Elliptic gradient estimates and Liouville theorems for a weighted nonlinear parabolic equation

TL;DR

This paper establishes gradient estimates and Liouville theorems for positive solutions of a weighted nonlinear parabolic equation on smooth metric measure spaces with lower Ricci bounds, extending classical results to weighted settings.

Contribution

It introduces new elliptic gradient estimates for weighted nonlinear parabolic equations on metric measure spaces with Ricci curvature bounds.

Findings

Proves Liouville-type theorems for positive ancient solutions.

Derives Harnack inequalities for positive bounded solutions.

Extends classical gradient estimates to weighted nonlinear parabolic equations.

Abstract

Let $(M^N, g, e^{-f}dv)$ be a complete smooth metric measure space with $\infty$-Bakry-\'Emery Ricci tensor bounded from below. We derive elliptic gradient estimates for positive solutions of a weighted nonlinear parabolic equation \begin{align*} \displaystyle \Big(\Delta_f - \frac{\partial}{\partial t}\Big) u(x,t) +q(x,t)u^\alpha(x,t) = 0, \end{align*} where $(x,t) \in M^N \times (-\infty, \infty)$ and $\alpha$ is an arbitrary constant. As Applications we prove a Liouville-type theorem for positive ancient solutions and Harnack-type inequalities for positive bounded solutions.