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August  2019, 24(8): 3525-3535. doi: 10.3934/dcdsb.2018276

## Convergences of asymptotically autonomous pullback attractors towards semigroup attractors

 School of Mathematics and Statistics, Huazhong University of Science & Technology, Wuhan 430074, China

Dedicated to Professor Peter Kloeden on his 70th birthday

Received  November 2017 Revised  May 2018 Published  October 2018

For pullback attractors of asymptotically autonomous dynamical systems we study the convergences of their components towards the global attractors of the limiting semigroups. We use some conditions of uniform boundedness of pullback attractors, instead of uniform compactness conditions used in the literature. Both forward convergence and backward convergence are studied.

Citation: Hongyong Cui. Convergences of asymptotically autonomous pullback attractors towards semigroup attractors. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3525-3535. doi: 10.3934/dcdsb.2018276
##### References:
 [1] A. N. Carvalho, J. A. Langa and J. C. Robinson, Non-autonomous dynamical systems, Discrete and Continuous Dynamical Systems-Series B, 20 (2015), 703-747. doi: 10.3934/dcdsb.2015.20.703. Google Scholar [2] A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, Vol. 182, Springer, 2013. doi: 10.1007/978-1-4614-4581-4. Google Scholar [3] H. Cui, M. M. Freitas and J. A. Langa, On random cocycle attractors with autonomous attraction universes, Discrete and Continuous Dynamical Systems - Series B, 22 (2017), 3379-3407. doi: 10.3934/dcdsb.2017142. Google Scholar [4] H. Cui and P. E. Kloeden, Tail convergences of pullback attractors for asymptotically converging multi-valued dynamical systems, submitted.Google Scholar [5] H. Cui, P. E. Kloeden and F. Wu, Pathwise upper semi-continuity of random pullback attractors along the time axis, Physica D: Nonlinear Phenomena, 374-375 (2018), 21-34. doi: 10.1016/j.physd.2018.03.002. Google Scholar [6] H. Cui, P. E. Kloeden and M. Yang, Forward omega limit sets of nonautonomous dynamical systems, Discrete and Continuous Dynamical Systems - Series S. Page in press.Google Scholar [7] H. Cui and J. A. Langa, Uniform attractors for non-autonomous random dynamical systems, Journal of Differential Equations, 263 (2017), 1225–1268. http://www.sciencedirect.com/science/article/pii/S0022039617301535.Google Scholar [8] H. Cui, J. A. Langa, Y. Li and J. Valero, Attractors for multi-valued non-autonomous dynamical systems: Relationship, characterization and robustness, Set-Valued and Variational Analysis, 26 (2018), 493-530. doi: 10.1007/s11228-016-0395-2. Google Scholar [9] P. Kloeden and T. Lorenz, Construction of nonautonomous forward attractors, Proceedings of the American Mathematical Society, 144 (2016), 259-268. doi: 10.1090/proc/12735. Google Scholar [10] P. E. Kloeden, T. Lorenz and M. Yang, Forward Attractors in Discrete Time Nonautonomous Dynamical Systems, in Differential and Difference Equations with Application, Springer International Publishing, 2015,313-322. doi: 10.1007/978-3-319-32857-7_29. Google Scholar [11] P. E. Kloeden and P. MarÃ­n-Rubio, Negatively invariant sets and entire solutions, Journal of Dynamics and Differential Equations, 23 (2011), 437-450. doi: 10.1007/s10884-010-9196-8. Google Scholar [12] P. E. Kloeden, C. Pötzsche and M. Rasmussen, Limitations of pullback attractors for processes, Journal of Difference Equations and Applications, 18 (2012), 693-701. doi: 10.1080/10236198.2011.578070. Google Scholar [13] P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, Number 176, American Mathematical Soc., 2011. doi: 10.1090/surv/176. Google Scholar [14] P. E. Kloeden and J. Simsen, Attractors of asymptotically autonomous quasi-linear parabolic equation with spatially variable exponents, Journal of Mathematical Analysis and Applications, 425 (2015), 911-918. doi: 10.1016/j.jmaa.2014.12.069. Google Scholar [15] P. E. Kloeden, J. Simsen and M. S. Simsen, Asymptotically autonomous multivalued cauchy problems with spatially variable exponents, Journal of Mathematical Analysis and Applications, 445 (2017), 513-531. doi: 10.1016/j.jmaa.2016.08.004. Google Scholar [16] P. E. Kloeden and M. Yang, Forward attraction in nonautonomous difference equations, Journal of Difference Equations & Applications, 22 (2015), 513-525. doi: 10.1080/10236198.2015.1107550. Google Scholar [17] Y. Li, L. She and R. Wang, Asymptotically autonomous dynamics for parabolic equations, Journal of Mathematical Analysis and Applications, 459 (2018), 1106-1123. doi: 10.1016/j.jmaa.2017.11.033. Google Scholar [18] Y. Li, L. She and J. Yin, Longtime robustness and semi-uniform compactness of a pullback attractor via nonautonomous PDE, Discrete and Continuous Dynamical Systems - Series B, 23 (2018), 1535-1557. doi: 10.3934/dcdsb.2018058. Google Scholar [19] J. C. Robinson, Infinite-Dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors, Vol. 28, Cambridge University Press, 2001. doi: 10.1007/978-94-010-0732-0. Google Scholar [20] G. R. Sell, Topological Dynamics and Ordinary Differential Equations, Van Nostrand-Reinhold, London, 1971. Google Scholar [21] B. Wang, Pullback attractors for non-autonomous reaction-diffusion equations on $\mathbb{R}^n$, Frontiers of Mathematics in China, 4 (2009), 563-583. doi: 10.1007/s11464-009-0033-5. Google Scholar [22] Y. Wang, D. Li and P. E. Kloeden, On the asymptotical behavior of nonautonomous dynamical systems, Nonlinear Analysis: Theory Methods & Applications, 59 (2004), 35-53. doi: 10.1016/j.na.2004.03.035. Google Scholar [23] Y. Wang, L. Wang and W. Zhao, Pullback attractors for nonautonomous reaction-diffusion equations in unbounded domains, Journal of Mathematical Analysis & Applications, 336 (2007), 330-347. doi: 10.1016/j.jmaa.2007.02.081. Google Scholar [24] B. Zhu, L. Liu and Y. Wu, Local and global existence of mild solutions for a class of nonlinear fractional reaction-diffusion equations with delay, Applied Mathematics Letters, 61 (2016), 73-79. doi: 10.1016/j.aml.2016.05.010. Google Scholar

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##### References:
 [1] A. N. Carvalho, J. A. Langa and J. C. Robinson, Non-autonomous dynamical systems, Discrete and Continuous Dynamical Systems-Series B, 20 (2015), 703-747. doi: 10.3934/dcdsb.2015.20.703. Google Scholar [2] A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, Vol. 182, Springer, 2013. doi: 10.1007/978-1-4614-4581-4. Google Scholar [3] H. Cui, M. M. Freitas and J. A. Langa, On random cocycle attractors with autonomous attraction universes, Discrete and Continuous Dynamical Systems - Series B, 22 (2017), 3379-3407. doi: 10.3934/dcdsb.2017142. Google Scholar [4] H. Cui and P. E. Kloeden, Tail convergences of pullback attractors for asymptotically converging multi-valued dynamical systems, submitted.Google Scholar [5] H. Cui, P. E. Kloeden and F. Wu, Pathwise upper semi-continuity of random pullback attractors along the time axis, Physica D: Nonlinear Phenomena, 374-375 (2018), 21-34. doi: 10.1016/j.physd.2018.03.002. Google Scholar [6] H. Cui, P. E. Kloeden and M. Yang, Forward omega limit sets of nonautonomous dynamical systems, Discrete and Continuous Dynamical Systems - Series S. Page in press.Google Scholar [7] H. Cui and J. A. Langa, Uniform attractors for non-autonomous random dynamical systems, Journal of Differential Equations, 263 (2017), 1225–1268. http://www.sciencedirect.com/science/article/pii/S0022039617301535.Google Scholar [8] H. Cui, J. A. Langa, Y. Li and J. Valero, Attractors for multi-valued non-autonomous dynamical systems: Relationship, characterization and robustness, Set-Valued and Variational Analysis, 26 (2018), 493-530. doi: 10.1007/s11228-016-0395-2. Google Scholar [9] P. Kloeden and T. Lorenz, Construction of nonautonomous forward attractors, Proceedings of the American Mathematical Society, 144 (2016), 259-268. doi: 10.1090/proc/12735. Google Scholar [10] P. E. Kloeden, T. Lorenz and M. Yang, Forward Attractors in Discrete Time Nonautonomous Dynamical Systems, in Differential and Difference Equations with Application, Springer International Publishing, 2015,313-322. doi: 10.1007/978-3-319-32857-7_29. Google Scholar [11] P. E. Kloeden and P. MarÃ­n-Rubio, Negatively invariant sets and entire solutions, Journal of Dynamics and Differential Equations, 23 (2011), 437-450. doi: 10.1007/s10884-010-9196-8. Google Scholar [12] P. E. Kloeden, C. Pötzsche and M. Rasmussen, Limitations of pullback attractors for processes, Journal of Difference Equations and Applications, 18 (2012), 693-701. doi: 10.1080/10236198.2011.578070. Google Scholar [13] P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, Number 176, American Mathematical Soc., 2011. doi: 10.1090/surv/176. Google Scholar [14] P. E. Kloeden and J. Simsen, Attractors of asymptotically autonomous quasi-linear parabolic equation with spatially variable exponents, Journal of Mathematical Analysis and Applications, 425 (2015), 911-918. doi: 10.1016/j.jmaa.2014.12.069. Google Scholar [15] P. E. Kloeden, J. Simsen and M. S. Simsen, Asymptotically autonomous multivalued cauchy problems with spatially variable exponents, Journal of Mathematical Analysis and Applications, 445 (2017), 513-531. doi: 10.1016/j.jmaa.2016.08.004. Google Scholar [16] P. E. Kloeden and M. Yang, Forward attraction in nonautonomous difference equations, Journal of Difference Equations & Applications, 22 (2015), 513-525. doi: 10.1080/10236198.2015.1107550. Google Scholar [17] Y. Li, L. She and R. Wang, Asymptotically autonomous dynamics for parabolic equations, Journal of Mathematical Analysis and Applications, 459 (2018), 1106-1123. doi: 10.1016/j.jmaa.2017.11.033. Google Scholar [18] Y. Li, L. She and J. Yin, Longtime robustness and semi-uniform compactness of a pullback attractor via nonautonomous PDE, Discrete and Continuous Dynamical Systems - Series B, 23 (2018), 1535-1557. doi: 10.3934/dcdsb.2018058. Google Scholar [19] J. C. Robinson, Infinite-Dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors, Vol. 28, Cambridge University Press, 2001. doi: 10.1007/978-94-010-0732-0. Google Scholar [20] G. R. Sell, Topological Dynamics and Ordinary Differential Equations, Van Nostrand-Reinhold, London, 1971. Google Scholar [21] B. Wang, Pullback attractors for non-autonomous reaction-diffusion equations on $\mathbb{R}^n$, Frontiers of Mathematics in China, 4 (2009), 563-583. doi: 10.1007/s11464-009-0033-5. Google Scholar [22] Y. Wang, D. Li and P. E. Kloeden, On the asymptotical behavior of nonautonomous dynamical systems, Nonlinear Analysis: Theory Methods & Applications, 59 (2004), 35-53. doi: 10.1016/j.na.2004.03.035. Google Scholar [23] Y. Wang, L. Wang and W. Zhao, Pullback attractors for nonautonomous reaction-diffusion equations in unbounded domains, Journal of Mathematical Analysis & Applications, 336 (2007), 330-347. doi: 10.1016/j.jmaa.2007.02.081. Google Scholar [24] B. Zhu, L. Liu and Y. Wu, Local and global existence of mild solutions for a class of nonlinear fractional reaction-diffusion equations with delay, Applied Mathematics Letters, 61 (2016), 73-79. doi: 10.1016/j.aml.2016.05.010. Google Scholar
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