In this paper, we are concerned with a superlinear parabolic equation
\[
{
∂
u
∂
t
−
Δ
u
=
u
p
+
h
(
t
,
x
)
,
a
m
p
;
(
t
,
x
)
∈
R
+
×
Ω
,
u
=
0
,
a
m
p
;
(
t
,
x
)
∈
R
+
×
∂
Ω
,
u
>
0
,
a
m
p
;
(
t
,
x
)
∈
R
+
×
∂
Ω
,
\left \{ {\begin {array}{*{20}{c}} {\frac {{\partial u}}{{\partial t}} - \Delta u = {u^p} + h(t,x),} \hfill & {(t,x) \in {{\mathbf {R}}_ + } \times \Omega ,} \hfill \\ {{u = 0,}} \hfill & {(t,x) \in {{\mathbf {R}}_ + } \times \partial \Omega ,} \hfill \\ {{u > 0,}} \hfill & {(t,x) \in {{\mathbf {R}}_ + } \times \partial \Omega ,} \hfill \\ \end {array} } \right .
\]
where
Ω
⊂
R
N
\Omega \subset {{\mathbf {R}}^N}
is a bounded domain with smooth boundary
∂
Ω
\partial \Omega
, h is T-periodic with respect to the first variable, and
1
>
p
>
N
+
2
N
−
2
1 > p > \frac {{N + 2}}{{N - 2}}
if
N
≥
3
N \geq 3
and
1
>
p
>
+
∞
1 > p > + \infty
if
N
≤
2
N \leq 2
. It is shown that there exist a stable and an unstable positive T-periodic solution for this problem if h is sufficiently small in
L
∞
{L^\infty }
.