Gradient estimates for the porous medium equations on Riemannian manifolds

Details Gradient estimates for the porous medium equations on Riemannian manifolds Gradient estimates for the porous medium equations on Riemannian manifolds

2011-06-13

arxiv.org/abs/1106.2373v1

Mathematics

Differential Geometry

21 pages

Scientific paper

In this paper we study gradient estimates for the positive solutions of the porous medium equation: $$u_t=\Delta u^m$$ where $m>1$, which is a nonlinear version of the heat equation. We derive local gradient estimates of the Li-Yau type for positive solutions of porous medium equations on Riemannian manifolds with Ricci curvature bounded from below. As applications, several parabolic Harnack inequalities are obtained. In particular, our results improve the ones of Lu, Ni, V\'{a}zquez and Villani in [10]. Moreover, our results recover the ones of Davies in [4], Hamilton in [5] and Li and Xu in [7].

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