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

TL;DR

This paper introduces the concept of -attractors in infinite-dimensional systems, providing criteria for their existence and estimates of their attraction speed, with applications to semilinear wave equations.

Contribution

It defines -attractors based on decay functions, establishes conditions for their existence, and applies these results to semilinear wave equations with critical nonlinearities.

Findings

-decay ensures -attractor existence

Criteria for -decay with respect to noncompactness measure

Existence and estimate of exponential attractors for wave equations

Abstract

This paper is devoted to the quantitative study of the attractive velocity of generalized attractors for infinite-dimensional dynamical systems. We introduce the notion of~$\varphi$-attractor 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 dissipitive system to have a~$\varphi$-attractor. Furthermore, several criteria for~$\varphi$-decay with respect to noncompactness measure are provided. Finally, as an application, we establish the existence of a generalized exponential attractor and the specific estimate of its attractive velocity for a semilinear wave equation with a critical nonlinearity.