Local parabolic and elliptic gradient estimates for a generalized heat-type equation under the Yamabe flow

Abstract

In this paper, we investigate local parabolic (Li-Yau's type and J. Li's type) and local elliptic (Hamilton's type Shouplet-Zhang's type) gradient estimates for positive solutions to a generalized heat-type equation of (Δϕ−∂t)u=huq under the Yamabe flow. As applications of local parabolic gradient estimates, we establish associated Harnack inequalities.