  We discuss the dynamics of intersecting $p$-branes with cosmological
constants in the higher-dimensional gravity theories. For the delocalized brane
case, these solutions describe an asymptotically de Sitter or power-law
expanding universe, while for the partially localized intersecting branes, they
describe homogeneous and isotropic universes at each position of the overall
transverse space. We then apply these time-dependent branes to the study on the
collision of two 0-branes and show that the 0$-$8-brane system or the smeared
$0-p_I$-brane system can provide an example of colliding branes if they have
the same brane charges and only one overall transverse space. Finally, we argue
some applications of the solutions in supergravity models.
