A Li-Yau gradient estimate for the Finslerian logarithmic Schr\"{o}dinger equation

TL;DR

This paper establishes a Li-Yau gradient estimate for the Finslerian logarithmic Schrödinger equation on non-compact Finsler manifolds, leading to Harnack inequalities and a priori bounds for solutions.

Contribution

It introduces a new Laplacian comparison theorem and derives gradient estimates for a nonlinear parabolic equation in Finsler geometry, extending classical results to this setting.

Findings

Derived a Li-Yau type gradient estimate for the Finslerian logarithmic Schrödinger equation.

Established a Harnack inequality for positive solutions.

Provided an a priori estimate for solutions.

Abstract

By leveraging a new Laplacian comparison theorem, we derive a Li-Yau type gradient estimate for a particular nonlinear parabolic equation, namely, the Finslerian logarithmic Schrodinger equation on a non-compact, complete Finsler manifold with mixed weighted Ricci curvature bounded from below. In our framework, all coefficients are time-dependent functions defined on the manifold. As applications, we establish both a Harnack inequality and an a priori estimate for the positive solutions of this specific equation