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

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

(ΔV t q(x,t))u(x,t) = F(u(x,t))

on an n-dimensional complete Riemannian manifold with the Bakry–Émery Ricci curvature RicV bounded below, which is weaker than the m-Bakry–Émery Ricci curvature RicV m bounded below considered by Chen and Zhao (2018). As applications, we obtain the local elliptic gradient estimates for the cases that F(u) = aulnu and auγ. 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(u) = 0.

gradient estimate, Liouville theorem, $V$-Laplacian, Bakry–Émery Ricci curvature, parabolic equation
Mathematical Subject Classification 2010
Primary: 58J35
Secondary: 35B53, 35K05
Received: 12 October 2018
Revised: 12 December 2019
Accepted: 10 July 2020
Published: 14 January 2021
Jian-hong Wang
School of Statistics and Mathematics
Shanghai Lixin University of Accounting and Finance
Yu Zheng
School of Mathematical Sciences
East China Normal University