Liouville type theorems and gradient estimates for nonlinear heat equations along ancient K-super Ricci flow via reduced geometry

Abstract

In this paper, we use the techniques of Kunikawa-Sakurai in [11] and the approach of Dung-Dung in [6] to study gradient estimates for the positive bounded solutions to the nonlinear parabolic equation concerning Perelman's reduced distance∂∂tu(x,t)=Δu(x,t)+au(x,t)ln⁡u(x,t) along ancient K-super Ricci flow, where a∈R. As applications, we prove Liouville type results of a nonlinear backward heat equation. Our results generalize and improve many previous works.