UDC 517.9
We prove the existence and attraction property for admissible invariant unstable and center-unstable manifolds of admissible classes of solutions to the partial neutral functional-differential equation in Banach space
X
of the form
&
∂
∂
t
F
u
t
=
A
(
t
)
F
u
t
+
f
(
t
,
u
t
)
,
t
≥
s
,
t
,
s
∈
ℝ
,
&
u
s
=
ϕ
∈
𝒞
:
=
C
(
[
-
r
,0
]
,
X
)
under the conditions that the family of linear partial differential operators
(
A
(
t
)
)
t
∈
ℝ
generates the evolution family
(
U
(
t
,
s
)
)
t
≥
s
with an exponential dichotomy on the whole line
ℝ
;
the difference operator
F
:
𝒞
→
X
is bounded and linear, and the nonlinear delay operator
f
satisfies the
φ
-Lipschitz condition, i.e.,
‖
f
(
t
,
ϕ
)
-
f
(
t
,
ψ
)
‖
≤
φ
(
t
)
‖
ϕ
-
ψ
‖
𝒞
for
ϕ
,
ψ
∈
𝒞
,
where
φ
(
⋅
)
belongs to an admissible function space defined on
ℝ
.
We also prove that an unstable manifold of the admissible class attracts all other solutions with exponential rates. Our main method is based on the Lyapunov – Perron equation combined with the admissibility of function spaces. We apply our results to the finite-delayed heat equation for a material with memory.
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