The Harnack Estimate for a Nonlinear Parabolic Equation under the Ricci Flow

Let (M, g(t)), 0 <= t <= T, be an n-dimensional closed manifold with nonnegative Ricci curvature, |Rc| <= C/t for some constant C > 0 and g(t) evolving by the Ricci flow partial derivative g(ij)/partial derivative t = -2R(ij). In this paper, we derive a differential Harnack estimate for positive solutions to parabolic equations of the type u(t) = Delta u - aulogu - bu on M x (0, T], where a > 0 and b is an element of R.