GRADIENT ESTIMATES FOR A NONLINEAR PARABOLIC EQUATION WITH POTENTIAL UNDER GEOMETRIC FLOW

Let (M, g) be an n dimensional complete Riemannian manifold. In this article we prove local Li-Yau type gradient estimates for all positive solutions to the nonlinear parabolic equation (partial derivative(t) - Delta(g) + R)u(x, t) = -au(x,t) log u(x, t) along the generalised geometric flow on M. Here R = R(x, t) is a smooth potential function and a is an arbitrary constant. As an application we derive a global estimate and a space-time Harnack inequality.