---
_id: '59432'
abstract:
- lang: eng
  text: <jats:title>Zusammenfassung</jats:title><jats:p>In stochastischen Situationen
    mit zwei dichotomen Merkmalen erlauben weder die schulüblichen Baumdiagramme noch
    Vierfeldertafeln die simultane Darstellung sämtlicher in der Situation möglicher
    Wahrscheinlichkeiten. Das im vorliegenden Beitrag vorgestellte Netz hat die Kapazität,
    alle vier möglichen Randwahrscheinlichkeiten, alle vier Schnittwahrscheinlichkeiten
    sowie alle acht bedingten Wahrscheinlichkeiten<jats:italic> gleichzeitig</jats:italic>
    darzustellen. Darüber hinaus ist – aufgrund der Knoten-Ast-Struktur des Netzes –
    die simultane Darstellung von Wahrscheinlichkeiten <jats:italic>und</jats:italic>
    absoluten Häufigkeiten mit dieser Visualisierung ebenfalls möglich. Bei der sukzessiven
    Erweiterung des typischen Baumdiagramms zunächst zum Doppelbaum und schließlich
    zum Netz sinkt der Inferenzgrad (d. h. weniger kognitive Schritte sind erforderlich)
    z. B. für Fragen nach bedingten Wahrscheinlichkeiten, aber gleichzeitig steigt
    die Komplexität der Darstellung und somit die extrinsische kognitive Belastung.
    Im vorliegenden Artikel erfolgt zunächst ein theoretischer Vergleich dieser Knoten-Ast-Strukturen.
    Eine anschließende Studie illustriert, dass sich die sukzessive Erweiterung bereits
    vollständig ausgefüllter Diagramme positiv auf die Performanz von <jats:italic>N</jats:italic> = 269 Schülerinnen
    und Schülern auswirkt. Obwohl <jats:italic>Häufigkeitsdoppelbäume</jats:italic>
    und <jats:italic>Häufigkeitsnetze </jats:italic>den Schülerinnen und Schülern
    gänzlich unbekannt waren, unterstützten diese Visualisierungen die Schülerinnen
    und Schüler bei der Bearbeitung der Aufgaben am meisten.</jats:p>
article_type: original
author:
- first_name: Karin
  full_name: Binder, Karin
  id: '83381'
  last_name: Binder
- first_name: Nicole
  full_name: Steib, Nicole
  last_name: Steib
- first_name: Stefan
  full_name: Krauss, Stefan
  last_name: Krauss
citation:
  ama: Binder K, Steib N, Krauss S. Von Baumdiagrammen über Doppelbäume zu Häufigkeitsnetzen –
    kognitive Überlastung oder didaktische Unterstützung? Moving from tree diagrams
    to double trees to net diagrams—cognitively overwhelming or educationally supportive?
    <i>Journal für Mathematik-Didaktik</i>. 2022;44(2):471-503. doi:<a href="https://doi.org/10.1007/s13138-022-00215-9">10.1007/s13138-022-00215-9</a>
  apa: Binder, K., Steib, N., &#38; Krauss, S. (2022). Von Baumdiagrammen über Doppelbäume
    zu Häufigkeitsnetzen – kognitive Überlastung oder didaktische Unterstützung? Moving
    from tree diagrams to double trees to net diagrams—cognitively overwhelming or
    educationally supportive? <i>Journal für Mathematik-Didaktik</i>, <i>44</i>(2),
    471–503. <a href="https://doi.org/10.1007/s13138-022-00215-9">https://doi.org/10.1007/s13138-022-00215-9</a>
  bibtex: '@article{Binder_Steib_Krauss_2022, title={Von Baumdiagrammen über Doppelbäume
    zu Häufigkeitsnetzen – kognitive Überlastung oder didaktische Unterstützung? Moving
    from tree diagrams to double trees to net diagrams—cognitively overwhelming or
    educationally supportive?}, volume={44}, DOI={<a href="https://doi.org/10.1007/s13138-022-00215-9">10.1007/s13138-022-00215-9</a>},
    number={2}, journal={Journal für Mathematik-Didaktik}, publisher={Springer Science
    and Business Media LLC}, author={Binder, Karin and Steib, Nicole and Krauss, Stefan},
    year={2022}, pages={471–503} }'
  chicago: 'Binder, Karin, Nicole Steib, and Stefan Krauss. “Von Baumdiagrammen über
    Doppelbäume zu Häufigkeitsnetzen – kognitive Überlastung oder didaktische Unterstützung?
    Moving from tree diagrams to double trees to net diagrams—cognitively overwhelming
    or educationally supportive?” <i>Journal für Mathematik-Didaktik</i> 44, no. 2
    (2022): 471–503. <a href="https://doi.org/10.1007/s13138-022-00215-9">https://doi.org/10.1007/s13138-022-00215-9</a>.'
  ieee: 'K. Binder, N. Steib, and S. Krauss, “Von Baumdiagrammen über Doppelbäume
    zu Häufigkeitsnetzen – kognitive Überlastung oder didaktische Unterstützung? Moving
    from tree diagrams to double trees to net diagrams—cognitively overwhelming or
    educationally supportive?,” <i>Journal für Mathematik-Didaktik</i>, vol. 44, no.
    2, pp. 471–503, 2022, doi: <a href="https://doi.org/10.1007/s13138-022-00215-9">10.1007/s13138-022-00215-9</a>.'
  mla: Binder, Karin, et al. “Von Baumdiagrammen über Doppelbäume zu Häufigkeitsnetzen –
    kognitive Überlastung oder didaktische Unterstützung? Moving from tree diagrams
    to double trees to net diagrams—cognitively overwhelming or educationally supportive?”
    <i>Journal für Mathematik-Didaktik</i>, vol. 44, no. 2, Springer Science and Business
    Media LLC, 2022, pp. 471–503, doi:<a href="https://doi.org/10.1007/s13138-022-00215-9">10.1007/s13138-022-00215-9</a>.
  short: K. Binder, N. Steib, S. Krauss, Journal für Mathematik-Didaktik 44 (2022)
    471–503.
date_created: 2025-04-08T11:32:04Z
date_updated: 2025-05-17T18:37:30Z
ddc:
- '370'
doi: 10.1007/s13138-022-00215-9
file:
- access_level: closed
  content_type: application/pdf
  creator: binder
  date_created: 2025-05-17T18:37:16Z
  date_updated: 2025-05-17T18:37:16Z
  file_id: '59933'
  file_name: Binder Steib Krauss 2023 Von Baumdiagrammen über Doppelbäume zu Häufigkeitsnetzen
    - JMD.pdf
  file_size: 3105846
  relation: main_file
  success: 1
file_date_updated: 2025-05-17T18:37:16Z
intvolume: '        44'
issue: '2'
language:
- iso: ger
page: 471-503
publication: Journal für Mathematik-Didaktik
publication_identifier:
  issn:
  - 0173-5322
  - 1869-2699
publication_status: published
publisher: Springer Science and Business Media LLC
status: public
title: Von Baumdiagrammen über Doppelbäume zu Häufigkeitsnetzen – kognitive Überlastung
  oder didaktische Unterstützung? Moving from tree diagrams to double trees to net
  diagrams—cognitively overwhelming or educationally supportive?
type: journal_article
user_id: '83381'
volume: 44
year: '2022'
...
---
_id: '47327'
author:
- first_name: Anna Viktoria Katrin
  full_name: Plaksin, Anna Viktoria Katrin
  id: '102981'
  last_name: Plaksin
  orcid: 0000-0002-9969-0608
- first_name: D.
  full_name: Lewis, D.
  last_name: Lewis
- first_name: N.
  full_name: Şahin, N.
  last_name: Şahin
- first_name: Axel
  full_name: Berndt, Axel
  id: '55565'
  last_name: Berndt
citation:
  ama: 'Plaksin AVK, Lewis D, Şahin N, Berndt A. Sharing MEI: Common Semantics in
    Diverse Musics? In: <i>{Music Encoding Conference}</i>. ; 2022:157–161.'
  apa: 'Plaksin, A. V. K., Lewis, D., Şahin, N., &#38; Berndt, A. (2022). Sharing
    MEI: Common Semantics in Diverse Musics? <i>{Music Encoding Conference}</i>, 157–161.'
  bibtex: '@inproceedings{Plaksin_Lewis_Şahin_Berndt_2022, place={Halifax, Canada},
    title={Sharing MEI: Common Semantics in Diverse Musics?}, booktitle={{Music Encoding
    Conference}}, author={Plaksin, Anna Viktoria Katrin and Lewis, D. and Şahin, N.
    and Berndt, Axel}, year={2022}, pages={157–161} }'
  chicago: 'Plaksin, Anna Viktoria Katrin, D. Lewis, N. Şahin, and Axel Berndt. “Sharing
    MEI: Common Semantics in Diverse Musics?” In <i>{Music Encoding Conference}</i>,
    157–161. Halifax, Canada, 2022.'
  ieee: 'A. V. K. Plaksin, D. Lewis, N. Şahin, and A. Berndt, “Sharing MEI: Common
    Semantics in Diverse Musics?,” in <i>{Music Encoding Conference}</i>, 2022, pp.
    157–161.'
  mla: 'Plaksin, Anna Viktoria Katrin, et al. “Sharing MEI: Common Semantics in Diverse
    Musics?” <i>{Music Encoding Conference}</i>, 2022, pp. 157–161.'
  short: 'A.V.K. Plaksin, D. Lewis, N. Şahin, A. Berndt, in: {Music Encoding Conference},
    Halifax, Canada, 2022, pp. 157–161.'
date_created: 2023-09-25T10:44:08Z
date_updated: 2025-05-21T15:09:48Z
language:
- iso: eng
page: 157–161
place: Halifax, Canada
publication: '{Music Encoding Conference}'
status: public
title: 'Sharing MEI: Common Semantics in Diverse Musics?'
type: conference
user_id: '55565'
year: '2022'
...
---
_id: '51196'
article_number: '100113'
author:
- first_name: G.
  full_name: Meschut, G.
  last_name: Meschut
- first_name: M.
  full_name: Merklein, M.
  last_name: Merklein
- first_name: A.
  full_name: Brosius, A.
  last_name: Brosius
- first_name: D.
  full_name: Drummer, D.
  last_name: Drummer
- first_name: L.
  full_name: Fratini, L.
  last_name: Fratini
- first_name: U.
  full_name: Füssel, U.
  last_name: Füssel
- first_name: M.
  full_name: Gude, M.
  last_name: Gude
- first_name: W.
  full_name: Homberg, W.
  last_name: Homberg
- first_name: P.A.F.
  full_name: Martins, P.A.F.
  last_name: Martins
- first_name: M.
  full_name: Bobbert, M.
  last_name: Bobbert
- first_name: M.
  full_name: Lechner, M.
  last_name: Lechner
- first_name: R.
  full_name: Kupfer, R.
  last_name: Kupfer
- first_name: B.
  full_name: Gröger, B.
  last_name: Gröger
- first_name: D.
  full_name: Han, D.
  last_name: Han
- first_name: J.
  full_name: Kalich, J.
  last_name: Kalich
- first_name: F.
  full_name: Kappe, F.
  last_name: Kappe
- first_name: T.
  full_name: Kleffel, T.
  last_name: Kleffel
- first_name: D.
  full_name: Köhler, D.
  last_name: Köhler
- first_name: C.-M.
  full_name: Kuball, C.-M.
  last_name: Kuball
- first_name: J.
  full_name: Popp, J.
  last_name: Popp
- first_name: D.
  full_name: Römisch, D.
  last_name: Römisch
- first_name: J.
  full_name: Troschitz, J.
  last_name: Troschitz
- first_name: C.
  full_name: Wischer, C.
  last_name: Wischer
- first_name: S.
  full_name: Wituschek, S.
  last_name: Wituschek
- first_name: M.
  full_name: Wolf, M.
  last_name: Wolf
citation:
  ama: Meschut G, Merklein M, Brosius A, et al. Review on mechanical joining by plastic
    deformation. <i>Journal of Advanced Joining Processes</i>. 2022;5. doi:<a href="https://doi.org/10.1016/j.jajp.2022.100113">10.1016/j.jajp.2022.100113</a>
  apa: Meschut, G., Merklein, M., Brosius, A., Drummer, D., Fratini, L., Füssel, U.,
    Gude, M., Homberg, W., Martins, P. A. F., Bobbert, M., Lechner, M., Kupfer, R.,
    Gröger, B., Han, D., Kalich, J., Kappe, F., Kleffel, T., Köhler, D., Kuball, C.-M.,
    … Wolf, M. (2022). Review on mechanical joining by plastic deformation. <i>Journal
    of Advanced Joining Processes</i>, <i>5</i>, Article 100113. <a href="https://doi.org/10.1016/j.jajp.2022.100113">https://doi.org/10.1016/j.jajp.2022.100113</a>
  bibtex: '@article{Meschut_Merklein_Brosius_Drummer_Fratini_Füssel_Gude_Homberg_Martins_Bobbert_et
    al._2022, title={Review on mechanical joining by plastic deformation}, volume={5},
    DOI={<a href="https://doi.org/10.1016/j.jajp.2022.100113">10.1016/j.jajp.2022.100113</a>},
    number={100113}, journal={Journal of Advanced Joining Processes}, publisher={Elsevier
    BV}, author={Meschut, G. and Merklein, M. and Brosius, A. and Drummer, D. and
    Fratini, L. and Füssel, U. and Gude, M. and Homberg, W. and Martins, P.A.F. and
    Bobbert, M. and et al.}, year={2022} }'
  chicago: Meschut, G., M. Merklein, A. Brosius, D. Drummer, L. Fratini, U. Füssel,
    M. Gude, et al. “Review on Mechanical Joining by Plastic Deformation.” <i>Journal
    of Advanced Joining Processes</i> 5 (2022). <a href="https://doi.org/10.1016/j.jajp.2022.100113">https://doi.org/10.1016/j.jajp.2022.100113</a>.
  ieee: 'G. Meschut <i>et al.</i>, “Review on mechanical joining by plastic deformation,”
    <i>Journal of Advanced Joining Processes</i>, vol. 5, Art. no. 100113, 2022, doi:
    <a href="https://doi.org/10.1016/j.jajp.2022.100113">10.1016/j.jajp.2022.100113</a>.'
  mla: Meschut, G., et al. “Review on Mechanical Joining by Plastic Deformation.”
    <i>Journal of Advanced Joining Processes</i>, vol. 5, 100113, Elsevier BV, 2022,
    doi:<a href="https://doi.org/10.1016/j.jajp.2022.100113">10.1016/j.jajp.2022.100113</a>.
  short: G. Meschut, M. Merklein, A. Brosius, D. Drummer, L. Fratini, U. Füssel, M.
    Gude, W. Homberg, P.A.F. Martins, M. Bobbert, M. Lechner, R. Kupfer, B. Gröger,
    D. Han, J. Kalich, F. Kappe, T. Kleffel, D. Köhler, C.-M. Kuball, J. Popp, D.
    Römisch, J. Troschitz, C. Wischer, S. Wituschek, M. Wolf, Journal of Advanced
    Joining Processes 5 (2022).
date_created: 2024-02-06T15:03:43Z
date_updated: 2025-06-02T20:20:04Z
doi: 10.1016/j.jajp.2022.100113
intvolume: '         5'
keyword:
- Mechanical Engineering
- Mechanics of Materials
- Engineering (miscellaneous)
- Chemical Engineering (miscellaneous)
language:
- iso: eng
publication: Journal of Advanced Joining Processes
publication_identifier:
  issn:
  - 2666-3309
publication_status: published
publisher: Elsevier BV
status: public
title: Review on mechanical joining by plastic deformation
type: journal_article
user_id: '83408'
volume: 5
year: '2022'
...
---
_id: '63667'
author:
- first_name: G.
  full_name: Kullmer, G.
  last_name: Kullmer
- first_name: D.
  full_name: Weiss, D.
  last_name: Weiss
- first_name: S.
  full_name: Krome, S.
  last_name: Krome
- first_name: H.
  full_name: A. Richard, H.
  last_name: A. Richard
citation:
  ama: Kullmer G, Weiss D, Krome S, A. Richard H. <i>Entwicklung Einer Einspannvorrichtung
    Für Axialrissproben Zur Ermittlung von Bruchmechanischen Kennwerten Für Rohre</i>.
    LibreCat University; 2022. doi:<a href="https://doi.org/10.48447/BR-2022-011">10.48447/BR-2022-011</a>
  apa: Kullmer, G., Weiss, D., Krome, S., &#38; A. Richard, H. (2022). <i>Entwicklung
    einer Einspannvorrichtung für Axialrissproben zur Ermittlung von bruchmechanischen
    Kennwerten für Rohre</i>. LibreCat University. <a href="https://doi.org/10.48447/BR-2022-011">https://doi.org/10.48447/BR-2022-011</a>
  bibtex: '@book{Kullmer_Weiss_Krome_A. Richard_2022, title={Entwicklung einer Einspannvorrichtung
    für Axialrissproben zur Ermittlung von bruchmechanischen Kennwerten für Rohre},
    DOI={<a href="https://doi.org/10.48447/BR-2022-011">10.48447/BR-2022-011</a>},
    publisher={LibreCat University}, author={Kullmer, G. and Weiss, D. and Krome,
    S. and A. Richard, H.}, year={2022} }'
  chicago: Kullmer, G., D. Weiss, S. Krome, and H. A. Richard. <i>Entwicklung Einer
    Einspannvorrichtung Für Axialrissproben Zur Ermittlung von Bruchmechanischen Kennwerten
    Für Rohre</i>. LibreCat University, 2022. <a href="https://doi.org/10.48447/BR-2022-011">https://doi.org/10.48447/BR-2022-011</a>.
  ieee: G. Kullmer, D. Weiss, S. Krome, and H. A. Richard, <i>Entwicklung einer Einspannvorrichtung
    für Axialrissproben zur Ermittlung von bruchmechanischen Kennwerten für Rohre</i>.
    LibreCat University, 2022.
  mla: Kullmer, G., et al. <i>Entwicklung Einer Einspannvorrichtung Für Axialrissproben
    Zur Ermittlung von Bruchmechanischen Kennwerten Für Rohre</i>. LibreCat University,
    2022, doi:<a href="https://doi.org/10.48447/BR-2022-011">10.48447/BR-2022-011</a>.
  short: G. Kullmer, D. Weiss, S. Krome, H. A. Richard, Entwicklung Einer Einspannvorrichtung
    Für Axialrissproben Zur Ermittlung von Bruchmechanischen Kennwerten Für Rohre,
    LibreCat University, 2022.
date_created: 2026-01-20T13:23:23Z
date_updated: 2026-01-20T13:24:40Z
doi: 10.48447/BR-2022-011
publisher: LibreCat University
status: public
title: Entwicklung einer Einspannvorrichtung für Axialrissproben zur Ermittlung von
  bruchmechanischen Kennwerten für Rohre
type: research_data
user_id: '57245'
year: '2022'
...
---
_id: '64042'
abstract:
- lang: eng
  text: Abstract Polymer-derived silicon carbonitride ceramic (SiCN) is used as an
    electrode material to prepare cylindrical sodium/sodium ion cells for solid-state
    NMR investigations. During galvanostatic cycling structural changes of the environment
    of sodium/sodium ions are investigated by applying 23Na in-situ solid-state NMR.
    Changes of the signals assigned to sodium metal, intercalated sodium cation and
    sodium cation originating from the electrolyte are monitored as well as the occurrence
    of an additional signal in the region of metallic sodium. The intensity of this
    additional signal changes periodically with the cycling process indicating the
    reversibility of structures formed and deformed during the galvanostatic cycling.
    To identify interactions of sodium/sodium ions with the SiCN electrode materials,
    the cycled SiCN material is studied by 23Na ex-situ MAS NMR at high spinning rates
    of 20 and 50 kHz to obtain appropriate spectral resolution.
author:
- first_name: Edina
  full_name: Sic, Edina
  last_name: Sic
- first_name: Marco
  full_name: Melzi d’Eril, Marco
  last_name: Melzi d’Eril
- first_name: Konstantin
  full_name: Schutjajew, Konstantin
  last_name: Schutjajew
- first_name: Magdalena J.
  full_name: Graczyk-Zajac, Magdalena J.
  last_name: Graczyk-Zajac
- first_name: Hergen
  full_name: Breitzke, Hergen
  last_name: Breitzke
- first_name: Ralf
  full_name: Riedel, Ralf
  last_name: Riedel
- first_name: Martin
  full_name: Oschatz, Martin
  last_name: Oschatz
- first_name: Torsten
  full_name: Gutmann, Torsten
  id: '118165'
  last_name: Gutmann
- first_name: Gerd
  full_name: Buntkowsky, Gerd
  last_name: Buntkowsky
citation:
  ama: Sic E, Melzi d’Eril M, Schutjajew K, et al. SiCN Ceramics as Electrode Materials
    for Sodium/Sodium Ion Cells Insights from 23Na In-Situ Solid-State NMR. <i>Batteries
    and Supercaps</i>. 2022;5:e202200066. doi:<a href="https://doi.org/10.1002/batt.202200066">10.1002/batt.202200066</a>
  apa: Sic, E., Melzi d’Eril, M., Schutjajew, K., Graczyk-Zajac, M. J., Breitzke,
    H., Riedel, R., Oschatz, M., Gutmann, T., &#38; Buntkowsky, G. (2022). SiCN Ceramics
    as Electrode Materials for Sodium/Sodium Ion Cells Insights from 23Na In-Situ
    Solid-State NMR. <i>Batteries and Supercaps</i>, <i>5</i>, e202200066. <a href="https://doi.org/10.1002/batt.202200066">https://doi.org/10.1002/batt.202200066</a>
  bibtex: '@article{Sic_Melzi d’Eril_Schutjajew_Graczyk-Zajac_Breitzke_Riedel_Oschatz_Gutmann_Buntkowsky_2022,
    title={SiCN Ceramics as Electrode Materials for Sodium/Sodium Ion Cells Insights
    from 23Na In-Situ Solid-State NMR}, volume={5}, DOI={<a href="https://doi.org/10.1002/batt.202200066">10.1002/batt.202200066</a>},
    journal={Batteries and Supercaps}, author={Sic, Edina and Melzi d’Eril, Marco
    and Schutjajew, Konstantin and Graczyk-Zajac, Magdalena J. and Breitzke, Hergen
    and Riedel, Ralf and Oschatz, Martin and Gutmann, Torsten and Buntkowsky, Gerd},
    year={2022}, pages={e202200066} }'
  chicago: 'Sic, Edina, Marco Melzi d’Eril, Konstantin Schutjajew, Magdalena J. Graczyk-Zajac,
    Hergen Breitzke, Ralf Riedel, Martin Oschatz, Torsten Gutmann, and Gerd Buntkowsky.
    “SiCN Ceramics as Electrode Materials for Sodium/Sodium Ion Cells Insights from
    23Na In-Situ Solid-State NMR.” <i>Batteries and Supercaps</i> 5 (2022): e202200066.
    <a href="https://doi.org/10.1002/batt.202200066">https://doi.org/10.1002/batt.202200066</a>.'
  ieee: 'E. Sic <i>et al.</i>, “SiCN Ceramics as Electrode Materials for Sodium/Sodium
    Ion Cells Insights from 23Na In-Situ Solid-State NMR,” <i>Batteries and Supercaps</i>,
    vol. 5, p. e202200066, 2022, doi: <a href="https://doi.org/10.1002/batt.202200066">10.1002/batt.202200066</a>.'
  mla: Sic, Edina, et al. “SiCN Ceramics as Electrode Materials for Sodium/Sodium
    Ion Cells Insights from 23Na In-Situ Solid-State NMR.” <i>Batteries and Supercaps</i>,
    vol. 5, 2022, p. e202200066, doi:<a href="https://doi.org/10.1002/batt.202200066">10.1002/batt.202200066</a>.
  short: E. Sic, M. Melzi d’Eril, K. Schutjajew, M.J. Graczyk-Zajac, H. Breitzke,
    R. Riedel, M. Oschatz, T. Gutmann, G. Buntkowsky, Batteries and Supercaps 5 (2022)
    e202200066.
date_created: 2026-02-07T16:10:25Z
date_updated: 2026-02-17T16:13:15Z
doi: 10.1002/batt.202200066
extern: '1'
intvolume: '         5'
language:
- iso: eng
page: e202200066
publication: Batteries and Supercaps
status: public
title: SiCN Ceramics as Electrode Materials for Sodium/Sodium Ion Cells Insights from
  23Na In-Situ Solid-State NMR
type: journal_article
user_id: '100715'
volume: 5
year: '2022'
...
---
_id: '63983'
abstract:
- lang: eng
  text: Polyethylene glycol (PEG) is increasingly used as an alternative green chemical
    solvent. New experimental measurements on density, viscosity, and self-diffusion
    coefficient are presented for PEG200, PEG400, and several binary mixtures of tri-
    and hexaethylene glycol covering a temperature range from 298.15 to 358.15 K.
    Because PEGs are polydisperse, the exact compositions of PEG200 from six different
    vendors are analytically determined and found to be comparable. Thus, only two
    of the most differing PEG200 samples are further examined. The effects of water
    as the most common impurity on densities, viscosities, and self-diffusion coefficients
    are inspected as well as the results of the “dry” samples obtained by extrapolation
    to zero water content. The obtained results are carefully compared to the available
    literature data. The temperature dependence of these physical properties is investigated
    and found to be linear for density, while viscosity and self-diffusion coefficients
    follow the Arrhenius law. Attempts to calculate the properties of the binary mixtures
    and PEG200 samples from the mole fraction weighted average of the physical properties
    of the mixture components result in reasonable agreement. Agreement between calculated
    and measured molar volumes is within measurement uncertainty. Agreement of calculated
    and measured viscosities is mostly within a few percent but increases with decreasing
    temperature (largest viscosities) reaching values of up to 15%. Similarly, calculated
    and measured self-diffusion coefficients mostly agree within 20%, which is near
    the measurement uncertainty, but overestimates increase to 30% for the highest
    temperatures (largest self-diffusion coefficients).
author:
- first_name: Markus M.
  full_name: Hoffmann, Markus M.
  last_name: Hoffmann
- first_name: Joseph D.
  full_name: Kealy, Joseph D.
  last_name: Kealy
- first_name: Torsten
  full_name: Gutmann, Torsten
  id: '118165'
  last_name: Gutmann
- first_name: Gerd
  full_name: Buntkowsky, Gerd
  last_name: Buntkowsky
citation:
  ama: Hoffmann MM, Kealy JD, Gutmann T, Buntkowsky G. Densities, Viscosities, and
    Self-Diffusion Coefficients of Several Polyethylene Glycols. <i>Journal of Chemical
    and Engineering Data</i>. 2022;67(1):88–103. doi:<a href="https://doi.org/10.1021/acs.jced.1c00759">10.1021/acs.jced.1c00759</a>
  apa: Hoffmann, M. M., Kealy, J. D., Gutmann, T., &#38; Buntkowsky, G. (2022). Densities,
    Viscosities, and Self-Diffusion Coefficients of Several Polyethylene Glycols.
    <i>Journal of Chemical and Engineering Data</i>, <i>67</i>(1), 88–103. <a href="https://doi.org/10.1021/acs.jced.1c00759">https://doi.org/10.1021/acs.jced.1c00759</a>
  bibtex: '@article{Hoffmann_Kealy_Gutmann_Buntkowsky_2022, title={Densities, Viscosities,
    and Self-Diffusion Coefficients of Several Polyethylene Glycols}, volume={67},
    DOI={<a href="https://doi.org/10.1021/acs.jced.1c00759">10.1021/acs.jced.1c00759</a>},
    number={1}, journal={Journal of Chemical and Engineering Data}, publisher={American
    Chemical Society}, author={Hoffmann, Markus M. and Kealy, Joseph D. and Gutmann,
    Torsten and Buntkowsky, Gerd}, year={2022}, pages={88–103} }'
  chicago: 'Hoffmann, Markus M., Joseph D. Kealy, Torsten Gutmann, and Gerd Buntkowsky.
    “Densities, Viscosities, and Self-Diffusion Coefficients of Several Polyethylene
    Glycols.” <i>Journal of Chemical and Engineering Data</i> 67, no. 1 (2022): 88–103.
    <a href="https://doi.org/10.1021/acs.jced.1c00759">https://doi.org/10.1021/acs.jced.1c00759</a>.'
  ieee: 'M. M. Hoffmann, J. D. Kealy, T. Gutmann, and G. Buntkowsky, “Densities, Viscosities,
    and Self-Diffusion Coefficients of Several Polyethylene Glycols,” <i>Journal of
    Chemical and Engineering Data</i>, vol. 67, no. 1, pp. 88–103, 2022, doi: <a href="https://doi.org/10.1021/acs.jced.1c00759">10.1021/acs.jced.1c00759</a>.'
  mla: Hoffmann, Markus M., et al. “Densities, Viscosities, and Self-Diffusion Coefficients
    of Several Polyethylene Glycols.” <i>Journal of Chemical and Engineering Data</i>,
    vol. 67, no. 1, American Chemical Society, 2022, pp. 88–103, doi:<a href="https://doi.org/10.1021/acs.jced.1c00759">10.1021/acs.jced.1c00759</a>.
  short: M.M. Hoffmann, J.D. Kealy, T. Gutmann, G. Buntkowsky, Journal of Chemical
    and Engineering Data 67 (2022) 88–103.
date_created: 2026-02-07T15:44:52Z
date_updated: 2026-02-17T16:16:54Z
doi: 10.1021/acs.jced.1c00759
extern: '1'
intvolume: '        67'
issue: '1'
language:
- iso: eng
page: 88–103
publication: Journal of Chemical and Engineering Data
publisher: American Chemical Society
status: public
title: Densities, Viscosities, and Self-Diffusion Coefficients of Several Polyethylene
  Glycols
type: journal_article
user_id: '100715'
volume: 67
year: '2022'
...
---
_id: '63829'
abstract:
- lang: eng
  text: '<jats:p>The 3D shear deformation and failure behaviour of a glass fibre reinforced
    polypropylene in a shear strain rate range of γ˙=2.2×10−4 to 3.4 1s is investigated.
    An Iosipescu testing setup on a servo-hydraulic high speed testing unit is used
    to experimentally characterise the in-plane and out-of-plane behaviour utilising
    three specimen configurations (12-, 13- and 31-direction). The experimental procedure
    as well as the testing results are presented and discussed. The measured shear
    stress–shear strain relations indicate a highly nonlinear behaviour and a distinct
    rate dependency. Two methods are investigated to derive according material characteristics:
    a classical engineering approach based on moduli and strengths and a data driven
    approach based on the curve progression. In all cases a Johnson–Cook based formulation
    is used to describe rate dependency. The analysis methodologies as well as the
    derived model parameters are described and discussed in detail. It is shown that
    a phenomenologically enhanced regression can be used to obtain material characteristics
    for a generalising constitutive model based on the data driven approach.</jats:p>'
article_number: '318'
article_type: original
author:
- first_name: Johannes
  full_name: Gerritzen, Johannes
  id: '105344'
  last_name: Gerritzen
  orcid: 0000-0002-0169-8602
- first_name: Andreas
  full_name: Hornig, Andreas
  last_name: Hornig
- first_name: Benjamin
  full_name: Gröger, Benjamin
  last_name: Gröger
- first_name: Maik
  full_name: Gude, Maik
  last_name: Gude
citation:
  ama: 'Gerritzen J, Hornig A, Gröger B, Gude M. A Data Driven Modelling Approach
    for the Strain Rate Dependent 3D Shear Deformation and Failure of Thermoplastic
    Fibre Reinforced Composites: Experimental Characterisation and Deriving Modelling
    Parameters. <i>Journal of Composites Science</i>. 2022;6(10). doi:<a href="https://doi.org/10.3390/jcs6100318">10.3390/jcs6100318</a>'
  apa: 'Gerritzen, J., Hornig, A., Gröger, B., &#38; Gude, M. (2022). A Data Driven
    Modelling Approach for the Strain Rate Dependent 3D Shear Deformation and Failure
    of Thermoplastic Fibre Reinforced Composites: Experimental Characterisation and
    Deriving Modelling Parameters. <i>Journal of Composites Science</i>, <i>6</i>(10),
    Article 318. <a href="https://doi.org/10.3390/jcs6100318">https://doi.org/10.3390/jcs6100318</a>'
  bibtex: '@article{Gerritzen_Hornig_Gröger_Gude_2022, title={A Data Driven Modelling
    Approach for the Strain Rate Dependent 3D Shear Deformation and Failure of Thermoplastic
    Fibre Reinforced Composites: Experimental Characterisation and Deriving Modelling
    Parameters}, volume={6}, DOI={<a href="https://doi.org/10.3390/jcs6100318">10.3390/jcs6100318</a>},
    number={10318}, journal={Journal of Composites Science}, publisher={MDPI AG},
    author={Gerritzen, Johannes and Hornig, Andreas and Gröger, Benjamin and Gude,
    Maik}, year={2022} }'
  chicago: 'Gerritzen, Johannes, Andreas Hornig, Benjamin Gröger, and Maik Gude. “A
    Data Driven Modelling Approach for the Strain Rate Dependent 3D Shear Deformation
    and Failure of Thermoplastic Fibre Reinforced Composites: Experimental Characterisation
    and Deriving Modelling Parameters.” <i>Journal of Composites Science</i> 6, no.
    10 (2022). <a href="https://doi.org/10.3390/jcs6100318">https://doi.org/10.3390/jcs6100318</a>.'
  ieee: 'J. Gerritzen, A. Hornig, B. Gröger, and M. Gude, “A Data Driven Modelling
    Approach for the Strain Rate Dependent 3D Shear Deformation and Failure of Thermoplastic
    Fibre Reinforced Composites: Experimental Characterisation and Deriving Modelling
    Parameters,” <i>Journal of Composites Science</i>, vol. 6, no. 10, Art. no. 318,
    2022, doi: <a href="https://doi.org/10.3390/jcs6100318">10.3390/jcs6100318</a>.'
  mla: 'Gerritzen, Johannes, et al. “A Data Driven Modelling Approach for the Strain
    Rate Dependent 3D Shear Deformation and Failure of Thermoplastic Fibre Reinforced
    Composites: Experimental Characterisation and Deriving Modelling Parameters.”
    <i>Journal of Composites Science</i>, vol. 6, no. 10, 318, MDPI AG, 2022, doi:<a
    href="https://doi.org/10.3390/jcs6100318">10.3390/jcs6100318</a>.'
  short: J. Gerritzen, A. Hornig, B. Gröger, M. Gude, Journal of Composites Science
    6 (2022).
date_created: 2026-02-02T08:41:00Z
date_updated: 2026-02-27T06:47:18Z
doi: 10.3390/jcs6100318
intvolume: '         6'
issue: '10'
language:
- iso: eng
project:
- _id: '137'
  name: TRR 285 - Subproject A03
- _id: '131'
  name: TRR 285 - Project Area A
- _id: '130'
  name: 'TRR 285:  Methodenentwicklung zur mechanischen Fügbarkeit in wandlungsfähigen
    Prozessketten'
publication: Journal of Composites Science
publication_identifier:
  issn:
  - 2504-477X
publication_status: published
publisher: MDPI AG
status: public
title: 'A Data Driven Modelling Approach for the Strain Rate Dependent 3D Shear Deformation
  and Failure of Thermoplastic Fibre Reinforced Composites: Experimental Characterisation
  and Deriving Modelling Parameters'
type: journal_article
user_id: '105344'
volume: 6
year: '2022'
...
---
_id: '61495'
citation:
  ama: 'Katrin K, Katrin B, eds. <i>Christoph Schwöbel: Gott in Beziehung. Studien
    Zur Dogmatik</i>. 2nd ed. Mohr Siebeck; 2022. doi:<a href="https://doi.org/10.1628/978-3-16-156141-2">10.1628/978-3-16-156141-2</a>'
  apa: 'Katrin, K., &#38; Katrin, B. (Eds.). (2022). <i>Christoph Schwöbel: Gott in
    Beziehung. Studien zur Dogmatik</i> (2nd ed.). Mohr Siebeck. <a href="https://doi.org/10.1628/978-3-16-156141-2">https://doi.org/10.1628/978-3-16-156141-2</a>'
  bibtex: '@book{Katrin_Katrin_2022, place={Tübingen}, edition={2}, title={Christoph
    Schwöbel: Gott in Beziehung. Studien zur Dogmatik}, DOI={<a href="https://doi.org/10.1628/978-3-16-156141-2">10.1628/978-3-16-156141-2</a>},
    publisher={Mohr Siebeck}, year={2022} }'
  chicago: 'Katrin, König, and Bosse Katrin, eds. <i>Christoph Schwöbel: Gott in Beziehung.
    Studien Zur Dogmatik</i>. 2nd ed. Tübingen: Mohr Siebeck, 2022. <a href="https://doi.org/10.1628/978-3-16-156141-2">https://doi.org/10.1628/978-3-16-156141-2</a>.'
  ieee: 'K. Katrin and B. Katrin, Eds., <i>Christoph Schwöbel: Gott in Beziehung.
    Studien zur Dogmatik</i>, 2nd ed. Tübingen: Mohr Siebeck, 2022.'
  mla: 'Katrin, König, and Bosse Katrin, editors. <i>Christoph Schwöbel: Gott in Beziehung.
    Studien Zur Dogmatik</i>. 2nd ed., Mohr Siebeck, 2022, doi:<a href="https://doi.org/10.1628/978-3-16-156141-2">10.1628/978-3-16-156141-2</a>.'
  short: 'K. Katrin, B. Katrin, eds., Christoph Schwöbel: Gott in Beziehung. Studien
    Zur Dogmatik, 2nd ed., Mohr Siebeck, Tübingen, 2022.'
date_created: 2025-10-02T07:19:37Z
date_updated: 2025-10-02T09:52:34Z
department:
- _id: '20'
doi: 10.1628/978-3-16-156141-2
edition: '2'
editor:
- first_name: König
  full_name: Katrin, König
  last_name: Katrin
- first_name: Bosse
  full_name: Katrin, Bosse
  last_name: Katrin
extern: '1'
language:
- iso: eng
place: Tübingen
publisher: Mohr Siebeck
status: public
title: 'Christoph Schwöbel: Gott in Beziehung. Studien zur Dogmatik'
type: book_editor
user_id: '118819'
year: '2022'
...
---
_id: '62801'
abstract:
- lang: eng
  text: The three-dimensional (3D) distribution of individual atoms on the surface
    of catalyst nanoparticles plays a vital role in their activity and stability.
    Optimising the performance of electrocatalysts requires atomic-scale information,
    but it is difficult to obtain. Here, we use atom probe tomography to elucidate
    the 3D structure of 10 nm sized Co2FeO4 and CoFe2O4 nanoparticles during oxygen
    evolution reaction (OER). We reveal nanoscale spinodal decomposition in pristine
    Co2FeO4. The interfaces of Co-rich and Fe-rich nanodomains of Co2FeO4 become trapping
    sites for hydroxyl groups, contributing to a higher OER activity compared to that
    of CoFe2O4. However, the activity of Co2FeO4 drops considerably due to concurrent
    irreversible transformation towards CoIVO2 and pronounced Fe dissolution. In contrast,
    there is negligible elemental redistribution for CoFe2O4 after OER, except for
    surface structural transformation towards (FeIII, CoIII)2O3. Overall, our study
    provides a unique 3D compositional distribution of mixed Co-Fe spinel oxides,
    which gives atomic-scale insights into active sites and the deactivation of electrocatalysts
    during OER.
article_number: '179'
article_type: original
author:
- first_name: Weikai
  full_name: Xiang, Weikai
  last_name: Xiang
- first_name: Nating
  full_name: Yang, Nating
  last_name: Yang
- first_name: Xiaopeng
  full_name: Li, Xiaopeng
  last_name: Li
- first_name: Julia
  full_name: Linnemann, Julia
  id: '116779'
  last_name: Linnemann
  orcid: 0000-0001-6883-5424
- first_name: Ulrich
  full_name: Hagemann, Ulrich
  last_name: Hagemann
- first_name: Olaf
  full_name: Ruediger, Olaf
  last_name: Ruediger
- first_name: Markus
  full_name: Heidelmann, Markus
  last_name: Heidelmann
- first_name: Tobias
  full_name: Falk, Tobias
  last_name: Falk
- first_name: Matteo
  full_name: Aramini, Matteo
  last_name: Aramini
- first_name: Serena
  full_name: DeBeer, Serena
  last_name: DeBeer
- first_name: Martin
  full_name: Muhler, Martin
  last_name: Muhler
- first_name: Kristina
  full_name: Tschulik, Kristina
  last_name: Tschulik
- first_name: Tong
  full_name: Li, Tong
  last_name: Li
citation:
  ama: Xiang W, Yang N, Li X, et al. 3D atomic-scale imaging of mixed Co-Fe spinel
    oxide nanoparticles during oxygen evolution reaction. <i>Nature Communications</i>.
    2022;13(1). doi:<a href="https://doi.org/10.1038/s41467-021-27788-2">10.1038/s41467-021-27788-2</a>
  apa: Xiang, W., Yang, N., Li, X., Linnemann, J., Hagemann, U., Ruediger, O., Heidelmann,
    M., Falk, T., Aramini, M., DeBeer, S., Muhler, M., Tschulik, K., &#38; Li, T.
    (2022). 3D atomic-scale imaging of mixed Co-Fe spinel oxide nanoparticles during
    oxygen evolution reaction. <i>Nature Communications</i>, <i>13</i>(1), Article
    179. <a href="https://doi.org/10.1038/s41467-021-27788-2">https://doi.org/10.1038/s41467-021-27788-2</a>
  bibtex: '@article{Xiang_Yang_Li_Linnemann_Hagemann_Ruediger_Heidelmann_Falk_Aramini_DeBeer_et
    al._2022, title={3D atomic-scale imaging of mixed Co-Fe spinel oxide nanoparticles
    during oxygen evolution reaction}, volume={13}, DOI={<a href="https://doi.org/10.1038/s41467-021-27788-2">10.1038/s41467-021-27788-2</a>},
    number={1179}, journal={Nature Communications}, publisher={Springer Science and
    Business Media LLC}, author={Xiang, Weikai and Yang, Nating and Li, Xiaopeng and
    Linnemann, Julia and Hagemann, Ulrich and Ruediger, Olaf and Heidelmann, Markus
    and Falk, Tobias and Aramini, Matteo and DeBeer, Serena and et al.}, year={2022}
    }'
  chicago: Xiang, Weikai, Nating Yang, Xiaopeng Li, Julia Linnemann, Ulrich Hagemann,
    Olaf Ruediger, Markus Heidelmann, et al. “3D Atomic-Scale Imaging of Mixed Co-Fe
    Spinel Oxide Nanoparticles during Oxygen Evolution Reaction.” <i>Nature Communications</i>
    13, no. 1 (2022). <a href="https://doi.org/10.1038/s41467-021-27788-2">https://doi.org/10.1038/s41467-021-27788-2</a>.
  ieee: 'W. Xiang <i>et al.</i>, “3D atomic-scale imaging of mixed Co-Fe spinel oxide
    nanoparticles during oxygen evolution reaction,” <i>Nature Communications</i>,
    vol. 13, no. 1, Art. no. 179, 2022, doi: <a href="https://doi.org/10.1038/s41467-021-27788-2">10.1038/s41467-021-27788-2</a>.'
  mla: Xiang, Weikai, et al. “3D Atomic-Scale Imaging of Mixed Co-Fe Spinel Oxide
    Nanoparticles during Oxygen Evolution Reaction.” <i>Nature Communications</i>,
    vol. 13, no. 1, 179, Springer Science and Business Media LLC, 2022, doi:<a href="https://doi.org/10.1038/s41467-021-27788-2">10.1038/s41467-021-27788-2</a>.
  short: W. Xiang, N. Yang, X. Li, J. Linnemann, U. Hagemann, O. Ruediger, M. Heidelmann,
    T. Falk, M. Aramini, S. DeBeer, M. Muhler, K. Tschulik, T. Li, Nature Communications
    13 (2022).
date_created: 2025-12-03T15:22:16Z
date_updated: 2025-12-03T16:30:12Z
department:
- _id: '985'
doi: 10.1038/s41467-021-27788-2
extern: '1'
intvolume: '        13'
issue: '1'
keyword:
- electrocatalysis
- oxygen evolution reaction
- cobalt spinel
- electrochemical impedance spectroscopy
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://www.nature.com/articles/s41467-021-27788-2
oa: '1'
publication: Nature Communications
publication_identifier:
  issn:
  - 2041-1723
publication_status: published
publisher: Springer Science and Business Media LLC
quality_controlled: '1'
status: public
title: 3D atomic-scale imaging of mixed Co-Fe spinel oxide nanoparticles during oxygen
  evolution reaction
type: journal_article
user_id: '116779'
volume: 13
year: '2022'
...
---
_id: '41800'
author:
- first_name: M
  full_name: Sartison, M
  last_name: Sartison
- first_name: O
  full_name: ' Camacho Ibarra, O'
  last_name: ' Camacho Ibarra'
- first_name: Klaus D.
  full_name: Jöns, Klaus D.
  id: '85353'
  last_name: Jöns
- first_name: I
  full_name: Caltzidis, I
  last_name: Caltzidis
- first_name: Dirk
  full_name: Reuter, Dirk
  id: '37763'
  last_name: Reuter
citation:
  ama: Sartison M,  Camacho Ibarra O, Jöns KD, Caltzidis I, Reuter D. Scalable integration
    of quantum emitters into photonic integrated circuits. 2022;2. doi:<a href="https://doi.org/10.1088/2633-4356/ac6f3e">https://doi.org/10.1088/2633-4356/ac6f3e</a>
  apa: Sartison, M.,  Camacho Ibarra, O., Jöns, K. D., Caltzidis, I., &#38; Reuter,
    D. (2022). <i>Scalable integration of quantum emitters into photonic integrated
    circuits</i> (Vol. 2). <a href="https://doi.org/10.1088/2633-4356/ac6f3e">https://doi.org/10.1088/2633-4356/ac6f3e</a>
  bibtex: '@article{Sartison_ Camacho Ibarra_Jöns_Caltzidis_Reuter_2022, series={Materials
    for Quantum Technology}, title={Scalable integration of quantum emitters into
    photonic integrated circuits}, volume={2}, DOI={<a href="https://doi.org/10.1088/2633-4356/ac6f3e">https://doi.org/10.1088/2633-4356/ac6f3e</a>},
    author={Sartison, M and  Camacho Ibarra, O and Jöns, Klaus D. and Caltzidis, I
    and Reuter, Dirk}, year={2022}, collection={Materials for Quantum Technology}
    }'
  chicago: Sartison, M, O  Camacho Ibarra, Klaus D. Jöns, I Caltzidis, and Dirk Reuter.
    “Scalable integration of quantum emitters into photonic integrated circuits.”
    Materials for Quantum Technology, 2022. <a href="https://doi.org/10.1088/2633-4356/ac6f3e">https://doi.org/10.1088/2633-4356/ac6f3e</a>.
  ieee: 'M. Sartison, O.  Camacho Ibarra, K. D. Jöns, I. Caltzidis, and D. Reuter,
    “Scalable integration of quantum emitters into photonic integrated circuits,”
    vol. 2. 2022, doi: <a href="https://doi.org/10.1088/2633-4356/ac6f3e">https://doi.org/10.1088/2633-4356/ac6f3e</a>.'
  mla: Sartison, M., et al. <i>Scalable integration of quantum emitters into photonic
    integrated circuits</i>. 2022, doi:<a href="https://doi.org/10.1088/2633-4356/ac6f3e">https://doi.org/10.1088/2633-4356/ac6f3e</a>.
  short: M. Sartison, O.  Camacho Ibarra, K.D. Jöns, I. Caltzidis, D. Reuter, 2 (2022).
date_created: 2023-02-06T02:30:08Z
date_updated: 2025-12-11T13:09:55Z
department:
- _id: '623'
- _id: '15'
- _id: '429'
- _id: '642'
doi: https://doi.org/10.1088/2633-4356/ac6f3e
intvolume: '         2'
language:
- iso: ger
publication_status: published
series_title: Materials for Quantum Technology
status: public
title: Scalable integration of quantum emitters into photonic integrated circuits
type: conference
user_id: '48188'
volume: 2
year: '2022'
...
---
_id: '55135'
alternative_title:
- 'Themenheft zur 20. Jahrestagung der Gesellschaft für Aphasieforschung und –behandlung.
  Neurologie & Rehabilitation. '
citation:
  ama: Jonas K, Quinting J, Gerhards L, et al., eds. <i>Kommunikation – Sprache, Emotion,
    Kognition</i>.; 2022.
  apa: Jonas, K., Quinting, J., Gerhards, L., Hüsgen, A., Rubi-Fessen, I., Stenneken,
    P., &#38; Rosenkranz, A. (Eds.). (2022). <i>Kommunikation – Sprache, Emotion,
    Kognition</i>.
  bibtex: '@book{Jonas_Quinting_Gerhards_Hüsgen_Rubi-Fessen_Stenneken_Rosenkranz_2022,
    title={Kommunikation – Sprache, Emotion, Kognition}, year={2022} }'
  chicago: Jonas, Kristina, Jana Quinting, Lisa Gerhards, Anne Hüsgen, Ilona Rubi-Fessen,
    Prisca Stenneken, and Anna Rosenkranz, eds. <i>Kommunikation – Sprache, Emotion,
    Kognition</i>, 2022.
  ieee: K. Jonas <i>et al.</i>, Eds., <i>Kommunikation – Sprache, Emotion, Kognition</i>.
    2022.
  mla: Jonas, Kristina, et al., editors. <i>Kommunikation – Sprache, Emotion, Kognition</i>.
    2022.
  short: K. Jonas, J. Quinting, L. Gerhards, A. Hüsgen, I. Rubi-Fessen, P. Stenneken,
    A. Rosenkranz, eds., Kommunikation – Sprache, Emotion, Kognition, 2022.
date_created: 2024-07-08T11:24:56Z
date_updated: 2025-12-17T07:34:41Z
department:
- _id: '5'
- _id: '890'
editor:
- first_name: Kristina
  full_name: Jonas, Kristina
  id: '94540'
  last_name: Jonas
  orcid: 0000-0002-1067-9139
- first_name: Jana
  full_name: Quinting, Jana
  last_name: Quinting
- first_name: Lisa
  full_name: Gerhards, Lisa
  last_name: Gerhards
- first_name: Anne
  full_name: Hüsgen, Anne
  last_name: Hüsgen
- first_name: Ilona
  full_name: Rubi-Fessen, Ilona
  last_name: Rubi-Fessen
- first_name: Prisca
  full_name: Stenneken, Prisca
  last_name: Stenneken
- first_name: Anna
  full_name: Rosenkranz, Anna
  last_name: Rosenkranz
extern: '1'
language:
- iso: eng
status: public
title: Kommunikation – Sprache, Emotion, Kognition
type: conference_editor
user_id: '94540'
year: '2022'
...
---
_id: '63310'
abstract:
- lang: eng
  text: <jats:title>Abstract</jats:title><jats:p>The chemotaxis–Stokes system<jats:disp-formula
    id="j_ans-2022-0004_eq_001"><jats:alternatives><jats:graphic xmlns:xlink="http://www.w3.org/1999/xlink"
    xlink:href="graphic/j_ans-2022-0004_eq_001.png"/><m:math xmlns:m="http://www.w3.org/1998/Math/MathML"
    display="block"><m:mfenced open="{" close=""><m:mrow><m:mtable displaystyle="true"><m:mtr><m:mtd
    columnalign="left"><m:msub><m:mrow><m:mi>n</m:mi></m:mrow><m:mrow><m:mi>t</m:mi></m:mrow></m:msub><m:mo>+</m:mo><m:mi>u</m:mi><m:mo>⋅</m:mo><m:mrow><m:mo>∇</m:mo></m:mrow><m:mi>n</m:mi><m:mo>=</m:mo><m:mrow><m:mo>∇</m:mo></m:mrow><m:mo>⋅</m:mo><m:mrow><m:mo
    stretchy="false">(</m:mo><m:mrow/></m:mrow><m:mi>D</m:mi><m:mrow><m:mo>(</m:mo><m:mrow><m:mi>n</m:mi></m:mrow><m:mo>)</m:mo></m:mrow><m:mrow><m:mo>∇</m:mo></m:mrow><m:mi>n</m:mi><m:mrow><m:mo
    stretchy="false">)</m:mo><m:mrow/></m:mrow><m:mo>−</m:mo><m:mrow><m:mo>∇</m:mo></m:mrow><m:mo>⋅</m:mo><m:mrow><m:mo
    stretchy="false">(</m:mo><m:mrow/></m:mrow><m:mi>n</m:mi><m:mi>S</m:mi><m:mrow><m:mo>(</m:mo><m:mrow><m:mi>x</m:mi><m:mo>,</m:mo><m:mi>n</m:mi><m:mo>,</m:mo><m:mi>c</m:mi></m:mrow><m:mo>)</m:mo></m:mrow><m:mo>⋅</m:mo><m:mrow><m:mo>∇</m:mo></m:mrow><m:mi>c</m:mi><m:mrow><m:mo
    stretchy="false">)</m:mo><m:mrow/></m:mrow><m:mo>,</m:mo></m:mtd></m:mtr><m:mtr><m:mtd
    columnalign="left"><m:msub><m:mrow><m:mi>c</m:mi></m:mrow><m:mrow><m:mi>t</m:mi></m:mrow></m:msub><m:mo>+</m:mo><m:mi>u</m:mi><m:mo>⋅</m:mo><m:mrow><m:mo>∇</m:mo></m:mrow><m:mi>c</m:mi><m:mo>=</m:mo><m:mi
    mathvariant="normal">Δ</m:mi><m:mi>c</m:mi><m:mo>−</m:mo><m:mi>n</m:mi><m:mi>c</m:mi><m:mo>,</m:mo></m:mtd></m:mtr><m:mtr><m:mtd
    columnalign="left"><m:msub><m:mrow><m:mi>u</m:mi></m:mrow><m:mrow><m:mi>t</m:mi></m:mrow></m:msub><m:mo>=</m:mo><m:mi
    mathvariant="normal">Δ</m:mi><m:mi>u</m:mi><m:mo>+</m:mo><m:mrow><m:mo>∇</m:mo></m:mrow><m:mi>P</m:mi><m:mo>+</m:mo><m:mi>n</m:mi><m:mrow><m:mo>∇</m:mo></m:mrow><m:mi
    mathvariant="normal">Φ</m:mi><m:mo>,</m:mo><m:mspace width="1.0em"/><m:mrow><m:mo>∇</m:mo></m:mrow><m:mo>⋅</m:mo><m:mi>u</m:mi><m:mo>=</m:mo><m:mn>0</m:mn><m:mo>,</m:mo></m:mtd></m:mtr></m:mtable></m:mrow></m:mfenced></m:math><jats:tex-math>\left\{\begin{array}{l}{n}_{t}+u\cdot
    \nabla n=\nabla \cdot (D\left(n)\nabla n)-\nabla \cdot (nS\left(x,n,c)\cdot \nabla
    c),\\ {c}_{t}+u\cdot \nabla c=\Delta c-nc,\\ {u}_{t}=\Delta u+\nabla P+n\nabla
    \Phi ,\hspace{1.0em}\nabla \cdot u=0,\end{array}\right.</jats:tex-math></jats:alternatives></jats:disp-formula>is
    considered in a smoothly bounded convex domain<jats:inline-formula><jats:alternatives><jats:inline-graphic
    xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ans-2022-0004_eq_002.png"/><m:math
    xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mi mathvariant="normal">Ω</m:mi><m:mo>⊂</m:mo><m:msup><m:mrow><m:mi
    mathvariant="double-struck">R</m:mi></m:mrow><m:mrow><m:mn>3</m:mn></m:mrow></m:msup></m:math><jats:tex-math>\Omega
    \subset {{\mathbb{R}}}^{3}</jats:tex-math></jats:alternatives></jats:inline-formula>,
    with given suitably regular functions<jats:inline-formula><jats:alternatives><jats:inline-graphic
    xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ans-2022-0004_eq_003.png"/><m:math
    xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mi>D</m:mi><m:mo>:</m:mo><m:mrow><m:mo
    stretchy="false">[</m:mo><m:mrow><m:mn>0</m:mn><m:mo>,</m:mo><m:mi>∞</m:mi></m:mrow><m:mo
    stretchy="false">)</m:mo></m:mrow><m:mo>→</m:mo><m:mrow><m:mo stretchy="false">[</m:mo><m:mrow><m:mn>0</m:mn><m:mo>,</m:mo><m:mi>∞</m:mi></m:mrow><m:mo
    stretchy="false">)</m:mo></m:mrow></m:math><jats:tex-math>D:{[}0,\infty )\to {[}0,\infty
    )</jats:tex-math></jats:alternatives></jats:inline-formula>,<jats:inline-formula><jats:alternatives><jats:inline-graphic
    xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ans-2022-0004_eq_004.png"/><m:math
    xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mi>S</m:mi><m:mo>:</m:mo><m:mover
    accent="true"><m:mrow><m:mi mathvariant="normal">Ω</m:mi></m:mrow><m:mrow><m:mo
    stretchy="true">¯</m:mo></m:mrow></m:mover><m:mo>×</m:mo><m:mrow><m:mo stretchy="false">[</m:mo><m:mrow><m:mn>0</m:mn><m:mo>,</m:mo><m:mi>∞</m:mi></m:mrow><m:mo
    stretchy="false">)</m:mo></m:mrow><m:mo>×</m:mo><m:mrow><m:mo>(</m:mo><m:mrow><m:mn>0</m:mn><m:mo>,</m:mo><m:mi>∞</m:mi></m:mrow><m:mo>)</m:mo></m:mrow><m:mo>→</m:mo><m:msup><m:mrow><m:mi
    mathvariant="double-struck">R</m:mi></m:mrow><m:mrow><m:mn>3</m:mn><m:mo>×</m:mo><m:mn>3</m:mn></m:mrow></m:msup></m:math><jats:tex-math>S:\overline{\Omega
    }\times {[}0,\infty )\times \left(0,\infty )\to {{\mathbb{R}}}^{3\times 3}</jats:tex-math></jats:alternatives></jats:inline-formula>and<jats:inline-formula><jats:alternatives><jats:inline-graphic
    xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ans-2022-0004_eq_005.png"/><m:math
    xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mi mathvariant="normal">Φ</m:mi><m:mo>:</m:mo><m:mover
    accent="true"><m:mrow><m:mi mathvariant="normal">Ω</m:mi></m:mrow><m:mrow><m:mo
    stretchy="true">¯</m:mo></m:mrow></m:mover><m:mo>→</m:mo><m:mi mathvariant="double-struck">R</m:mi></m:math><jats:tex-math>\Phi
    :\overline{\Omega }\to {\mathbb{R}}</jats:tex-math></jats:alternatives></jats:inline-formula>such
    that<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink"
    xlink:href="graphic/j_ans-2022-0004_eq_006.png"/><m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mi>D</m:mi><m:mo>&gt;</m:mo><m:mn>0</m:mn></m:math><jats:tex-math>D\gt
    0</jats:tex-math></jats:alternatives></jats:inline-formula>on<jats:inline-formula><jats:alternatives><jats:inline-graphic
    xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ans-2022-0004_eq_007.png"/><m:math
    xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:mo>(</m:mo><m:mrow><m:mn>0</m:mn><m:mo>,</m:mo><m:mi>∞</m:mi></m:mrow><m:mo>)</m:mo></m:mrow></m:math><jats:tex-math>\left(0,\infty
    )</jats:tex-math></jats:alternatives></jats:inline-formula>. It is shown that
    if with some nondecreasing<jats:inline-formula><jats:alternatives><jats:inline-graphic
    xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ans-2022-0004_eq_008.png"/><m:math
    xmlns:m="http://www.w3.org/1998/Math/MathML"><m:msub><m:mrow><m:mi>S</m:mi></m:mrow><m:mrow><m:mn>0</m:mn></m:mrow></m:msub><m:mo>:</m:mo><m:mrow><m:mo>(</m:mo><m:mrow><m:mn>0</m:mn><m:mo>,</m:mo><m:mi>∞</m:mi></m:mrow><m:mo>)</m:mo></m:mrow><m:mo>→</m:mo><m:mrow><m:mo>(</m:mo><m:mrow><m:mn>0</m:mn><m:mo>,</m:mo><m:mi>∞</m:mi></m:mrow><m:mo>)</m:mo></m:mrow></m:math><jats:tex-math>{S}_{0}:\left(0,\infty
    )\to \left(0,\infty )</jats:tex-math></jats:alternatives></jats:inline-formula>we
    have<jats:disp-formula id="j_ans-2022-0004_eq_002"><jats:alternatives><jats:graphic
    xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ans-2022-0004_eq_009.png"/><m:math
    xmlns:m="http://www.w3.org/1998/Math/MathML" display="block"><m:mo>∣</m:mo><m:mi>S</m:mi><m:mrow><m:mo>(</m:mo><m:mrow><m:mi>x</m:mi><m:mo>,</m:mo><m:mi>n</m:mi><m:mo>,</m:mo><m:mi>c</m:mi></m:mrow><m:mo>)</m:mo></m:mrow><m:mo>∣</m:mo><m:mo>≤</m:mo><m:mfrac><m:mrow><m:msub><m:mrow><m:mi>S</m:mi></m:mrow><m:mrow><m:mn>0</m:mn></m:mrow></m:msub><m:mrow><m:mo>(</m:mo><m:mrow><m:mi>c</m:mi></m:mrow><m:mo>)</m:mo></m:mrow></m:mrow><m:mrow><m:msup><m:mrow><m:mi>c</m:mi></m:mrow><m:mrow><m:mstyle
    displaystyle="false"><m:mfrac><m:mrow><m:mn>1</m:mn></m:mrow><m:mrow><m:mn>2</m:mn></m:mrow></m:mfrac></m:mstyle></m:mrow></m:msup></m:mrow></m:mfrac><m:mspace
    width="1.0em"/><m:mspace width="0.1em"/><m:mtext>for all</m:mtext><m:mspace width="0.1em"/><m:mspace
    width="0.33em"/><m:mrow><m:mo>(</m:mo><m:mrow><m:mi>x</m:mi><m:mo>,</m:mo><m:mi>n</m:mi><m:mo>,</m:mo><m:mi>c</m:mi></m:mrow><m:mo>)</m:mo></m:mrow><m:mo>∈</m:mo><m:mover
    accent="true"><m:mrow><m:mi mathvariant="normal">Ω</m:mi></m:mrow><m:mrow><m:mo
    stretchy="true">¯</m:mo></m:mrow></m:mover><m:mo>×</m:mo><m:mrow><m:mo stretchy="false">[</m:mo><m:mrow><m:mn>0</m:mn><m:mo>,</m:mo><m:mi>∞</m:mi></m:mrow><m:mo
    stretchy="false">)</m:mo></m:mrow><m:mo>×</m:mo><m:mrow><m:mo>(</m:mo><m:mrow><m:mn>0</m:mn><m:mo>,</m:mo><m:mi>∞</m:mi></m:mrow><m:mo>)</m:mo></m:mrow><m:mo>,</m:mo></m:math><jats:tex-math>|
    S\left(x,n,c)| \le \frac{{S}_{0}\left(c)}{{c}^{\tfrac{1}{2}}}\hspace{1.0em}\hspace{0.1em}\text{for
    all}\hspace{0.1em}\hspace{0.33em}\left(x,n,c)\in \overline{\Omega }\times {[}0,\infty
    )\times \left(0,\infty ),</jats:tex-math></jats:alternatives></jats:disp-formula>then
    for all<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink"
    xlink:href="graphic/j_ans-2022-0004_eq_010.png"/><m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mi>M</m:mi><m:mo>&gt;</m:mo><m:mn>0</m:mn></m:math><jats:tex-math>M\gt
    0</jats:tex-math></jats:alternatives></jats:inline-formula>there exists<jats:inline-formula><jats:alternatives><jats:inline-graphic
    xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ans-2022-0004_eq_011.png"/><m:math
    xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mi>L</m:mi><m:mrow><m:mo>(</m:mo><m:mrow><m:mi>M</m:mi></m:mrow><m:mo>)</m:mo></m:mrow><m:mo>&gt;</m:mo><m:mn>0</m:mn></m:math><jats:tex-math>L\left(M)\gt
    0</jats:tex-math></jats:alternatives></jats:inline-formula>such that whenever<jats:disp-formula
    id="j_ans-2022-0004_eq_003"><jats:alternatives><jats:graphic xmlns:xlink="http://www.w3.org/1999/xlink"
    xlink:href="graphic/j_ans-2022-0004_eq_012.png"/><m:math xmlns:m="http://www.w3.org/1998/Math/MathML"
    display="block"><m:munder><m:mrow><m:mi>liminf</m:mi></m:mrow><m:mrow><m:mi>n</m:mi><m:mo>→</m:mo><m:mi>∞</m:mi></m:mrow></m:munder><m:mi>D</m:mi><m:mrow><m:mo>(</m:mo><m:mrow><m:mi>n</m:mi></m:mrow><m:mo>)</m:mo></m:mrow><m:mo>&gt;</m:mo><m:mi>L</m:mi><m:mrow><m:mo>(</m:mo><m:mrow><m:mi>M</m:mi></m:mrow><m:mo>)</m:mo></m:mrow><m:mspace
    width="1.0em"/><m:mspace width="0.1em"/><m:mtext>and</m:mtext><m:mspace width="0.1em"/><m:mspace
    width="1.0em"/><m:munder><m:mrow><m:mi>liminf</m:mi></m:mrow><m:mrow><m:mi>n</m:mi><m:mo>↘</m:mo><m:mn>0</m:mn></m:mrow></m:munder><m:mfrac><m:mrow><m:mi>D</m:mi><m:mrow><m:mo>(</m:mo><m:mrow><m:mi>n</m:mi></m:mrow><m:mo>)</m:mo></m:mrow></m:mrow><m:mrow><m:mi>n</m:mi></m:mrow></m:mfrac><m:mo>&gt;</m:mo><m:mn>0</m:mn><m:mo>,</m:mo></m:math><jats:tex-math>\mathop{\mathrm{liminf}}\limits_{n\to
    \infty }D\left(n)\gt L\left(M)\hspace{1.0em}\hspace{0.1em}\text{and}\hspace{0.1em}\hspace{1.0em}\mathop{\mathrm{liminf}}\limits_{n\searrow
    0}\frac{D\left(n)}{n}\gt 0,</jats:tex-math></jats:alternatives></jats:disp-formula>for
    all sufficiently regular initial data<jats:inline-formula><jats:alternatives><jats:inline-graphic
    xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ans-2022-0004_eq_013.png"/><m:math
    xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:mo>(</m:mo><m:mrow><m:msub><m:mrow><m:mi>n</m:mi></m:mrow><m:mrow><m:mn>0</m:mn></m:mrow></m:msub><m:mo>,</m:mo><m:msub><m:mrow><m:mi>c</m:mi></m:mrow><m:mrow><m:mn>0</m:mn></m:mrow></m:msub><m:mo>,</m:mo><m:msub><m:mrow><m:mi>u</m:mi></m:mrow><m:mrow><m:mn>0</m:mn></m:mrow></m:msub></m:mrow><m:mo>)</m:mo></m:mrow></m:math><jats:tex-math>\left({n}_{0},{c}_{0},{u}_{0})</jats:tex-math></jats:alternatives></jats:inline-formula>fulfilling<jats:inline-formula><jats:alternatives><jats:inline-graphic
    xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ans-2022-0004_eq_014.png"/><m:math
    xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mo>‖</m:mo><m:msub><m:mrow><m:mi>c</m:mi></m:mrow><m:mrow><m:mn>0</m:mn></m:mrow></m:msub><m:msub><m:mrow><m:mo>‖</m:mo></m:mrow><m:mrow><m:msup><m:mrow><m:mi>L</m:mi></m:mrow><m:mrow><m:mi>∞</m:mi></m:mrow></m:msup><m:mrow><m:mo>(</m:mo><m:mrow><m:mi
    mathvariant="normal">Ω</m:mi></m:mrow><m:mo>)</m:mo></m:mrow></m:mrow></m:msub><m:mo>≤</m:mo><m:mi>M</m:mi></m:math><jats:tex-math>\Vert
    {c}_{0}{\Vert }_{{L}^{\infty }\left(\Omega )}\le M</jats:tex-math></jats:alternatives></jats:inline-formula>an
    associated no-flux/no-flux/Dirichlet initial-boundary value problem admits a global
    bounded weak solution, classical if additionally<jats:inline-formula><jats:alternatives><jats:inline-graphic
    xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ans-2022-0004_eq_015.png"/><m:math
    xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mi>D</m:mi><m:mrow><m:mo>(</m:mo><m:mrow><m:mn>0</m:mn></m:mrow><m:mo>)</m:mo></m:mrow><m:mo>&gt;</m:mo><m:mn>0</m:mn></m:math><jats:tex-math>D\left(0)\gt
    0</jats:tex-math></jats:alternatives></jats:inline-formula>. When combined with
    previously known results, this particularly implies global existence of bounded
    solutions when<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink"
    xlink:href="graphic/j_ans-2022-0004_eq_016.png"/><m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mi>D</m:mi><m:mrow><m:mo>(</m:mo><m:mrow><m:mi>n</m:mi></m:mrow><m:mo>)</m:mo></m:mrow><m:mo>=</m:mo><m:msup><m:mrow><m:mi>n</m:mi></m:mrow><m:mrow><m:mi>m</m:mi><m:mo>−</m:mo><m:mn>1</m:mn></m:mrow></m:msup></m:math><jats:tex-math>D\left(n)={n}^{m-1}</jats:tex-math></jats:alternatives></jats:inline-formula>,<jats:inline-formula><jats:alternatives><jats:inline-graphic
    xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ans-2022-0004_eq_017.png"/><m:math
    xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mi>n</m:mi><m:mo>≥</m:mo><m:mn>0</m:mn></m:math><jats:tex-math>n\ge
    0</jats:tex-math></jats:alternatives></jats:inline-formula>, with arbitrary<jats:inline-formula><jats:alternatives><jats:inline-graphic
    xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ans-2022-0004_eq_018.png"/><m:math
    xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mi>m</m:mi><m:mo>&gt;</m:mo><m:mn>1</m:mn></m:math><jats:tex-math>m\gt
    1</jats:tex-math></jats:alternatives></jats:inline-formula>, but beyond this asserts
    global boundedness also in the presence of diffusivities which exhibit arbitrarily
    slow divergence to<jats:inline-formula><jats:alternatives><jats:inline-graphic
    xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ans-2022-0004_eq_019.png"/><m:math
    xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mo>+</m:mo><m:mi>∞</m:mi></m:math><jats:tex-math>+\infty</jats:tex-math></jats:alternatives></jats:inline-formula>at
    large densities and of possibly singular chemotactic sensitivities.</jats:p>
author:
- first_name: Michael
  full_name: Winkler, Michael
  id: '31496'
  last_name: Winkler
citation:
  ama: 'Winkler M. Chemotaxis-Stokes interaction with very weak diffusion enhancement:
    Blow-up exclusion via detection of absorption-induced entropy structures involving
    multiplicative couplings. <i>Advanced Nonlinear Studies</i>. 2022;22(1):88-117.
    doi:<a href="https://doi.org/10.1515/ans-2022-0004">10.1515/ans-2022-0004</a>'
  apa: 'Winkler, M. (2022). Chemotaxis-Stokes interaction with very weak diffusion
    enhancement: Blow-up exclusion via detection of absorption-induced entropy structures
    involving multiplicative couplings. <i>Advanced Nonlinear Studies</i>, <i>22</i>(1),
    88–117. <a href="https://doi.org/10.1515/ans-2022-0004">https://doi.org/10.1515/ans-2022-0004</a>'
  bibtex: '@article{Winkler_2022, title={Chemotaxis-Stokes interaction with very weak
    diffusion enhancement: Blow-up exclusion via detection of absorption-induced entropy
    structures involving multiplicative couplings}, volume={22}, DOI={<a href="https://doi.org/10.1515/ans-2022-0004">10.1515/ans-2022-0004</a>},
    number={1}, journal={Advanced Nonlinear Studies}, publisher={Walter de Gruyter
    GmbH}, author={Winkler, Michael}, year={2022}, pages={88–117} }'
  chicago: 'Winkler, Michael. “Chemotaxis-Stokes Interaction with Very Weak Diffusion
    Enhancement: Blow-up Exclusion via Detection of Absorption-Induced Entropy Structures
    Involving Multiplicative Couplings.” <i>Advanced Nonlinear Studies</i> 22, no.
    1 (2022): 88–117. <a href="https://doi.org/10.1515/ans-2022-0004">https://doi.org/10.1515/ans-2022-0004</a>.'
  ieee: 'M. Winkler, “Chemotaxis-Stokes interaction with very weak diffusion enhancement:
    Blow-up exclusion via detection of absorption-induced entropy structures involving
    multiplicative couplings,” <i>Advanced Nonlinear Studies</i>, vol. 22, no. 1,
    pp. 88–117, 2022, doi: <a href="https://doi.org/10.1515/ans-2022-0004">10.1515/ans-2022-0004</a>.'
  mla: 'Winkler, Michael. “Chemotaxis-Stokes Interaction with Very Weak Diffusion
    Enhancement: Blow-up Exclusion via Detection of Absorption-Induced Entropy Structures
    Involving Multiplicative Couplings.” <i>Advanced Nonlinear Studies</i>, vol. 22,
    no. 1, Walter de Gruyter GmbH, 2022, pp. 88–117, doi:<a href="https://doi.org/10.1515/ans-2022-0004">10.1515/ans-2022-0004</a>.'
  short: M. Winkler, Advanced Nonlinear Studies 22 (2022) 88–117.
date_created: 2025-12-18T19:29:40Z
date_updated: 2025-12-18T20:05:30Z
doi: 10.1515/ans-2022-0004
intvolume: '        22'
issue: '1'
language:
- iso: eng
page: 88-117
publication: Advanced Nonlinear Studies
publication_identifier:
  issn:
  - 2169-0375
publication_status: published
publisher: Walter de Gruyter GmbH
status: public
title: 'Chemotaxis-Stokes interaction with very weak diffusion enhancement: Blow-up
  exclusion via detection of absorption-induced entropy structures involving multiplicative
  couplings'
type: journal_article
user_id: '31496'
volume: 22
year: '2022'
...
---
_id: '63312'
abstract:
- lang: eng
  text: "<jats:p xml:lang=\"fr\">&lt;p style='text-indent:20px;'&gt;The chemotaxis
    system&lt;/p&gt;&lt;p style='text-indent:20px;'&gt;&lt;disp-formula&gt; &lt;label/&gt;
    &lt;tex-math id=\"FE1\"&gt; \\begin{document}$ \\begin{array}{l}\\left\\{ \\begin{array}{l}
    \tu_t = \\nabla \\cdot \\big( D(u) \\nabla u \\big) - \\nabla \\cdot \\big( uS(x,
    u, v)\\cdot \\nabla v\\big), \\\\ \tv_t = \\Delta v -uv, \\end{array} \\right.
    \\end{array} $\\end{document} &lt;/tex-math&gt;&lt;/disp-formula&gt;&lt;/p&gt;&lt;p
    style='text-indent:20px;'&gt;is considered in a bounded domain &lt;inline-formula&gt;&lt;tex-math
    id=\"M1\"&gt;\\begin{document}$ \\Omega\\subset \\mathbb{R}^n $\\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;,
    &lt;inline-formula&gt;&lt;tex-math id=\"M2\"&gt;\\begin{document}$ n\\ge 2 $\\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;,
    with smooth boundary.&lt;/p&gt;&lt;p style='text-indent:20px;'&gt;It is shown
    that if &lt;inline-formula&gt;&lt;tex-math id=\"M3\"&gt;\\begin{document}$ D:
    [0, \\infty) \\to [0, \\infty) $\\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;
    and &lt;inline-formula&gt;&lt;tex-math id=\"M4\"&gt;\\begin{document}$ S: \\overline{\\Omega}\\times
    [0, \\infty)\\times (0, \\infty)\\to \\mathbb{R}^{n\\times n} $\\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;
    are suitably smooth functions satisfying&lt;/p&gt;&lt;p style='text-indent:20px;'&gt;&lt;disp-formula&gt;
    &lt;label/&gt; &lt;tex-math id=\"FE2\"&gt; \\begin{document}$ \\begin{array}{l}D(u)
    \\ge k_D u^{m-1} \t\\qquad {\\rm{for\\; all}}\\; u\\ge 0 \\end{array} $\\end{document}
    &lt;/tex-math&gt;&lt;/disp-formula&gt;&lt;/p&gt;&lt;p style='text-indent:20px;'&gt;and&lt;/p&gt;&lt;p
    style='text-indent:20px;'&gt;&lt;disp-formula&gt; &lt;label/&gt; &lt;tex-math
    id=\"FE3\"&gt; \\begin{document}$ \\begin{array}{l}|S(x, u, v)| \\le \\frac{S_0(v)}{v^\\alpha}
    \\qquad {\\rm{for\\; all}}\\; (x, u, v)\\; \\in \\Omega\\times (0, \\infty)^2
    \\end{array} $\\end{document} &lt;/tex-math&gt;&lt;/disp-formula&gt;&lt;/p&gt;&lt;p
    style='text-indent:20px;'&gt;with some&lt;/p&gt;&lt;p style='text-indent:20px;'&gt;&lt;disp-formula&gt;
    &lt;label/&gt; &lt;tex-math id=\"FE4\"&gt; \\begin{document}$ \\begin{array}{l}m&amp;gt;\\frac{3n-2}{2n}
    \t\\qquad {\\rm{and}}\\;\\alpha\\in [0, 1), \\end{array} $\\end{document} &lt;/tex-math&gt;&lt;/disp-formula&gt;&lt;/p&gt;&lt;p
    style='text-indent:20px;'&gt;and with some &lt;inline-formula&gt;&lt;tex-math
    id=\"M5\"&gt;\\begin{document}$ k_D&amp;gt;0 $\\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;
    and nondecreasing &lt;inline-formula&gt;&lt;tex-math id=\"M6\"&gt;\\begin{document}$
    S_0: (0, \\infty)\\to (0, \\infty) $\\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;,
    then for all suitably regular initial data a corresponding no-flux type initial-boundary
    value problem admits a global bounded weak solution which actually is smooth and
    classical if &lt;inline-formula&gt;&lt;tex-math id=\"M7\"&gt;\\begin{document}$
    D(0)&amp;gt;0 $\\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;.&lt;/p&gt;</jats:p>"
article_number: '6565'
author:
- first_name: Michael
  full_name: Winkler, Michael
  id: '31496'
  last_name: Winkler
citation:
  ama: Winkler M. Approaching logarithmic singularities in quasilinear chemotaxis-consumption
    systems with signal-dependent sensitivities. <i>Discrete and Continuous Dynamical
    Systems - B</i>. 2022;27(11). doi:<a href="https://doi.org/10.3934/dcdsb.2022009">10.3934/dcdsb.2022009</a>
  apa: Winkler, M. (2022). Approaching logarithmic singularities in quasilinear chemotaxis-consumption
    systems with signal-dependent sensitivities. <i>Discrete and Continuous Dynamical
    Systems - B</i>, <i>27</i>(11), Article 6565. <a href="https://doi.org/10.3934/dcdsb.2022009">https://doi.org/10.3934/dcdsb.2022009</a>
  bibtex: '@article{Winkler_2022, title={Approaching logarithmic singularities in
    quasilinear chemotaxis-consumption systems with signal-dependent sensitivities},
    volume={27}, DOI={<a href="https://doi.org/10.3934/dcdsb.2022009">10.3934/dcdsb.2022009</a>},
    number={116565}, journal={Discrete and Continuous Dynamical Systems - B}, publisher={American
    Institute of Mathematical Sciences (AIMS)}, author={Winkler, Michael}, year={2022}
    }'
  chicago: Winkler, Michael. “Approaching Logarithmic Singularities in Quasilinear
    Chemotaxis-Consumption Systems with Signal-Dependent Sensitivities.” <i>Discrete
    and Continuous Dynamical Systems - B</i> 27, no. 11 (2022). <a href="https://doi.org/10.3934/dcdsb.2022009">https://doi.org/10.3934/dcdsb.2022009</a>.
  ieee: 'M. Winkler, “Approaching logarithmic singularities in quasilinear chemotaxis-consumption
    systems with signal-dependent sensitivities,” <i>Discrete and Continuous Dynamical
    Systems - B</i>, vol. 27, no. 11, Art. no. 6565, 2022, doi: <a href="https://doi.org/10.3934/dcdsb.2022009">10.3934/dcdsb.2022009</a>.'
  mla: Winkler, Michael. “Approaching Logarithmic Singularities in Quasilinear Chemotaxis-Consumption
    Systems with Signal-Dependent Sensitivities.” <i>Discrete and Continuous Dynamical
    Systems - B</i>, vol. 27, no. 11, 6565, American Institute of Mathematical Sciences
    (AIMS), 2022, doi:<a href="https://doi.org/10.3934/dcdsb.2022009">10.3934/dcdsb.2022009</a>.
  short: M. Winkler, Discrete and Continuous Dynamical Systems - B 27 (2022).
date_created: 2025-12-18T19:30:32Z
date_updated: 2025-12-18T20:05:47Z
doi: 10.3934/dcdsb.2022009
intvolume: '        27'
issue: '11'
language:
- iso: eng
publication: Discrete and Continuous Dynamical Systems - B
publication_identifier:
  issn:
  - 1531-3492
  - 1553-524X
publication_status: published
publisher: American Institute of Mathematical Sciences (AIMS)
status: public
title: Approaching logarithmic singularities in quasilinear chemotaxis-consumption
  systems with signal-dependent sensitivities
type: journal_article
user_id: '31496'
volume: 27
year: '2022'
...
---
_id: '63266'
abstract:
- lang: eng
  text: "<jats:title>Abstract</jats:title><jats:p>In a ball <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\Omega
    =B_R(0)\\subset \\mathbb {R}^n$$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n
    \                 <mml:mrow>\r\n                    <mml:mi>Ω</mml:mi>\r\n                    <mml:mo>=</mml:mo>\r\n
    \                   <mml:msub>\r\n                      <mml:mi>B</mml:mi>\r\n
    \                     <mml:mi>R</mml:mi>\r\n                    </mml:msub>\r\n
    \                   <mml:mrow>\r\n                      <mml:mo>(</mml:mo>\r\n
    \                     <mml:mn>0</mml:mn>\r\n                      <mml:mo>)</mml:mo>\r\n
    \                   </mml:mrow>\r\n                    <mml:mo>⊂</mml:mo>\r\n
    \                   <mml:msup>\r\n                      <mml:mrow>\r\n                        <mml:mi>R</mml:mi>\r\n
    \                     </mml:mrow>\r\n                      <mml:mi>n</mml:mi>\r\n
    \                   </mml:msup>\r\n                  </mml:mrow>\r\n                </mml:math></jats:alternatives></jats:inline-formula>,
    <jats:inline-formula><jats:alternatives><jats:tex-math>$$n\\ge 2$$</jats:tex-math><mml:math
    xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n                  <mml:mrow>\r\n
    \                   <mml:mi>n</mml:mi>\r\n                    <mml:mo>≥</mml:mo>\r\n
    \                   <mml:mn>2</mml:mn>\r\n                  </mml:mrow>\r\n                </mml:math></jats:alternatives></jats:inline-formula>,
    the chemotaxis system <jats:disp-formula><jats:alternatives><jats:tex-math>$$\\begin{aligned}
    \\left\\{ \\begin{array}{l}u_t = \\nabla \\cdot \\big ( D(u) \\nabla u \\big )
    - \\nabla \\cdot \\big ( uS(u)\\nabla v\\big ), \\\\ 0 = \\Delta v - \\mu + u,
    \\qquad \\mu =\\frac{1}{|\\Omega |} \\int _\\Omega u, \\end{array} \\right. \\qquad
    \\qquad (\\star ) \\end{aligned}$$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n
    \                 <mml:mrow>\r\n                    <mml:mtable>\r\n                      <mml:mtr>\r\n
    \                       <mml:mtd>\r\n                          <mml:mrow>\r\n
    \                           <mml:mfenced>\r\n                              <mml:mrow>\r\n
    \                               <mml:mtable>\r\n                                  <mml:mtr>\r\n
    \                                   <mml:mtd>\r\n                                      <mml:mrow>\r\n
    \                                       <mml:msub>\r\n                                          <mml:mi>u</mml:mi>\r\n
    \                                         <mml:mi>t</mml:mi>\r\n                                        </mml:msub>\r\n
    \                                       <mml:mo>=</mml:mo>\r\n                                        <mml:mi>∇</mml:mi>\r\n
    \                                       <mml:mo>·</mml:mo>\r\n                                        <mml:mrow>\r\n
    \                                         <mml:mo>(</mml:mo>\r\n                                        </mml:mrow>\r\n
    \                                       <mml:mi>D</mml:mi>\r\n                                        <mml:mrow>\r\n
    \                                         <mml:mo>(</mml:mo>\r\n                                          <mml:mi>u</mml:mi>\r\n
    \                                         <mml:mo>)</mml:mo>\r\n                                        </mml:mrow>\r\n
    \                                       <mml:mi>∇</mml:mi>\r\n                                        <mml:mi>u</mml:mi>\r\n
    \                                       <mml:mrow>\r\n                                          <mml:mo>)</mml:mo>\r\n
    \                                       </mml:mrow>\r\n                                        <mml:mo>-</mml:mo>\r\n
    \                                       <mml:mi>∇</mml:mi>\r\n                                        <mml:mo>·</mml:mo>\r\n
    \                                       <mml:mrow>\r\n                                          <mml:mo>(</mml:mo>\r\n
    \                                       </mml:mrow>\r\n                                        <mml:mi>u</mml:mi>\r\n
    \                                       <mml:mi>S</mml:mi>\r\n                                        <mml:mrow>\r\n
    \                                         <mml:mo>(</mml:mo>\r\n                                          <mml:mi>u</mml:mi>\r\n
    \                                         <mml:mo>)</mml:mo>\r\n                                        </mml:mrow>\r\n
    \                                       <mml:mi>∇</mml:mi>\r\n                                        <mml:mi>v</mml:mi>\r\n
    \                                       <mml:mrow>\r\n                                          <mml:mo>)</mml:mo>\r\n
    \                                       </mml:mrow>\r\n                                        <mml:mo>,</mml:mo>\r\n
    \                                     </mml:mrow>\r\n                                    </mml:mtd>\r\n
    \                                 </mml:mtr>\r\n                                  <mml:mtr>\r\n
    \                                   <mml:mtd>\r\n                                      <mml:mrow>\r\n
    \                                       <mml:mrow/>\r\n                                        <mml:mn>0</mml:mn>\r\n
    \                                       <mml:mo>=</mml:mo>\r\n                                        <mml:mi>Δ</mml:mi>\r\n
    \                                       <mml:mi>v</mml:mi>\r\n                                        <mml:mo>-</mml:mo>\r\n
    \                                       <mml:mi>μ</mml:mi>\r\n                                        <mml:mo>+</mml:mo>\r\n
    \                                       <mml:mi>u</mml:mi>\r\n                                        <mml:mo>,</mml:mo>\r\n
    \                                       <mml:mspace/>\r\n                                        <mml:mi>μ</mml:mi>\r\n
    \                                       <mml:mo>=</mml:mo>\r\n                                        <mml:mfrac>\r\n
    \                                         <mml:mn>1</mml:mn>\r\n                                          <mml:mrow>\r\n
    \                                           <mml:mo>|</mml:mo>\r\n                                            <mml:mi>Ω</mml:mi>\r\n
    \                                           <mml:mo>|</mml:mo>\r\n                                          </mml:mrow>\r\n
    \                                       </mml:mfrac>\r\n                                        <mml:msub>\r\n
    \                                         <mml:mo>∫</mml:mo>\r\n                                          <mml:mi>Ω</mml:mi>\r\n
    \                                       </mml:msub>\r\n                                        <mml:mi>u</mml:mi>\r\n
    \                                       <mml:mo>,</mml:mo>\r\n                                      </mml:mrow>\r\n
    \                                   </mml:mtd>\r\n                                  </mml:mtr>\r\n
    \                               </mml:mtable>\r\n                              </mml:mrow>\r\n
    \                           </mml:mfenced>\r\n                            <mml:mspace/>\r\n
    \                           <mml:mspace/>\r\n                            <mml:mrow>\r\n
    \                             <mml:mo>(</mml:mo>\r\n                              <mml:mo>⋆</mml:mo>\r\n
    \                             <mml:mo>)</mml:mo>\r\n                            </mml:mrow>\r\n
    \                         </mml:mrow>\r\n                        </mml:mtd>\r\n
    \                     </mml:mtr>\r\n                    </mml:mtable>\r\n                  </mml:mrow>\r\n
    \               </mml:math></jats:alternatives></jats:disp-formula>is considered
    under no-flux boundary conditions, with a focus on nonlinearities <jats:inline-formula><jats:alternatives><jats:tex-math>$$S\\in
    C^2([0,\\infty ))$$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n
    \                 <mml:mrow>\r\n                    <mml:mi>S</mml:mi>\r\n                    <mml:mo>∈</mml:mo>\r\n
    \                   <mml:msup>\r\n                      <mml:mi>C</mml:mi>\r\n
    \                     <mml:mn>2</mml:mn>\r\n                    </mml:msup>\r\n
    \                   <mml:mrow>\r\n                      <mml:mo>(</mml:mo>\r\n
    \                     <mml:mrow>\r\n                        <mml:mo>[</mml:mo>\r\n
    \                       <mml:mn>0</mml:mn>\r\n                        <mml:mo>,</mml:mo>\r\n
    \                       <mml:mi>∞</mml:mi>\r\n                        <mml:mo>)</mml:mo>\r\n
    \                     </mml:mrow>\r\n                      <mml:mo>)</mml:mo>\r\n
    \                   </mml:mrow>\r\n                  </mml:mrow>\r\n                </mml:math></jats:alternatives></jats:inline-formula>
    which exhibit super-algebraically fast decay in the sense that with some <jats:inline-formula><jats:alternatives><jats:tex-math>$$K_S&gt;0,
    \\beta \\in [0,1)$$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n
    \                 <mml:mrow>\r\n                    <mml:msub>\r\n                      <mml:mi>K</mml:mi>\r\n
    \                     <mml:mi>S</mml:mi>\r\n                    </mml:msub>\r\n
    \                   <mml:mo>&gt;</mml:mo>\r\n                    <mml:mn>0</mml:mn>\r\n
    \                   <mml:mo>,</mml:mo>\r\n                    <mml:mi>β</mml:mi>\r\n
    \                   <mml:mo>∈</mml:mo>\r\n                    <mml:mrow>\r\n                      <mml:mo>[</mml:mo>\r\n
    \                     <mml:mn>0</mml:mn>\r\n                      <mml:mo>,</mml:mo>\r\n
    \                     <mml:mn>1</mml:mn>\r\n                      <mml:mo>)</mml:mo>\r\n
    \                   </mml:mrow>\r\n                  </mml:mrow>\r\n                </mml:math></jats:alternatives></jats:inline-formula>
    and <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\xi _0&gt;0$$</jats:tex-math><mml:math
    xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n                  <mml:mrow>\r\n
    \                   <mml:msub>\r\n                      <mml:mi>ξ</mml:mi>\r\n
    \                     <mml:mn>0</mml:mn>\r\n                    </mml:msub>\r\n
    \                   <mml:mo>&gt;</mml:mo>\r\n                    <mml:mn>0</mml:mn>\r\n
    \                 </mml:mrow>\r\n                </mml:math></jats:alternatives></jats:inline-formula>,
    <jats:disp-formula><jats:alternatives><jats:tex-math>$$\\begin{aligned} S(\\xi
    )&gt;0 \\quad \\text{ and } \\quad S'(\\xi ) \\le -K_S\\xi ^{-\\beta } S(\\xi
    ) \\qquad \\text{ for } \\text{ all } \\xi \\ge \\xi _0. \\end{aligned}$$</jats:tex-math><mml:math
    xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n                  <mml:mrow>\r\n
    \                   <mml:mtable>\r\n                      <mml:mtr>\r\n                        <mml:mtd>\r\n
    \                         <mml:mrow>\r\n                            <mml:mi>S</mml:mi>\r\n
    \                           <mml:mrow>\r\n                              <mml:mo>(</mml:mo>\r\n
    \                             <mml:mi>ξ</mml:mi>\r\n                              <mml:mo>)</mml:mo>\r\n
    \                           </mml:mrow>\r\n                            <mml:mo>&gt;</mml:mo>\r\n
    \                           <mml:mn>0</mml:mn>\r\n                            <mml:mspace/>\r\n
    \                           <mml:mspace/>\r\n                            <mml:mtext>and</mml:mtext>\r\n
    \                           <mml:mspace/>\r\n                            <mml:mspace/>\r\n
    \                           <mml:msup>\r\n                              <mml:mi>S</mml:mi>\r\n
    \                             <mml:mo>′</mml:mo>\r\n                            </mml:msup>\r\n
    \                           <mml:mrow>\r\n                              <mml:mo>(</mml:mo>\r\n
    \                             <mml:mi>ξ</mml:mi>\r\n                              <mml:mo>)</mml:mo>\r\n
    \                           </mml:mrow>\r\n                            <mml:mo>≤</mml:mo>\r\n
    \                           <mml:mo>-</mml:mo>\r\n                            <mml:msub>\r\n
    \                             <mml:mi>K</mml:mi>\r\n                              <mml:mi>S</mml:mi>\r\n
    \                           </mml:msub>\r\n                            <mml:msup>\r\n
    \                             <mml:mi>ξ</mml:mi>\r\n                              <mml:mrow>\r\n
    \                               <mml:mo>-</mml:mo>\r\n                                <mml:mi>β</mml:mi>\r\n
    \                             </mml:mrow>\r\n                            </mml:msup>\r\n
    \                           <mml:mi>S</mml:mi>\r\n                            <mml:mrow>\r\n
    \                             <mml:mo>(</mml:mo>\r\n                              <mml:mi>ξ</mml:mi>\r\n
    \                             <mml:mo>)</mml:mo>\r\n                            </mml:mrow>\r\n
    \                           <mml:mspace/>\r\n                            <mml:mspace/>\r\n
    \                           <mml:mtext>for</mml:mtext>\r\n                            <mml:mspace/>\r\n
    \                           <mml:mspace/>\r\n                            <mml:mtext>all</mml:mtext>\r\n
    \                           <mml:mspace/>\r\n                            <mml:mi>ξ</mml:mi>\r\n
    \                           <mml:mo>≥</mml:mo>\r\n                            <mml:msub>\r\n
    \                             <mml:mi>ξ</mml:mi>\r\n                              <mml:mn>0</mml:mn>\r\n
    \                           </mml:msub>\r\n                            <mml:mo>.</mml:mo>\r\n
    \                         </mml:mrow>\r\n                        </mml:mtd>\r\n
    \                     </mml:mtr>\r\n                    </mml:mtable>\r\n                  </mml:mrow>\r\n
    \               </mml:math></jats:alternatives></jats:disp-formula>It is, inter
    alia, shown that if furthermore <jats:inline-formula><jats:alternatives><jats:tex-math>$$D\\in
    C^2((0,\\infty ))$$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n
    \                 <mml:mrow>\r\n                    <mml:mi>D</mml:mi>\r\n                    <mml:mo>∈</mml:mo>\r\n
    \                   <mml:msup>\r\n                      <mml:mi>C</mml:mi>\r\n
    \                     <mml:mn>2</mml:mn>\r\n                    </mml:msup>\r\n
    \                   <mml:mrow>\r\n                      <mml:mo>(</mml:mo>\r\n
    \                     <mml:mrow>\r\n                        <mml:mo>(</mml:mo>\r\n
    \                       <mml:mn>0</mml:mn>\r\n                        <mml:mo>,</mml:mo>\r\n
    \                       <mml:mi>∞</mml:mi>\r\n                        <mml:mo>)</mml:mo>\r\n
    \                     </mml:mrow>\r\n                      <mml:mo>)</mml:mo>\r\n
    \                   </mml:mrow>\r\n                  </mml:mrow>\r\n                </mml:math></jats:alternatives></jats:inline-formula>
    is positive and suitably small in relation to <jats:italic>S</jats:italic> by
    satisfying <jats:disp-formula><jats:alternatives><jats:tex-math>$$\\begin{aligned}
    \\frac{\\xi S(\\xi )}{D(\\xi )} \\ge K_{SD}\\xi ^\\lambda \\qquad \\text{ for
    } \\text{ all } \\xi \\ge \\xi _0 \\end{aligned}$$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n
    \                 <mml:mrow>\r\n                    <mml:mtable>\r\n                      <mml:mtr>\r\n
    \                       <mml:mtd>\r\n                          <mml:mrow>\r\n
    \                           <mml:mfrac>\r\n                              <mml:mrow>\r\n
    \                               <mml:mi>ξ</mml:mi>\r\n                                <mml:mi>S</mml:mi>\r\n
    \                               <mml:mo>(</mml:mo>\r\n                                <mml:mi>ξ</mml:mi>\r\n
    \                               <mml:mo>)</mml:mo>\r\n                              </mml:mrow>\r\n
    \                             <mml:mrow>\r\n                                <mml:mi>D</mml:mi>\r\n
    \                               <mml:mo>(</mml:mo>\r\n                                <mml:mi>ξ</mml:mi>\r\n
    \                               <mml:mo>)</mml:mo>\r\n                              </mml:mrow>\r\n
    \                           </mml:mfrac>\r\n                            <mml:mo>≥</mml:mo>\r\n
    \                           <mml:msub>\r\n                              <mml:mi>K</mml:mi>\r\n
    \                             <mml:mrow>\r\n                                <mml:mi>SD</mml:mi>\r\n
    \                             </mml:mrow>\r\n                            </mml:msub>\r\n
    \                           <mml:msup>\r\n                              <mml:mi>ξ</mml:mi>\r\n
    \                             <mml:mi>λ</mml:mi>\r\n                            </mml:msup>\r\n
    \                           <mml:mspace/>\r\n                            <mml:mspace/>\r\n
    \                           <mml:mtext>for</mml:mtext>\r\n                            <mml:mspace/>\r\n
    \                           <mml:mspace/>\r\n                            <mml:mtext>all</mml:mtext>\r\n
    \                           <mml:mspace/>\r\n                            <mml:mi>ξ</mml:mi>\r\n
    \                           <mml:mo>≥</mml:mo>\r\n                            <mml:msub>\r\n
    \                             <mml:mi>ξ</mml:mi>\r\n                              <mml:mn>0</mml:mn>\r\n
    \                           </mml:msub>\r\n                          </mml:mrow>\r\n
    \                       </mml:mtd>\r\n                      </mml:mtr>\r\n                    </mml:mtable>\r\n
    \                 </mml:mrow>\r\n                </mml:math></jats:alternatives></jats:disp-formula>with
    some <jats:inline-formula><jats:alternatives><jats:tex-math>$$K_{SD}&gt;0$$</jats:tex-math><mml:math
    xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n                  <mml:mrow>\r\n
    \                   <mml:msub>\r\n                      <mml:mi>K</mml:mi>\r\n
    \                     <mml:mrow>\r\n                        <mml:mi>SD</mml:mi>\r\n
    \                     </mml:mrow>\r\n                    </mml:msub>\r\n                    <mml:mo>&gt;</mml:mo>\r\n
    \                   <mml:mn>0</mml:mn>\r\n                  </mml:mrow>\r\n                </mml:math></jats:alternatives></jats:inline-formula>
    and <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\lambda &gt;\\frac{2}{n}$$</jats:tex-math><mml:math
    xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n                  <mml:mrow>\r\n
    \                   <mml:mi>λ</mml:mi>\r\n                    <mml:mo>&gt;</mml:mo>\r\n
    \                   <mml:mfrac>\r\n                      <mml:mn>2</mml:mn>\r\n
    \                     <mml:mi>n</mml:mi>\r\n                    </mml:mfrac>\r\n
    \                 </mml:mrow>\r\n                </mml:math></jats:alternatives></jats:inline-formula>,
    then throughout a considerably large set of initial data, (<jats:inline-formula><jats:alternatives><jats:tex-math>$$\\star
    $$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n
    \                 <mml:mo>⋆</mml:mo>\r\n                </mml:math></jats:alternatives></jats:inline-formula>)
    admits global classical solutions (<jats:italic>u</jats:italic>, <jats:italic>v</jats:italic>)
    fulfilling <jats:disp-formula><jats:alternatives><jats:tex-math>$$\\begin{aligned}
    \\frac{z(t)}{C} \\le \\Vert u(\\cdot ,t)\\Vert _{L^\\infty (\\Omega )} \\le Cz(t)
    \\qquad \\text{ for } \\text{ all } t&gt;0, \\end{aligned}$$</jats:tex-math><mml:math
    xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n                  <mml:mrow>\r\n
    \                   <mml:mtable>\r\n                      <mml:mtr>\r\n                        <mml:mtd>\r\n
    \                         <mml:mrow>\r\n                            <mml:mfrac>\r\n
    \                             <mml:mrow>\r\n                                <mml:mi>z</mml:mi>\r\n
    \                               <mml:mo>(</mml:mo>\r\n                                <mml:mi>t</mml:mi>\r\n
    \                               <mml:mo>)</mml:mo>\r\n                              </mml:mrow>\r\n
    \                             <mml:mi>C</mml:mi>\r\n                            </mml:mfrac>\r\n
    \                           <mml:mo>≤</mml:mo>\r\n                            <mml:msub>\r\n
    \                             <mml:mrow>\r\n                                <mml:mo>‖</mml:mo>\r\n
    \                               <mml:mi>u</mml:mi>\r\n                                <mml:mrow>\r\n
    \                                 <mml:mo>(</mml:mo>\r\n                                  <mml:mo>·</mml:mo>\r\n
    \                                 <mml:mo>,</mml:mo>\r\n                                  <mml:mi>t</mml:mi>\r\n
    \                                 <mml:mo>)</mml:mo>\r\n                                </mml:mrow>\r\n
    \                               <mml:mo>‖</mml:mo>\r\n                              </mml:mrow>\r\n
    \                             <mml:mrow>\r\n                                <mml:msup>\r\n
    \                                 <mml:mi>L</mml:mi>\r\n                                  <mml:mi>∞</mml:mi>\r\n
    \                               </mml:msup>\r\n                                <mml:mrow>\r\n
    \                                 <mml:mo>(</mml:mo>\r\n                                  <mml:mi>Ω</mml:mi>\r\n
    \                                 <mml:mo>)</mml:mo>\r\n                                </mml:mrow>\r\n
    \                             </mml:mrow>\r\n                            </mml:msub>\r\n
    \                           <mml:mo>≤</mml:mo>\r\n                            <mml:mi>C</mml:mi>\r\n
    \                           <mml:mi>z</mml:mi>\r\n                            <mml:mrow>\r\n
    \                             <mml:mo>(</mml:mo>\r\n                              <mml:mi>t</mml:mi>\r\n
    \                             <mml:mo>)</mml:mo>\r\n                            </mml:mrow>\r\n
    \                           <mml:mspace/>\r\n                            <mml:mspace/>\r\n
    \                           <mml:mtext>for</mml:mtext>\r\n                            <mml:mspace/>\r\n
    \                           <mml:mspace/>\r\n                            <mml:mtext>all</mml:mtext>\r\n
    \                           <mml:mspace/>\r\n                            <mml:mi>t</mml:mi>\r\n
    \                           <mml:mo>&gt;</mml:mo>\r\n                            <mml:mn>0</mml:mn>\r\n
    \                           <mml:mo>,</mml:mo>\r\n                          </mml:mrow>\r\n
    \                       </mml:mtd>\r\n                      </mml:mtr>\r\n                    </mml:mtable>\r\n
    \                 </mml:mrow>\r\n                </mml:math></jats:alternatives></jats:disp-formula>with
    some <jats:inline-formula><jats:alternatives><jats:tex-math>$$C=C^{(u,v)}\\ge
    1$$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n
    \                 <mml:mrow>\r\n                    <mml:mi>C</mml:mi>\r\n                    <mml:mo>=</mml:mo>\r\n
    \                   <mml:msup>\r\n                      <mml:mi>C</mml:mi>\r\n
    \                     <mml:mrow>\r\n                        <mml:mo>(</mml:mo>\r\n
    \                       <mml:mi>u</mml:mi>\r\n                        <mml:mo>,</mml:mo>\r\n
    \                       <mml:mi>v</mml:mi>\r\n                        <mml:mo>)</mml:mo>\r\n
    \                     </mml:mrow>\r\n                    </mml:msup>\r\n                    <mml:mo>≥</mml:mo>\r\n
    \                   <mml:mn>1</mml:mn>\r\n                  </mml:mrow>\r\n                </mml:math></jats:alternatives></jats:inline-formula>,
    where <jats:italic>z</jats:italic> denotes the solution of <jats:disp-formula><jats:alternatives><jats:tex-math>$$\\begin{aligned}
    \\left\\{ \\begin{array}{l}z'(t) = z^2(t) \\cdot S\\big ( z(t)\\big ), \\qquad
    t&gt;0, \\\\ z(0)=\\xi _0, \\end{array} \\right. \\end{aligned}$$</jats:tex-math><mml:math
    xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n                  <mml:mrow>\r\n
    \                   <mml:mtable>\r\n                      <mml:mtr>\r\n                        <mml:mtd>\r\n
    \                         <mml:mfenced>\r\n                            <mml:mrow>\r\n
    \                             <mml:mtable>\r\n                                <mml:mtr>\r\n
    \                                 <mml:mtd>\r\n                                    <mml:mrow>\r\n
    \                                     <mml:msup>\r\n                                        <mml:mi>z</mml:mi>\r\n
    \                                       <mml:mo>′</mml:mo>\r\n                                      </mml:msup>\r\n
    \                                     <mml:mrow>\r\n                                        <mml:mo>(</mml:mo>\r\n
    \                                       <mml:mi>t</mml:mi>\r\n                                        <mml:mo>)</mml:mo>\r\n
    \                                     </mml:mrow>\r\n                                      <mml:mo>=</mml:mo>\r\n
    \                                     <mml:msup>\r\n                                        <mml:mi>z</mml:mi>\r\n
    \                                       <mml:mn>2</mml:mn>\r\n                                      </mml:msup>\r\n
    \                                     <mml:mrow>\r\n                                        <mml:mo>(</mml:mo>\r\n
    \                                       <mml:mi>t</mml:mi>\r\n                                        <mml:mo>)</mml:mo>\r\n
    \                                     </mml:mrow>\r\n                                      <mml:mo>·</mml:mo>\r\n
    \                                     <mml:mi>S</mml:mi>\r\n                                      <mml:mrow>\r\n
    \                                       <mml:mo>(</mml:mo>\r\n                                      </mml:mrow>\r\n
    \                                     <mml:mi>z</mml:mi>\r\n                                      <mml:mrow>\r\n
    \                                       <mml:mo>(</mml:mo>\r\n                                        <mml:mi>t</mml:mi>\r\n
    \                                       <mml:mo>)</mml:mo>\r\n                                      </mml:mrow>\r\n
    \                                     <mml:mrow>\r\n                                        <mml:mo>)</mml:mo>\r\n
    \                                     </mml:mrow>\r\n                                      <mml:mo>,</mml:mo>\r\n
    \                                     <mml:mspace/>\r\n                                      <mml:mi>t</mml:mi>\r\n
    \                                     <mml:mo>&gt;</mml:mo>\r\n                                      <mml:mn>0</mml:mn>\r\n
    \                                     <mml:mo>,</mml:mo>\r\n                                    </mml:mrow>\r\n
    \                                 </mml:mtd>\r\n                                </mml:mtr>\r\n
    \                               <mml:mtr>\r\n                                  <mml:mtd>\r\n
    \                                   <mml:mrow>\r\n                                      <mml:mrow/>\r\n
    \                                     <mml:mi>z</mml:mi>\r\n                                      <mml:mrow>\r\n
    \                                       <mml:mo>(</mml:mo>\r\n                                        <mml:mn>0</mml:mn>\r\n
    \                                       <mml:mo>)</mml:mo>\r\n                                      </mml:mrow>\r\n
    \                                     <mml:mo>=</mml:mo>\r\n                                      <mml:msub>\r\n
    \                                       <mml:mi>ξ</mml:mi>\r\n                                        <mml:mn>0</mml:mn>\r\n
    \                                     </mml:msub>\r\n                                      <mml:mo>,</mml:mo>\r\n
    \                                   </mml:mrow>\r\n                                  </mml:mtd>\r\n
    \                               </mml:mtr>\r\n                              </mml:mtable>\r\n
    \                           </mml:mrow>\r\n                          </mml:mfenced>\r\n
    \                       </mml:mtd>\r\n                      </mml:mtr>\r\n                    </mml:mtable>\r\n
    \                 </mml:mrow>\r\n                </mml:math></jats:alternatives></jats:disp-formula>which
    is seen to exist globally, and to satisfy <jats:inline-formula><jats:alternatives><jats:tex-math>$$z(t)\\rightarrow
    +\\infty $$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n
    \                 <mml:mrow>\r\n                    <mml:mi>z</mml:mi>\r\n                    <mml:mo>(</mml:mo>\r\n
    \                   <mml:mi>t</mml:mi>\r\n                    <mml:mo>)</mml:mo>\r\n
    \                   <mml:mo>→</mml:mo>\r\n                    <mml:mo>+</mml:mo>\r\n
    \                   <mml:mi>∞</mml:mi>\r\n                  </mml:mrow>\r\n                </mml:math></jats:alternatives></jats:inline-formula>
    as <jats:inline-formula><jats:alternatives><jats:tex-math>$$t\\rightarrow \\infty
    $$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n
    \                 <mml:mrow>\r\n                    <mml:mi>t</mml:mi>\r\n                    <mml:mo>→</mml:mo>\r\n
    \                   <mml:mi>∞</mml:mi>\r\n                  </mml:mrow>\r\n                </mml:math></jats:alternatives></jats:inline-formula>.
    As particular examples, exponentially and doubly exponentially decaying <jats:italic>S</jats:italic>
    are found to imply corresponding infinite-time blow-up properties in (<jats:inline-formula><jats:alternatives><jats:tex-math>$$\\star
    $$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n
    \                 <mml:mo>⋆</mml:mo>\r\n                </mml:math></jats:alternatives></jats:inline-formula>)
    at logarithmic and doubly logarithmic rates, respectively.</jats:p>"
author:
- first_name: Michael
  full_name: Winkler, Michael
  id: '31496'
  last_name: Winkler
citation:
  ama: Winkler M. Slow Grow-up in a Quasilinear Keller–Segel System. <i>Journal of
    Dynamics and Differential Equations</i>. 2022;36(2):1677-1702. doi:<a href="https://doi.org/10.1007/s10884-022-10167-w">10.1007/s10884-022-10167-w</a>
  apa: Winkler, M. (2022). Slow Grow-up in a Quasilinear Keller–Segel System. <i>Journal
    of Dynamics and Differential Equations</i>, <i>36</i>(2), 1677–1702. <a href="https://doi.org/10.1007/s10884-022-10167-w">https://doi.org/10.1007/s10884-022-10167-w</a>
  bibtex: '@article{Winkler_2022, title={Slow Grow-up in a Quasilinear Keller–Segel
    System}, volume={36}, DOI={<a href="https://doi.org/10.1007/s10884-022-10167-w">10.1007/s10884-022-10167-w</a>},
    number={2}, journal={Journal of Dynamics and Differential Equations}, publisher={Springer
    Science and Business Media LLC}, author={Winkler, Michael}, year={2022}, pages={1677–1702}
    }'
  chicago: 'Winkler, Michael. “Slow Grow-up in a Quasilinear Keller–Segel System.”
    <i>Journal of Dynamics and Differential Equations</i> 36, no. 2 (2022): 1677–1702.
    <a href="https://doi.org/10.1007/s10884-022-10167-w">https://doi.org/10.1007/s10884-022-10167-w</a>.'
  ieee: 'M. Winkler, “Slow Grow-up in a Quasilinear Keller–Segel System,” <i>Journal
    of Dynamics and Differential Equations</i>, vol. 36, no. 2, pp. 1677–1702, 2022,
    doi: <a href="https://doi.org/10.1007/s10884-022-10167-w">10.1007/s10884-022-10167-w</a>.'
  mla: Winkler, Michael. “Slow Grow-up in a Quasilinear Keller–Segel System.” <i>Journal
    of Dynamics and Differential Equations</i>, vol. 36, no. 2, Springer Science and
    Business Media LLC, 2022, pp. 1677–702, doi:<a href="https://doi.org/10.1007/s10884-022-10167-w">10.1007/s10884-022-10167-w</a>.
  short: M. Winkler, Journal of Dynamics and Differential Equations 36 (2022) 1677–1702.
date_created: 2025-12-18T19:10:32Z
date_updated: 2025-12-18T20:10:14Z
doi: 10.1007/s10884-022-10167-w
intvolume: '        36'
issue: '2'
language:
- iso: eng
page: 1677-1702
publication: Journal of Dynamics and Differential Equations
publication_identifier:
  issn:
  - 1040-7294
  - 1572-9222
publication_status: published
publisher: Springer Science and Business Media LLC
status: public
title: Slow Grow-up in a Quasilinear Keller–Segel System
type: journal_article
user_id: '31496'
volume: 36
year: '2022'
...
---
_id: '63278'
abstract:
- lang: eng
  text: "<jats:title>Abstract</jats:title>\r\n               <jats:p>The Neumann problem
    for (0.1)$$ \\begin{align}&amp; V_t = \\Delta V-aV+f(x,t) \\end{align}$$is considered
    in bounded domains $\\Omega \\subset {\\mathbb {R}}^n$ with smooth boundary, where
    $n\\ge 1$ and $a\\in {\\mathbb {R}}$. By means of a variational approach, a statement
    on boundedness of the quantities $$ \\begin{eqnarray*} \\sup_{t\\in (0,T)} \\int_\\Omega
    \\big|\\nabla V(\\cdot,t)\\big|^p L^{\\frac{n+p}{n+2}} \\Big( \\big|\\nabla V(\\cdot,t)\\big|
    \\Big) \\end{eqnarray*}$$in dependence on the expressions (0.2)$$ \\begin{align}&amp;
    \\sup_{t\\in (0,T-\\tau)} \\int_t^{t+\\tau} \\int_\\Omega |f|^{\\frac{(n+2)p}{n+p}}
    L\\big( |f|\\big) \\end{align}$$is derived for $p\\ge 2$, $\\tau&amp;gt;0$, and
    $T\\ge 2\\tau $, provided that $L\\in C^0([0,\\infty ))$ is positive, strictly
    increasing, unbounded, and slowly growing in the sense that $\\limsup _{s\\to
    \\infty } \\frac {L(s^{\\lambda _0})}{L(s)} &amp;lt;\\infty $ for some $\\lambda
    _0&amp;gt;1$. In the particular case when $p=n\\ge 2$, an additional condition
    on growth of $L$, particularly satisfied by $L(\\xi ):=\\ln ^\\alpha (\\xi +b)$
    whenever $b&amp;gt;0$ and $\\alpha&amp;gt;\\frac {(n+2)(n-1)}{2n}$, is identified
    as sufficient to ensure that as a consequence of the above, bounds for theintegrals
    in (0.2) even imply estimates for the spatio-temporal modulus of continuity of
    solutions to (0.1). A subsequent application to the Keller–Segel system $$ \\begin{eqnarray*}
    \\left\\{ \\begin{array}{l} u_t = \\nabla \\cdot \\big( D(v)\\nabla u\\big) -
    \\nabla \\cdot \\big( uS(v)\\nabla v\\big) + ru - \\mu u^2, \\\\[1mm] v_t = \\Delta
    v-v+u, \\end{array} \\right. \\end{eqnarray*}$$shows that when $n=2$, $r\\in {\\mathbb
    {R}}$, $0&amp;lt;D\\in C^2([0,\\infty ))$, and $S\\in C^2([0,\\infty )) \\cap
    W^{1,\\infty }((0,\\infty ))$ and thus especially in the presence of arbitrarily
    strong diffusion degeneracies implied by rapid decay of $D$, any choice of $\\mu&amp;gt;0$
    excludes blowup in the sense that for all suitably regular nonnegative initial
    data, an associated initial-boundary value problem admits a global bounded classical
    solution.</jats:p>"
author:
- first_name: Michael
  full_name: Winkler, Michael
  id: '31496'
  last_name: Winkler
citation:
  ama: Winkler M. A Result on Parabolic Gradient Regularity in Orlicz Spaces and Application
    to Absorption-Induced Blow-Up Prevention in a Keller–Segel-Type Cross-Diffusion
    System. <i>International Mathematics Research Notices</i>. 2022;2023(19):16336-16393.
    doi:<a href="https://doi.org/10.1093/imrn/rnac286">10.1093/imrn/rnac286</a>
  apa: Winkler, M. (2022). A Result on Parabolic Gradient Regularity in Orlicz Spaces
    and Application to Absorption-Induced Blow-Up Prevention in a Keller–Segel-Type
    Cross-Diffusion System. <i>International Mathematics Research Notices</i>, <i>2023</i>(19),
    16336–16393. <a href="https://doi.org/10.1093/imrn/rnac286">https://doi.org/10.1093/imrn/rnac286</a>
  bibtex: '@article{Winkler_2022, title={A Result on Parabolic Gradient Regularity
    in Orlicz Spaces and Application to Absorption-Induced Blow-Up Prevention in a
    Keller–Segel-Type Cross-Diffusion System}, volume={2023}, DOI={<a href="https://doi.org/10.1093/imrn/rnac286">10.1093/imrn/rnac286</a>},
    number={19}, journal={International Mathematics Research Notices}, publisher={Oxford
    University Press (OUP)}, author={Winkler, Michael}, year={2022}, pages={16336–16393}
    }'
  chicago: 'Winkler, Michael. “A Result on Parabolic Gradient Regularity in Orlicz
    Spaces and Application to Absorption-Induced Blow-Up Prevention in a Keller–Segel-Type
    Cross-Diffusion System.” <i>International Mathematics Research Notices</i> 2023,
    no. 19 (2022): 16336–93. <a href="https://doi.org/10.1093/imrn/rnac286">https://doi.org/10.1093/imrn/rnac286</a>.'
  ieee: 'M. Winkler, “A Result on Parabolic Gradient Regularity in Orlicz Spaces and
    Application to Absorption-Induced Blow-Up Prevention in a Keller–Segel-Type Cross-Diffusion
    System,” <i>International Mathematics Research Notices</i>, vol. 2023, no. 19,
    pp. 16336–16393, 2022, doi: <a href="https://doi.org/10.1093/imrn/rnac286">10.1093/imrn/rnac286</a>.'
  mla: Winkler, Michael. “A Result on Parabolic Gradient Regularity in Orlicz Spaces
    and Application to Absorption-Induced Blow-Up Prevention in a Keller–Segel-Type
    Cross-Diffusion System.” <i>International Mathematics Research Notices</i>, vol.
    2023, no. 19, Oxford University Press (OUP), 2022, pp. 16336–93, doi:<a href="https://doi.org/10.1093/imrn/rnac286">10.1093/imrn/rnac286</a>.
  short: M. Winkler, International Mathematics Research Notices 2023 (2022) 16336–16393.
date_created: 2025-12-18T19:15:52Z
date_updated: 2025-12-18T20:11:43Z
doi: 10.1093/imrn/rnac286
intvolume: '      2023'
issue: '19'
language:
- iso: eng
page: 16336-16393
publication: International Mathematics Research Notices
publication_identifier:
  issn:
  - 1073-7928
  - 1687-0247
publication_status: published
publisher: Oxford University Press (OUP)
status: public
title: A Result on Parabolic Gradient Regularity in Orlicz Spaces and Application
  to Absorption-Induced Blow-Up Prevention in a Keller–Segel-Type Cross-Diffusion
  System
type: journal_article
user_id: '31496'
volume: 2023
year: '2022'
...
---
_id: '63274'
abstract:
- lang: eng
  text: "<jats:p>In a ball <jats:inline-formula><jats:alternatives><jats:tex-math>$\\Omega
    \\subset \\mathbb {R}^{n}$</jats:tex-math><jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\"
    mime-subtype=\"png\" xlink:href=\"S0308210522000397_inline1.png\" /></jats:alternatives></jats:inline-formula>
    with <jats:inline-formula><jats:alternatives><jats:tex-math>$n\\ge 2$</jats:tex-math><jats:inline-graphic
    xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210522000397_inline2.png\"
    /></jats:alternatives></jats:inline-formula>, the chemotaxis system\r\n<jats:disp-formula><jats:alternatives><jats:tex-math>\\[
    \\left\\{ \\begin{array}{@{}l} u_t = \\nabla \\cdot \\big( D(u)\\nabla u\\big)
    + \\nabla\\cdot \\big(\\dfrac{u}{v} \\nabla v\\big), \\\\ 0=\\Delta v - uv \\end{array}
    \\right. \\]</jats:tex-math><jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\"
    mime-subtype=\"png\" mimetype=\"image\" position=\"float\" xlink:href=\"S0308210522000397_eqnU1.png\"
    /></jats:alternatives></jats:disp-formula>is considered along with no-flux boundary
    conditions for <jats:inline-formula><jats:alternatives><jats:tex-math>$u$</jats:tex-math><jats:inline-graphic
    xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210522000397_inline3.png\"
    /></jats:alternatives></jats:inline-formula> and with prescribed constant positive
    Dirichlet boundary data for <jats:inline-formula><jats:alternatives><jats:tex-math>$v$</jats:tex-math><jats:inline-graphic
    xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210522000397_inline4.png\"
    /></jats:alternatives></jats:inline-formula>. It is shown that if <jats:inline-formula><jats:alternatives><jats:tex-math>$D\\in
    C^{3}([0,\\infty ))$</jats:tex-math><jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\"
    mime-subtype=\"png\" xlink:href=\"S0308210522000397_inline5.png\" /></jats:alternatives></jats:inline-formula>
    is such that <jats:inline-formula><jats:alternatives><jats:tex-math>$0&lt; D(\\xi
    ) \\le {K_D} (\\xi +1)^{-\\alpha }$</jats:tex-math><jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\"
    mime-subtype=\"png\" xlink:href=\"S0308210522000397_inline6.png\" /></jats:alternatives></jats:inline-formula>
    for all <jats:inline-formula><jats:alternatives><jats:tex-math>$\\xi &gt;0$</jats:tex-math><jats:inline-graphic
    xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210522000397_inline7.png\"
    /></jats:alternatives></jats:inline-formula> with some <jats:inline-formula><jats:alternatives><jats:tex-math>${K_D}&gt;0$</jats:tex-math><jats:inline-graphic
    xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210522000397_inline8.png\"
    /></jats:alternatives></jats:inline-formula> and <jats:inline-formula><jats:alternatives><jats:tex-math>$\\alpha
    &gt;0$</jats:tex-math><jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\"
    mime-subtype=\"png\" xlink:href=\"S0308210522000397_inline9.png\" /></jats:alternatives></jats:inline-formula>,
    then for all initial data from a considerably large set of radial functions on
    <jats:inline-formula><jats:alternatives><jats:tex-math>$\\Omega$</jats:tex-math><jats:inline-graphic
    xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210522000397_inline10.png\"
    /></jats:alternatives></jats:inline-formula>, the corresponding initial-boundary
    value problem admits a solution blowing up in finite time.</jats:p>"
author:
- first_name: Yulan
  full_name: Wang, Yulan
  last_name: Wang
- first_name: Michael
  full_name: Winkler, Michael
  id: '31496'
  last_name: Winkler
citation:
  ama: 'Wang Y, Winkler M. Finite-time blow-up in a repulsive chemotaxis-consumption
    system. <i>Proceedings of the Royal Society of Edinburgh: Section A Mathematics</i>.
    2022;153(4):1150-1166. doi:<a href="https://doi.org/10.1017/prm.2022.39">10.1017/prm.2022.39</a>'
  apa: 'Wang, Y., &#38; Winkler, M. (2022). Finite-time blow-up in a repulsive chemotaxis-consumption
    system. <i>Proceedings of the Royal Society of Edinburgh: Section A Mathematics</i>,
    <i>153</i>(4), 1150–1166. <a href="https://doi.org/10.1017/prm.2022.39">https://doi.org/10.1017/prm.2022.39</a>'
  bibtex: '@article{Wang_Winkler_2022, title={Finite-time blow-up in a repulsive chemotaxis-consumption
    system}, volume={153}, DOI={<a href="https://doi.org/10.1017/prm.2022.39">10.1017/prm.2022.39</a>},
    number={4}, journal={Proceedings of the Royal Society of Edinburgh: Section A
    Mathematics}, publisher={Cambridge University Press (CUP)}, author={Wang, Yulan
    and Winkler, Michael}, year={2022}, pages={1150–1166} }'
  chicago: 'Wang, Yulan, and Michael Winkler. “Finite-Time Blow-up in a Repulsive
    Chemotaxis-Consumption System.” <i>Proceedings of the Royal Society of Edinburgh:
    Section A Mathematics</i> 153, no. 4 (2022): 1150–66. <a href="https://doi.org/10.1017/prm.2022.39">https://doi.org/10.1017/prm.2022.39</a>.'
  ieee: 'Y. Wang and M. Winkler, “Finite-time blow-up in a repulsive chemotaxis-consumption
    system,” <i>Proceedings of the Royal Society of Edinburgh: Section A Mathematics</i>,
    vol. 153, no. 4, pp. 1150–1166, 2022, doi: <a href="https://doi.org/10.1017/prm.2022.39">10.1017/prm.2022.39</a>.'
  mla: 'Wang, Yulan, and Michael Winkler. “Finite-Time Blow-up in a Repulsive Chemotaxis-Consumption
    System.” <i>Proceedings of the Royal Society of Edinburgh: Section A Mathematics</i>,
    vol. 153, no. 4, Cambridge University Press (CUP), 2022, pp. 1150–66, doi:<a href="https://doi.org/10.1017/prm.2022.39">10.1017/prm.2022.39</a>.'
  short: 'Y. Wang, M. Winkler, Proceedings of the Royal Society of Edinburgh: Section
    A Mathematics 153 (2022) 1150–1166.'
date_created: 2025-12-18T19:14:20Z
date_updated: 2025-12-18T20:11:15Z
doi: 10.1017/prm.2022.39
intvolume: '       153'
issue: '4'
language:
- iso: eng
page: 1150-1166
publication: 'Proceedings of the Royal Society of Edinburgh: Section A Mathematics'
publication_identifier:
  issn:
  - 0308-2105
  - 1473-7124
publication_status: published
publisher: Cambridge University Press (CUP)
status: public
title: Finite-time blow-up in a repulsive chemotaxis-consumption system
type: journal_article
user_id: '31496'
volume: 153
year: '2022'
...
---
_id: '54311'
author:
- first_name: Richard
  full_name: Janus, Richard
  id: '31047'
  last_name: Janus
  orcid: https://orcid.org/0009-0006-0365-4546
citation:
  ama: 'Janus R. Kremers, Hermann (1860-1934). In: <i>Bauks</i>. ; 2022.'
  apa: Janus, R. (2022). Kremers, Hermann (1860-1934). In <i>Bauks</i>.
  bibtex: '@inbook{Janus_2022, title={Kremers, Hermann (1860-1934)}, booktitle={Bauks},
    author={Janus, Richard}, year={2022} }'
  chicago: Janus, Richard. “Kremers, Hermann (1860-1934).” In <i>Bauks</i>, 2022.
  ieee: R. Janus, “Kremers, Hermann (1860-1934),” in <i>Bauks</i>, 2022.
  mla: Janus, Richard. “Kremers, Hermann (1860-1934).” <i>Bauks</i>, 2022.
  short: 'R. Janus, in: Bauks, 2022.'
date_created: 2024-05-16T11:31:07Z
date_updated: 2024-11-18T17:19:17Z
department:
- _id: '36'
- _id: '20'
- _id: '500'
language:
- iso: eng
publication: Bauks
status: public
title: Kremers, Hermann (1860-1934)
type: encyclopedia_article
user_id: '31047'
year: '2022'
...
---
_id: '44488'
author:
- first_name: Nils Tobias
  full_name: Böer, Nils Tobias
  id: '52000'
  last_name: Böer
  orcid: 0000-0002-0236-7282
- first_name: Matthias
  full_name: Weigelt, Matthias
  id: '36388'
  last_name: Weigelt
- first_name: Christoph
  full_name: Schütz, Christoph
  last_name: Schütz
- first_name: Iris
  full_name: Güldenpenning, Iris
  id: '52931'
  last_name: Güldenpenning
  orcid: 0000-0003-0549-5543
citation:
  ama: 'Böer NT, Weigelt M, Schütz C, Güldenpenning I. Produktionskosten von Täuschungshandlungen
    im Sport: Der Einfluss von Übung auf die Durchführung von Blicktäuschungen im
    Basketball. In: Voelcker-Rehage C, Pixa NH, Rudisch J, et al., eds. <i>Ein Gehirn,
    viel Bewegung – Variabilität und Plastizität über die Lebensspanne. Abstractband
    der 54. Jahrestagung der Arbeitsgemeinschaft für Sportpsychologie (asp)</i>. Westfälische
    Wilhelms Universität Münster; 2022:60.'
  apa: 'Böer, N. T., Weigelt, M., Schütz, C., &#38; Güldenpenning, I. (2022). Produktionskosten
    von Täuschungshandlungen im Sport: Der Einfluss von Übung auf die Durchführung
    von Blicktäuschungen im Basketball. In C. Voelcker-Rehage, N. H. Pixa, J. Rudisch,
    V. Belkin, E. Eils, S. Fröhlich, T. Göcking, M. Hendricks, T. Janssen, R. Julian,
    L. Kopnarski, D. F. Kutz, M. Mack, L. Mendler, R. Stojan, &#38; L. Thorwesten
    (Eds.), <i>Ein Gehirn, viel Bewegung – Variabilität und Plastizität über die Lebensspanne.
    Abstractband der 54. Jahrestagung der Arbeitsgemeinschaft für Sportpsychologie
    (asp)</i> (p. 60). Westfälische Wilhelms Universität Münster.'
  bibtex: '@inproceedings{Böer_Weigelt_Schütz_Güldenpenning_2022, place={Münster},
    title={Produktionskosten von Täuschungshandlungen im Sport: Der Einfluss von Übung
    auf die Durchführung von Blicktäuschungen im Basketball}, booktitle={Ein Gehirn,
    viel Bewegung – Variabilität und Plastizität über die Lebensspanne. Abstractband
    der 54. Jahrestagung der Arbeitsgemeinschaft für Sportpsychologie (asp)}, publisher={Westfälische
    Wilhelms Universität Münster}, author={Böer, Nils Tobias and Weigelt, Matthias
    and Schütz, Christoph and Güldenpenning, Iris}, editor={Voelcker-Rehage, C. and
    Pixa, N. H. and Rudisch, J. and Belkin, V. and Eils, E. and Fröhlich, S. and Göcking,
    T. and Hendricks, M. and Janssen, T. and Julian, R. and et al.}, year={2022},
    pages={60} }'
  chicago: 'Böer, Nils Tobias, Matthias Weigelt, Christoph Schütz, and Iris Güldenpenning.
    “Produktionskosten von Täuschungshandlungen im Sport: Der Einfluss von Übung auf
    die Durchführung von Blicktäuschungen im Basketball.” In <i>Ein Gehirn, viel Bewegung
    – Variabilität und Plastizität über die Lebensspanne. Abstractband der 54. Jahrestagung
    der Arbeitsgemeinschaft für Sportpsychologie (asp)</i>, edited by C. Voelcker-Rehage,
    N. H. Pixa, J. Rudisch, V. Belkin, E. Eils, S. Fröhlich, T. Göcking, et al., 60.
    Münster: Westfälische Wilhelms Universität Münster, 2022.'
  ieee: 'N. T. Böer, M. Weigelt, C. Schütz, and I. Güldenpenning, “Produktionskosten
    von Täuschungshandlungen im Sport: Der Einfluss von Übung auf die Durchführung
    von Blicktäuschungen im Basketball,” in <i>Ein Gehirn, viel Bewegung – Variabilität
    und Plastizität über die Lebensspanne. Abstractband der 54. Jahrestagung der Arbeitsgemeinschaft
    für Sportpsychologie (asp)</i>, Münster, 2022, p. 60.'
  mla: 'Böer, Nils Tobias, et al. “Produktionskosten von Täuschungshandlungen im Sport:
    Der Einfluss von Übung auf die Durchführung von Blicktäuschungen im Basketball.”
    <i>Ein Gehirn, viel Bewegung – Variabilität und Plastizität über die Lebensspanne.
    Abstractband der 54. Jahrestagung der Arbeitsgemeinschaft für Sportpsychologie
    (asp)</i>, edited by C. Voelcker-Rehage et al., Westfälische Wilhelms Universität
    Münster, 2022, p. 60.'
  short: 'N.T. Böer, M. Weigelt, C. Schütz, I. Güldenpenning, in: C. Voelcker-Rehage,
    N.H. Pixa, J. Rudisch, V. Belkin, E. Eils, S. Fröhlich, T. Göcking, M. Hendricks,
    T. Janssen, R. Julian, L. Kopnarski, D.F. Kutz, M. Mack, L. Mendler, R. Stojan,
    L. Thorwesten (Eds.), Ein Gehirn, viel Bewegung – Variabilität und Plastizität
    über die Lebensspanne. Abstractband der 54. Jahrestagung der Arbeitsgemeinschaft
    für Sportpsychologie (asp), Westfälische Wilhelms Universität Münster, Münster,
    2022, p. 60.'
conference:
  end_date: 2022-06-18
  location: Münster
  name: 54. Jahrestagung der Arbeitsgemeinschaft für Sportpsychologie (asp)
  start_date: 2022-06-16
date_created: 2023-05-04T13:25:03Z
date_updated: 2024-11-29T11:06:40Z
ddc:
- '150'
department:
- _id: '17'
- _id: '266'
editor:
- first_name: C.
  full_name: Voelcker-Rehage, C.
  last_name: Voelcker-Rehage
- first_name: N. H.
  full_name: Pixa, N. H.
  last_name: Pixa
- first_name: J.
  full_name: Rudisch, J.
  last_name: Rudisch
- first_name: V.
  full_name: Belkin, V.
  last_name: Belkin
- first_name: E.
  full_name: Eils, E.
  last_name: Eils
- first_name: S.
  full_name: Fröhlich, S.
  last_name: Fröhlich
- first_name: T.
  full_name: Göcking, T.
  last_name: Göcking
- first_name: M.
  full_name: Hendricks, M.
  last_name: Hendricks
- first_name: T.
  full_name: Janssen, T.
  last_name: Janssen
- first_name: R.
  full_name: Julian, R.
  last_name: Julian
- first_name: L.
  full_name: Kopnarski, L.
  last_name: Kopnarski
- first_name: D. F.
  full_name: Kutz, D. F.
  last_name: Kutz
- first_name: M.
  full_name: Mack, M.
  last_name: Mack
- first_name: L.
  full_name: Mendler, L.
  last_name: Mendler
- first_name: R.
  full_name: Stojan, R.
  last_name: Stojan
- first_name: L.
  full_name: Thorwesten, L.
  last_name: Thorwesten
file:
- access_level: closed
  content_type: application/pdf
  creator: nboeer
  date_created: 2024-11-29T11:06:30Z
  date_updated: 2024-11-29T11:06:30Z
  file_id: '57503'
  file_name: asp2022 Böer et al.pdf
  file_size: 207167
  relation: main_file
  success: 1
file_date_updated: 2024-11-29T11:06:30Z
has_accepted_license: '1'
language:
- iso: ger
page: '60'
place: Münster
publication: Ein Gehirn, viel Bewegung – Variabilität und Plastizität über die Lebensspanne.
  Abstractband der 54. Jahrestagung der Arbeitsgemeinschaft für Sportpsychologie (asp)
publication_status: published
publisher: Westfälische Wilhelms Universität Münster
quality_controlled: '1'
status: public
title: 'Produktionskosten von Täuschungshandlungen im Sport: Der Einfluss von Übung
  auf die Durchführung von Blicktäuschungen im Basketball'
type: conference_abstract
user_id: '52000'
year: '2022'
...
---
_id: '57766'
author:
- first_name: Beate
  full_name: Flath, Beate
  id: '58896'
  last_name: Flath
  orcid: https://orcid.org/0000-0002-1648-0796
- first_name: Christoph
  full_name: Jacke, Christoph
  id: '20747'
  last_name: Jacke
- first_name: Manuel
  full_name: Troike, Manuel
  id: '33226'
  last_name: Troike
  orcid: 0009-0002-8606-3864
citation:
  ama: 'Flath B, Jacke C, Troike M. Transformational POP: Transitions, Breaks, and
    Crises in Popular Music (Studies) – Introduction. In: Flath B, Jacke C, Troike  M,
    eds. <i>Transformational POP: Transitions, Breaks, and Crises in Popular Music
    (Studies)</i>. Vol 2. Vibes The IASPM D-A-CH Series. ; 2022:7-15.'
  apa: 'Flath, B., Jacke, C., &#38; Troike, M. (2022). Transformational POP: Transitions,
    Breaks, and Crises in Popular Music (Studies) – Introduction. In B. Flath, C.
    Jacke, &#38; M. Troike  (Eds.), <i>Transformational POP: Transitions, Breaks,
    and Crises in Popular Music (Studies)</i> (Vol. 2, pp. 7–15).'
  bibtex: '@inbook{Flath_Jacke_Troike_2022, series={Vibes The IASPM D-A-CH Series},
    title={Transformational POP: Transitions, Breaks, and Crises in Popular Music
    (Studies) – Introduction}, volume={2}, booktitle={Transformational POP: Transitions,
    Breaks, and Crises in Popular Music (Studies)}, author={Flath, Beate and Jacke,
    Christoph and Troike, Manuel}, editor={Flath, Beate and Jacke, Christoph  and
    Troike , Manuel }, year={2022}, pages={7–15}, collection={Vibes The IASPM D-A-CH
    Series} }'
  chicago: 'Flath, Beate, Christoph Jacke, and Manuel Troike. “Transformational POP:
    Transitions, Breaks, and Crises in Popular Music (Studies) – Introduction.” In
    <i>Transformational POP: Transitions, Breaks, and Crises in Popular Music (Studies)</i>,
    edited by Beate Flath, Christoph  Jacke, and Manuel  Troike , 2:7–15. Vibes The
    IASPM D-A-CH Series, 2022.'
  ieee: 'B. Flath, C. Jacke, and M. Troike, “Transformational POP: Transitions, Breaks,
    and Crises in Popular Music (Studies) – Introduction,” in <i>Transformational
    POP: Transitions, Breaks, and Crises in Popular Music (Studies)</i>, vol. 2, B.
    Flath, C. Jacke, and M. Troike , Eds. 2022, pp. 7–15.'
  mla: 'Flath, Beate, et al. “Transformational POP: Transitions, Breaks, and Crises
    in Popular Music (Studies) – Introduction.” <i>Transformational POP: Transitions,
    Breaks, and Crises in Popular Music (Studies)</i>, edited by Beate Flath et al.,
    vol. 2, 2022, pp. 7–15.'
  short: 'B. Flath, C. Jacke, M. Troike, in: B. Flath, C. Jacke, M. Troike  (Eds.),
    Transformational POP: Transitions, Breaks, and Crises in Popular Music (Studies),
    2022, pp. 7–15.'
date_created: 2024-12-13T15:44:35Z
date_updated: 2024-12-13T15:44:56Z
department:
- _id: '36'
editor:
- first_name: Beate
  full_name: Flath, Beate
  last_name: Flath
- first_name: 'Christoph '
  full_name: 'Jacke, Christoph '
  last_name: Jacke
- first_name: 'Manuel '
  full_name: 'Troike , Manuel '
  last_name: 'Troike '
intvolume: '         2'
language:
- iso: eng
main_file_link:
- url: http://vibes-theseries.org/flath-jacke-troike-introduction/
page: 7-15
publication: 'Transformational POP: Transitions, Breaks, and Crises in Popular Music
  (Studies)'
publication_identifier:
  unknown:
  - 2748-6389
publication_status: published
series_title: Vibes The IASPM D-A-CH Series
status: public
title: 'Transformational POP: Transitions, Breaks, and Crises in Popular Music (Studies)
  – Introduction'
type: book_chapter
user_id: '58896'
volume: 2
year: '2022'
...
---
_id: '46660'
author:
- first_name: Eva
  full_name: Böhm, Eva
  id: '3043'
  last_name: Böhm
  orcid: 0000-0001-6053-1012
- first_name: A.
  full_name: Eggert, A.
  last_name: Eggert
- first_name: I.
  full_name: Garnefeld, I.
  last_name: Garnefeld
- first_name: H.
  full_name: Holzmüller, H.
  last_name: Holzmüller
- first_name: T
  full_name: Schaefers, T
  last_name: Schaefers
- first_name: Lena
  full_name: Steinhoff, Lena
  id: '4336'
  last_name: Steinhoff
- first_name: D.
  full_name: Woisetschläger, D.
  last_name: Woisetschläger
citation:
  ama: 'Böhm E, Eggert A, Garnefeld I, et al. Exploring the customer journey of voice
    commerce: A research agenda. <i>SMR - Journal of Service Management Research</i>.
    2022;6(4):216-231.'
  apa: 'Böhm, E., Eggert, A., Garnefeld, I., Holzmüller, H., Schaefers, T., Steinhoff,
    L., &#38; Woisetschläger, D. (2022). Exploring the customer journey of voice commerce:
    A research agenda. <i>SMR - Journal of Service Management Research</i>, <i>6</i>(4),
    216–231.'
  bibtex: '@article{Böhm_Eggert_Garnefeld_Holzmüller_Schaefers_Steinhoff_Woisetschläger_2022,
    title={Exploring the customer journey of voice commerce: A research agenda}, volume={6},
    number={4}, journal={SMR - Journal of Service Management Research}, author={Böhm,
    Eva and Eggert, A. and Garnefeld, I. and Holzmüller, H. and Schaefers, T and Steinhoff,
    Lena and Woisetschläger, D.}, year={2022}, pages={216–231} }'
  chicago: 'Böhm, Eva, A. Eggert, I. Garnefeld, H. Holzmüller, T Schaefers, Lena Steinhoff,
    and D. Woisetschläger. “Exploring the Customer Journey of Voice Commerce: A Research
    Agenda.” <i>SMR - Journal of Service Management Research</i> 6, no. 4 (2022):
    216–31.'
  ieee: 'E. Böhm <i>et al.</i>, “Exploring the customer journey of voice commerce:
    A research agenda,” <i>SMR - Journal of Service Management Research</i>, vol.
    6, no. 4, pp. 216–231, 2022.'
  mla: 'Böhm, Eva, et al. “Exploring the Customer Journey of Voice Commerce: A Research
    Agenda.” <i>SMR - Journal of Service Management Research</i>, vol. 6, no. 4, 2022,
    pp. 216–31.'
  short: E. Böhm, A. Eggert, I. Garnefeld, H. Holzmüller, T. Schaefers, L. Steinhoff,
    D. Woisetschläger, SMR - Journal of Service Management Research 6 (2022) 216–231.
date_created: 2023-08-25T07:17:47Z
date_updated: 2024-12-13T11:00:02Z
department:
- _id: '785'
extern: '1'
intvolume: '         6'
issue: '4'
language:
- iso: eng
page: 216-231
publication: SMR - Journal of Service Management Research
publication_status: published
status: public
title: 'Exploring the customer journey of voice commerce: A research agenda'
type: journal_article
user_id: '3043'
volume: 6
year: '2022'
...