  We investigate the virialization of cosmic structures in the framework of
flat FLRW cosmological models, in which the vacuum energy density evolves with
time. In particular, our analysis focuses on the study of spherical matter
perturbations, as they decouple from the background expansion, "turn around"
and finally collapse. We generalize the spherical collapse model in the case
when the vacuum energy is a running function of the Hubble rate,
$\Lambda=\Lambda(H)$. A particularly well motivated model of this type is the
so-called quantum field vacuum, in which $\Lambda(H)$ is a quadratic function,
$\Lambda(H)=n_0+n_2\,H^2$, with $n_0\neq 0$. This model was previously studied
by our team using the latest high quality cosmological data to constrain its
free parameters, as well as the predicted cluster formation rate. It turns out
that the corresponding Hubble expansion history resembles that of the
traditional $\Lambda$CDM cosmology. We use this $\Lambda(t)$CDM framework to
illustrate the fact that the properties of the spherical collapse model (virial
density, collapse factor, etc.) depend on the choice of the considered vacuum
energy (homogeneous or clustered). In particular, if the distribution of the
vacuum energy is clustered, then, under specific conditions, we can produce
more concentrated structures with respect to the homogeneous vacuum energy
case.
