  (Abridged) We present results from a JCMT/SCUBA 850 um search for cold dust
around nearby young stars belonging to the beta Pic (t~12 Myr) and the Local
Association (t~50 Myr) moving groups. Unlike most past sub-mm studies, our
sample was chosen on the basis of stellar age. Our observations achieve about
an order of magnitude greater sensitivity in dust mass compared to previous
work in this age range. We detected two of the three M dwarfs in our sample at
850 um, GJ 182 and GJ 803. GJ 182 may also possess a 25 um excess, indicative
of warm dust in the inner few AU of its disk. For GJ 803 (AU Mic), sub-mm
mapping finds that the 850 um emission is unresolved. A non-detection of the CO
3-2 line indicates the system is gas-poor, and the SED suggests the presence of
a large inner disk hole (~17 AU = 1.7 arcsec in radius). These are possible
indications that planets at large separations can form around M dwarfs within
\~10 Myr. In a companion paper (Kalas, Liu & Matthews 2004), we confirm the
existence of a dust disk around GJ 803 using optical coronagraphic imaging.
Given its youthfulness, proximity, and detectability, the GJ 803 disk will be a
valuable system for studying disk, and perhaps planet, formation in great
detail. Overall, sub-mm measurements of debris disks point to a drop in dust
mass by a factor of about 10^3 within the first ~10 Myr, with the subsequent
decline in the masses of sub-mm detected disks consistent with t^{-0.5} to
t^{-1}.
