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 $u_t=\Delta u+au\log u$ in complete noncompact Riemannian manifolds. As its application, we show that if $u$ is a positive solution of the equation $u_t=\Delta u$ 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 $u_t=\Delta u$.