Differential Harnack estimates for a nonlinear heat equation

We consider positive solutions to the semilinear heat equation wt=Δw+awlogw, a≠0a≠0, on complete Riemannian manifolds without boundary. This equation has applications to studying Ricci flow and gradient Ricci solitons. We derive several differential Harnack inequalities which improve on those of Y. Yang (2008) [13]. We use these inequalities to derive bounds on gradient Ricci solitons.