors for non-autonomous lattice systems, J. Dynam. Differential Equations, 24 (2012), 601-631. doi: 10.1007/s10884-012-9260-7.
[28]	S. Zhou and W. Shi, Attractors and dimension of dissipative lattice systems, J. Differential Equations, 224 (2006), 172-204. doi: 10.1016/j.jde.2005.06.024.