In this paper, we prove sharp gradient estimates for a positive solution to the heat equation
u
t
=
Δ
u
+
a
u
log
u
u_t=\Delta u+au\log u
in complete noncompact Riemannian manifolds. As its application, we show that if
u
u
is a positive solution of the equation
u
t
=
Δ
u
u_t=\Delta u
and
log
u
\log u
is of sublinear growth in both spatial and time directions, then
u
u
must be constant. This gradient estimate is sharp since it is well known that
u
(
x
,
t
)
=
e
x
+
t
u(x,t)=e^{x+t}
satisfying
u
t
=
Δ
u
u_t=\Delta u
.