Gradient estimates of a parabolic equation under the Finsler-geometric flow

Abstract

Let [Formula: see text], [Formula: see text], be a compact Finsler manifold. In this paper, we consider a Finsler manifold evolving by the Finsler-geometric flow [Formula: see text], where [Formula: see text] is the symmetric metric tensor associated with [Formula: see text], and [Formula: see text] is a symmetric [Formula: see text]-tensor, and the study parabolic equation [Formula: see text] where [Formula: see text] is a function on [Formula: see text] of [Formula: see text] in [Formula: see text]-variables and [Formula: see text] in [Formula: see text]-variable and [Formula: see text] is a function of [Formula: see text] in [Formula: see text]. We obtain a global estimate and a Harnack estimate for positive solutions. Our results are also natural extension of similar results on Riemannian-geometric flow.