The Super-Diffusive Singular Perturbation Problem
2020
In this paper we study a class of singularly perturbed defined abstract Cauchy problems. We investigate the singular perturbation problem (
P
ϵ
)
ϵ
α
D
t
α
u
ϵ
(
t
)
+
u
ϵ
′
(
t
)
=
A
u
ϵ
(
t
)
,
t
∈
[
0
,
T
]
, 1
<
α
<
2
,
ϵ
>
0
,
for the parabolic equation (
P
)
u
0
′
(
t
)
=
A
u
0
(
t
)
,
t
∈
[
0
,
T
]
,
in a Banach space, as the singular parameter goes to zero. Under the assumption that A is the generator of a bounded analytic semigroup and under some regularity conditions we show that problem (
P
ϵ
)
has a unique solution u
ϵ
(
t
)
for each small ϵ
>
0
.
Moreover u
ϵ
(
t
)
converges to u
0
(
t
)
as ϵ
→
0
+
,
the unique solution of equation (
P
)
.
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