  We have studied the evolution of globular cluster systems (GCS) in elliptical
galaxies with a power-law initial GCS mass function (GCMF) ($f(M) \propto
M^{-\alpha}$) similar to that predicted by some theoretical studies of globular
clusters formation and to that of young cluster systems observed in merging
galaxies. We have carried out a survey over a large number of different host
galaxies and we have considered different values for the index, $\alpha$, of
the initial power-law GCMF ($\alpha=1.5,1.8,2.0$); we show the dependence of
the main GCS final properties (mean mass and dispersion of the final GCMF,
fraction of surviving clusters, radial gradient of the GCMF parameters) on the
structure of the host galaxy and on the slope of the initial GCMF. For a
subsample of host galaxies with values of effective masses and radii equal to
those determined using observational data for a number of giant, normal and
dwarf galaxies our results show that the relation between the final GCMF
properties and those of the host galaxies as well as the dependence of the
final GCMF parameters on the galactocentric distance within individual galaxies
differ from those observed in old GCS: the values of the final GCS mean mass
are in general smaller (4.2 < log M_final < 5.0) than those observed and the
galaxy-to-galaxy dispersion of logM_final is larger than that reported by
observational analyses. The results are compared with those of a companion
paper in which we investigated the evolution of GCS with a log-normal initial
GCMF and in which the final GCS properties were perfectly consistent with
observations.
