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: on M x [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 GlobAnalGeom 43: 209-232, 2013), Li andXu (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: 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).