Gradient estimation of a generalized non-linear heat type equation along Super-Perelman Ricci flow on weighted Riemannian manifolds

TL;DR

This paper develops gradient estimates for solutions to a generalized nonlinear heat equation on weighted Riemannian manifolds evolving under super Perelman-Ricci flow, leading to Harnack inequalities and Liouville theorems.

Contribution

It introduces new gradient estimation techniques for nonlinear heat equations on evolving weighted manifolds under super Perelman-Ricci flow.

Findings

Established gradient estimates for positive solutions.

Derived Harnack inequalities along the flow.

Proved Liouville type theorems for solutions.

Abstract

In this article we derive gradient estimation for positive solution of the equation \begin{equation*} (\partial_t-\Delta_f)u = A(u)p(x,t) + B(u)q(x,t) + \mathcal{G}(u) \end{equation*} on a weighted Riemannian manifold evolving along the $(k,m)$ super Perelman-Ricci flow \begin{equation*} \frac{\partial g}{\partial t}(x,t)+2Ric_f^m(g)(x,t)\ge -2kg(x,t). \end{equation*} As an application of gradient estimation we derive a Harnack type inequality along with a Liouville type theorem.