Journal article
The Nonlinear Schrödinger Equation with Combined Power-Type Nonlinearities
Communications in partial differential equations, Vol.32(8), pp.1281-1343
08/08/2007
DOI: 10.1080/03605300701588805
Abstract
We undertake a comprehensive study of the nonlinear Schrödinger equation
where u(t, x) is a complex-valued function in spacetime
, λ
1
and λ
2
are nonzero real constants, and
. We address questions related to local and global well-posedness, finite time blowup, and asymptotic behaviour. Scattering is considered both in the energy space H
1
(ℝ
n
) and in the pseudoconformal space Σ := {f ∈ H
1
(ℝ
n
); xf ∈ L
2
(ℝ
n
)}. Of particular interest is the case when both nonlinearities are defocusing and correspond to the
-critical, respectively
-critical NLS, that is, λ
1
, λ
2
> 0 and
,
. The results at the endpoint
are conditional on a conjectured global existence and spacetime estimate for the
-critical nonlinear Schrödinger equation, which has been verified in dimensions n ≥ 2 for radial data in Tao et al. (Tao et al. to appear a,b) and Killip et al. (preprint).
As an off-shoot of our analysis, we also obtain a new, simpler proof of scattering in
for solutions to the nonlinear Schrödinger equation
with
, which was first obtained by Ginibre and Velo (
1985
).
Details
- Title: Subtitle
- The Nonlinear Schrödinger Equation with Combined Power-Type Nonlinearities
- Creators
- Terence Tao - University of California, Los AngelesMonica Visan - Institute for Advanced StudyXiaoyi Zhang - Chinese Academy of Sciences
- Resource Type
- Journal article
- Publication Details
- Communications in partial differential equations, Vol.32(8), pp.1281-1343
- Publisher
- Taylor & Francis Group
- DOI
- 10.1080/03605300701588805
- ISSN
- 0360-5302
- eISSN
- 1532-4133
- Language
- English
- Date published
- 08/08/2007
- Academic Unit
- Mathematics
- Record Identifier
- 9984240778302771
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