Sharp gradient estimate and Yau's Liouville theorem for the heat equation on noncompact manifolds
Philippe Souplet, Qi S. Zhang
TL;DR
The paper develops a sharp, localized elliptic-type gradient estimate for positive solutions of the heat equation on noncompact manifolds, featuring a logarithmic correction that extends Cheng–Yau-type estimates for the Laplace equation and aligns with Hamilton’s results for bounded solutions on compact manifolds. Under Ricci curvature bounds and boundedness within a parabolic region, it proves: $\frac{|\nabla u|}{u} \le c\left(\frac{1}{R}+\frac{1}{\sqrt{T}}+\sqrt{k}\right)\left(1+\ln\frac{M}{u}\right)$ in $Q_{R/2,T/2}$, and, when $\mathrm{Ric} \ge 0$, $\frac{|\nabla u|}{u} \le c_1\frac{1}{\sqrt{t}}\left(c_2+\ln\frac{u(x,2t)}{u(x,t)}\right)$. As applications, the authors establish a time-dependent Liouville-type theorem for positive eternal solutions with subexponential growth on complete noncompact manifolds with nonnegative Ricci curvature, and a sharpened long-time gradient bound for the log of the heat kernel: $\frac{|\nabla_x G(x,y,t)|}{G(x,y,t)} \le c\,t^{-1/2}\left(1+\frac{d(x,y)^2}{t}\right)$. The proofs employ a parabolic gradient bound using $f=\ln u$ and $w=|\nabla \ln(1-f)|^2$ with a Li–Yau cutoff, and the results are shown to be sharp (e.g., via the example $u=e^{ax+a^2 t}$), highlighting their relevance in noncompact and large-time regimes.
Abstract
We derive a sharp, localized version of elliptic type gradient estimates for positive solutions (bounded or not) to the heat equation. These estimates are akin to the Cheng-Yau estimate for the Laplace equation and Hamilton's estimate for bounded solutions to the heat equation on compact manifolds. As applications, we generalize Yau's celebrated Liouville theorem for positive harmonic functions to positive eternal solutions of the heat equation, under certain growth condition. Surprisingly, this Liouville theorem for the heat equation does not hold even in ${\bf R}^n$ without such a condition. We also prove a sharpened long time gradient estimate for the log of heat kernel on noncompact manifolds. This has been an open problem in view of the well known estimates in the compact, short time case.