In this paper, for the parabolic equation
u
t
=
Δ
u
+
g
(
x
,
u
)
,
(
x
,
t
)
∈
Ω
×
(
0
,
T
)
{u_t} = \Delta u + g\left ( {x, u} \right ), \left ( {x, t} \right ) \in \\ \Omega \times \left ( {0, T} \right )
, with nonlocal boundary conditions
u
|
∂
Ω
=
∫
Ω
f
(
x
,
y
)
u
(
y
,
t
)
d
y
u\left | {_{\partial \Omega }} \right . = \int _{\Omega } f\left ( {x, y} \right )u\left ( {y, t} \right )dy
, we establish the comparison theorem and local existence of the solution. We also discuss its long time behavior.