  The formation and stability of collisionless self-gravitating systems is a
long standing problem, which dates back to the work of D. Lynden-Bell on
violent relaxation, and extends to the issue of virialization of dark matter
(DM) halos. An important prediction of such a relaxation process is that
spherical equilibrium states can be described by a Fermi-Dirac phase-space
distribution, when the extremization of a coarse-grained entropy is reached. In
the case of DM fermions, the most general solution develops a degenerate
compact core surrounded by a diluted halo. As shown recently, the latter is
able to explain the galaxy rotation curves while the DM core can mimic the
central black hole. A yet open problem is whether this kind of astrophysical
core-halo configurations can form at all, and if they remain stable within
cosmological timescales. We assess these issues by performing a thermodynamic
stability analysis in the microcanonical ensemble for solutions with given
particle number at halo virialization in a cosmological framework. For the
first time we demonstrate that the above core-halo DM profiles are stable (i.e.
maxima of entropy) and extremely long lived. We find the existence of a
critical point at the onset of instability of the core-halo solutions, where
the fermion-core collapses towards a supermassive black hole. For particle
masses in the keV range, the core-collapse can only occur for $M_{\rm vir}
\gtrsim 10^9 M_\odot$ starting at $z_{\rm vir}\approx 10$ in the given
cosmological framework. Our results prove that DM halos with a core-halo
morphology are a very plausible outcome within nonlinear stages of structure
formation.
