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

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.