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−Δpu+|∇u|q=0in (0,∞)×RN, where N⩾1N⩾1, p∈(1,2)p∈(1,2), and q>0q>0. Based on gradient estimates for the solutions, we classify the behavior of the solutions for large times, obtaining either positivity as t→∞t→∞ for q>p−N/(N+1)q>p−N/(N+1), optimal decay estimates as t→∞t→∞ for p/2⩽q⩽p−N/(N+1)p/2⩽q⩽p−N/(N+1), or extinction in finite time for 0