Hamilton-Type Gradient Estimates for Porous Medium and Fast Diffusion Equations on Riemannian Manifolds for all exponents

We establish local Hamilton-type gradient estimates for positive $C^{2,1}$ solutions of $u_t=Δu^m$ on complete Riemannian manifolds whose Ricci curvature is bounded from below. For every fixed $m>1$ and every fixed $00$ and $C=C(m,n)>0$ such that a solution $0<u\leqslant