Journal article
The focusing energy-critical Hartree equation
Journal of Differential Equations, Vol.246(3), pp.1139-1163
2009
DOI: 10.1016/j.jde.2008.05.013
Abstract
We consider the focusing energy-critical nonlinear Hartree equation
i
u
t
+
Δ
u
=
−
(
|
x
|
−4
∗
|
u
|
2
)
u
. We proved that if a maximal-lifespan solution
u
:
I
×
R
d
→
C
satisfies
sup
t
∈
I
‖
∇
u
(
t
)
‖
2
<
‖
∇
W
‖
2
, where
W is the static solution of the equation, then the maximal-lifespan
I
=
R
, moreover, the solution scatters in both time directions. For spherically symmetric initial data, similar result has been obtained in [C. Miao, G. Xu, L. Zhao, Global wellposedness, scattering and blowup for the energy-critical, focusing Hartree equation in the radial case, Colloq. Math., in press]. The argument is an adaptation of the recent work of R. Killip and M. Visan [R. Killip, M. Visan, The focusing energy-critical nonlinear Schrödinger equation in dimensions five and higher, preprint] on energy-critical nonlinear Schrödinger equations.
Details
- Title: Subtitle
- The focusing energy-critical Hartree equation
- Creators
- Dong Li - Institute for Advanced StudyChangxing Miao - Institute of Applied PhysicsXiaoyi Zhang - Institute for Advanced Study
- Resource Type
- Journal article
- Publication Details
- Journal of Differential Equations, Vol.246(3), pp.1139-1163
- DOI
- 10.1016/j.jde.2008.05.013
- ISSN
- 0022-0396
- eISSN
- 1090-2732
- Publisher
- Elsevier Inc
- Language
- English
- Date published
- 2009
- Academic Unit
- Mathematics
- Record Identifier
- 9984241038802771
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