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A Note on the Polynomially Attracting Sets for Dynamical Systems

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Abstract

It is well-known that exponential attractor is an important concept to study the exponentially attracting rate of a compact set for dynamical systems. But recently, Zhao (Appl Math Lett, 2022. https://doi.org/10.1016/j.aml.2021.107791) studied the polynomially attracting sets for a class of wave equations:

$$\begin{aligned} u_{tt}-\Delta u+k\Vert u_{t}\Vert ^{p}u_{t}+f(u)=\int _{\Omega }K(x,y)u_{t}(t,y)dy+h(x). \end{aligned}$$

Clearly, exponential attracting can imply the polynomial attracting. While the converse is unknown. In this note, we give an example which illustrates the existence of a polynomially attracting set but nonexistence of any exponential attractor.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing, 210044, People’s Republic of China

    Xiangming Zhu

  2. Department of Mathematics, Nanjing University, Nanjing, 210093, People’s Republic of China

    Xiangming Zhu & Chengkui Zhong

Authors
  1. Xiangming Zhu
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  2. Chengkui Zhong
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Correspondence to Xiangming Zhu.

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The work is supported by the NSFC (11731005).

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Zhu, X., Zhong, C. A Note on the Polynomially Attracting Sets for Dynamical Systems. J Dyn Diff Equat (2023). https://doi.org/10.1007/s10884-023-10322-x

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  • DOI: https://doi.org/10.1007/s10884-023-10322-x

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