Elliptic gradient estimates for a nonlinear equation with Dirichlet boundary condition

Abstract

We prove local elliptic type gradient estimates for a nonlinear parabolic equation (including the fast diffusion equation and the porous medium equation) with Dirichlet boundary condition on Riemannian manifolds with compact boundary when the Ricci curvature and the mean curvature of boundary are bounded below. As an application, we obtain Liouville type results for certain ancient solutions to the nonlinear parabolic equation on Riemannian manifolds with compact boundary.