Gradient estimates of positive solutions for the weighted nonlinear parabolic equation

In this paper, we prove a Li–Yau type gradient estimate for a positive solution to the weighted nonlinear parabolic type equation (Δϕ-∂t)u(x,t)+a(x,t)u(x,t)lnu(x,t)+b(x,t)u(x,t)=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} (\Delta _{\phi }-\partial _{t})u(x,t) +a(x,t)u(x,t)\ln u(x,t)+b(x,t)u(x,t)=0 \end{aligned}$$\end{document}on the complete smooth metric measure space under integral Bakry–Émery Ricci curvature bounds. This estimates optimize the obtained conclusions by Zhang and Zhu (J Funct Anal 275:478–515, 2018).