Gradient estimates and Harnack inequalities of a nonlinear parabolic equation for the V-Laplacian

In this paper, we consider gradient estimates for the positive solutions to the following nonlinear parabolic equation: ut=ΔVu+aulogu\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} u_t=\Delta _V u + au \log u \end{aligned}$$\end{document}on M×[0,T]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M \times [0, T]$$\end{document}, where a is a real constant. We obtain the Li-Yau type bounds of the above equation, which cover the estimates in Davies (Heat kernels and spectral theory 1989), Huang et al. (Ann Glob Anal Geom 43:209–232, 2013), Li and Xu (Adv Math 226:4456–4491, 2011) and Qian (J Math Anal Appl 409:556–566, 2014). Besides, as a corollary, we give a gradient estimate for the corresponding elliptic case: ΔVu+aulogu=0,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Delta _V u + au \log u = 0, \end{aligned}$$\end{document}which improves the estimates in Chen and Chen (Ann Glob Anal Geom 35:397–404, 2009) and Yang ( Proc AMS 136(11):4095–4102, 2008).