  In this paper we analyze a nonlinear parabolic equation characterized by a
singular diffusion term describing very fast diffusion effects. The equation is
settled in a smooth bounded three-dimensional domain and complemented with a
general boundary condition of dynamic type. This type of condition prescribes
some kind of mass conservation; hence extinction effects are not expected for
solutions that emanate from strictly positive initial data. Our main results
regard existence of weak solutions, instantaneous regularization properties,
long-time behavior, and, under special conditions, uniqueness.
