Parabolic Gradient Estimates and Harnack Inequalities for a Nonlinear Equation Under The Ricci Flow

When the Riemannian metric evolves under the Ricci flow, we investigate parabolic gradient estimates (Li–Yau’s type and J. Li’s type) for positive solutions to the nonlinear parabolic equation (Δ-∂t)u=(p+1)|∇u|2u+qu\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\Delta -\partial _t)u=(p+1)\frac{|\nabla u|^2}{u}+qu$$\end{document} on the underlying manifold. Based on these gradient estimates, we derive associated Harnack inequalities, respectively.