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September  2022, 27(9): 5317-5342. doi: 10.3934/dcdsb.2021276

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 Published  September 2022 Early access  November 2021

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: Lattice dynamics, pullback global attractor, pullback exponential attractor, fractal dimension, quasi-stability.
Mathematics Subject Classification: Primary: 37L30; Secondary: 37L60, 34D45.
Citation: Radosław Czaja. Pullback attractors via quasi-stability for non-autonomous lattice dynamical systems. Discrete and Continuous Dynamical Systems - B, 2022, 27 (9) : 5317-5342. doi: 10.3934/dcdsb.2021276
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.

show all references

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.

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