Gradient estimates for some $f$-heat equations driven by Lichnerowicz's equation on complete smooth metric measure spaces

TL;DR

This paper derives gradient estimates for solutions to complex $f$-heat equations on complete smooth metric measure spaces with lower bounds on Bakry-Émery Ricci curvature, leading to Liouville and Harnack inequalities for various nonlinear equations.

Contribution

It introduces new gradient estimates for $f$-heat equations driven by Lichnerowicz's equation, extending analysis on metric measure spaces with curvature bounds.

Findings

Established gradient estimates for $f$-heat equations.

Derived Liouville-type theorems for nonlinear equations.

Obtained Harnack inequalities for positive solutions.

Abstract

Given a complete, smooth metric measure space $(M,g,e^{-f}dv)$ with the Bakry-\'Emery Ricci curvature bounded from below, various gradient estimates for solutions of the following general $f$-heat equations $$ u_t=\Delta_f u+au\log u+bu +Au^p+Bu^{-q} $$ and \[ u_t=\Delta_f u+Ae^{pu}+Be^{-pu}+D \] are studied. As by-product, we obtain some Liouville-type theorems and Harnack-type inequalities for positive solutions of several nonlinear equations including the Schr\"{o}dinger equation, the Yamabe equation, and Lichnerowicz-type equations as special cases.