On the global well-posedness of focusing energy-critical inhomogeneous NLS

• Published in: Journal of evolution equations Vol. 20; no. 4; pp. 1349 - 1380
• Main Authors: Cho, Yonggeun, Hong, Seokchang, Lee, Kiyeon
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01-12-2020
• Subjects: Analysis; Mathematics; Mathematics and Statistics; GWP; Focusing energy-critical nonlinearity; Scattering; Inhomogeneous NLS; Blowup; 35Q40; Kenig–Merle argument; M35Q55
• ISSN: 1424-3199; 1424-3202
• DOI: 10.1007/s00028-020-00558-1

We consider the focusing energy-critical inhomogeneous nonlinear Schrödinger equation:
i
u
t
+
Δ
u
+
g
|
u
|
p
-
1
u
=
0
,
u
(
0
)
=
φ
∈
H
˙
1
,
where
0
≤
g
i
≤
|
x
|
b
g
≤
g
s
,
0
<
b
<
4
3
, and
p
=
5
-
2
b
. On the road map of Kenig and Merle (Invent Math 166:645–675, 2006), we show the global well-posedness and scattering of radial solutions under energy condition
E
g
(
φ
)
<
E
g
(
Q
b
)
,
and
g
s
1
p
0
‖
φ
‖
H
˙
1
2
<
‖
Q
b
‖
H
˙
1
2
,
where
Q
b
is the solution of
Δ
Q
b
+
|
x
|
-
b
Q
b
p
=
0
, together with scaling condition
|
x
|
|
∇
g
(
x
)
|
≲
|
x
|
-
b
, variational condition
g
s
2
p
-
1
(
p
+
1
2
-
g
i
)
≤
p
-
1
2
, and rigidity condition
-
b
g
(
x
)
≤
x
·
∇
g
(
x
)
. We also provide sharp finite time blow-up results for non-radial and radial solutions. For this, we utilize the localized virial identity.