  We study the global structure of some exact scalar hairy dynamical black
holes which were constructed in Einstein gravity either minimally or
non-minimally coupled to a scalar field. We find that both the apparent horizon
and the local event horizon (measured in luminosity coordinate) monotonically
increase with the advanced time as well as the Vaidya mass. At late advanced
times, the apparent horizon approaches the event horizon and gradually becomes
future outer. Correspondingly, the space-time arrives at stationary black hole
states with the relaxation time inversely proportional to the $1/(n-1)$ power
of the final black hole mass, where $n$ is the space-time dimension. These
results strongly support the solutions describing the formation of black holes
with scalar hair. We also obtain new charged dynamical solutions in the
non-minimal theory by introducing an Maxwell field which is non-minimally
coupled to the scalar.
  The presence of the electric charge strongly modifies the dynamical evolution
of the space-time.
