  We study the gravitational collapse of compact objects in the Brane-World. We
begin by arguing that the regularity of the five-dimensional geodesics does not
allow the energy-momentum tensor of matter on the brane to have (step-like)
discontinuities, which are instead admitted in the four-dimensional General
Relativistic case, and compact sources must therefore have an atmosphere. Under
the simplifying assumption that matter is a spherically symmetric cloud of dust
without dissipation, we can find the conditions for which the collapsing star
generically ``evaporates'' and approaches the Hawking behavior as the
(apparent) horizon is being formed. Subsequently, the apparent horizon evolves
into the atmosphere and the back-reaction on the brane metric reduces the
evaporation, which continues until the effective energy of the star vanishes.
This occurs at a finite radius, and the star afterwards re-expands and
``anti-evaporates''. We clarify that the Israel junction conditions across the
brane (holographically related to the matter trace anomaly) and the projection
of the Weyl tensor on the brane (holographically interpreted as the quantum
back-reaction on the brane metric) contribute to the total energy as,
respectively, an ``anti-evaporation'' and an ``evaporation'' term. Concluding,
we comment on the possible effects of dissipation and obtain a new stringent
bound for the brane tension.
