Differential Harnack Estimates for the Filtration Equations on Riemannian Manifolds via Nash--Moser Iteration

We prove local differential Harnack estimates for smooth positive solutions of the Filtration Equations $u_t=\mathrm{\Delta }F(u)$ on complete Riemannian manifolds in uniformly parabolic ranges. A Bochner–discriminant argument gives a coercive positive-part inequality, which is closed by Nash–Moser iteration. We derive Harnack and Liouville consequences, recover the Aronson-Bénilan estimates for porous medium equation and fast diffusion equation, and describe a class of genuinely non-power filtration laws.