Blow-up Results for Nonlinear Parabolic Equations on Manifolds

1. Introduction. The aim of this paper is threefold. First, by a unified approach, we prove that several classical blow-up results obtained over the last three decades for semilinear and quasilinear parabolic problems in R n are valid on noncompact, complete Riemannian manifolds, which include those with nonnegative Ricci curvatures. Next, we remove some unnecessary a priori growth conditions on solutions of the quasilinear case, which are assumed in the existing literature. Finally, we demonstrate a new critical phenomenon for some inhomogeneous, quasilinear parabolic equations. We also hope that this paper serves as a link for the many other papers on this subject, which lie scattered in several journals over a period of three decades. Specifically, we study the blow-up properties of the following homogeneous and inhomogeneous, semilinear parabolic equations and of the porous medium equations with nonlinear source: u − ∂ t u + V (x)u p = 0 in M n × (0, ∞), u(x, 0) = u 0 (x) in M n , u ≥ 0, (1.1)