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 $u_t=\Delta u$. We also emphasize that our results are better than those given by Jiang (\cite{ XJ16}), Souplet-Zhang (\cite{ SZ06}), Wu (\cite{ Wu15, Wu17}), and others.
Categories: math.DG