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Let M be a complete noncompact Riemannian manifold of dimension n. In this paper, we derive a local gradient estimate for positive solutions of fast diffusion equations ∂ t u = Δu α , 1 ― 2 n < α < 1 n on M x (―∞, 0]. We also obtain a theorem of Liouville type for positive solutions of the fast diffusion equation.

Hamilton’s gradient estimates and Liouville theorems for fast diffusion equations on noncompact Riemannian manifolds