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

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