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,∞)× R , 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 → ∞ for q > p − N/(N + 1), optimal decay estimates as t → ∞ for p/2 ≤ q ≤ p − N/(N + 1), or extinction in finite time for 0 < q < p/2. In addition, we show how the diffusion prevents extinction in finite time in some ranges of exponents where extinction occurs for the non-diffusive Hamilton-Jacobi equation. AMS Subject Classification: 35B40, 35K67, 35K92, 35K10, 35B33, 49L25.