Hamilton type estimates for heat equations on manifolds

2010/09/03 by Ma, Li
#Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1009.0603

Abstract

In this paper, we study the gradient estimates of Li-Yau-Hamilton type for positive solutions to both drifting heat equation and the simple nonlinear heat equation problem u_t-Δu=aulog u, ugt;0 on the compact Riemannian manifold (M,g) of dimension n and with non-negative (Bakry-Emery)-Ricci curvature. Here a≤ 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. We derive various versions of gradient estimates which generalize Hamilton's gradient estimate. An question concerning the Hamilton type gradient estimate for the simple nonlinear heat equation is addressed.

Related

• Gradient estimates for a simple nonlinear heat equation on manifolds
• Gradient estimates of Hamilton - Souplet - Zhang type for a general heat\n equation on Riemannian manifolds
• Hamilton's gradient estimate for the heat kernel on complete manifolds
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• Sharp gradient estimate and Yau's Liouville theorem for the heat equation on noncompact manifolds