Abstract

In this paper, we establish a local elliptic gradient estimate for positive bounded solutions to a parabolic equation concerning the $V$-Laplacian

on an $n$-dimensional complete Riemannian manifold with the Bakry–Émery Ricci curvature ${Ric}_V$ bounded below, which is weaker than the $m$-Bakry–Émery Ricci curvature ${Ric}_V^m$ bounded below considered by Chen and Zhao (2018). As applications, we obtain the local elliptic gradient estimates for the cases that $F\left(u\right)=aulnu$ and $a{u}^{\gamma }$. Moreover, we prove parabolic Liouville theorems for the solutions satisfying some growth restriction near infinity and study the problem about conformal deformation of the scalar curvature. In the end, we also derive a global Bernstein-type gradient estimate for the above equation with $F\left(u\right)=0$.

Keywords

Mathematical Subject Classification 2010

Milestones

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