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The Cauchy Problem for Non-linear Higher Order Hartree Type Equation in Modulation Spaces
We study the Cauchy problem for Hartree equation with cubic convolution nonlinearity F ( u ) = ( K ⋆ | u | 2 k ) u under a specified condition on potential K with Cauchy data in modulation spaces M p , q ( R n ) . We establish global well-posedness results in M 1 , 1 ( R n ) , when K ( x ) = λ | x |...
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| Published in: | The Journal of fourier analysis and applications 2019-08, Vol.25 (4), p.1319-1349 |
|---|---|
| Main Author: | |
| 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: | We study the Cauchy problem for Hartree equation with cubic convolution nonlinearity
F
(
u
)
=
(
K
⋆
|
u
|
2
k
)
u
under a specified condition on potential
K
with Cauchy data in modulation spaces
M
p
,
q
(
R
n
)
. We establish global well-posedness results in
M
1
,
1
(
R
n
)
, when
K
(
x
)
=
λ
|
x
|
ν
(
λ
∈
R
,
0
<
ν
<
m
i
n
{
2
,
n
2
}
)
, for
k
<
n
+
2
-
ν
n
;
and local well-posedness results in
M
1
,
1
(
R
n
)
, when
K
(
x
)
=
λ
|
x
|
ν
(
λ
∈
R
,
0
<
ν
<
n
)
,
for
k
∈
N
;
in
M
p
,
q
(
R
n
)
with
1
≤
p
≤
4
,
1
≤
q
≤
2
2
k
-
2
2
2
k
-
2
-
1
,
k
∈
N
, when
K
∈
M
∞
,
1
(
R
n
)
. Moreover, we also consider the Cauchy problem for the non-linear higher order Hartree equations on modulation spaces
M
p
,
1
(
R
n
)
,
when
K
∈
M
1
,
∞
(
R
n
)
and show the existence of a unique global solution by using integrability of time decay factors of Strichartz estimates. As a consequence, we are able to deal with wider classes of a nonlinearity and a solution space. |
|---|---|
| ISSN: | 1069-5869 1531-5851 |
| DOI: | 10.1007/s00041-018-9629-z |