Gradient estimates for the elliptic and parabolic Lichnerowicz equations on compact manifolds

Let (M, g) be a smooth compact Riemannian manifold of dimension n ≥ 3. Denote \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Delta_g=-{\rm div}_g\nabla}$$\end{document} the Laplace–Beltrami operator. We establish some local gradient estimates for the positive solutions of the Lichnerowicz equation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta_gu(x)+h(x)u(x)=A(x)u^p(x)+\frac{B(x)}{u^q(x)}$$\end{document} on (M, g). Here, p, q ≥ 0, A(x), B(x) and h(x) are smooth functions on (M, g). We also derive the Harnack differential inequality for the positive solutions of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_t(x,t)+\Delta_gu(x,t)+h(x)u(x,t)=A(x)u^p(x,t)+\frac{B(x)}{u^q(x,t)}$$\end{document} on (M, g) with initial data u(x, 0) > 0.