Gradient estimates for a weighted parabolic equation under geometric flow

Abstract

Let $(M^n,g,e^{-\varphi }dv)$ be a weighted Riemannian manifold evolving by geometric flow $\frac{\partial g}{\partial t}=2h(t),\frac{\partial \varphi }{\partial t}=\Delta \varphi$. In this paper, we obtain a series of space-time gradient estimates for positive solutions of a parabolic partial equation $$({\Delta }_{\varphi }-{\partial }_t)u(x,t)=q(x,t)u^{a+1}(x,t)+p(x,t)A(u(x,t))),(x,t)\in M\times [0,T]$$ on a weighted Riemannian manifold under geometric flow. By integrating the gradient estimates, we find the corresponding Harnack inequalities.