Methods and Applications of Analysis

Volume 22 (2015)

Number 4

Optimal decay rate for degenerate parabolic equations on noncompact manifolds

Pages: 359 – 376

DOI: http://dx.doi.org/10.4310/MAA.2015.v22.n4.a2

Authors

Daniele Andreucci (Dipartimento di Scienze di Base e Applicate per l’Ingegneria, Sapienza Università di Roma, Italy)

Anatoli F. Tedeev (South Mathematical Institute of VSC RAS, Vladikavkaz, Markusa, Russia)

Abstract

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.

Keywords

doubly degenerate parabolic equation, noncompact Riemannian manifold, relative Faber–Krahn inequality, optimal global and local bounds

2010 Mathematics Subject Classification

35B40, 35K55, 35K65

Full Text (PDF format)