UDC 517.9
Integral manifolds are very useful in studying dynamics of nonlinear evolution equations. In this paper, we consider the nondensely-defined partial differential equation
ⅆ
u
ⅆ
t
=
(
A
+
B
(
t
)
)
u
(
t
)
+
f
(
t
,
u
t
)
,
t
∈
R
,
(
1
)
where
(
A
,
D
(
A
)
)
satisfies the Hille – Yosida condition,
(
B
(
t
)
)
t
∈
R
is a family of operators in
ℒ
(
D
(
A
)
¯
,
X
)
satisfying some measurability and boundedness conditions, and the nonlinear forcing term
f
satisfies
‖
f
(
t
,
ϕ
)
-
f
(
t
,
ψ
)
‖
≤
φ
(
t
)
‖
ϕ
-
ψ
‖
𝒞
,
here,
φ
belongs to some admissible spaces and
ϕ
,
ψ
∈
𝒞
:
=
C
(
[
-
r
,0
]
,
X
)
. We first present an exponential convergence result between the stable manifold and every mild solution of (1). Then we prove the existence of center-unstable manifolds for such solutions.Our main methods are invoked by the extrapolation theory and the Lyapunov – Perron method based on the admissible functions properties.