Positivity, decay, and extinction for a singular diffusion equation with gradient absorption (Q408147)

The Cauchy problem in \(\mathbb{R}^N\times (0,\infty)\) is studied for the equation NEWLINE\[NEWLINE{\partial u\over\partial t}- \text{div}(|\nabla_u|^{p-2}\nabla u)+ |\nabla u|^q= 0NEWLINE\]NEWLINE in the fast diffusion case \(1< p< 2\). Here \(q> 0\). Depending on the range of the parameters, one encounters extinction in finite time \((0< q<{p\over 2})\), preserved positivity and optimal decay estimates. One of the main tools are gradient estimates. A notion of viscosity solutions is used, although the equation is singular when \(\nabla u= 0\).