Li-Yau Harnack Estimates for a Heat-Type Equation Under the Geometric Flow

In this paper, we consider the gradient estimates for a postive solution of the nonlinear parabolic equation ∂tu = Δtu + hup on a Riemannian manifold whose metrics evolve under the geometric flow ∂tg(t) = − 2Sg(t). To obtain these estimate, we introduce a quantity S̲\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\underline {\boldsymbol {S}}$\end{document} along the flow which measures whether the tensor Sij satisfies the second contracted Bianchi identity. Under conditions on Ricg(t),Sg(t), and S̲\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\underline {\boldsymbol {S}}$\end{document}, we obtain the gradient estimates.