  We describe the transition of a graviationally radiating, axially and
reflection symmetric dissipative fluid, to a non--radiating state. It is shown
that very shortly after the end of the radiating regime, at a time scale of the
order the thermal relaxation time, the thermal adjustment time and the
hydrostatic time (whichever is larger), the system reaches the equilibrium
state. This result is at variance with all the studies carried out in the past,
on gravitational radiation outside the source, which strongly suggest that
after a radiating period, the conditions for a return to a static case, look
rather forbidding. As we shall see, the reason for such a discrepancy resides
in the fact that some elementary, but essential, physical properties of the
source, have been overlooked in these latter studies.
