  Qualitative properties of non-negative solutions to a quasilinear degenerate
parabolic equation with an absorption term depending solely on the gradient are
shown, providing information on the competition between the nonlinear diffusion
and the nonlinear absorption. In particular, the limit as time goes to infinity
of the mass of integrable solutions is identified, together with the rate of
expansion of the support for compactly supported initial data. The persistence
of dead cores is also shown. The proof of these results strongly relies on
gradient estimates which are first established.
