  Context: The induced gravitational collapse (IGC) paradigm addresses the very
energetic (10^{52}-10^{54}erg) long gamma-ray bursts (GRBs) associated to
supernovae (SNe). Unlike the traditional "collapsar" model, an evolved FeCO
core with a companion neutron star (NS) in a tight binary system is considered
as the progenitor. This special class of sources, here named "binary driven
hypernovae" (BdHNe), presents a composite sequence composed of four different
episodes [...]. Aims: We first compare and contrast the steep decay, the
plateau and the power-law decay of the X-ray luminosities of three selected
BdHNe [...]. Second, to explain the different sizes and Lorentz factors of the
emitting regions of the four Episodes, [...]. Finally, we show the possible
role of r-process, which originates in the binary system of the progenitor..
Methods: We compare and contrast the late X-ray luminosity of the above three
BdHNe. We examine correlations between the time at the starting point of the
constant late power-law decay, t^*_a, the average prompt luminosity, <L_{iso}>,
and the luminosity at the end of the plateau, L_a. We analyze a thermal
emission (~0.97-0.29 keV), observed during the X-ray steep decay phase of GRB
090618. Results: The late X-ray luminosities of the three BdHNe [...] show a
precisely constrained "nested" structure [...]. Conclusions: We confirm a
constant slope power-law behavior for the late X-ray luminosity in the source
rest-frame, which may lead to a new distance indicator for BdHNe. These
results, as well as the emitter size and Lorentz factor, appear to be
inconsistent with the traditional afterglow model based on synchrotron emission
from an ultra-relativistic [...] collimated jet outflow. We argue, instead, for
the possible role of r-process, originating in the binary system, to power the
mildly relativistic X-ray source.
