---
OA_place: repository
OA_type: green
_id: '22047'
abstract:
- lang: eng
  text: "We undertake a comprehensive study of the nonlinear Schrödinger equation
    (mathematical formular) where u(t, x) is a complex-valued function in spacetime
    R, xRn/x, λ1 and λ2 are nonzero real constants, and (mathematical formular). 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 L2/x-critical, respectively H1/x-critical NLS, that is, λ1, λ2 > 0 and (mathematical
    formular) . The results at the endpoint p1= 4/n are conditional on a conjectured
    global existence and spacetime estimate for the L2/x-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).\r\nAs an off-shoot
    of our analysis, we also obtain a new, simpler proof of scattering in H1/x for
    solutions to the nonlinear Schrödinger equation (mathematical formular) with 4/n
    < p < 4/n-2, which was first obtained by Ginibre and Velo (Citation1985)."
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Terence
  full_name: Tao, Terence
  last_name: Tao
- first_name: Monica
  full_name: Visan, Monica
  id: 056daca0-b8d1-11f0-964f-f91054abf8ca
  last_name: Visan
- first_name: Xiaoyi
  full_name: Zhang, Xiaoyi
  last_name: Zhang
citation:
  ama: Tao T, Vişan M, Zhang X. The nonlinear Schrödinger equation with combined power-type
    nonlinearities. <i>Communications in Partial Differential Equations</i>. 2007;32(8):1281-1343.
    doi:<a href="https://doi.org/10.1080/03605300701588805">10.1080/03605300701588805</a>
  apa: Tao, T., Vişan, M., &#38; Zhang, X. (2007). The nonlinear Schrödinger equation
    with combined power-type nonlinearities. <i>Communications in Partial Differential
    Equations</i>. Informa UK Limited. <a href="https://doi.org/10.1080/03605300701588805">https://doi.org/10.1080/03605300701588805</a>
  chicago: Tao, Terence, Monica Vişan, and Xiaoyi Zhang. “The Nonlinear Schrödinger
    Equation with Combined Power-Type Nonlinearities.” <i>Communications in Partial
    Differential Equations</i>. Informa UK Limited, 2007. <a href="https://doi.org/10.1080/03605300701588805">https://doi.org/10.1080/03605300701588805</a>.
  ieee: T. Tao, M. Vişan, and X. Zhang, “The nonlinear Schrödinger equation with combined
    power-type nonlinearities,” <i>Communications in Partial Differential Equations</i>,
    vol. 32, no. 8. Informa UK Limited, pp. 1281–1343, 2007.
  ista: Tao T, Vişan M, Zhang X. 2007. The nonlinear Schrödinger equation with combined
    power-type nonlinearities. Communications in Partial Differential Equations. 32(8),
    1281–1343.
  mla: Tao, Terence, et al. “The Nonlinear Schrödinger Equation with Combined Power-Type
    Nonlinearities.” <i>Communications in Partial Differential Equations</i>, vol.
    32, no. 8, Informa UK Limited, 2007, pp. 1281–343, doi:<a href="https://doi.org/10.1080/03605300701588805">10.1080/03605300701588805</a>.
  short: T. Tao, M. Vişan, X. Zhang, Communications in Partial Differential Equations
    32 (2007) 1281–1343.
das_tickbox: '1'
date_created: 2026-06-19T07:49:46Z
date_published: 2007-08-29T00:00:00Z
date_updated: 2026-06-25T08:04:20Z
day: '29'
doi: 10.1080/03605300701588805
extern: '1'
external_id:
  arxiv:
  - math/0511070
intvolume: '        32'
issue: '8'
keyword:
- Energy-critical
- Mass-critical
- Nonlinear Schrödinger equation
- Wellposedness
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.48550/arXiv.math/0511070
mathsc:
- 35Q55
month: '08'
oa: 1
oa_version: Preprint
page: 1281-1343
publication: Communications in Partial Differential Equations
publication_identifier:
  eissn:
  - 1532-4133
  issn:
  - 0360-5302
publication_status: published
publisher: Informa UK Limited
quality_controlled: '1'
scopus_import: '1'
status: public
title: The nonlinear Schrödinger equation with combined power-type nonlinearities
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 32
year: '2007'
...