Local Hamilton type Gradient Estimates and Harnack Inequalities for Nonlinear Parabolic Equations on Riemannian Manifolds

In this paper, we prove a local Hamilton type gradient estimate for positive solution of the nonlinear parabolic equation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${u_t}(x,t) = \Delta u(x,t) + au(x,t)\ln \,u(x,t) + b{u<^>\alpha }(x,t),$$\end{document} on M x (-infinity, infinity) with alpha is an element of R, where a and b are constants. As application, the Harnack inequalities are derived.