Gradient estimates for a weighted nonlinear parabolic equation

Let (Mn,g,e-fdv)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(M^{n},g,e^{-f}dv)$$\end{document} be a complete smooth metric measure space. We prove elliptic gradient estimates for positive solutions of a weighted nonlinear parabolic equation Δf-∂∂tu(x,t)+q(x,t)u(x,t)+au(x,t)(lnu(x,t))α=0,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left( \varDelta _{f}-\frac{\partial }{\partial t}\right) u(x,t)+q(x,t)u(x,t)+au(x,t)(\ln u(x,t))^{\alpha }=0, \end{aligned}$$\end{document}where (x,t)∈M×(-∞,∞)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(x,t)\in M\times (-\infty ,\infty )$$\end{document} 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 arbitrary constants. Under the assumption that the ∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\infty $$\end{document}-Bakry-Émery Ricci curvature is bounded from below, we obtain a local elliptic (Hamilton’s type and Souplet–Zhang’s type) gradient estimates to positive smooth solutions of this equation.