---
_id: '12304'
abstract:
- lang: eng
  text: 'We establish sharp criteria for the instantaneous propagation of free boundaries
    in solutions to the thin-film equation. The criteria are formulated in terms of
    the initial distribution of mass (as opposed to previous almost-optimal results),
    reflecting the fact that mass is a locally conserved quantity for the thin-film
    equation. In the regime of weak slippage, our criteria are at the same time necessary
    and sufficient. The proof of our upper bounds on free boundary propagation is
    based on a strategy of “propagation of degeneracy” down to arbitrarily small spatial
    scales: We combine estimates on the local mass and estimates on energies to show
    that “degeneracy” on a certain space-time cylinder entails “degeneracy” on a spatially
    smaller space-time cylinder with the same time horizon. The derivation of our
    lower bounds on free boundary propagation is based on a combination of a monotone
    quantity and almost optimal estimates established previously by the second author
    with a new estimate connecting motion of mass to entropy production.'
acknowledgement: N. De Nitti acknowledges the kind hospitality of IST Austria within
  the framework of the ISTernship Summer Program 2018, during which most of the present
  article was written. N. DeNitti has received funding by The Austrian Agency for
  International Cooperation in Education &Research (OeAD-GmbH) via its financial support
  of the ISTernship Summer Program 2018. N.De Nitti would also like to thank Giuseppe
  Coclite, Giuseppe Devillanova, Giuseppe Florio, Sebastian Hensel, and Francesco
  Maddalena for several helpful conversations on topics related to this work.
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Nicola
  full_name: De Nitti, Nicola
  last_name: De Nitti
- first_name: Julian L
  full_name: Fischer, Julian L
  id: 2C12A0B0-F248-11E8-B48F-1D18A9856A87
  last_name: Fischer
  orcid: 0000-0002-0479-558X
citation:
  ama: De Nitti N, Fischer JL. Sharp criteria for the waiting time phenomenon in solutions
    to the thin-film equation. <i>Communications in Partial Differential Equations</i>.
    2022;47(7):1394-1434. doi:<a href="https://doi.org/10.1080/03605302.2022.2056702">10.1080/03605302.2022.2056702</a>
  apa: De Nitti, N., &#38; Fischer, J. L. (2022). Sharp criteria for the waiting time
    phenomenon in solutions to the thin-film equation. <i>Communications in Partial
    Differential Equations</i>. Taylor &#38; Francis. <a href="https://doi.org/10.1080/03605302.2022.2056702">https://doi.org/10.1080/03605302.2022.2056702</a>
  chicago: De Nitti, Nicola, and Julian L Fischer. “Sharp Criteria for the Waiting
    Time Phenomenon in Solutions to the Thin-Film Equation.” <i>Communications in
    Partial Differential Equations</i>. Taylor &#38; Francis, 2022. <a href="https://doi.org/10.1080/03605302.2022.2056702">https://doi.org/10.1080/03605302.2022.2056702</a>.
  ieee: N. De Nitti and J. L. Fischer, “Sharp criteria for the waiting time phenomenon
    in solutions to the thin-film equation,” <i>Communications in Partial Differential
    Equations</i>, vol. 47, no. 7. Taylor &#38; Francis, pp. 1394–1434, 2022.
  ista: De Nitti N, Fischer JL. 2022. Sharp criteria for the waiting time phenomenon
    in solutions to the thin-film equation. Communications in Partial Differential
    Equations. 47(7), 1394–1434.
  mla: De Nitti, Nicola, and Julian L. Fischer. “Sharp Criteria for the Waiting Time
    Phenomenon in Solutions to the Thin-Film Equation.” <i>Communications in Partial
    Differential Equations</i>, vol. 47, no. 7, Taylor &#38; Francis, 2022, pp. 1394–434,
    doi:<a href="https://doi.org/10.1080/03605302.2022.2056702">10.1080/03605302.2022.2056702</a>.
  short: N. De Nitti, J.L. Fischer, Communications in Partial Differential Equations
    47 (2022) 1394–1434.
corr_author: '1'
date_created: 2023-01-16T10:06:50Z
date_published: 2022-07-01T00:00:00Z
date_updated: 2024-10-09T21:03:57Z
day: '01'
department:
- _id: JuFi
doi: 10.1080/03605302.2022.2056702
external_id:
  arxiv:
  - '1907.05342'
  isi:
  - '000805689800001'
intvolume: '        47'
isi: 1
issue: '7'
keyword:
- Applied Mathematics
- Analysis
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: ' https://doi.org/10.48550/arXiv.1907.05342'
month: '07'
oa: 1
oa_version: Preprint
page: 1394-1434
publication: Communications in Partial Differential Equations
publication_identifier:
  eissn:
  - 1532-4133
  issn:
  - 0360-5302
publication_status: published
publisher: Taylor & Francis
quality_controlled: '1'
scopus_import: '1'
status: public
title: Sharp criteria for the waiting time phenomenon in solutions to the thin-film
  equation
type: journal_article
user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 47
year: '2022'
...
---
OA_place: repository
OA_type: green
_id: '22066'
abstract:
- lang: eng
  text: "We prove almost sure global existence and scattering for the energy-critical
    nonlinear Schrödinger equation with randomized spherically symmetric initial data
    in \U0001D43B\U0001D460⁡(ℝ4) with \r\n5/6<\U0001D460<1. We were inspired to consider
    this problem by the recent work of Dodson–Lührmann–Mendelson, which treated the
    analogous problem for the energy-critical wave equation."
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
citation:
  ama: Killip R, Murphy J, Vişan M. Almost sure scattering for the energy-critical
    NLS with radial data below H1(R4). <i>Communications in Partial Differential Equations</i>.
    2019;44(1):51-71. doi:<a href="https://doi.org/10.1080/03605302.2018.1541904">10.1080/03605302.2018.1541904</a>
  apa: Killip, R., Murphy, J., &#38; Vişan, M. (2019). Almost sure scattering for
    the energy-critical NLS with radial data below H1(R4). <i>Communications in Partial
    Differential Equations</i>. Informa UK Limited. <a href="https://doi.org/10.1080/03605302.2018.1541904">https://doi.org/10.1080/03605302.2018.1541904</a>
  chicago: Killip, Rowan, Jason Murphy, and Monica Vişan. “Almost Sure Scattering
    for the Energy-Critical NLS with Radial Data below H1(R4).” <i>Communications
    in Partial Differential Equations</i>. Informa UK Limited, 2019. <a href="https://doi.org/10.1080/03605302.2018.1541904">https://doi.org/10.1080/03605302.2018.1541904</a>.
  ieee: R. Killip, J. Murphy, and M. Vişan, “Almost sure scattering for the energy-critical
    NLS with radial data below H1(R4),” <i>Communications in Partial Differential
    Equations</i>, vol. 44, no. 1. Informa UK Limited, pp. 51–71, 2019.
  ista: Killip R, Murphy J, Vişan M. 2019. Almost sure scattering for the energy-critical
    NLS with radial data below H1(R4). Communications in Partial Differential Equations.
    44(1), 51–71.
  mla: Killip, Rowan, et al. “Almost Sure Scattering for the Energy-Critical NLS with
    Radial Data below H1(R4).” <i>Communications in Partial Differential Equations</i>,
    vol. 44, no. 1, Informa UK Limited, 2019, pp. 51–71, doi:<a href="https://doi.org/10.1080/03605302.2018.1541904">10.1080/03605302.2018.1541904</a>.
  short: R. Killip, J. Murphy, M. Vişan, Communications in Partial Differential Equations
    44 (2019) 51–71.
das_tickbox: '1'
date_created: 2026-06-19T08:13:26Z
date_published: 2019-02-15T00:00:00Z
date_updated: 2026-06-30T07:16:06Z
day: '15'
doi: 10.1080/03605302.2018.1541904
extern: '1'
external_id:
  arxiv:
  - '1707.09051'
intvolume: '        44'
issue: '1'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.48550/arXiv.1707.09051
mathsc:
- 35Q55
month: '02'
oa: 1
oa_version: Preprint
page: 51-71
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: Almost sure scattering for the energy-critical NLS with radial data below H1(R4)
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 44
year: '2019'
...
---
OA_place: repository
OA_type: green
_id: '22029'
abstract:
- lang: eng
  text: We consider two classes of defocusing energy-supercritical nonlinear Schrödinger
    equations in dimensions d ≥ 5. We prove that if the solution u is a priori bounded
    in the critical Sobolev space, that is, {mathematical formular} , then u is global
    and scatters.
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Rowan
  full_name: Killip, Rowan
  last_name: Killip
- first_name: Monica
  full_name: Visan, Monica
  id: 056daca0-b8d1-11f0-964f-f91054abf8ca
  last_name: Visan
citation:
  ama: 'Killip R, Vişan M. Energy-supercritical NLS: Critical [Hdot]^s-bounds imply
    scattering. <i>Communications in Partial Differential Equations</i>. 2010;35(6):945-987.
    doi:<a href="https://doi.org/10.1080/03605301003717084">10.1080/03605301003717084</a>'
  apa: 'Killip, R., &#38; Vişan, M. (2010). Energy-supercritical NLS: Critical [Hdot]^s-bounds
    imply scattering. <i>Communications in Partial Differential Equations</i>. Informa
    UK Limited. <a href="https://doi.org/10.1080/03605301003717084">https://doi.org/10.1080/03605301003717084</a>'
  chicago: 'Killip, Rowan, and Monica Vişan. “Energy-Supercritical NLS: Critical [Hdot]^s-Bounds
    Imply Scattering.” <i>Communications in Partial Differential Equations</i>. Informa
    UK Limited, 2010. <a href="https://doi.org/10.1080/03605301003717084">https://doi.org/10.1080/03605301003717084</a>.'
  ieee: 'R. Killip and M. Vişan, “Energy-supercritical NLS: Critical [Hdot]^s-bounds
    imply scattering,” <i>Communications in Partial Differential Equations</i>, vol.
    35, no. 6. Informa UK Limited, pp. 945–987, 2010.'
  ista: 'Killip R, Vişan M. 2010. Energy-supercritical NLS: Critical [Hdot]^s-bounds
    imply scattering. Communications in Partial Differential Equations. 35(6), 945–987.'
  mla: 'Killip, Rowan, and Monica Vişan. “Energy-Supercritical NLS: Critical [Hdot]^s-Bounds
    Imply Scattering.” <i>Communications in Partial Differential Equations</i>, vol.
    35, no. 6, Informa UK Limited, 2010, pp. 945–87, doi:<a href="https://doi.org/10.1080/03605301003717084">10.1080/03605301003717084</a>.'
  short: R. Killip, M. Vişan, Communications in Partial Differential Equations 35
    (2010) 945–987.
date_created: 2026-06-19T07:41:22Z
date_published: 2010-06-01T00:00:00Z
date_updated: 2026-06-29T06:32:07Z
day: '01'
doi: 10.1080/03605301003717084
extern: '1'
external_id:
  arxiv:
  - '0812.2084'
intvolume: '        35'
issue: '6'
keyword:
- critical regularity
- Nonlinear Schrödinger equations
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.48550/arXiv.0812.2084
month: '06'
oa: 1
oa_version: Preprint
page: 945-987
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: 'Energy-supercritical NLS: Critical [Hdot]^s-bounds imply scattering'
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 35
year: '2010'
...
---
OA_place: repository
OA_type: green
_id: '22058'
abstract:
- lang: eng
  text: "We consider the defocusing H˙ 1-critical nonlinear Schr¨odinger equation
    in all dimensions (n ≥ 3) with a quadratic potential V (x) = ± 1/2|x|^2. We show
    global well-posedness for radial initial data obeying ∇u0(x), xu0(x) ∈L^2. In
    view of the potential V , this is the natural energy space. In the repulsive case,
    we also prove scattering. We follow the approach pioneered by Bourgain and Tao
    in the case of no potential; indeed, we include a proof of their results that
    incorporates a\r\ncouple of simplifications discovered while treating the problem
    with quadratic potential."
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Rowan
  full_name: Killip, Rowan
  last_name: Killip
- 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: Killip R, Vişan M, Zhang X. Energy-critical NLS with quadratic potentials.
    <i>Communications in Partial Differential Equations</i>. 2009;34(12):1531-1565.
    doi:<a href="https://doi.org/10.1080/03605300903328109">10.1080/03605300903328109</a>
  apa: Killip, R., Vişan, M., &#38; Zhang, X. (2009). Energy-critical NLS with quadratic
    potentials. <i>Communications in Partial Differential Equations</i>. Informa UK
    Limited. <a href="https://doi.org/10.1080/03605300903328109">https://doi.org/10.1080/03605300903328109</a>
  chicago: Killip, Rowan, Monica Vişan, and Xiaoyi Zhang. “Energy-Critical NLS with
    Quadratic Potentials.” <i>Communications in Partial Differential Equations</i>.
    Informa UK Limited, 2009. <a href="https://doi.org/10.1080/03605300903328109">https://doi.org/10.1080/03605300903328109</a>.
  ieee: R. Killip, M. Vişan, and X. Zhang, “Energy-critical NLS with quadratic potentials,”
    <i>Communications in Partial Differential Equations</i>, vol. 34, no. 12. Informa
    UK Limited, pp. 1531–1565, 2009.
  ista: Killip R, Vişan M, Zhang X. 2009. Energy-critical NLS with quadratic potentials.
    Communications in Partial Differential Equations. 34(12), 1531–1565.
  mla: Killip, Rowan, et al. “Energy-Critical NLS with Quadratic Potentials.” <i>Communications
    in Partial Differential Equations</i>, vol. 34, no. 12, Informa UK Limited, 2009,
    pp. 1531–65, doi:<a href="https://doi.org/10.1080/03605300903328109">10.1080/03605300903328109</a>.
  short: R. Killip, M. Vişan, X. Zhang, Communications in Partial Differential Equations
    34 (2009) 1531–1565.
das_tickbox: '1'
date_created: 2026-06-19T07:57:55Z
date_published: 2009-11-01T00:00:00Z
date_updated: 2026-06-29T10:20:21Z
day: '01'
doi: 10.1080/03605300903328109
extern: '1'
external_id:
  arxiv:
  - math/0611394
intvolume: '        34'
issue: '12'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.48550/arXiv.math/0611394
mathsc:
- 35Q55
- 35B33
month: '11'
oa: 1
oa_version: Preprint
page: 1531-1565
publication: Communications in Partial Differential Equations
publication_identifier:
  issn:
  - 0360-5302
  - 1532-4133
publication_status: published
publisher: Informa UK Limited
quality_controlled: '1'
scopus_import: '1'
status: public
title: Energy-critical NLS with quadratic potentials
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 34
year: '2009'
...
---
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'
...