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Pullback attractors via quasi-stability for non-autonomous lattice dynamical systems
Institute of Mathematics, University of Silesia in Katowice, Bankowa 14, 40-007 Katowice, Poland |
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.
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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