  We study the dynamics of the FLRW flat cosmological models in which the
vacuum energy varies with time, $\Lambda(t)$. In this model we find that the
main cosmological functions such as the scale factor of the universe and the
Hubble flow are defined in terms of exponential functions. Applying a joint
likelihood analysis of the recent supernovae type Ia data, the Cosmic Microwave
Background shift parameter and the Baryonic Acoustic Oscillations traced by the
Sloan Digital Sky Survey (SDSS) galaxies, we place tight constraints on the
main cosmological parameters of the $\Lambda(t)$ scenario. Also, we compare the
$\Lambda(t)$ model with the traditional $\Lambda$ cosmology and we find that
the former model provides a Hubble expansion which compares well with that of
the $\Lambda$ cosmology. However, the $\Lambda(t)$ scenario predicts stronger
small scale dynamics, which implies a faster growth rate of perturbations with
respect to the usual $\Lambda$-cosmology, despite the fact that they share the
same equation of state parameter. In this framework, we find that galaxy
clusters in the $\Lambda(t)$ model appear to form earlier than in the $\Lambda$
model.
