  We present numerical time-dependent calculations for fall-back disks relevant
to GRBs in which the disk of material surrounding the black hole (BH) powering
the GRB jet modulates the mass flow, and hence the strength of the jet. Given
the initial existence of a small mass <10^{-4} msun near the progenitor with a
circularization radius ~10^{10}-10^{11} cm, an unavoidable consequence will be
the formation of an "external disk" whose outer edge continually moves to
larger radii due to angular momentum transport and lack of a confining torque.
For long GRBs, if the mass distribution in the initial fall-back disk traces
the progenitor envelope, then a radius ~10^{11} cm gives a time scale ~10^4 s
for the X-ray plateau. For late times t>10^7 s a steepening due to a cooling
front in the disk may have observational support in GRB 060729. For short GRBs,
one expects most of the mass initially to lie at small radii <10^8 cm; however
the presence of even a trace amount ~10^{-9} msun of high angular momentum
material can give a brief plateau in the light curve. By studying the plateaus
in the X-ray decay of GRBs, which can last up to ~10^4 s after the prompt
emission, Dainotti et al. find an apparent inverse relation between the X-ray
luminosity at the end of the plateau and the duration of the plateau. We show
that this relation may simply represent the fact that one is biased against
detecting faint plateaus, and therefore preferentially sampling the more
energetic GRBs. If, however, there were a standard reservoir in fall-back mass,
our model can reproduce the inverse X-ray luminosity-duration relation. We
emphasize that we do not address the very steep, initial decays immediately
following the prompt emission, which have been modeled by Lindner et al. as
fall-back of the progenitor core, and may entail the accretion of > 1 msun.
