Gradient estimates and liouville theorems for linear and nonlinear parabolic equations on riemannian manifolds

In this article, we will derive local elliptic type gradient estimates for positive solutions of linear parabolic equations and nonlinear parabolic equations on Riemannian manifolds. As applications, we obtain some theorems of Liouville type for positive ancient solutions of such equations. Our results generalize that of Souplet-Zhang ([1], Bull. London Math. Soc. 38(2006), 1045–1053) and the author ([2], Nonlinear Anal. 74 (2011), 5141–5146).