# GRADIENT ESTIMATES AND HARNACK INEQUALITES OF NONLINEAR HEAT EQUATIONS FOR THE V -LAPLACIAN

• Dung, Ha Tuan
• Accepted : 2018.08.29
• Published : 2018.11.01
• 193 40

#### Abstract

This note is motivated by gradient estimates of Li-Yau, Hamilton, and Souplet-Zhang for heat equations. In this paper, our aim is to investigate Yamabe equations and a non linear heat equation arising from gradient Ricci soliton. We will apply Bochner technique and maximal principle to derive gradient estimates of the general non-linear heat equation on Riemannian manifolds. As their consequence, we give several applications to study heat equation and Yamabe equation such as Harnack type inequalities, gradient estimates, Liouville type results.

#### Keywords

gradient estimates;Bakry-${\acute{E}}mery$ curvature;Bochner's technique;Harnack-type inequalities;Liouville-type theorems

#### Acknowledgement

Supported by : Hanoi Pedagogical University No. 2