Journal of the
Korean Mathematical Society JKMS

This note is motivated by gradient estimates of Li-Yau, Hami\-lton, 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-\'{E}mery curvature, Bochner's technique, Harnack-type inequalities, Liouville-type theorems

MSC numbers: Primary 32M05; Secondary 32H02