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

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.