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

  • *Corresponding author: Chengkui Zhong

    *Corresponding author: Chengkui Zhong 

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

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  • 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.

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


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