Hamilton’s gradient estimate for fast diffusion equations under geometric flow

Abstract

Suppose that \lt i \gt M \lt /i \gt is a complete noncompact Riemannian manifold of dimension \lt i \gt n \lt /i \gt . In the present paper, we obtain a Hamilton's gradient estimate for positive solutions of the fast diffusion equations $ \dfrac{\partial u}{\partial t} = \Delta u^m ,\qquad 1-\dfrac{4}{n+8} \lt m \lt 1 $ on $M\times (-\infty, 0]$ under the geometric flow.