Methods and Applications of Analysis
Volume 22 (2015)
Optimal decay rate for degenerate parabolic equations on noncompact manifolds
Pages: 359 – 376
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.
doubly degenerate parabolic equation, noncompact Riemannian manifold, relative Faber–Krahn inequality, optimal global and local bounds
2010 Mathematics Subject Classification
35B40, 35K55, 35K65