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

## Nguyen Ngoc Khanh

Address: newline Faculty of Mathematics, Mechanics, and Informatics (MIM), Hanoi University of Science (HUS-VNU), Vietnam National University, No. 334, Nguyen Trai Road, Thanh Xuan, Hanoi

E-mail: khanh.mimhus@gmail.com

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-Émery curvature and the $N$-dimensional Bakry-Émery 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.

AMSclassification: primary 58J35; secondary 35B53.

Keywords: gradient estimates, general heat equation, Laplacian comparison theorem, V-Bochner-Weitzenböck, Bakry-Emery Ricci curvature.

DOI: 10.5817/AM2016-4-207