Gradient estimates for a nonlinear parabolic equation with Dirichlet boundary condition

TL;DR

This paper establishes gradient estimates for nonlinear parabolic equations on smooth metric measure spaces with boundary, extending previous results from linear to nonlinear cases and deriving new Liouville type theorems.

Contribution

It provides the first Souplet-Zhang type gradient estimates for nonlinear parabolic equations with Dirichlet boundary conditions on smooth metric measure spaces.

Findings

Gradient estimates are proved under bounded Bakry-Emery Ricci tensor and mean curvature.

A new Liouville type theorem for space-time functions is derived.

Results generalize linear cases to nonlinear equations.

Abstract

In this paper, we prove Souplet-Zhang type gradient estimates for a nonlinear parabolic equation on smooth metric measure spaces with the compact boundary under the Dirichlet boundary condition when the Bakry-Emery Ricci tensor and the weighted mean curvature are both bounded below. As an application, we obtain a new Liouville type result for some space-time functions on such smooth metric measure spaces. These results generalize previous linear equations to a nonlinear case.