  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.
