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Existence of local solutions of the Kirchhoff–Carrier equation in Banach spaces
This paper is concerned with the study of the existence of a local solution of the problem ( ∗ ) | B u ″ ( t ) + M ( ‖ u ( t ) ‖ W β ) A u ( t ) = 0 , in V ′ , t > 0 , u ( 0 ) = u 0 , u ′ ( 0 ) = u 1 ( u 0 ≠ 0 ) , where V is a Hilbert space with dual V ′ ; A and B symmetric linear operators from...
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| Published in: | Nonlinear analysis 2008-06, Vol.68 (11), p.3565-3580 |
|---|---|
| Main Authors: | , , |
| Format: | Article |
| Language: | English |
| Subjects: | |
| Citations: | Items that this one cites Items that cite this one |
| Online Access: | Get full text |
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| Summary: | This paper is concerned with the study of the existence of a local solution of the problem
(
∗
)
|
B
u
″
(
t
)
+
M
(
‖
u
(
t
)
‖
W
β
)
A
u
(
t
)
=
0
,
in
V
′
,
t
>
0
,
u
(
0
)
=
u
0
,
u
′
(
0
)
=
u
1
(
u
0
≠
0
)
,
where
V
is a Hilbert space with dual
V
′
;
A
and
B
symmetric linear operators from
V
into
V
′
with
〈
A
v
,
v
〉
≥
0
and
〈
B
v
,
v
〉
>
0
,
v
≠
0
;
W
a Banach space with
V
continuously embedding in
W
;
β
a real number with
β
≥
1
; and
M
(
ξ
)
a function with
M
(
ξ
)
≥
0
,
M
(
‖
u
0
‖
W
β
)
>
0
and smooth in a neighborhood of
‖
u
0
‖
W
β
.
The characterization of the derivative of the nonlinear term of the equation of (∗) and the Arzela–Ascoli Theorem allow us to obtain a solution
u
of (∗) defined in
[
0
,
T
0
]
where
T
0
depends on
u
0
,
u
1
and
M
(
ξ
)
. |
|---|---|
| ISSN: | 0362-546X 1873-5215 |
| DOI: | 10.1016/j.na.2007.03.047 |