Pointwise Gradient Estimates in Multi-dimensional Slow Diffusion Equations with a Singular Quenching Term

We consider the high-dimensional equation , 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 , with , , we prove some pointwise gradient estimates for a certain range of the dimension , and , mainly when the absorption dominates over the diffusion ( ). In particular, a new kind of universal gradient estimate is proved when . 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.