Gradient estimates for a weighted parabolic equation under geometric flow

TL;DR

This paper derives space-time gradient estimates and Harnack inequalities for positive solutions of a weighted parabolic equation on a manifold evolving under geometric flow, extending classical results to a dynamic geometric setting.

Contribution

It introduces new gradient estimates and Harnack inequalities for parabolic equations on weighted manifolds under geometric flow, a novel extension of existing static results.

Findings

Established space-time gradient estimates for solutions.

Derived Harnack inequalities from the gradient estimates.

Extended classical results to evolving weighted manifolds.

Abstract

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