Sharp gradient estimates for a heat equation in Riemannian manifolds

Abstract

In this paper, we prove sharp gradient estimates for a positive solution to the heat equation u t = ∆u + au log u in complete noncompact Riemannian manifolds. As its application, we show that if u is a positive solution of the equation u t = ∆u and log u is of sublinear growth in both spatial and time directions then u must be constant. This gradient estimate is sharp since it is well-known that u(x, t) = e x+t satisfying u t = ∆u. We also emphasize that our results are better than those given by Jiang ([Jia16]), Souplet-Zhang ([SZ06]), Wu ([Wu15, Wu17]), and others.

Sharp gradient estimates on weighted manifolds with compact boundary

“…For the f -heat equation, we have another useful inequality, which was established in [3] for the manifold case.…”

“…In fact, they proved that a gradient estimate of Brighton type is valid under only a lower bound of Ric f . Second, we prove elliptic gradient estimates for the f -heat equation with Dirichlet boundary condition on (M, g, e −f dv) by improving the argument of [3], which seems to be new even for manifolds.…”

“…We also would like to mention that the main differences between the proof of our sharp results and Kunikawa-Sakurai results are due to a choice of the gradient quantity in the Bochner formula in (2.3). This choice was first used by the first two authors in the manifold case [3] and leads to a sharper estimate in Theorem 1.4 as compared with Theorem 1.2.…”

Gradient estimates for a nonlinear parabolic equation with Dirichlet boundary condition

“…As in [5], we have the following Bochner type formula, which holds on smooth metric measure spaces without any assumption on f . Lemma 2.4.…”

“…Meanwhile, we need an important Bochner type formula for equation (1.1) in the proof of our result, which was similarly discussed in [5]. Let 0 < u ≤ B for some positive constant B be a solution to (1.1) in Q R,T (∂M).…”