Elliptic gradient estimates for a parabolic equation with $V$-Laplacian and applications

Jian-hong Wang and Yu Zheng

Vol. 309 (2020), No. 2, 453–474

DOI: 10.2140/pjm.2020.309.453

Abstract

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

$$\left({\Delta }_V-{\partial }_t-q\left(x,t\right)\right)u\left(x,t\right)=F\left(u\left(x,t\right)\right)$$

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

gradient estimate, Liouville theorem, $V$-Laplacian, Bakry–Émery Ricci curvature, parabolic equation

Mathematical Subject Classification 2010

Primary: 58J35

Secondary: 35B53, 35K05

Milestones

Received: 12 October 2018

Revised: 12 December 2019

Accepted: 10 July 2020

Published: 14 January 2021