SHARP GRADIENT ESTIMATES FOR A HEAT EQUATION IN RIEMANNIAN MANIFOLDS.

In this paper, we prove sharp gradient estimates for a positive solution to the heat equation ut = Δu + au log u in complete noncompact Riemannian manifolds. As its application, we show that if u is a positive solution of the equation ut = Δu 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) = ex+t satisfying ut = Δu [ABSTRACT FROM AUTHOR]