Local Hamilton Type Gradient Estimates for Nonlinear Parabolic Equations on Riemannian Manifolds

In this paper, we present a new proof and attain some new local Hamilton type gradient estimates for positive solutions to the parabolic equation ut=Delta Vup+aulogu+bu\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} u_{t}=\Delta _{V}u<^>p+au\log u+bu \end{aligned}$$\end{document}on a complete noncompact Riemannian manifold with k-Bakry-& Eacute;mery Ricci curvature bounded from below, where p, a and b are some given constants. As applications, related local Hamilton type gradient estimates, some parabolic type Liouville theorems and Harnack inequalities for porous media type equation and fast diffusion type equation are established. Some known results were generalized by our results.