  In this work, we study the stellar-dynamical hardening of unequal mass
supermassive black hole (SMBH) binaries in the central regions of merging
galactic nuclei. We present a comprehensive set of direct $N$-body simulations
of the problem, varying both the total mass and the mass ratio of the SMBH
binary (SMBHB). Simulations were carried out with the $\varphi-$GPU $N$-body
code, which enabled us to fully exploit supercomputers equipped with graphic
processing units (GPUs). As a model for the galactic nuclei, we adopted initial
axisymmetric, rotating models, aimed at reproducing the properties of a
galactic nucleus emerging from a galaxy merger event, containing two SMBHs
which were unbound initially. We found no 'final-parsec problem', as our SMBHs
tend to pair and shrink without showing significant signs of stalling. This
confirms earlier results and extends them to large particle numbers and
rotating systems. We find that the SMBHB hardening depends on the
binary-reduced mass ratio via a single parameter function. Our results suggest
that, at a fixed value for the SMBHB primary mass, the merger time of highly
asymmetric binaries is up to four order of magnitudes smaller than the
equal-mass binaries. This can significantly affect the population of SMBHs
potentially detectable as gravitational wave sources.
