Harnack differential inequalities for the parabolic equation u t = ℒF(u) on Riemannian manifolds and applications

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In this paper, let (Mn, g) be an n-dimensional complete Riemannian manifold with the m-dimensional Bakry–Emery Ricci curvature bounded below. By using the maximum principle, we first prove a Li–Yau type Harnack differential inequality for positive solutions to the parabolic equation $${u_t} = LF\left( u \right) = \Delta F\left( u \right) - \nabla f \cdot \nabla F\left( u \right),$$ on compact Riemannian manifolds Mn, where F ∈ C2(0,∞), F′ > 0 and f is a C2-smooth function defined on Mn. As application, the Harnack differential inequalities for fast diffusion type equation and porous media type equation are derived. On the other hand, we derive a local Hamilton type gradient estimate for positive solutions of the degenerate parabolic equation on complete Riemannian manifolds. As application, related local Hamilton type gradient estimate and Harnack inequality for fast dfiffusion type equation are established. Our results generalize some known results.