Differential Harnack Inequality for a Parabolic Equation Under the Finsler-Geometric Flow

Let (M-n, F(t), m), t is an element of [0, T], be a compact Finsler manifold. In this paper, we obtain global differential Harnack estimates for positive solutions of parabolic equation partial derivative(t)u(x, t) = triangle(m)u(x, t) - R(x, t)u(x, t) - a(u(x, t))(b+1) - cu(x, t)(log u(x, t))(d), onM x [0, T] under the Finsler-geometric flow partial derivative g(x,t)/partial derivative t = 2h(x, t), where R is a smooth function, a, b, c, d are real constants, g(t) is the symmetric metric tensor associated with F, and h(t) is a symmetric (0, 2)-tensor. We use the estimates to construct classical Harnack inequalities.