  We study false vacuum decay catalyzed by black holes. We consider a toy
two-dimensional model of a scalar field with an unstable potential in the
background of a dilaton black hole. A realistic black hole in four dimensions
possesses the potential barrier for linear field perturbations. We model this
barrier -- the greybody factor -- for spherically-symmetric perturbations in
the toy model by adding a coupling between the scalar field and dilaton. We
compute analytically the decay rate for the black hole in thermal equilibrium
(Hartle-Hawking state) and for the radiating black hole in empty space (Unruh
state). Our results show that, contrary to the Hartle-Hawking vacuum, the decay
probability of the Unruh vacuum remains exponentially suppressed at all black
hole temperatures. We argue that this result holds also in four dimensions.
