Gradient estimates for a general type of nonlinear parabolic equations under geometric conditions and related problems

Abstract

In this paper, we establish gradient estimates for the positive bounded solutions to a general type of nonlinear parabolic equation concerning the weighted Laplacian ∂∂t−a(x,t)−Δfu(x,t)=F(u(x,t))on a smooth metric measure space with the metric evolving under the (k,∞)-super Perelman–Ricci flow and the Yamabe flow. Applications of our results include Liouville type results and gradient estimates for some important geometric partial differential equations such as the equations involving gradient Ricci solitons and the Einstein-scalar field Lichnerowicz type equations. Our results generalize and improve many previous works.