  We show that a standard Shakura-Sunyaev accretion disc around a black hole
with an accretion rate lower than the critical Eddington limit does show the
instability in the radiation pressure dominated zone. We obtain this result
performing time-dependent simulations of accretion disks for a set of values of
the viscosity parameter and accretion rate. In particular we always find the
occurrence of the collapse of the disc: the instability develops always towards
a collapsed gas pressure dominated disc and not towards the expansion. This
result is valid for all initial configurations we tested. We find significant
convective heat flux that increases the instability development time, but is
not strong enough to inhibit the disc collapse. A physical explanation of the
lack of the expansion phase is proposed considering the role of the radial heat
advection. Our finding is relevant since it excludes the formation of the hot
comptonizing corona -often suggested to be present- around the central object
by the mechanism of the Shakura-Sunyaev instability. We also show that, in the
parameters range we simulated, accretion disks are crossed by significant
amplitude acoustic waves.
