Differential Harnack Estimates for a Nonlinear Evolution Equation of Allen-Cahn Type

We discuss local gradient estimates of Li and Yau type on the smooth bounded positive solutions w : M x [0, infinity) -> R to a nonlinear evolution equation w(t) = Delta w+ a(w-w(3)), where a > 0 is a constant, on a complete Riemannian manifold M. Global estimates are obtained from the local ones, the consequence of which will eventually yield classical Harnack inequalities for Parabolic Allen-Cahn equation and a Liouville type result for steady state solutions under the hypothesis of nonnegative Ricci curvature tensor.