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

Abstract

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

25 pages