  We consider the Schrodinger equation on a compact manifold, in the presence
of a nonlinear damping term, which is homogeneous and sublinear. For initial
data in the energy space, we construct a weak solution, defined for all
positive time, which is shown to be unique. In the one-dimensional case, we
show that it becomes zero in finite time. In the two and three-dimensional
cases, we prove the same result under the assumption of extra regularity on the
initial datum.
