  The PBHs clusters can be source of gravitational waves, and the merger rate
depends on the spatial distribution of PBHs in the cluster which changes over
time. It is well known that gravitational collisional systems experience the
core collapse, that leads to significantly increase of the central density and
shrinking of the core. After core collapse, the cluster expands almost
self-similarly (i.e. density profile extends in size without changing its
shape). These dynamic processes affect at the merger rate of PBHs. In this
paper the dynamics of the PBH cluster is considered using the Fokker-Planck
equation. We calculate the merger rate of PBHs on cosmic time scales and show
that its time dependence has a unique signature. Namely, it grows by about an
order of magnitude at the moment of core collapse which depends on the
characteristic of the cluster, and then decreases according to the dependence
$\mathcal{R} \propto t^{-1.48}$. It was obtained for monochromatic and
power-law PBH mass distributions with some fixed parameters. Obtained results
can be used to test the model of the PBH clusters via observation of
gravitational waves at high redshift.
