Journal of Mathematical Physics, Analysis, Geometry 2021, vol. 17, No 4, pp. 521-548   https://doi.org/10.15407/mag17.04.521     ( to contents , go back )

### Gradient Estimates and Harnack Inequalities for a Nonlinear Heat Equation with the Finsler Laplacian

Fanqi Zeng

Xinyang Normal University, 237 Nanhu Road, Xinyang, 464000, P.R. China
E-mail: fanzeng10@126.com

Received October 7, 2020, revised April 30, 2021.

Abstract

Let $(M^{n}, F, m)$ be an $n$-dimensional compact Finsler manifold. In this paper, we study the nonlinear heat equation
$$\partial_{t}u=\Delta_{m} u\quad\text{on}\ M^n\times[0, T],$$
where $\Delta_{m}$ is the Finsler-Laplacian. We derive Li-Yau type gradient estimates for positive global solutions of this equation on static Finsler manifolds, as well as under the Finsler-Ricci flow. As corollaries, in both cases, the corresponding Harnack inequalities are also obtained.

Mathematics Subject Classification 2010: 35K55, 53C21
Key words: Li-Yau type gradient estimates, Harnack inequality, nonlinear heat equation, Finsler-Ricci flow