  In the local universe, black holes of $10^{5-6}$ M$_{\odot}$ are hosted in
galaxies displaying a variety of stellar profiles and morphologies. These black
holes are the anticipated targets of LISA, the Laser Interferometer Space
Antenna that will detect the low-frequency gravitational-wave signal emitted by
binary black holes in this mass interval. In this paper, we infer upper limits
on the lifetime of binary black holes of $10^{5-6}$ M$_{\odot}$ and up to
$10^8$ M$_{\odot}$, forming in galaxy mergers, exploring two underlying stellar
density profiles, by Dehnen and by Prugniel & Simien, and by exploiting local
scaling relations between the mass of the black holes and several quantities of
their hosts. We focus on the phase of the dynamical evolution when the binary
is transitioning from the hardening phase ruled by the interaction with single
stars to the phase driven by the emission of gravitational waves. We find that
different stellar profiles predict very distinct trends with binary mass, with
lifetimes ranging between fractions of a Gyr to more than 10 Gyr, and with a
spread of about one order of magnitude, given by the uncertainties in the
observed correlations, which are larger in the low-mass tail of the observed
black hole population.
