#### Vol. 309, No. 2, 2020

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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
##### 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