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 (Delta-partial derivative/partial derivative t)u(x,t) q(x, t)u(x, t) = 0 and nonlinear parabolic equations (Delta-partial derivative/partial derivative t)u(x,t) h(x, t)u(p) (x, t) = 0 (p > 1) on Riemannian manifolds. As applications, we obtain some theorems of Lionville 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).