Gradient Estimates for a Nonlinear Parabolic Equation on Complete Smooth Metric Measure Spaces

Let (M,g,e-ϕdν)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(M,g,e^{-\phi } d\nu )$$\end{document} be a complete smooth metric measure space with the m-Bakry–Émery Ricci curvature bounded from below. In the first part of this article, we establish a local Hamilton-type gradient estimate for positive solutions to a nonlinear parabolic equation (Δϕ-q-∂t)u=au(lnu)α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} (\Delta _{\phi }-q-\partial _{t})u=au(\ln u)^{\alpha } \end{aligned}$$\end{document}on (M,g,e-ϕdν)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(M,g,e^{-\phi } d\nu )$$\end{document}, where q(x, t) is a smooth space-time function and a, α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} are real constants. By integrating the gradient estimates, we derive the corresponding Harnack inequalities. In the second part of this article, we obtain a refined global gradient estimates for the positive solutions to the above nonlinear parabolic equation with a smooth function q(x) and α=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha =1$$\end{document}. As its application, we derive a uniform bound for the solutions to the corresponding elliptic equation. Our results generalize some recent works on this direction.