  The gravitational instability of kinks is scrutinized in the context of
information entropy, including the black hole formation near the scalar field
critical collapse. The differential configurational entropy (DCE) is computed
and examined for globally perturbed static kinks and the far-from-equilibrium
kink solutions as well. In the case of the first kink, the DCE determines a
critical value of the perturbation parameter regulating points of bifurcation
that characterize a phase transition, whose supercritical range leads to
collapse and subsequent black hole formation. For the far-from-equilibrium kink
solutions, the DCE supports black hole formation.
