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

Optimal decay rate for degenerate parabolic equations on noncompact manifolds