  We show that thin accretion disks made of Carbon or Oxygen are subject to the
same thermal ionization instability as Hydrogen and Helium disks. We argue that
the instability applies to disks of any metal content. The relevance of the
instability to supernova fallback disks probably means that their power-law
evolution breaks down when they first become neutral. We construct simple
analytical models for the viscous evolution of fallback disks to show that it
is possible for these disks to become neutral when they are still young (ages
of a few 10^3 to 10^4 years), compact in size (a few 10^9 cm to 10^11 cm) and
generally accreting at sub-Eddington rates (Mdot ~ a few 10^14 - 10^18 g/s).
Based on recent results on the nature of viscosity in the disks of close
binaries, we argue that this time may also correspond to the end of the disk
activity period. Indeed, in the absence of a significant source of viscosity in
the neutral phase, the entire disk will likely turn to dust and become passive.
We discuss various applications of the evolutionary model, including anomalous
X-ray pulsars and young radio pulsars. Our analysis indicates that metal-rich
fallback disks around newly-born neutron stars and black holes become neutral
generally inside the tidal truncation radius (Roche limit) for planets, at
\~10^11 cm. Consequently, the efficiency of the planetary formation process in
this context will mostly depend on the ability of the resulting disk of rocks
to spread via collisions beyond the Roche limit. It appears easier for the
merger product of a doubly degenerate binary, whether it is a massive white
dwarf or a neutron star, to harbor planets because it can spread beyond the
Roche limit before becoming neutral.[Abridged]
