Differential Harnack estimates for a nonlinear evolution equation of Allen-Cahn type (Q821505)

The author proves local gradient estimates of Li and Yau type for smooth bounded positive solutions to the evolution equation \[ w_t = \Delta w+a(w-w^3), \] where \(a > 0\) is a constant, on a complete Riemannian manifold \(M.\) Global estimates come in a straightforward way from the local ones. This result is important in the direction of deriving classical Harnack inequalities for parabolic Allen-Cahn equations and for a Liouville type result for steady state solutions under the hypothesis of nonnegative Ricci curvature tensor.