Existence of compact $ \varphi $-attracting sets and estimate of their attractive velocity for infinite-dimensional dynamical systems

Chunyan Zhao; Chengkui Zhong; Xiangming Zhu

doi:10.3934/dcdsb.2022051

Article Contents

2022, Volume 27, Issue 12: 7493-7520. 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
^*Corresponding author: Chengkui Zhong

Received: September 2021

Revised: January 2022

Early access: March 2022

Published: December 2022

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

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Abstract

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.

Keywords:

Compact φ-attracting set,
attractive velocity,
φ-decay with respect to noncompactness measure,
wave equation,
critical nonlinearity.

Mathematics Subject Classification: Primary: 35B40, 35B41; Secondary: 35L05.

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