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

Abstract

In this paper, we consider gradient estimates for the positive solutions to the following nonlinear parabolic equation: $$\begin{aligned} u_t=\Delta _V u + au \log u \end{aligned}$$ on \(M \times [0, T]\), 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: $$\begin{aligned} \Delta _V u + au \log u = 0, \end{aligned}$$ 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).