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

Abstract

<p style='text-indent:20px;'>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 <inline-formula><tex-math id="M2">\begin{document}$ \varphi $\end{document}</tex-math></inline-formula>-attracting set whose attractive speed is characterized by a general non-negative decay function <inline-formula><tex-math id="M3">\begin{document}$ \varphi $\end{document}</tex-math></inline-formula>, and prove that <inline-formula><tex-math id="M4">\begin{document}$ \varphi $\end{document}</tex-math></inline-formula>-decay with respect to noncompactness measure is a sufficient condition for a dissipative system to have a compact <inline-formula><tex-math id="M5">\begin{document}$ \varphi $\end{document}</tex-math></inline-formula>-attracting set. Furthermore, several criteria for <inline-formula><tex-math id="M6">\begin{document}$ \varphi $\end{document}</tex-math></inline-formula>-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.</p>