  We consider a slow diffusion equation with a singular quenching term,
extending the mathematical treatment made in 1992 by B. Kawohl and R. Kersner
for the one-dimensional case. Besides the existence of a very weak solution, we
prove some pointwise gradient estimates, mainly when the absorption dominates
over the diffusion. In particular, a new kind of universal gradient estimate is
proved. Several qualitative properties (such as the finite time quenching
phenomena and the finite speed of propagation) and the study of the Cauchy
problem are also considered.
