Gradient estimates of Li Yau type for a general heat equation on Riemannian manifolds

Abstract

In this paper, we consider gradient estimates on complete noncompact Riemannian manifolds $(M,g)$ for the following general heat equation \[ u_t=\Delta _V u+au\log u+bu \] where $a$ is a constant and $b$ is a differentiable function defined on $M\times [0, \infty )$. We suppose that the Bakry-Emery curvature and the $N$-dimensional Bakry-Emery curvature are bounded from below, respectively. Then we obtain the gradient estimate of Li-Yau type for the above general heat equation. Our results generalize the work of Huang-Ma ([4]) and Y. Li ([6]), recently.