---
OA_place: repository
OA_type: green
_id: '22088'
abstract:
- lang: eng
  text: We consider the focusing cubic nonlinear Schrödinger equation with inverse-square
    potential in three space dimensions. We identify a sharp threshold between scattering
    and blowup, establishing a result analogous to that of Duyckaerts, Holmer, and
    Roudenko for the standard focusing cubic NLS. We also prove failure of uniform
    space-time bounds at the threshold.
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Rowan
  full_name: Killip, Rowan
  last_name: Killip
- first_name: Jason
  full_name: Murphy, Jason
  last_name: Murphy
- first_name: Monica
  full_name: Visan, Monica
  id: 056daca0-b8d1-11f0-964f-f91054abf8ca
  last_name: Visan
- first_name: Jiqiang
  full_name: Zheng, Jiqiang
  last_name: Zheng
citation:
  ama: Killip R, Murphy J, Vişan M, Zheng J. The focusing cubic NLS with inverse-square
    potential in three space dimensions. <i>Differential and Integral Equations</i>.
    2017;30(3/4):161-206. doi:<a href="https://doi.org/10.57262/die/1487386822">10.57262/die/1487386822</a>
  apa: Killip, R., Murphy, J., Vişan, M., &#38; Zheng, J. (2017). The focusing cubic
    NLS with inverse-square potential in three space dimensions. <i>Differential and
    Integral Equations</i>. Khayyam Publishing. <a href="https://doi.org/10.57262/die/1487386822">https://doi.org/10.57262/die/1487386822</a>
  chicago: Killip, Rowan, Jason Murphy, Monica Vişan, and Jiqiang Zheng. “The Focusing
    Cubic NLS with Inverse-Square Potential in Three Space Dimensions.” <i>Differential
    and Integral Equations</i>. Khayyam Publishing, 2017. <a href="https://doi.org/10.57262/die/1487386822">https://doi.org/10.57262/die/1487386822</a>.
  ieee: R. Killip, J. Murphy, M. Vişan, and J. Zheng, “The focusing cubic NLS with
    inverse-square potential in three space dimensions,” <i>Differential and Integral
    Equations</i>, vol. 30, no. 3/4. Khayyam Publishing, pp. 161–206, 2017.
  ista: Killip R, Murphy J, Vişan M, Zheng J. 2017. The focusing cubic NLS with inverse-square
    potential in three space dimensions. Differential and Integral Equations. 30(3/4),
    161–206.
  mla: Killip, Rowan, et al. “The Focusing Cubic NLS with Inverse-Square Potential
    in Three Space Dimensions.” <i>Differential and Integral Equations</i>, vol. 30,
    no. 3/4, Khayyam Publishing, 2017, pp. 161–206, doi:<a href="https://doi.org/10.57262/die/1487386822">10.57262/die/1487386822</a>.
  short: R. Killip, J. Murphy, M. Vişan, J. Zheng, Differential and Integral Equations
    30 (2017) 161–206.
das_tickbox: '1'
date_created: 2026-06-19T08:47:48Z
date_published: 2017-03-01T00:00:00Z
date_updated: 2026-07-01T12:42:38Z
day: '01'
doi: 10.57262/die/1487386822
extern: '1'
external_id:
  arxiv:
  - '1603.08912'
intvolume: '        30'
issue: 3/4
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.48550/arXiv.1603.08912
mathsc:
- 35Q55
month: '03'
oa: 1
oa_version: Preprint
page: 161-206
publication: Differential and Integral Equations
publication_identifier:
  issn:
  - 0893-4983
publication_status: published
publisher: Khayyam Publishing
quality_controlled: '1'
scopus_import: '1'
status: public
title: The focusing cubic NLS with inverse-square potential in three space dimensions
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 30
year: '2017'
...
---
OA_place: repository
OA_type: green
_id: '22085'
abstract:
- lang: eng
  text: "We prove global well posedness and scattering for the nonlinear Schröodinger
    equation with power-type nonlinearity (mathematical formular) below the energy
    space, i.e., for s<1. In [15], J. Colliander, M. Keel, G. Staffilani, H. Takaoka,
    and T. Tao established polynomial growth of the \r\nHs/x-norm of the solution,
    and hence global well posedness for initial data in Hs/x, provided \r\ns is sufficiently
    close to 1. However, their bounds are insufficient to yield scattering. In this
    paper, we use the a priori interaction Morawetz inequality to show that scattering
    holds in H^s(R^n)\r\n whenever s is larger than some value 0<s0(n,p)<1."
article_processing_charge: No
article_type: original
arxiv: 1
author:
- 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: Vişan M, Zhang X. Global well-posedness and scattering for a class of nonlinear
    Schröodinger equations below the energy space. <i>Differential and Integral Equations</i>.
    2009;22(1/2):99-124. doi:<a href="https://doi.org/10.57262/die/1356038556">10.57262/die/1356038556</a>
  apa: Vişan, M., &#38; Zhang, X. (2009). Global well-posedness and scattering for
    a class of nonlinear Schröodinger equations below the energy space. <i>Differential
    and Integral Equations</i>. Khayyam Publishing. <a href="https://doi.org/10.57262/die/1356038556">https://doi.org/10.57262/die/1356038556</a>
  chicago: Vişan, Monica, and Xiaoyi Zhang. “Global Well-Posedness and Scattering
    for a Class of Nonlinear Schröodinger Equations below the Energy Space.” <i>Differential
    and Integral Equations</i>. Khayyam Publishing, 2009. <a href="https://doi.org/10.57262/die/1356038556">https://doi.org/10.57262/die/1356038556</a>.
  ieee: M. Vişan and X. Zhang, “Global well-posedness and scattering for a class of
    nonlinear Schröodinger equations below the energy space,” <i>Differential and
    Integral Equations</i>, vol. 22, no. 1/2. Khayyam Publishing, pp. 99–124, 2009.
  ista: Vişan M, Zhang X. 2009. Global well-posedness and scattering for a class of
    nonlinear Schröodinger equations below the energy space. Differential and Integral
    Equations. 22(1/2), 99–124.
  mla: Vişan, Monica, and Xiaoyi Zhang. “Global Well-Posedness and Scattering for
    a Class of Nonlinear Schröodinger Equations below the Energy Space.” <i>Differential
    and Integral Equations</i>, vol. 22, no. 1/2, Khayyam Publishing, 2009, pp. 99–124,
    doi:<a href="https://doi.org/10.57262/die/1356038556">10.57262/die/1356038556</a>.
  short: M. Vişan, X. Zhang, Differential and Integral Equations 22 (2009) 99–124.
das_tickbox: '1'
date_created: 2026-06-19T08:34:09Z
date_published: 2009-01-01T00:00:00Z
date_updated: 2026-07-01T08:33:52Z
day: '01'
doi: 10.57262/die/1356038556
extern: '1'
external_id:
  arxiv:
  - math/0606611
intvolume: '        22'
issue: 1/2
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.48550/arXiv.math/0606611
mathsc:
- 35Q55
month: '01'
oa: 1
oa_version: Preprint
page: 99-124
publication: Differential and Integral Equations
publication_identifier:
  issn:
  - 0893-4983
publication_status: published
publisher: Khayyam Publishing
quality_controlled: '1'
scopus_import: '1'
status: public
title: Global well-posedness and scattering for a class of nonlinear Schröodinger
  equations below the energy space
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 22
year: '2009'
...