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This paper is devoted to the quantitative study of the attractive velocity ofgeneralized attractors for infinite-dimensional dynamical systems. We introducethe notion of~$\varphi$-attractor whose attractive speed is characterized by ageneral non-negative decay function~$\varphi$, and prove that~$\varphi$-decaywith respect to noncompactness measure is a sufficient condition for adissipitive system to have a~$\varphi$-attractor. Furthermore, several criteriafor~$\varphi$-decay with respect to noncompactness measure are provided.Finally, as an application, we establish the existence of a generalizedexponential attractor and the specific estimate of its attractive velocity fora semilinear wave equation with a critical nonlinearity.

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