  This article is devoted to a generalized version of Smoluchowski's
coagulation equation. This model describes the time evolution of a system of
aggregating particles under the effect of external input and output particles.
We show that for a large class of coagulation kernels, output rates, and
exponentially decaying input rates, there is a weak solution. Moreover, the
solution satisfies the mass-conservation property for linear coagulation rate
and an additional condition on input and output rates. The uniqueness of weak
solutions is also established by applying additional restrictions on the rates.
