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

Let (M n , F (t) , m) , t ∈ [ 0 , T ] , be a compact Finsler manifold. In this paper, we consider a Finsler manifold evolving by the Finsler-geometric flow ∂ g (x , t) ∂ t = 2 h (x , t) , where g (t) is the symmetric metric tensor associated with F , and h (t) is a symmetric (0 , 2) -tensor, and the study parabolic equation ∂ t u (x , t) = Δ m u (x , t) − A (x , t) u (x , t) − B (u (x , t)) , (x , t) ∈ M × [ 0 , T ] , where A is a function on M × [ 0 , T ] of C 2 in x -variables and C 1 in t -variable and B (u) is a function of C 2 in u. 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. [ABSTRACT FROM AUTHOR]