Hamilton's gradient estimates for fast diffusion equations under the Ricci flow

Let M be a complete noncompact Riemannian manifold of dimension n. We derive a Hamilton's gradient estimate for positive solutions of the fast diffusion equations ∂ u ∂ t = Δ u m , 1 − 4 n + 8 m 1 on M × ( − ∞ , 0 ] under the Ricci flow.