  We theoretically propose a finite-size quasi-one-dimensional Bose-Einstein
condensate with coherent source and drain placed at its two ends, which can in
principle sustain a stationary sonic black hole with a single event horizon.
Our analysis is focused on the condensate persistence against quantum
fluctuations. We show that similar to white hole-black hole pairs, dynamical
instabilities occur. Investigating in detail the instabilities' dependence on
the system parameters, we also identify windows of formally infinite black hole
lifetimes. By using quantum depletion of the condensate as a diagnostic tool,
we validate the usage of Bogoliubov theory to describe the analogue Hawking
process, and establish novel signatures of Hawking radiation in the depleted
cloud, both inside and outside the event horizon.
