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doi: 10.3934/dcdsb.2022051
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## Existence of compact $\varphi$-attracting sets and estimate of their attractive velocity for infinite-dimensional dynamical systems

 1 School of Mathematics and Big Data, Anhui University of Science and Technology, Huainan, 232001, China 2 Department of Mathematics, Nanjing University, Nanjing, 210093, China

*Corresponding author: Chengkui Zhong

Received  September 2021 Revised  January 2022 Early access March 2022

Fund Project: The work is supported by National Natural Science Foundation of China (No.11731005; No.11801071)

This paper is devoted to the quantitative study of the attractive velocity of compact semi-invariant attracting sets for infinite-dimensional dynamical systems. We introduce the notion of compact $\varphi$-attracting set whose attractive speed is characterized by a general non-negative decay function $\varphi$, and prove that $\varphi$-decay with respect to noncompactness measure is a sufficient condition for a dissipative system to have a compact $\varphi$-attracting set. Furthermore, several criteria for $\varphi$-decay with respect to noncompactness measure are provided. Finally, as an application, we establish the existence of a compact exponential attracting set and the specific estimate of its attractive velocity for a semilinear wave equation with a critical nonlinearity.

Citation: Chunyan Zhao, Chengkui Zhong, Xiangming Zhu. Existence of compact $\varphi$-attracting sets and estimate of their attractive velocity for infinite-dimensional dynamical systems. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022051
##### References:
 [1] R. R. Akhmerov, M. I. Kamenskiĭ, A. S. Potapov, A. E. Rodkina and B. N. Sadovskiĭ, Measures of Noncompactness and Condensing Operators, Birkhäuser Verlag, Basel, 1992. doi: 10.1007/978-3-0348-5727-7. [2] I. Chueshov and I. Lasiecka, Long-time Behavior of Second Order Evolution Equations with Nonlinear Damping, Mem. Amer. Math. Soc., 2008. doi: 10.1090/memo/0912. [3] P. Constantin, C. Foias, B. Nicolaenko and R. Temam, Integral Manifolds and Inertial Manifolds for Dissipative Partial Differential Equations, Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4612-3506-4. [4] K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, 1985. doi: 10.1007/978-3-662-00547-7. [5] A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential Attractors for Dissipative Evolution Equations, Masson, Paris, 1994. [6] C. Foias and E. Olson, Finite fractal dimension and Hölder-Lipshitz parametrization, Indiana Univ. Math. J., 45 (1996), 603-616.  doi: 10.1512/iumj.1996.45.1326. [7] C. Foias, G. R. Sell and R. Temam, Inertial manifolds for nonlinear evolution equations, J. Diff. Eqns., 73 (1988), 309-353.  doi: 10.1016/0022-0396(88)90110-6. [8] B. R. Hunt and V. Y. Kaloshin, Regularity of embeddings of infinite-dimensional fractal sets into finite-dimensional spaces, Nonlinearity, 12 (1999), 1263-1275.  doi: 10.1088/0951-7715/12/5/303. [9] P. Lax, Functional Analysis, Wiley-Interscience [John Wiley & Sons], New York, 2002. [10] Q. Ma, S. Wang and C. Zhong, Necessary and sufficient conditions for the existence of global attractors for semigroups and applications, Indiana Univ. Math. J., 51 (2002), 1541-1559.  doi: 10.1512/iumj.2002.51.2255. [11] A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains, In Handbook of Differential Equations: Evolutionary Equations., Elsevier/North-Holland, Amsterdam, 4 (2008), 103–200. doi: 10.1016/S1874-5717(08)00003-0. [12] X. Mora and J. Solà-Morales, Existence and nonexistence of finite-dimensional globally attracting invariant manifolds in semilinear damped wave equations, In Dynamics of Infinite-dimensional Systems, Springer, Berlin, 37 (1987), 187–210. [13] J. C. Robinson, Infinite-Dimensional Dynamical Systems, Cambridge University Press, Cambridge, 2001. [14] G. R. Sell and Y. You, Dynamics of Evolutionary Equations, Springer, New York, 2002. doi: 10.1007/978-1-4757-5037-9. [15] J. Simon, Compact sets in the space $L^p(0, T;B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.  doi: 10.1007/BF01762360. [16] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, 2nd edition, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3. [17] S. Zelik, The attractor for a nonlinear hyperbolic equation in the unbounded domain, Discrete Cont. Dyn. Systems, 7 (2001), 593-641.  doi: 10.3934/dcds.2001.7.593. [18] S. Zelik, The attractor for a nonlinear reaction-diffusion system in the unbounded domain and Kolmogorov's $\epsilon$-entropy, Math. Nachr., 232 (2001), 129-179.  doi: 10.1002/1522-2616(200112)232:1<129::AID-MANA129>3.0.CO;2-T. [19] S. Zelik, Attractors of reaction-diffusion systems in unbounded domains and their spatial complexity, Commun. Pure Appl. Math., 56 (2003), 584-637.  doi: 10.1002/cpa.10068. [20] J. Zhang, P. Kloeden, M. Yang and C. Zhong, Global exponential $\kappa-$dissipative semigroups and exponential attraction, Discrete Contin. Dyn. Syst., 37 (2017), 3487-3502.  doi: 10.3934/dcds.2017148. [21] C. Zhao, C. Zhao and C. Zhong, Asymptotic behaviour of the wave equation with nonlocal weak damping and anti-damping, J. Math. Anal. Appl., 490 (2020), 124186, 16 pp. doi: 10.1016/j.jmaa.2020.124186. [22] C. Zhao, C. Zhong and Z. Tang, Asymptotic behaviour of the wave equation with nonlocal weak damping, anti-damping and critical nonlinearity, preprint, arXiv: 2108.07395. [23] C. Zhao, C. Zhong and S. Yan, Existence of a generalized polynomial attractor for the wave equation with nonlocal weak damping, anti-damping and critical nonlinearity, Appl. Math. Lett., 128 (2022), Paper No. 107791, 9 pp. doi: 10.1016/j.aml.2021.107791. [24] C. Zhao, C. Zhong and C. Zhao, Estimate of the attractive velocity of attractors for some dynamical systems (in Chinese), Sci. Sin. Math., 52 (2022), 1–20, arXiv: 2108.07410. [25] C. Zhong and W. Niu, On the $Z_2$ index of the global attractor for a class of $p$-Laplacian equations, Nonlinear Anal., 73 (2010), 3698-3704.  doi: 10.1016/j.na.2010.07.022.

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##### References:
 [1] R. R. Akhmerov, M. I. Kamenskiĭ, A. S. Potapov, A. E. Rodkina and B. N. Sadovskiĭ, Measures of Noncompactness and Condensing Operators, Birkhäuser Verlag, Basel, 1992. doi: 10.1007/978-3-0348-5727-7. [2] I. Chueshov and I. Lasiecka, Long-time Behavior of Second Order Evolution Equations with Nonlinear Damping, Mem. Amer. Math. Soc., 2008. doi: 10.1090/memo/0912. [3] P. Constantin, C. Foias, B. Nicolaenko and R. Temam, Integral Manifolds and Inertial Manifolds for Dissipative Partial Differential Equations, Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4612-3506-4. [4] K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, 1985. doi: 10.1007/978-3-662-00547-7. [5] A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential Attractors for Dissipative Evolution Equations, Masson, Paris, 1994. [6] C. Foias and E. Olson, Finite fractal dimension and Hölder-Lipshitz parametrization, Indiana Univ. Math. J., 45 (1996), 603-616.  doi: 10.1512/iumj.1996.45.1326. [7] C. Foias, G. R. Sell and R. Temam, Inertial manifolds for nonlinear evolution equations, J. Diff. Eqns., 73 (1988), 309-353.  doi: 10.1016/0022-0396(88)90110-6. [8] B. R. Hunt and V. Y. Kaloshin, Regularity of embeddings of infinite-dimensional fractal sets into finite-dimensional spaces, Nonlinearity, 12 (1999), 1263-1275.  doi: 10.1088/0951-7715/12/5/303. [9] P. Lax, Functional Analysis, Wiley-Interscience [John Wiley & Sons], New York, 2002. [10] Q. Ma, S. Wang and C. Zhong, Necessary and sufficient conditions for the existence of global attractors for semigroups and applications, Indiana Univ. Math. J., 51 (2002), 1541-1559.  doi: 10.1512/iumj.2002.51.2255. [11] A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains, In Handbook of Differential Equations: Evolutionary Equations., Elsevier/North-Holland, Amsterdam, 4 (2008), 103–200. doi: 10.1016/S1874-5717(08)00003-0. [12] X. Mora and J. Solà-Morales, Existence and nonexistence of finite-dimensional globally attracting invariant manifolds in semilinear damped wave equations, In Dynamics of Infinite-dimensional Systems, Springer, Berlin, 37 (1987), 187–210. [13] J. C. Robinson, Infinite-Dimensional Dynamical Systems, Cambridge University Press, Cambridge, 2001. [14] G. R. Sell and Y. You, Dynamics of Evolutionary Equations, Springer, New York, 2002. doi: 10.1007/978-1-4757-5037-9. [15] J. Simon, Compact sets in the space $L^p(0, T;B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.  doi: 10.1007/BF01762360. [16] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, 2nd edition, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3. [17] S. Zelik, The attractor for a nonlinear hyperbolic equation in the unbounded domain, Discrete Cont. Dyn. Systems, 7 (2001), 593-641.  doi: 10.3934/dcds.2001.7.593. [18] S. Zelik, The attractor for a nonlinear reaction-diffusion system in the unbounded domain and Kolmogorov's $\epsilon$-entropy, Math. Nachr., 232 (2001), 129-179.  doi: 10.1002/1522-2616(200112)232:1<129::AID-MANA129>3.0.CO;2-T. [19] S. Zelik, Attractors of reaction-diffusion systems in unbounded domains and their spatial complexity, Commun. Pure Appl. Math., 56 (2003), 584-637.  doi: 10.1002/cpa.10068. [20] J. Zhang, P. Kloeden, M. Yang and C. Zhong, Global exponential $\kappa-$dissipative semigroups and exponential attraction, Discrete Contin. Dyn. Syst., 37 (2017), 3487-3502.  doi: 10.3934/dcds.2017148. [21] C. Zhao, C. Zhao and C. Zhong, Asymptotic behaviour of the wave equation with nonlocal weak damping and anti-damping, J. Math. Anal. Appl., 490 (2020), 124186, 16 pp. doi: 10.1016/j.jmaa.2020.124186. [22] C. Zhao, C. Zhong and Z. Tang, Asymptotic behaviour of the wave equation with nonlocal weak damping, anti-damping and critical nonlinearity, preprint, arXiv: 2108.07395. [23] C. Zhao, C. Zhong and S. Yan, Existence of a generalized polynomial attractor for the wave equation with nonlocal weak damping, anti-damping and critical nonlinearity, Appl. Math. Lett., 128 (2022), Paper No. 107791, 9 pp. doi: 10.1016/j.aml.2021.107791. [24] C. Zhao, C. Zhong and C. Zhao, Estimate of the attractive velocity of attractors for some dynamical systems (in Chinese), Sci. Sin. Math., 52 (2022), 1–20, arXiv: 2108.07410. [25] C. Zhong and W. Niu, On the $Z_2$ index of the global attractor for a class of $p$-Laplacian equations, Nonlinear Anal., 73 (2010), 3698-3704.  doi: 10.1016/j.na.2010.07.022.
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