Gradient estimates and Liouville theorems for nonlinear parabolic equations on noncompact Riemannian manifolds

Let M be a complete noncompact Riemannian manifold. In this paper, we derive a local gradient estimate for positive solutions of the nonlinear parabolic equation. (Delta - partial derivative/t) u(x, t) + lambda(x, t)u(alpha) (x, t) = 0 on M x (-infinity, 0]. We also obtain a theorem of Liouville type for positive solutions of this nonlinear equation. This paper extends the result of Souplet and Zhang [P. Souplet, Qi S. Zhang, Sharp gradient estimate and Yau's Liouville theorem for the heat equation on noncompact manifolds, Bull. Lond. Math. Soc. 38 (2006) 1045-1053]. (C) 2011 Elsevier Ltd. All rights reserved.