Differential Harnack inequalities for Fisher-KPP type equations on Riemannian manifolds

We obtain almost optimal differential Harnack inequalities for a class of nonlinear parabolic equations on Riemannian manifolds with Bakry-Émery Ricci curvature bounded below, which includes the classical Fisher-KPP equation and Newell-Whitehead equation. Compared to existing research, we do not impose any additional conditions on the positive solutions. As its application, we derive some optimal Liouville properties.