  We study Hawking radiation on a Vaidya space-time with a gravitational
collapse followed by evaporation. The collapsing body is a null thin-shell and
the evaporation is induced by a negative energy collapsing null-shell. This
mimics the back-reaction to the Hawking radiation. Using Hawking's original
method of Bogoliubov transformations we characterize the radiated spectrum as
dominated by a thermal emission with an increasing effective temperature. We
compute this time dependent temperature and find numerical agreement with
results obtained by other techniques. The known divergences at the evaporation
time are explained by the divergent nature of the effective temperature. As a
consistency check, we re-derived the results from a zero mass limit of a
remnant BH scenario.
