A Note on Li-Yau-Type Gradient Estimate

Abstract

In this paper, we obtain Li-Yau type gradient estimates with time dependent parameter for positive solutions of the heat equation that are different with the estimates by Li-Xu [12] and Qian [16]. As an application of the estimate, we also obtained slight improvements of Davies' Li-Yau type gradient estimate.2010 Mathematics Subject Classification. Primary 35K05; Secondary 53C44.

Gradient Estimates and Harnack Inequalities of a Nonlinear Heat Equation for the Finsler-Laplacian

“…Proof of Corollary 1.5. This can be proved using arguments similar to those for Corollary 2.1 and Corollary 2.3 in [27]. We omit the proofs here.…”

“…The estimates (1.5) and (1.6) were later generalized by Qian [23]. And more recently, Yu and Zhao [27] obtained a Li-Yau type gradient estimate for positive solutions of (1.1) which is different with the estimates by Li-Xu [15] and Qian [23] as follows:…”

“…Remark 1.9. In the proof of above theorems, compared with the case of compact Riemannian manifolds as in [8,9,27], in our case, we need to overcome three obstructions. First, because of the lack of higher order regularity, we need to modify the arguments in [8,9,27].…”

“…In the proof of above theorems, compared with the case of compact Riemannian manifolds as in [8,9,27], in our case, we need to overcome three obstructions. First, because of the lack of higher order regularity, we need to modify the arguments in [8,9,27]. Second, in the Finsler case, ∆ m u has no definition at the maximum point of u, and thus the maximum principle can not be suitable for the Finsler-Laplacian.…”