Hamilton type estimates for heat equations on manifolds

Details Hamilton type estimates for heat equations on manifolds Hamilton type estimates for heat equations on manifolds

2010-09-03

arxiv.org/abs/1009.0603v1

Mathematics

Differential Geometry

6 pages

Scientific paper

In this paper, we study the gradient estimates of Hamilton type for positive solutions to both drifting heat equation and the simple nonlinear heat equation problem $$ u_t-\Delta u=au\log u, u>0 $$ on the compact Riemannian manifold $(M,g)$ of dimension $n$ and with non-negative (Bakry-Emery)-Ricci curvature. Here $a\leq 0$ is a constant. The latter heat equation is a basic evolution equation which is the negative gradient heat flow to the functional of Log-Sobolev inequality on the Riemannian manifold. An open question concerning the Hamilton type gradient estimate is proposed.

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