Optimal decay rate for degenerate parabolic equations on noncompact manifolds

We consider an initial value problem for a doubly degenerate parabolic equation in a noncompact Riemannian manifold. The geometrical features of the manifold are coded in either a Faber-Krahn inequality or a relative Faber-Krahn inequality. We prove optimal decay and space-time local estimates of solutions. We employ a simplified version of the by now classical local approach by DeGiorgi, Ladyzhenskaya-Uraltseva, DiBenedetto which is of independent interest even in the euclidean case.