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

We study qualitative properties of non-negative solutions to the Cauchy problem for the fast diffusion equation with gradient absorption ∂tu − pu + |∇u|q =0 in(0,∞)×RN, where N 1, p ∈ (1, 2), and q >0. Based on gradient estimates for the solutions, we classify the behavior of the solutions for large times, obtaining either positivity as t→∞forq >p−N/(N +1), optimal decay estimates as t→∞for p/2 q p−N/(N +1), or extinction in finite time for 0