Some gradient estimates for nonlinear heat-type equations on smooth metric measure spaces with compact boundary

TL;DR

This paper establishes new Hamilton and Li-Yau type gradient estimates for positive solutions to nonlinear heat equations on smooth metric measure spaces with boundary, linking geometry to solution behavior.

Contribution

It introduces generalized gradient estimates on such spaces considering curvature bounds, extending classical results to more complex geometric and nonlinear settings.

Findings

Derived local and global gradient estimates.

Established parabolic Harnack inequalities.

Proved Liouville type results for solutions.

Abstract

In this paper we prove some Hamilton type and Li-Yau type gradient estimates on positive solutions to generalized nonlinear parabolic equations on smooth metric measure space with compact boundary. The geometry of the space in terms of lower bounds on the weighted Bakry-Emery Ricci curvature tensor and weighted mean curvature of the boundary are key in proving generalized local and global gradient estimates. Various applications of these gradient estimates in terms of parabolic Harnack inequalities and Liouville type results are discussed. Further consequences to some specific models informed by the nature of the nonlinearities are highlighted.