Let
(
M
,
g
)
(M,g)
be a complete noncompact Riemannian manifold. In this paper, we derive a local gradient estimate for positive solutions to a simple nonlinear parabolic equation
\[
∂
u
∂
t
=
Δ
u
+
a
u
log
u
+
b
u
\frac {\partial u}{\partial t}=\Delta u+au\log u+bu
\]
on
M
×
[
0
,
+
∞
)
M\times [0,+\infty )
, where
a
a
,
b
b
are two real constants. This equation is closely related to the gradient Ricci soliton. We extend the result of L. Ma (Journal of Functional Analysis 241 (2006) 374-382).