  In this work, we propose a model of the gravitational collapse of dark matter
in the presence of quintessence or phantom-like scalar fields. Our treatment is
based on the principles of general relativity up to virialization. We have
chosen a spherical patch that starts to collapse gravitationally as it happens
in top-hat collapse. It is seen that although the dark matter sector collapses
the dark energy sector does keep a profile that is almost similar to the dark
energy profile for the background expanding Friedmann-Lemaitre-Robertson-Walker
(FLRW) universe for suitable model parameters. It is observed that in order to
formulate the problem in the general relativistic setting one has to abandon
the idea of a closed FLRW isolated collapsing patch. General relativity
requires an external generalized Vaidya spacetime to be matched with the
internal spherical patch whose dynamics is guided by the FLRW metric. It is
shown that almost all collapses are accompanied by some flux of matter and
radiation in the generalized Vaidya spacetime. Some of the spherical regions of
the universe are seen not to collapse but expand eternally, producing void-like
structures. Whether a spherical region will collapse or expand depends upon the
initial values of the system and other model parameters. As this work shows
that collapsing structures must emit some form of radiation, this may be taken
as an observational signature of our proposal.
