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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. FoiasG. 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. MaS. 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. ZhangP. KloedenM. 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.

show all references

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. FoiasG. 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. MaS. 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. ZhangP. KloedenM. 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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