---
_id: '53262'
author:
- first_name: Ignacio
  full_name: Santamaria, Ignacio
  last_name: Santamaria
- first_name: Mohammad
  full_name: Soleymani, Mohammad
  last_name: Soleymani
- first_name: Eduard
  full_name: Jorswieck, Eduard
  last_name: Jorswieck
- first_name: Jesús
  full_name: Gutiérrez, Jesús
  last_name: Gutiérrez
citation:
  ama: Santamaria I, Soleymani M, Jorswieck E, Gutiérrez J. SNR Maximization in Beyond
    Diagonal RIS-Assisted Single and Multiple Antenna Links. <i>IEEE Signal Processing
    Letters</i>. 2023;30:923-926. doi:<a href="https://doi.org/10.1109/lsp.2023.3296902">10.1109/lsp.2023.3296902</a>
  apa: Santamaria, I., Soleymani, M., Jorswieck, E., &#38; Gutiérrez, J. (2023). SNR
    Maximization in Beyond Diagonal RIS-Assisted Single and Multiple Antenna Links.
    <i>IEEE Signal Processing Letters</i>, <i>30</i>, 923–926. <a href="https://doi.org/10.1109/lsp.2023.3296902">https://doi.org/10.1109/lsp.2023.3296902</a>
  bibtex: '@article{Santamaria_Soleymani_Jorswieck_Gutiérrez_2023, title={SNR Maximization
    in Beyond Diagonal RIS-Assisted Single and Multiple Antenna Links}, volume={30},
    DOI={<a href="https://doi.org/10.1109/lsp.2023.3296902">10.1109/lsp.2023.3296902</a>},
    journal={IEEE Signal Processing Letters}, publisher={Institute of Electrical and
    Electronics Engineers (IEEE)}, author={Santamaria, Ignacio and Soleymani, Mohammad
    and Jorswieck, Eduard and Gutiérrez, Jesús}, year={2023}, pages={923–926} }'
  chicago: 'Santamaria, Ignacio, Mohammad Soleymani, Eduard Jorswieck, and Jesús Gutiérrez.
    “SNR Maximization in Beyond Diagonal RIS-Assisted Single and Multiple Antenna
    Links.” <i>IEEE Signal Processing Letters</i> 30 (2023): 923–26. <a href="https://doi.org/10.1109/lsp.2023.3296902">https://doi.org/10.1109/lsp.2023.3296902</a>.'
  ieee: 'I. Santamaria, M. Soleymani, E. Jorswieck, and J. Gutiérrez, “SNR Maximization
    in Beyond Diagonal RIS-Assisted Single and Multiple Antenna Links,” <i>IEEE Signal
    Processing Letters</i>, vol. 30, pp. 923–926, 2023, doi: <a href="https://doi.org/10.1109/lsp.2023.3296902">10.1109/lsp.2023.3296902</a>.'
  mla: Santamaria, Ignacio, et al. “SNR Maximization in Beyond Diagonal RIS-Assisted
    Single and Multiple Antenna Links.” <i>IEEE Signal Processing Letters</i>, vol.
    30, Institute of Electrical and Electronics Engineers (IEEE), 2023, pp. 923–26,
    doi:<a href="https://doi.org/10.1109/lsp.2023.3296902">10.1109/lsp.2023.3296902</a>.
  short: I. Santamaria, M. Soleymani, E. Jorswieck, J. Gutiérrez, IEEE Signal Processing
    Letters 30 (2023) 923–926.
date_created: 2024-04-05T09:01:21Z
date_updated: 2024-04-05T13:20:51Z
department:
- _id: '263'
doi: 10.1109/lsp.2023.3296902
intvolume: '        30'
keyword:
- Applied Mathematics
- Electrical and Electronic Engineering
- Signal Processing
language:
- iso: eng
page: 923-926
publication: IEEE Signal Processing Letters
publication_identifier:
  issn:
  - 1070-9908
  - 1558-2361
publication_status: published
publisher: Institute of Electrical and Electronics Engineers (IEEE)
status: public
title: SNR Maximization in Beyond Diagonal RIS-Assisted Single and Multiple Antenna
  Links
type: journal_article
user_id: '67076'
volume: 30
year: '2023'
...
---
_id: '53261'
author:
- first_name: Mohammad
  full_name: Soleymani, Mohammad
  last_name: Soleymani
- first_name: Ignacio
  full_name: Santamaria, Ignacio
  last_name: Santamaria
- first_name: Eduard
  full_name: Jorswieck, Eduard
  last_name: Jorswieck
- first_name: Bruno
  full_name: Clerckx, Bruno
  last_name: Clerckx
citation:
  ama: Soleymani M, Santamaria I, Jorswieck E, Clerckx B. Optimization of Rate-Splitting
    Multiple Access in Beyond Diagonal RIS-assisted URLLC Systems. <i>IEEE Transactions
    on Wireless Communications</i>. Published online 2023:1-1. doi:<a href="https://doi.org/10.1109/twc.2023.3324190">10.1109/twc.2023.3324190</a>
  apa: Soleymani, M., Santamaria, I., Jorswieck, E., &#38; Clerckx, B. (2023). Optimization
    of Rate-Splitting Multiple Access in Beyond Diagonal RIS-assisted URLLC Systems.
    <i>IEEE Transactions on Wireless Communications</i>, 1–1. <a href="https://doi.org/10.1109/twc.2023.3324190">https://doi.org/10.1109/twc.2023.3324190</a>
  bibtex: '@article{Soleymani_Santamaria_Jorswieck_Clerckx_2023, title={Optimization
    of Rate-Splitting Multiple Access in Beyond Diagonal RIS-assisted URLLC Systems},
    DOI={<a href="https://doi.org/10.1109/twc.2023.3324190">10.1109/twc.2023.3324190</a>},
    journal={IEEE Transactions on Wireless Communications}, publisher={Institute of
    Electrical and Electronics Engineers (IEEE)}, author={Soleymani, Mohammad and
    Santamaria, Ignacio and Jorswieck, Eduard and Clerckx, Bruno}, year={2023}, pages={1–1}
    }'
  chicago: Soleymani, Mohammad, Ignacio Santamaria, Eduard Jorswieck, and Bruno Clerckx.
    “Optimization of Rate-Splitting Multiple Access in Beyond Diagonal RIS-Assisted
    URLLC Systems.” <i>IEEE Transactions on Wireless Communications</i>, 2023, 1–1.
    <a href="https://doi.org/10.1109/twc.2023.3324190">https://doi.org/10.1109/twc.2023.3324190</a>.
  ieee: 'M. Soleymani, I. Santamaria, E. Jorswieck, and B. Clerckx, “Optimization
    of Rate-Splitting Multiple Access in Beyond Diagonal RIS-assisted URLLC Systems,”
    <i>IEEE Transactions on Wireless Communications</i>, pp. 1–1, 2023, doi: <a href="https://doi.org/10.1109/twc.2023.3324190">10.1109/twc.2023.3324190</a>.'
  mla: Soleymani, Mohammad, et al. “Optimization of Rate-Splitting Multiple Access
    in Beyond Diagonal RIS-Assisted URLLC Systems.” <i>IEEE Transactions on Wireless
    Communications</i>, Institute of Electrical and Electronics Engineers (IEEE),
    2023, pp. 1–1, doi:<a href="https://doi.org/10.1109/twc.2023.3324190">10.1109/twc.2023.3324190</a>.
  short: M. Soleymani, I. Santamaria, E. Jorswieck, B. Clerckx, IEEE Transactions
    on Wireless Communications (2023) 1–1.
date_created: 2024-04-05T09:01:04Z
date_updated: 2024-04-05T13:20:40Z
department:
- _id: '263'
doi: 10.1109/twc.2023.3324190
keyword:
- Applied Mathematics
- Electrical and Electronic Engineering
- Computer Science Applications
language:
- iso: eng
page: 1-1
publication: IEEE Transactions on Wireless Communications
publication_identifier:
  issn:
  - 1536-1276
  - 1558-2248
publication_status: published
publisher: Institute of Electrical and Electronics Engineers (IEEE)
status: public
title: Optimization of Rate-Splitting Multiple Access in Beyond Diagonal RIS-assisted
  URLLC Systems
type: journal_article
user_id: '67076'
year: '2023'
...
---
_id: '53317'
author:
- first_name: Youshan
  full_name: Tao, Youshan
  last_name: Tao
- first_name: Michael
  full_name: Winkler, Michael
  last_name: Winkler
citation:
  ama: Tao Y, Winkler M. Global smooth solutions in a three-dimensional cross-diffusive
    SIS epidemic model with saturated taxis at large densities. <i>Evolution Equations
    and Control Theory</i>. 2023;12(6):1676-1687. doi:<a href="https://doi.org/10.3934/eect.2023031">10.3934/eect.2023031</a>
  apa: Tao, Y., &#38; Winkler, M. (2023). Global smooth solutions in a three-dimensional
    cross-diffusive SIS epidemic model with saturated taxis at large densities. <i>Evolution
    Equations and Control Theory</i>, <i>12</i>(6), 1676–1687. <a href="https://doi.org/10.3934/eect.2023031">https://doi.org/10.3934/eect.2023031</a>
  bibtex: '@article{Tao_Winkler_2023, title={Global smooth solutions in a three-dimensional
    cross-diffusive SIS epidemic model with saturated taxis at large densities}, volume={12},
    DOI={<a href="https://doi.org/10.3934/eect.2023031">10.3934/eect.2023031</a>},
    number={6}, journal={Evolution Equations and Control Theory}, publisher={American
    Institute of Mathematical Sciences (AIMS)}, author={Tao, Youshan and Winkler,
    Michael}, year={2023}, pages={1676–1687} }'
  chicago: 'Tao, Youshan, and Michael Winkler. “Global Smooth Solutions in a Three-Dimensional
    Cross-Diffusive SIS Epidemic Model with Saturated Taxis at Large Densities.” <i>Evolution
    Equations and Control Theory</i> 12, no. 6 (2023): 1676–87. <a href="https://doi.org/10.3934/eect.2023031">https://doi.org/10.3934/eect.2023031</a>.'
  ieee: 'Y. Tao and M. Winkler, “Global smooth solutions in a three-dimensional cross-diffusive
    SIS epidemic model with saturated taxis at large densities,” <i>Evolution Equations
    and Control Theory</i>, vol. 12, no. 6, pp. 1676–1687, 2023, doi: <a href="https://doi.org/10.3934/eect.2023031">10.3934/eect.2023031</a>.'
  mla: Tao, Youshan, and Michael Winkler. “Global Smooth Solutions in a Three-Dimensional
    Cross-Diffusive SIS Epidemic Model with Saturated Taxis at Large Densities.” <i>Evolution
    Equations and Control Theory</i>, vol. 12, no. 6, American Institute of Mathematical
    Sciences (AIMS), 2023, pp. 1676–87, doi:<a href="https://doi.org/10.3934/eect.2023031">10.3934/eect.2023031</a>.
  short: Y. Tao, M. Winkler, Evolution Equations and Control Theory 12 (2023) 1676–1687.
date_created: 2024-04-07T12:30:25Z
date_updated: 2024-04-07T12:36:17Z
doi: 10.3934/eect.2023031
intvolume: '        12'
issue: '6'
keyword:
- Applied Mathematics
- Control and Optimization
- Modeling and Simulation
language:
- iso: eng
page: 1676-1687
publication: Evolution Equations and Control Theory
publication_identifier:
  issn:
  - 2163-2480
publication_status: published
publisher: American Institute of Mathematical Sciences (AIMS)
status: public
title: Global smooth solutions in a three-dimensional cross-diffusive SIS epidemic
  model with saturated taxis at large densities
type: journal_article
user_id: '31496'
volume: 12
year: '2023'
...
---
_id: '53320'
author:
- first_name: Michael
  full_name: Winkler, Michael
  last_name: Winkler
citation:
  ama: Winkler M. A quantitative strong parabolic maximum principle and application
    to a taxis-type migration–consumption model involving signal-dependent degenerate
    diffusion. <i>Annales de l’Institut Henri Poincaré C, Analyse non linéaire</i>.
    Published online 2023. doi:<a href="https://doi.org/10.4171/aihpc/73">10.4171/aihpc/73</a>
  apa: Winkler, M. (2023). A quantitative strong parabolic maximum principle and application
    to a taxis-type migration–consumption model involving signal-dependent degenerate
    diffusion. <i>Annales de l’Institut Henri Poincaré C, Analyse Non Linéaire</i>.
    <a href="https://doi.org/10.4171/aihpc/73">https://doi.org/10.4171/aihpc/73</a>
  bibtex: '@article{Winkler_2023, title={A quantitative strong parabolic maximum principle
    and application to a taxis-type migration–consumption model involving signal-dependent
    degenerate diffusion}, DOI={<a href="https://doi.org/10.4171/aihpc/73">10.4171/aihpc/73</a>},
    journal={Annales de l’Institut Henri Poincaré C, Analyse non linéaire}, publisher={European
    Mathematical Society - EMS - Publishing House GmbH}, author={Winkler, Michael},
    year={2023} }'
  chicago: Winkler, Michael. “A Quantitative Strong Parabolic Maximum Principle and
    Application to a Taxis-Type Migration–Consumption Model Involving Signal-Dependent
    Degenerate Diffusion.” <i>Annales de l’Institut Henri Poincaré C, Analyse Non
    Linéaire</i>, 2023. <a href="https://doi.org/10.4171/aihpc/73">https://doi.org/10.4171/aihpc/73</a>.
  ieee: 'M. Winkler, “A quantitative strong parabolic maximum principle and application
    to a taxis-type migration–consumption model involving signal-dependent degenerate
    diffusion,” <i>Annales de l’Institut Henri Poincaré C, Analyse non linéaire</i>,
    2023, doi: <a href="https://doi.org/10.4171/aihpc/73">10.4171/aihpc/73</a>.'
  mla: Winkler, Michael. “A Quantitative Strong Parabolic Maximum Principle and Application
    to a Taxis-Type Migration–Consumption Model Involving Signal-Dependent Degenerate
    Diffusion.” <i>Annales de l’Institut Henri Poincaré C, Analyse Non Linéaire</i>,
    European Mathematical Society - EMS - Publishing House GmbH, 2023, doi:<a href="https://doi.org/10.4171/aihpc/73">10.4171/aihpc/73</a>.
  short: M. Winkler, Annales de l’Institut Henri Poincaré C, Analyse Non Linéaire
    (2023).
date_created: 2024-04-07T12:34:35Z
date_updated: 2024-04-07T12:36:00Z
doi: 10.4171/aihpc/73
keyword:
- Mathematical Physics
- Analysis
- Applied Mathematics
language:
- iso: eng
publication: Annales de l'Institut Henri Poincaré C, Analyse non linéaire
publication_identifier:
  issn:
  - 0294-1449
  - 1873-1430
publication_status: published
publisher: European Mathematical Society - EMS - Publishing House GmbH
status: public
title: A quantitative strong parabolic maximum principle and application to a taxis-type
  migration–consumption model involving signal-dependent degenerate diffusion
type: journal_article
user_id: '31496'
year: '2023'
...
---
_id: '53318'
author:
- first_name: Genglin
  full_name: Li, Genglin
  last_name: Li
- first_name: Michael
  full_name: Winkler, Michael
  last_name: Winkler
citation:
  ama: Li G, Winkler M. Refined regularity analysis for a Keller-Segel-consumption
    system involving signal-dependent motilities. <i>Applicable Analysis</i>. 2023;103(1):45-64.
    doi:<a href="https://doi.org/10.1080/00036811.2023.2173183">10.1080/00036811.2023.2173183</a>
  apa: Li, G., &#38; Winkler, M. (2023). Refined regularity analysis for a Keller-Segel-consumption
    system involving signal-dependent motilities. <i>Applicable Analysis</i>, <i>103</i>(1),
    45–64. <a href="https://doi.org/10.1080/00036811.2023.2173183">https://doi.org/10.1080/00036811.2023.2173183</a>
  bibtex: '@article{Li_Winkler_2023, title={Refined regularity analysis for a Keller-Segel-consumption
    system involving signal-dependent motilities}, volume={103}, DOI={<a href="https://doi.org/10.1080/00036811.2023.2173183">10.1080/00036811.2023.2173183</a>},
    number={1}, journal={Applicable Analysis}, publisher={Informa UK Limited}, author={Li,
    Genglin and Winkler, Michael}, year={2023}, pages={45–64} }'
  chicago: 'Li, Genglin, and Michael Winkler. “Refined Regularity Analysis for a Keller-Segel-Consumption
    System Involving Signal-Dependent Motilities.” <i>Applicable Analysis</i> 103,
    no. 1 (2023): 45–64. <a href="https://doi.org/10.1080/00036811.2023.2173183">https://doi.org/10.1080/00036811.2023.2173183</a>.'
  ieee: 'G. Li and M. Winkler, “Refined regularity analysis for a Keller-Segel-consumption
    system involving signal-dependent motilities,” <i>Applicable Analysis</i>, vol.
    103, no. 1, pp. 45–64, 2023, doi: <a href="https://doi.org/10.1080/00036811.2023.2173183">10.1080/00036811.2023.2173183</a>.'
  mla: Li, Genglin, and Michael Winkler. “Refined Regularity Analysis for a Keller-Segel-Consumption
    System Involving Signal-Dependent Motilities.” <i>Applicable Analysis</i>, vol.
    103, no. 1, Informa UK Limited, 2023, pp. 45–64, doi:<a href="https://doi.org/10.1080/00036811.2023.2173183">10.1080/00036811.2023.2173183</a>.
  short: G. Li, M. Winkler, Applicable Analysis 103 (2023) 45–64.
date_created: 2024-04-07T12:32:55Z
date_updated: 2024-04-07T12:36:11Z
doi: 10.1080/00036811.2023.2173183
intvolume: '       103'
issue: '1'
keyword:
- Applied Mathematics
- Analysis
language:
- iso: eng
page: 45-64
publication: Applicable Analysis
publication_identifier:
  issn:
  - 0003-6811
  - 1563-504X
publication_status: published
publisher: Informa UK Limited
status: public
title: Refined regularity analysis for a Keller-Segel-consumption system involving
  signal-dependent motilities
type: journal_article
user_id: '31496'
volume: 103
year: '2023'
...
---
_id: '53328'
abstract:
- lang: eng
  text: <jats:p> As a simplified version of a three-component taxis cascade model
    accounting for different migration strategies of two population groups in search
    of food, a two-component nonlocal nutrient taxis system is considered in a two-dimensional
    bounded convex domain with smooth boundary. For any given conveniently regular
    and biologically meaningful initial data, smallness conditions on the prescribed
    resource growth and on the initial nutrient signal concentration are identified
    which ensure the global existence of a global classical solution to the corresponding
    no-flux initial-boundary value problem. Moreover, under additional assumptions
    on the food production source these solutions are shown to be bounded, and to
    stabilize toward semi-trivial equilibria in the large time limit, respectively.
    </jats:p>
author:
- first_name: Youshan
  full_name: Tao, Youshan
  last_name: Tao
- first_name: Michael
  full_name: Winkler, Michael
  last_name: Winkler
citation:
  ama: Tao Y, Winkler M. Small-signal solutions to a nonlocal cross-diffusion model
    for interaction of scroungers with rapidly diffusing foragers. <i>Mathematical
    Models and Methods in Applied Sciences</i>. 2023;33(01):103-138. doi:<a href="https://doi.org/10.1142/s0218202523500045">10.1142/s0218202523500045</a>
  apa: Tao, Y., &#38; Winkler, M. (2023). Small-signal solutions to a nonlocal cross-diffusion
    model for interaction of scroungers with rapidly diffusing foragers. <i>Mathematical
    Models and Methods in Applied Sciences</i>, <i>33</i>(01), 103–138. <a href="https://doi.org/10.1142/s0218202523500045">https://doi.org/10.1142/s0218202523500045</a>
  bibtex: '@article{Tao_Winkler_2023, title={Small-signal solutions to a nonlocal
    cross-diffusion model for interaction of scroungers with rapidly diffusing foragers},
    volume={33}, DOI={<a href="https://doi.org/10.1142/s0218202523500045">10.1142/s0218202523500045</a>},
    number={01}, journal={Mathematical Models and Methods in Applied Sciences}, publisher={World
    Scientific Pub Co Pte Ltd}, author={Tao, Youshan and Winkler, Michael}, year={2023},
    pages={103–138} }'
  chicago: 'Tao, Youshan, and Michael Winkler. “Small-Signal Solutions to a Nonlocal
    Cross-Diffusion Model for Interaction of Scroungers with Rapidly Diffusing Foragers.”
    <i>Mathematical Models and Methods in Applied Sciences</i> 33, no. 01 (2023):
    103–38. <a href="https://doi.org/10.1142/s0218202523500045">https://doi.org/10.1142/s0218202523500045</a>.'
  ieee: 'Y. Tao and M. Winkler, “Small-signal solutions to a nonlocal cross-diffusion
    model for interaction of scroungers with rapidly diffusing foragers,” <i>Mathematical
    Models and Methods in Applied Sciences</i>, vol. 33, no. 01, pp. 103–138, 2023,
    doi: <a href="https://doi.org/10.1142/s0218202523500045">10.1142/s0218202523500045</a>.'
  mla: Tao, Youshan, and Michael Winkler. “Small-Signal Solutions to a Nonlocal Cross-Diffusion
    Model for Interaction of Scroungers with Rapidly Diffusing Foragers.” <i>Mathematical
    Models and Methods in Applied Sciences</i>, vol. 33, no. 01, World Scientific
    Pub Co Pte Ltd, 2023, pp. 103–38, doi:<a href="https://doi.org/10.1142/s0218202523500045">10.1142/s0218202523500045</a>.
  short: Y. Tao, M. Winkler, Mathematical Models and Methods in Applied Sciences 33
    (2023) 103–138.
date_created: 2024-04-07T12:43:13Z
date_updated: 2024-04-07T12:43:17Z
doi: 10.1142/s0218202523500045
intvolume: '        33'
issue: '01'
keyword:
- Applied Mathematics
- Modeling and Simulation
language:
- iso: eng
page: 103-138
publication: Mathematical Models and Methods in Applied Sciences
publication_identifier:
  issn:
  - 0218-2025
  - 1793-6314
publication_status: published
publisher: World Scientific Pub Co Pte Ltd
status: public
title: Small-signal solutions to a nonlocal cross-diffusion model for interaction
  of scroungers with rapidly diffusing foragers
type: journal_article
user_id: '31496'
volume: 33
year: '2023'
...
---
_id: '53324'
article_number: '180'
author:
- first_name: Jaewook
  full_name: Ahn, Jaewook
  last_name: Ahn
- first_name: Michael
  full_name: Winkler, Michael
  last_name: Winkler
citation:
  ama: Ahn J, Winkler M. A critical exponent for blow-up in a two-dimensional chemotaxis-consumption
    system. <i>Calculus of Variations and Partial Differential Equations</i>. 2023;62(6).
    doi:<a href="https://doi.org/10.1007/s00526-023-02523-5">10.1007/s00526-023-02523-5</a>
  apa: Ahn, J., &#38; Winkler, M. (2023). A critical exponent for blow-up in a two-dimensional
    chemotaxis-consumption system. <i>Calculus of Variations and Partial Differential
    Equations</i>, <i>62</i>(6), Article 180. <a href="https://doi.org/10.1007/s00526-023-02523-5">https://doi.org/10.1007/s00526-023-02523-5</a>
  bibtex: '@article{Ahn_Winkler_2023, title={A critical exponent for blow-up in a
    two-dimensional chemotaxis-consumption system}, volume={62}, DOI={<a href="https://doi.org/10.1007/s00526-023-02523-5">10.1007/s00526-023-02523-5</a>},
    number={6180}, journal={Calculus of Variations and Partial Differential Equations},
    publisher={Springer Science and Business Media LLC}, author={Ahn, Jaewook and
    Winkler, Michael}, year={2023} }'
  chicago: Ahn, Jaewook, and Michael Winkler. “A Critical Exponent for Blow-up in
    a Two-Dimensional Chemotaxis-Consumption System.” <i>Calculus of Variations and
    Partial Differential Equations</i> 62, no. 6 (2023). <a href="https://doi.org/10.1007/s00526-023-02523-5">https://doi.org/10.1007/s00526-023-02523-5</a>.
  ieee: 'J. Ahn and M. Winkler, “A critical exponent for blow-up in a two-dimensional
    chemotaxis-consumption system,” <i>Calculus of Variations and Partial Differential
    Equations</i>, vol. 62, no. 6, Art. no. 180, 2023, doi: <a href="https://doi.org/10.1007/s00526-023-02523-5">10.1007/s00526-023-02523-5</a>.'
  mla: Ahn, Jaewook, and Michael Winkler. “A Critical Exponent for Blow-up in a Two-Dimensional
    Chemotaxis-Consumption System.” <i>Calculus of Variations and Partial Differential
    Equations</i>, vol. 62, no. 6, 180, Springer Science and Business Media LLC, 2023,
    doi:<a href="https://doi.org/10.1007/s00526-023-02523-5">10.1007/s00526-023-02523-5</a>.
  short: J. Ahn, M. Winkler, Calculus of Variations and Partial Differential Equations
    62 (2023).
date_created: 2024-04-07T12:40:02Z
date_updated: 2024-04-07T12:40:06Z
doi: 10.1007/s00526-023-02523-5
intvolume: '        62'
issue: '6'
keyword:
- Applied Mathematics
- Analysis
language:
- iso: eng
publication: Calculus of Variations and Partial Differential Equations
publication_identifier:
  issn:
  - 0944-2669
  - 1432-0835
publication_status: published
publisher: Springer Science and Business Media LLC
status: public
title: A critical exponent for blow-up in a two-dimensional chemotaxis-consumption
  system
type: journal_article
user_id: '31496'
volume: 62
year: '2023'
...
---
_id: '53329'
article_number: '103820'
author:
- first_name: Youshan
  full_name: Tao, Youshan
  last_name: Tao
- first_name: Michael
  full_name: Winkler, Michael
  last_name: Winkler
citation:
  ama: 'Tao Y, Winkler M. Analysis of a chemotaxis-SIS epidemic model with unbounded
    infection force. <i>Nonlinear Analysis: Real World Applications</i>. 2023;71.
    doi:<a href="https://doi.org/10.1016/j.nonrwa.2022.103820">10.1016/j.nonrwa.2022.103820</a>'
  apa: 'Tao, Y., &#38; Winkler, M. (2023). Analysis of a chemotaxis-SIS epidemic model
    with unbounded infection force. <i>Nonlinear Analysis: Real World Applications</i>,
    <i>71</i>, Article 103820. <a href="https://doi.org/10.1016/j.nonrwa.2022.103820">https://doi.org/10.1016/j.nonrwa.2022.103820</a>'
  bibtex: '@article{Tao_Winkler_2023, title={Analysis of a chemotaxis-SIS epidemic
    model with unbounded infection force}, volume={71}, DOI={<a href="https://doi.org/10.1016/j.nonrwa.2022.103820">10.1016/j.nonrwa.2022.103820</a>},
    number={103820}, journal={Nonlinear Analysis: Real World Applications}, publisher={Elsevier
    BV}, author={Tao, Youshan and Winkler, Michael}, year={2023} }'
  chicago: 'Tao, Youshan, and Michael Winkler. “Analysis of a Chemotaxis-SIS Epidemic
    Model with Unbounded Infection Force.” <i>Nonlinear Analysis: Real World Applications</i>
    71 (2023). <a href="https://doi.org/10.1016/j.nonrwa.2022.103820">https://doi.org/10.1016/j.nonrwa.2022.103820</a>.'
  ieee: 'Y. Tao and M. Winkler, “Analysis of a chemotaxis-SIS epidemic model with
    unbounded infection force,” <i>Nonlinear Analysis: Real World Applications</i>,
    vol. 71, Art. no. 103820, 2023, doi: <a href="https://doi.org/10.1016/j.nonrwa.2022.103820">10.1016/j.nonrwa.2022.103820</a>.'
  mla: 'Tao, Youshan, and Michael Winkler. “Analysis of a Chemotaxis-SIS Epidemic
    Model with Unbounded Infection Force.” <i>Nonlinear Analysis: Real World Applications</i>,
    vol. 71, 103820, Elsevier BV, 2023, doi:<a href="https://doi.org/10.1016/j.nonrwa.2022.103820">10.1016/j.nonrwa.2022.103820</a>.'
  short: 'Y. Tao, M. Winkler, Nonlinear Analysis: Real World Applications 71 (2023).'
date_created: 2024-04-07T12:43:49Z
date_updated: 2024-04-07T12:43:53Z
doi: 10.1016/j.nonrwa.2022.103820
intvolume: '        71'
keyword:
- Applied Mathematics
- Computational Mathematics
- General Economics
- Econometrics and Finance
- General Engineering
- General Medicine
- Analysis
language:
- iso: eng
publication: 'Nonlinear Analysis: Real World Applications'
publication_identifier:
  issn:
  - 1468-1218
publication_status: published
publisher: Elsevier BV
status: public
title: Analysis of a chemotaxis-SIS epidemic model with unbounded infection force
type: journal_article
user_id: '31496'
volume: 71
year: '2023'
...
---
_id: '53326'
author:
- first_name: Genglin
  full_name: Li, Genglin
  last_name: Li
- first_name: Michael
  full_name: Winkler, Michael
  last_name: Winkler
citation:
  ama: Li G, Winkler M. Relaxation in a Keller-Segel-consumption system involving
    signal-dependent motilities. <i>Communications in Mathematical Sciences</i>. 2023;21(2):299-322.
    doi:<a href="https://doi.org/10.4310/cms.2023.v21.n2.a1">10.4310/cms.2023.v21.n2.a1</a>
  apa: Li, G., &#38; Winkler, M. (2023). Relaxation in a Keller-Segel-consumption
    system involving signal-dependent motilities. <i>Communications in Mathematical
    Sciences</i>, <i>21</i>(2), 299–322. <a href="https://doi.org/10.4310/cms.2023.v21.n2.a1">https://doi.org/10.4310/cms.2023.v21.n2.a1</a>
  bibtex: '@article{Li_Winkler_2023, title={Relaxation in a Keller-Segel-consumption
    system involving signal-dependent motilities}, volume={21}, DOI={<a href="https://doi.org/10.4310/cms.2023.v21.n2.a1">10.4310/cms.2023.v21.n2.a1</a>},
    number={2}, journal={Communications in Mathematical Sciences}, publisher={International
    Press of Boston}, author={Li, Genglin and Winkler, Michael}, year={2023}, pages={299–322}
    }'
  chicago: 'Li, Genglin, and Michael Winkler. “Relaxation in a Keller-Segel-Consumption
    System Involving Signal-Dependent Motilities.” <i>Communications in Mathematical
    Sciences</i> 21, no. 2 (2023): 299–322. <a href="https://doi.org/10.4310/cms.2023.v21.n2.a1">https://doi.org/10.4310/cms.2023.v21.n2.a1</a>.'
  ieee: 'G. Li and M. Winkler, “Relaxation in a Keller-Segel-consumption system involving
    signal-dependent motilities,” <i>Communications in Mathematical Sciences</i>,
    vol. 21, no. 2, pp. 299–322, 2023, doi: <a href="https://doi.org/10.4310/cms.2023.v21.n2.a1">10.4310/cms.2023.v21.n2.a1</a>.'
  mla: Li, Genglin, and Michael Winkler. “Relaxation in a Keller-Segel-Consumption
    System Involving Signal-Dependent Motilities.” <i>Communications in Mathematical
    Sciences</i>, vol. 21, no. 2, International Press of Boston, 2023, pp. 299–322,
    doi:<a href="https://doi.org/10.4310/cms.2023.v21.n2.a1">10.4310/cms.2023.v21.n2.a1</a>.
  short: G. Li, M. Winkler, Communications in Mathematical Sciences 21 (2023) 299–322.
date_created: 2024-04-07T12:41:49Z
date_updated: 2024-04-07T12:41:54Z
doi: 10.4310/cms.2023.v21.n2.a1
intvolume: '        21'
issue: '2'
keyword:
- Applied Mathematics
- General Mathematics
language:
- iso: eng
page: 299-322
publication: Communications in Mathematical Sciences
publication_identifier:
  issn:
  - 1539-6746
  - 1945-0796
publication_status: published
publisher: International Press of Boston
status: public
title: Relaxation in a Keller-Segel-consumption system involving signal-dependent
  motilities
type: journal_article
user_id: '31496'
volume: 21
year: '2023'
...
---
_id: '53343'
abstract:
- lang: eng
  text: "<jats:title>Abstract</jats:title>\r\n               <jats:p>The Cauchy problem
    in <jats:inline-formula>\r\n                     <jats:alternatives>\r\n                        <jats:inline-graphic
    xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2022-0578_eq_001.png\"
    />\r\n                        <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\r\n
    \                          <m:msup>\r\n                              <m:mrow>\r\n
    \                                <m:mi mathvariant=\"double-struck\">R</m:mi>\r\n
    \                             </m:mrow>\r\n                              <m:mrow>\r\n
    \                                <m:mi>n</m:mi>\r\n                              </m:mrow>\r\n
    \                          </m:msup>\r\n                        </m:math>\r\n
    \                       <jats:tex-math>{{\\mathbb{R}}}^{n}</jats:tex-math>\r\n
    \                    </jats:alternatives>\r\n                  </jats:inline-formula>,
    <jats:inline-formula>\r\n                     <jats:alternatives>\r\n                        <jats:inline-graphic
    xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2022-0578_eq_002.png\"
    />\r\n                        <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\r\n
    \                          <m:mi>n</m:mi>\r\n                           <m:mo>≥</m:mo>\r\n
    \                          <m:mn>2</m:mn>\r\n                        </m:math>\r\n
    \                       <jats:tex-math>n\\ge 2</jats:tex-math>\r\n                     </jats:alternatives>\r\n
    \                 </jats:inline-formula>, for <jats:disp-formula id=\"j_math-2022-0578_eq_001\">\r\n
    \                    <jats:alternatives>\r\n                        <jats:graphic
    xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2022-0578_eq_003.png\"
    />\r\n                        <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"
    display=\"block\">\r\n                           <m:mtable displaystyle=\"true\">\r\n
    \                             <m:mtr>\r\n                                 <m:mtd
    columnalign=\"right\">\r\n                                    <m:mfenced open=\"{\"
    close=\"\">\r\n                                       <m:mrow>\r\n                                          <m:mspace
    depth=\"1.25em\" />\r\n                                          <m:mtable displaystyle=\"true\">\r\n
    \                                            <m:mtr>\r\n                                                <m:mtd
    columnalign=\"left\">\r\n                                                   <m:msub>\r\n
    \                                                     <m:mrow>\r\n                                                         <m:mi>u</m:mi>\r\n
    \                                                     </m:mrow>\r\n                                                      <m:mrow>\r\n
    \                                                        <m:mi>t</m:mi>\r\n                                                      </m:mrow>\r\n
    \                                                  </m:msub>\r\n                                                   <m:mo>=</m:mo>\r\n
    \                                                  <m:mi mathvariant=\"normal\">Δ</m:mi>\r\n
    \                                                  <m:mi>u</m:mi>\r\n                                                   <m:mo>−</m:mo>\r\n
    \                                                  <m:mrow>\r\n                                                      <m:mo>∇</m:mo>\r\n
    \                                                  </m:mrow>\r\n                                                   <m:mo>⋅</m:mo>\r\n
    \                                                  <m:mrow>\r\n                                                      <m:mo>(</m:mo>\r\n
    \                                                     <m:mrow>\r\n                                                         <m:mi>u</m:mi>\r\n
    \                                                        <m:mi>S</m:mi>\r\n                                                         <m:mo>⋅</m:mo>\r\n
    \                                                        <m:mrow>\r\n                                                            <m:mo>∇</m:mo>\r\n
    \                                                        </m:mrow>\r\n                                                         <m:mi>v</m:mi>\r\n
    \                                                     </m:mrow>\r\n                                                      <m:mo>)</m:mo>\r\n
    \                                                  </m:mrow>\r\n                                                   <m:mo>,</m:mo>\r\n
    \                                               </m:mtd>\r\n                                             </m:mtr>\r\n
    \                                            <m:mtr>\r\n                                                <m:mtd
    columnalign=\"left\">\r\n                                                   <m:mn>0</m:mn>\r\n
    \                                                  <m:mo>=</m:mo>\r\n                                                   <m:mi
    mathvariant=\"normal\">Δ</m:mi>\r\n                                                   <m:mi>v</m:mi>\r\n
    \                                                  <m:mo>+</m:mo>\r\n                                                   <m:mi>u</m:mi>\r\n
    \                                                  <m:mo>,</m:mo>\r\n                                                </m:mtd>\r\n
    \                                            </m:mtr>\r\n                                          </m:mtable>\r\n
    \                                      </m:mrow>\r\n                                    </m:mfenced>\r\n
    \                                   <m:mspace width=\"2.0em\" />\r\n                                    <m:mspace
    width=\"2.0em\" />\r\n                                    <m:mspace width=\"2.0em\"
    />\r\n                                    <m:mrow>\r\n                                       <m:mo>(</m:mo>\r\n
    \                                      <m:mrow>\r\n                                          <m:mo>⋆</m:mo>\r\n
    \                                      </m:mrow>\r\n                                       <m:mo>)</m:mo>\r\n
    \                                   </m:mrow>\r\n                                 </m:mtd>\r\n
    \                             </m:mtr>\r\n                           </m:mtable>\r\n
    \                       </m:math>\r\n                        <jats:tex-math>\\begin{array}{r}\\left\\{\\phantom{\\rule[-1.25em]{}{0ex}}\\begin{array}{l}{u}_{t}=\\Delta
    u-\\nabla \\cdot \\left(uS\\cdot \\nabla v),\\\\ 0=\\Delta v+u,\\end{array}\\right.\\hspace{2.0em}\\hspace{2.0em}\\hspace{2.0em}\\left(\\star
    )\\end{array}</jats:tex-math>\r\n                     </jats:alternatives>\r\n
    \                 </jats:disp-formula> is considered for general matrices <jats:inline-formula>\r\n
    \                    <jats:alternatives>\r\n                        <jats:inline-graphic
    xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2022-0578_eq_004.png\"
    />\r\n                        <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\r\n
    \                          <m:mi>S</m:mi>\r\n                           <m:mo>∈</m:mo>\r\n
    \                          <m:msup>\r\n                              <m:mrow>\r\n
    \                                <m:mi mathvariant=\"double-struck\">R</m:mi>\r\n
    \                             </m:mrow>\r\n                              <m:mrow>\r\n
    \                                <m:mi>n</m:mi>\r\n                                 <m:mo>×</m:mo>\r\n
    \                                <m:mi>n</m:mi>\r\n                              </m:mrow>\r\n
    \                          </m:msup>\r\n                        </m:math>\r\n
    \                       <jats:tex-math>S\\in {{\\mathbb{R}}}^{n\\times n}</jats:tex-math>\r\n
    \                    </jats:alternatives>\r\n                  </jats:inline-formula>.
    A theory of local-in-time classical existence and extensibility is developed in
    a framework that differs from those considered in large parts of the literature
    by involving bounded classical solutions. Specifically, it is shown that for all
    non-negative initial data belonging to <jats:inline-formula>\r\n                     <jats:alternatives>\r\n
    \                       <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\"
    xlink:href=\"graphic/j_math-2022-0578_eq_005.png\" />\r\n                        <m:math
    xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\r\n                           <m:mi
    mathvariant=\"normal\">BUC</m:mi>\r\n                           <m:mrow>\r\n                              <m:mo>(</m:mo>\r\n
    \                             <m:mrow>\r\n                                 <m:msup>\r\n
    \                                   <m:mrow>\r\n                                       <m:mi
    mathvariant=\"double-struck\">R</m:mi>\r\n                                    </m:mrow>\r\n
    \                                   <m:mrow>\r\n                                       <m:mi>n</m:mi>\r\n
    \                                   </m:mrow>\r\n                                 </m:msup>\r\n
    \                             </m:mrow>\r\n                              <m:mo>)</m:mo>\r\n
    \                          </m:mrow>\r\n                           <m:mo>∩</m:mo>\r\n
    \                          <m:msup>\r\n                              <m:mrow>\r\n
    \                                <m:mi>L</m:mi>\r\n                              </m:mrow>\r\n
    \                             <m:mrow>\r\n                                 <m:mi>p</m:mi>\r\n
    \                             </m:mrow>\r\n                           </m:msup>\r\n
    \                          <m:mrow>\r\n                              <m:mo>(</m:mo>\r\n
    \                             <m:mrow>\r\n                                 <m:msup>\r\n
    \                                   <m:mrow>\r\n                                       <m:mi
    mathvariant=\"double-struck\">R</m:mi>\r\n                                    </m:mrow>\r\n
    \                                   <m:mrow>\r\n                                       <m:mi>n</m:mi>\r\n
    \                                   </m:mrow>\r\n                                 </m:msup>\r\n
    \                             </m:mrow>\r\n                              <m:mo>)</m:mo>\r\n
    \                          </m:mrow>\r\n                        </m:math>\r\n
    \                       <jats:tex-math>{\\rm{BUC}}\\left({{\\mathbb{R}}}^{n})\\cap
    {L}^{p}\\left({{\\mathbb{R}}}^{n})</jats:tex-math>\r\n                     </jats:alternatives>\r\n
    \                 </jats:inline-formula> with some <jats:inline-formula>\r\n                     <jats:alternatives>\r\n
    \                       <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\"
    xlink:href=\"graphic/j_math-2022-0578_eq_006.png\" />\r\n                        <m:math
    xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\r\n                           <m:mi>p</m:mi>\r\n
    \                          <m:mo>∈</m:mo>\r\n                           <m:mrow>\r\n
    \                             <m:mo>[</m:mo>\r\n                              <m:mrow>\r\n
    \                                <m:mn>1</m:mn>\r\n                                 <m:mo>,</m:mo>\r\n
    \                                <m:mi>n</m:mi>\r\n                              </m:mrow>\r\n
    \                             <m:mo>)</m:mo>\r\n                           </m:mrow>\r\n
    \                       </m:math>\r\n                        <jats:tex-math>p\\in
    \\left[1,n)</jats:tex-math>\r\n                     </jats:alternatives>\r\n                  </jats:inline-formula>,
    there exist <jats:inline-formula>\r\n                     <jats:alternatives>\r\n
    \                       <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\"
    xlink:href=\"graphic/j_math-2022-0578_eq_007.png\" />\r\n                        <m:math
    xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\r\n                           <m:msub>\r\n
    \                             <m:mrow>\r\n                                 <m:mi>T</m:mi>\r\n
    \                             </m:mrow>\r\n                              <m:mrow>\r\n
    \                                <m:mi>max</m:mi>\r\n                              </m:mrow>\r\n
    \                          </m:msub>\r\n                           <m:mo>∈</m:mo>\r\n
    \                          <m:mrow>\r\n                              <m:mo>(</m:mo>\r\n
    \                             <m:mrow>\r\n                                 <m:mn>0</m:mn>\r\n
    \                                <m:mo>,</m:mo>\r\n                                 <m:mi>∞</m:mi>\r\n
    \                             </m:mrow>\r\n                              <m:mo>]</m:mo>\r\n
    \                          </m:mrow>\r\n                        </m:math>\r\n
    \                       <jats:tex-math>{T}_{\\max }\\in \\left(0,\\infty ]</jats:tex-math>\r\n
    \                    </jats:alternatives>\r\n                  </jats:inline-formula>
    and a uniquely determined <jats:inline-formula>\r\n                     <jats:alternatives>\r\n
    \                       <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\"
    xlink:href=\"graphic/j_math-2022-0578_eq_008.png\" />\r\n                        <m:math
    xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\r\n                           <m:mi>u</m:mi>\r\n
    \                          <m:mo>∈</m:mo>\r\n                           <m:msup>\r\n
    \                             <m:mrow>\r\n                                 <m:mi>C</m:mi>\r\n
    \                             </m:mrow>\r\n                              <m:mrow>\r\n
    \                                <m:mn>0</m:mn>\r\n                              </m:mrow>\r\n
    \                          </m:msup>\r\n                           <m:mrow>\r\n
    \                             <m:mo>(</m:mo>\r\n                              <m:mrow>\r\n
    \                                <m:mrow>\r\n                                    <m:mo>[</m:mo>\r\n
    \                                   <m:mrow>\r\n                                       <m:mn>0</m:mn>\r\n
    \                                      <m:mo>,</m:mo>\r\n                                       <m:msub>\r\n
    \                                         <m:mrow>\r\n                                             <m:mi>T</m:mi>\r\n
    \                                         </m:mrow>\r\n                                          <m:mrow>\r\n
    \                                            <m:mi>max</m:mi>\r\n                                          </m:mrow>\r\n
    \                                      </m:msub>\r\n                                    </m:mrow>\r\n
    \                                   <m:mo>)</m:mo>\r\n                                 </m:mrow>\r\n
    \                                <m:mo>;</m:mo>\r\n                                 <m:mspace
    width=\"0.33em\" />\r\n                                 <m:mi mathvariant=\"normal\">BUC</m:mi>\r\n
    \                                <m:mrow>\r\n                                    <m:mo>(</m:mo>\r\n
    \                                   <m:mrow>\r\n                                       <m:msup>\r\n
    \                                         <m:mrow>\r\n                                             <m:mi
    mathvariant=\"double-struck\">R</m:mi>\r\n                                          </m:mrow>\r\n
    \                                         <m:mrow>\r\n                                             <m:mi>n</m:mi>\r\n
    \                                         </m:mrow>\r\n                                       </m:msup>\r\n
    \                                   </m:mrow>\r\n                                    <m:mo>)</m:mo>\r\n
    \                                </m:mrow>\r\n                              </m:mrow>\r\n
    \                             <m:mo>)</m:mo>\r\n                           </m:mrow>\r\n
    \                          <m:mo>∩</m:mo>\r\n                           <m:msup>\r\n
    \                             <m:mrow>\r\n                                 <m:mi>C</m:mi>\r\n
    \                             </m:mrow>\r\n                              <m:mrow>\r\n
    \                                <m:mn>0</m:mn>\r\n                              </m:mrow>\r\n
    \                          </m:msup>\r\n                           <m:mrow>\r\n
    \                             <m:mo>(</m:mo>\r\n                              <m:mrow>\r\n
    \                                <m:mrow>\r\n                                    <m:mo>[</m:mo>\r\n
    \                                   <m:mrow>\r\n                                       <m:mn>0</m:mn>\r\n
    \                                      <m:mo>,</m:mo>\r\n                                       <m:msub>\r\n
    \                                         <m:mrow>\r\n                                             <m:mi>T</m:mi>\r\n
    \                                         </m:mrow>\r\n                                          <m:mrow>\r\n
    \                                            <m:mi>max</m:mi>\r\n                                          </m:mrow>\r\n
    \                                      </m:msub>\r\n                                    </m:mrow>\r\n
    \                                   <m:mo>)</m:mo>\r\n                                 </m:mrow>\r\n
    \                                <m:mo>;</m:mo>\r\n                                 <m:mspace
    width=\"0.33em\" />\r\n                                 <m:msup>\r\n                                    <m:mrow>\r\n
    \                                      <m:mi>L</m:mi>\r\n                                    </m:mrow>\r\n
    \                                   <m:mrow>\r\n                                       <m:mi>p</m:mi>\r\n
    \                                   </m:mrow>\r\n                                 </m:msup>\r\n
    \                                <m:mrow>\r\n                                    <m:mo>(</m:mo>\r\n
    \                                   <m:mrow>\r\n                                       <m:msup>\r\n
    \                                         <m:mrow>\r\n                                             <m:mi
    mathvariant=\"double-struck\">R</m:mi>\r\n                                          </m:mrow>\r\n
    \                                         <m:mrow>\r\n                                             <m:mi>n</m:mi>\r\n
    \                                         </m:mrow>\r\n                                       </m:msup>\r\n
    \                                   </m:mrow>\r\n                                    <m:mo>)</m:mo>\r\n
    \                                </m:mrow>\r\n                              </m:mrow>\r\n
    \                             <m:mo>)</m:mo>\r\n                           </m:mrow>\r\n
    \                          <m:mo>∩</m:mo>\r\n                           <m:msup>\r\n
    \                             <m:mrow>\r\n                                 <m:mi>C</m:mi>\r\n
    \                             </m:mrow>\r\n                              <m:mrow>\r\n
    \                                <m:mi>∞</m:mi>\r\n                              </m:mrow>\r\n
    \                          </m:msup>\r\n                           <m:mrow>\r\n
    \                             <m:mo>(</m:mo>\r\n                              <m:mrow>\r\n
    \                                <m:msup>\r\n                                    <m:mrow>\r\n
    \                                      <m:mi mathvariant=\"double-struck\">R</m:mi>\r\n
    \                                   </m:mrow>\r\n                                    <m:mrow>\r\n
    \                                      <m:mi>n</m:mi>\r\n                                    </m:mrow>\r\n
    \                                </m:msup>\r\n                                 <m:mo>×</m:mo>\r\n
    \                                <m:mrow>\r\n                                    <m:mo>(</m:mo>\r\n
    \                                   <m:mrow>\r\n                                       <m:mn>0</m:mn>\r\n
    \                                      <m:mo>,</m:mo>\r\n                                       <m:msub>\r\n
    \                                         <m:mrow>\r\n                                             <m:mi>T</m:mi>\r\n
    \                                         </m:mrow>\r\n                                          <m:mrow>\r\n
    \                                            <m:mi>max</m:mi>\r\n                                          </m:mrow>\r\n
    \                                      </m:msub>\r\n                                    </m:mrow>\r\n
    \                                   <m:mo>)</m:mo>\r\n                                 </m:mrow>\r\n
    \                             </m:mrow>\r\n                              <m:mo>)</m:mo>\r\n
    \                          </m:mrow>\r\n                        </m:math>\r\n
    \                       <jats:tex-math>u\\in {C}^{0}\\left(\\left[0,{T}_{\\max
    });\\hspace{0.33em}{\\rm{BUC}}\\left({{\\mathbb{R}}}^{n}))\\cap {C}^{0}\\left(\\left[0,{T}_{\\max
    });\\hspace{0.33em}{L}^{p}\\left({{\\mathbb{R}}}^{n}))\\cap {C}^{\\infty }\\left({{\\mathbb{R}}}^{n}\\times
    \\left(0,{T}_{\\max }))</jats:tex-math>\r\n                     </jats:alternatives>\r\n
    \                 </jats:inline-formula> such that with <jats:inline-formula>\r\n
    \                    <jats:alternatives>\r\n                        <jats:inline-graphic
    xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2022-0578_eq_009.png\"
    />\r\n                        <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\r\n
    \                          <m:mi>v</m:mi>\r\n                           <m:mo>≔</m:mo>\r\n
    \                          <m:mi mathvariant=\"normal\">Γ</m:mi>\r\n                           <m:mo>⋆</m:mo>\r\n
    \                          <m:mi>u</m:mi>\r\n                        </m:math>\r\n
    \                       <jats:tex-math>v:= \\Gamma \\star u</jats:tex-math>\r\n
    \                    </jats:alternatives>\r\n                  </jats:inline-formula>,
    and with <jats:inline-formula>\r\n                     <jats:alternatives>\r\n
    \                       <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\"
    xlink:href=\"graphic/j_math-2022-0578_eq_010.png\" />\r\n                        <m:math
    xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\r\n                           <m:mi
    mathvariant=\"normal\">Γ</m:mi>\r\n                        </m:math>\r\n                        <jats:tex-math>\\Gamma
    </jats:tex-math>\r\n                     </jats:alternatives>\r\n                  </jats:inline-formula>
    denoting the Newtonian kernel on <jats:inline-formula>\r\n                     <jats:alternatives>\r\n
    \                       <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\"
    xlink:href=\"graphic/j_math-2022-0578_eq_011.png\" />\r\n                        <m:math
    xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\r\n                           <m:msup>\r\n
    \                             <m:mrow>\r\n                                 <m:mi
    mathvariant=\"double-struck\">R</m:mi>\r\n                              </m:mrow>\r\n
    \                             <m:mrow>\r\n                                 <m:mi>n</m:mi>\r\n
    \                             </m:mrow>\r\n                           </m:msup>\r\n
    \                       </m:math>\r\n                        <jats:tex-math>{{\\mathbb{R}}}^{n}</jats:tex-math>\r\n
    \                    </jats:alternatives>\r\n                  </jats:inline-formula>,
    the pair <jats:inline-formula>\r\n                     <jats:alternatives>\r\n
    \                       <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\"
    xlink:href=\"graphic/j_math-2022-0578_eq_012.png\" />\r\n                        <m:math
    xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\r\n                           <m:mrow>\r\n
    \                             <m:mo>(</m:mo>\r\n                              <m:mrow>\r\n
    \                                <m:mi>u</m:mi>\r\n                                 <m:mo>,</m:mo>\r\n
    \                                <m:mi>v</m:mi>\r\n                              </m:mrow>\r\n
    \                             <m:mo>)</m:mo>\r\n                           </m:mrow>\r\n
    \                       </m:math>\r\n                        <jats:tex-math>\\left(u,v)</jats:tex-math>\r\n
    \                    </jats:alternatives>\r\n                  </jats:inline-formula>
    forms a classical solution of (<jats:inline-formula>\r\n                     <jats:alternatives>\r\n
    \                       <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\"
    xlink:href=\"graphic/j_math-2022-0578_eq_013.png\" />\r\n                        <m:math
    xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\r\n                           <m:mo>⋆</m:mo>\r\n
    \                       </m:math>\r\n                        <jats:tex-math>\\star
    </jats:tex-math>\r\n                     </jats:alternatives>\r\n                  </jats:inline-formula>)
    in <jats:inline-formula>\r\n                     <jats:alternatives>\r\n                        <jats:inline-graphic
    xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2022-0578_eq_014.png\"
    />\r\n                        <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\r\n
    \                          <m:msup>\r\n                              <m:mrow>\r\n
    \                                <m:mi mathvariant=\"double-struck\">R</m:mi>\r\n
    \                             </m:mrow>\r\n                              <m:mrow>\r\n
    \                                <m:mi>n</m:mi>\r\n                              </m:mrow>\r\n
    \                          </m:msup>\r\n                           <m:mo>×</m:mo>\r\n
    \                          <m:mrow>\r\n                              <m:mo>(</m:mo>\r\n
    \                             <m:mrow>\r\n                                 <m:mn>0</m:mn>\r\n
    \                                <m:mo>,</m:mo>\r\n                                 <m:msub>\r\n
    \                                   <m:mrow>\r\n                                       <m:mi>T</m:mi>\r\n
    \                                   </m:mrow>\r\n                                    <m:mrow>\r\n
    \                                      <m:mi>max</m:mi>\r\n                                    </m:mrow>\r\n
    \                                </m:msub>\r\n                              </m:mrow>\r\n
    \                             <m:mo>)</m:mo>\r\n                           </m:mrow>\r\n
    \                       </m:math>\r\n                        <jats:tex-math>{{\\mathbb{R}}}^{n}\\times
    \\left(0,{T}_{\\max })</jats:tex-math>\r\n                     </jats:alternatives>\r\n
    \                 </jats:inline-formula>, which has the property that <jats:disp-formula
    id=\"j_math-2022-0578_eq_002\">\r\n                     <jats:alternatives>\r\n
    \                       <jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\"
    xlink:href=\"graphic/j_math-2022-0578_eq_015.png\" />\r\n                        <m:math
    xmlns:m=\"http://www.w3.org/1998/Math/MathML\" display=\"block\">\r\n                           <m:mspace
    width=\"0.1em\" />\r\n                           <m:mtext>if</m:mtext>\r\n                           <m:mspace
    width=\"0.1em\" />\r\n                           <m:mspace width=\"0.33em\" />\r\n
    \                          <m:msub>\r\n                              <m:mrow>\r\n
    \                                <m:mi>T</m:mi>\r\n                              </m:mrow>\r\n
    \                             <m:mrow>\r\n                                 <m:mi>max</m:mi>\r\n
    \                             </m:mrow>\r\n                           </m:msub>\r\n
    \                          <m:mo>&lt;</m:mo>\r\n                           <m:mi>∞</m:mi>\r\n
    \                          <m:mo>,</m:mo>\r\n                           <m:mspace
    width=\"1.0em\" />\r\n                           <m:mstyle>\r\n                              <m:mspace
    width=\"0.1em\" />\r\n                              <m:mtext>then both</m:mtext>\r\n
    \                             <m:mspace width=\"0.1em\" />\r\n                           </m:mstyle>\r\n
    \                          <m:mspace width=\"0.33em\" />\r\n                           <m:munder>\r\n
    \                             <m:mrow>\r\n                                 <m:mi>limsup</m:mi>\r\n
    \                             </m:mrow>\r\n                              <m:mrow>\r\n
    \                                <m:mi>t</m:mi>\r\n                                 <m:mo>↗</m:mo>\r\n
    \                                <m:msub>\r\n                                    <m:mrow>\r\n
    \                                      <m:mi>T</m:mi>\r\n                                    </m:mrow>\r\n
    \                                   <m:mrow>\r\n                                       <m:mi>max</m:mi>\r\n
    \                                   </m:mrow>\r\n                                 </m:msub>\r\n
    \                             </m:mrow>\r\n                           </m:munder>\r\n
    \                          <m:msub>\r\n                              <m:mrow>\r\n
    \                                <m:mo>‖</m:mo>\r\n                                 <m:mi>u</m:mi>\r\n
    \                                <m:mrow>\r\n                                    <m:mo>(</m:mo>\r\n
    \                                   <m:mrow>\r\n                                       <m:mo>⋅</m:mo>\r\n
    \                                      <m:mo>,</m:mo>\r\n                                       <m:mi>t</m:mi>\r\n
    \                                   </m:mrow>\r\n                                    <m:mo>)</m:mo>\r\n
    \                                </m:mrow>\r\n                                 <m:mo>‖</m:mo>\r\n
    \                             </m:mrow>\r\n                              <m:mrow>\r\n
    \                                <m:msup>\r\n                                    <m:mrow>\r\n
    \                                      <m:mi>L</m:mi>\r\n                                    </m:mrow>\r\n
    \                                   <m:mrow>\r\n                                       <m:mi>∞</m:mi>\r\n
    \                                   </m:mrow>\r\n                                 </m:msup>\r\n
    \                                <m:mrow>\r\n                                    <m:mo>(</m:mo>\r\n
    \                                   <m:mrow>\r\n                                       <m:msup>\r\n
    \                                         <m:mrow>\r\n                                             <m:mi
    mathvariant=\"double-struck\">R</m:mi>\r\n                                          </m:mrow>\r\n
    \                                         <m:mrow>\r\n                                             <m:mi>n</m:mi>\r\n
    \                                         </m:mrow>\r\n                                       </m:msup>\r\n
    \                                   </m:mrow>\r\n                                    <m:mo>)</m:mo>\r\n
    \                                </m:mrow>\r\n                              </m:mrow>\r\n
    \                          </m:msub>\r\n                           <m:mo>=</m:mo>\r\n
    \                          <m:mi>∞</m:mi>\r\n                           <m:mspace
    width=\"1.0em\" />\r\n                           <m:mspace width=\"0.1em\" />\r\n
    \                          <m:mtext>and</m:mtext>\r\n                           <m:mspace
    width=\"0.1em\" />\r\n                           <m:mspace width=\"1.0em\" />\r\n
    \                          <m:munder>\r\n                              <m:mrow>\r\n
    \                                <m:mi>limsup</m:mi>\r\n                              </m:mrow>\r\n
    \                             <m:mrow>\r\n                                 <m:mi>t</m:mi>\r\n
    \                                <m:mo>↗</m:mo>\r\n                                 <m:msub>\r\n
    \                                   <m:mrow>\r\n                                       <m:mi>T</m:mi>\r\n
    \                                   </m:mrow>\r\n                                    <m:mrow>\r\n
    \                                      <m:mi>max</m:mi>\r\n                                    </m:mrow>\r\n
    \                                </m:msub>\r\n                              </m:mrow>\r\n
    \                          </m:munder>\r\n                           <m:msub>\r\n
    \                             <m:mrow>\r\n                                 <m:mo>‖</m:mo>\r\n
    \                                <m:mrow>\r\n                                    <m:mo>∇</m:mo>\r\n
    \                                </m:mrow>\r\n                                 <m:mi>v</m:mi>\r\n
    \                                <m:mrow>\r\n                                    <m:mo>(</m:mo>\r\n
    \                                   <m:mrow>\r\n                                       <m:mo>⋅</m:mo>\r\n
    \                                      <m:mo>,</m:mo>\r\n                                       <m:mi>t</m:mi>\r\n
    \                                   </m:mrow>\r\n                                    <m:mo>)</m:mo>\r\n
    \                                </m:mrow>\r\n                                 <m:mo>‖</m:mo>\r\n
    \                             </m:mrow>\r\n                              <m:mrow>\r\n
    \                                <m:msup>\r\n                                    <m:mrow>\r\n
    \                                      <m:mi>L</m:mi>\r\n                                    </m:mrow>\r\n
    \                                   <m:mrow>\r\n                                       <m:mi>∞</m:mi>\r\n
    \                                   </m:mrow>\r\n                                 </m:msup>\r\n
    \                                <m:mrow>\r\n                                    <m:mo>(</m:mo>\r\n
    \                                   <m:mrow>\r\n                                       <m:msup>\r\n
    \                                         <m:mrow>\r\n                                             <m:mi
    mathvariant=\"double-struck\">R</m:mi>\r\n                                          </m:mrow>\r\n
    \                                         <m:mrow>\r\n                                             <m:mi>n</m:mi>\r\n
    \                                         </m:mrow>\r\n                                       </m:msup>\r\n
    \                                   </m:mrow>\r\n                                    <m:mo>)</m:mo>\r\n
    \                                </m:mrow>\r\n                              </m:mrow>\r\n
    \                          </m:msub>\r\n                           <m:mo>=</m:mo>\r\n
    \                          <m:mi>∞</m:mi>\r\n                           <m:mo>.</m:mo>\r\n
    \                       </m:math>\r\n                        <jats:tex-math>\\hspace{0.1em}\\text{if}\\hspace{0.1em}\\hspace{0.33em}{T}_{\\max
    }\\lt \\infty ,\\hspace{1.0em}\\hspace{0.1em}\\text{then both}\\hspace{0.1em}\\hspace{0.33em}\\mathop{\\mathrm{limsup}}\\limits_{t\\nearrow
    {T}_{\\max }}\\Vert u\\left(\\cdot ,t){\\Vert }_{{L}^{\\infty }\\left({{\\mathbb{R}}}^{n})}=\\infty
    \\hspace{1.0em}\\hspace{0.1em}\\text{and}\\hspace{0.1em}\\hspace{1.0em}\\mathop{\\mathrm{limsup}}\\limits_{t\\nearrow
    {T}_{\\max }}\\Vert \\nabla v\\left(\\cdot ,t){\\Vert }_{{L}^{\\infty }\\left({{\\mathbb{R}}}^{n})}=\\infty
    .</jats:tex-math>\r\n                     </jats:alternatives>\r\n                  </jats:disp-formula>
    An exemplary application of this provides a result on global classical solvability
    in cases when <jats:inline-formula>\r\n                     <jats:alternatives>\r\n
    \                       <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\"
    xlink:href=\"graphic/j_math-2022-0578_eq_016.png\" />\r\n                        <m:math
    xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\r\n                           <m:mo>∣</m:mo>\r\n
    \                          <m:mi>S</m:mi>\r\n                           <m:mo>+</m:mo>\r\n
    \                          <m:mn mathvariant=\"bold\">1</m:mn>\r\n                           <m:mo>∣</m:mo>\r\n
    \                       </m:math>\r\n                        <jats:tex-math>|
    S+{\\bf{1}}| </jats:tex-math>\r\n                     </jats:alternatives>\r\n
    \                 </jats:inline-formula> is sufficiently small, where <jats:inline-formula>\r\n
    \                    <jats:alternatives>\r\n                        <jats:inline-graphic
    xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2022-0578_eq_017.png\"
    />\r\n                        <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\r\n
    \                          <m:mn mathvariant=\"bold\">1</m:mn>\r\n                           <m:mo>=</m:mo>\r\n
    \                          <m:mi mathvariant=\"normal\">diag</m:mi>\r\n                           <m:mspace
    width=\"0.33em\" />\r\n                           <m:mrow>\r\n                              <m:mo>(</m:mo>\r\n
    \                             <m:mrow>\r\n                                 <m:mn>1</m:mn>\r\n
    \                                <m:mo>,</m:mo>\r\n                                 <m:mrow>\r\n
    \                                   <m:mo>…</m:mo>\r\n                                 </m:mrow>\r\n
    \                                <m:mo>,</m:mo>\r\n                                 <m:mn>1</m:mn>\r\n
    \                             </m:mrow>\r\n                              <m:mo>)</m:mo>\r\n
    \                          </m:mrow>\r\n                        </m:math>\r\n
    \                       <jats:tex-math>{\\bf{1}}={\\rm{diag}}\\hspace{0.33em}\\left(1,\\ldots
    ,1)</jats:tex-math>\r\n                     </jats:alternatives>\r\n                  </jats:inline-formula>.</jats:p>"
author:
- first_name: Michael
  full_name: Winkler, Michael
  last_name: Winkler
citation:
  ama: Winkler M. Classical solutions to Cauchy problems for parabolic–elliptic systems
    of Keller-Segel type. <i>Open Mathematics</i>. 2023;21(1). doi:<a href="https://doi.org/10.1515/math-2022-0578">10.1515/math-2022-0578</a>
  apa: Winkler, M. (2023). Classical solutions to Cauchy problems for parabolic–elliptic
    systems of Keller-Segel type. <i>Open Mathematics</i>, <i>21</i>(1). <a href="https://doi.org/10.1515/math-2022-0578">https://doi.org/10.1515/math-2022-0578</a>
  bibtex: '@article{Winkler_2023, title={Classical solutions to Cauchy problems for
    parabolic–elliptic systems of Keller-Segel type}, volume={21}, DOI={<a href="https://doi.org/10.1515/math-2022-0578">10.1515/math-2022-0578</a>},
    number={1}, journal={Open Mathematics}, publisher={Walter de Gruyter GmbH}, author={Winkler,
    Michael}, year={2023} }'
  chicago: Winkler, Michael. “Classical Solutions to Cauchy Problems for Parabolic–Elliptic
    Systems of Keller-Segel Type.” <i>Open Mathematics</i> 21, no. 1 (2023). <a href="https://doi.org/10.1515/math-2022-0578">https://doi.org/10.1515/math-2022-0578</a>.
  ieee: 'M. Winkler, “Classical solutions to Cauchy problems for parabolic–elliptic
    systems of Keller-Segel type,” <i>Open Mathematics</i>, vol. 21, no. 1, 2023,
    doi: <a href="https://doi.org/10.1515/math-2022-0578">10.1515/math-2022-0578</a>.'
  mla: Winkler, Michael. “Classical Solutions to Cauchy Problems for Parabolic–Elliptic
    Systems of Keller-Segel Type.” <i>Open Mathematics</i>, vol. 21, no. 1, Walter
    de Gruyter GmbH, 2023, doi:<a href="https://doi.org/10.1515/math-2022-0578">10.1515/math-2022-0578</a>.
  short: M. Winkler, Open Mathematics 21 (2023).
date_created: 2024-04-07T12:54:31Z
date_updated: 2024-04-07T12:54:34Z
doi: 10.1515/math-2022-0578
intvolume: '        21'
issue: '1'
keyword:
- General Mathematics
language:
- iso: eng
publication: Open Mathematics
publication_identifier:
  issn:
  - 2391-5455
publication_status: published
publisher: Walter de Gruyter GmbH
status: public
title: Classical solutions to Cauchy problems for parabolic–elliptic systems of Keller-Segel
  type
type: journal_article
user_id: '31496'
volume: 21
year: '2023'
...
---
_id: '53345'
abstract:
- lang: eng
  text: '<jats:title>Abstract</jats:title><jats:p>A no-flux initial-boundary value
    problem for<jats:disp-formula id="nonace22eueqn1"><jats:tex-math><?CDATA \begin{align*}
    \begin{cases} u_t = \Delta \big(u\phi(v)\big), \\[1mm] v_t = \Delta v-uv, \end{cases}
    \qquad \qquad (\star) \end{align*}?></jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"
    display="block" overflow="scroll"><mml:mtable columnalign="right left right left
    right left right left right left right left" columnspacing="0.2777777777777778em
    2em 0.2777777777777778em 2em 0.2777777777777778em 2em 0.2777777777777778em 2em
    0.2777777777777778em 2em 0.2777777777777778em" rowspacing="3pt"><mml:mtr><mml:mtd><mml:mfenced
    close="" open="{"><mml:mtable columnalign="left left" columnspacing="1em" rowspacing=".1em"><mml:mtr><mml:mtd><mml:msub><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mi
    mathvariant="normal">Δ</mml:mi><mml:mrow><mml:mo maxsize="1.2em" minsize="1.2em">(</mml:mo></mml:mrow><mml:mi>u</mml:mi><mml:mi>ϕ</mml:mi><mml:mo
    stretchy="false">(</mml:mo><mml:mi>v</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mrow><mml:mo
    maxsize="1.2em" minsize="1.2em">)</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:msub><mml:mi>v</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mi
    mathvariant="normal">Δ</mml:mi><mml:mi>v</mml:mi><mml:mo>−</mml:mo><mml:mi>u</mml:mi><mml:mi>v</mml:mi><mml:mo>,</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:mfenced><mml:mo
    stretchy="false">(</mml:mo><mml:mo>⋆</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math><jats:graphic
    xmlns:xlink="http://www.w3.org/1999/xlink" orientation="portrait" position="float"
    xlink:href="nonace22eueqn1.gif" xlink:type="simple" /></jats:disp-formula>is considered
    in smoothly bounded subdomains of<jats:inline-formula><jats:tex-math><?CDATA $\mathbb{R}^n$?></jats:tex-math><mml:math
    xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:msup><mml:mrow><mml:mi
    mathvariant="double-struck">R</mml:mi></mml:mrow><mml:mi>n</mml:mi></mml:msup></mml:math><jats:inline-graphic
    xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="nonace22eieqn1.gif" xlink:type="simple"
    /></jats:inline-formula>with<jats:inline-formula><jats:tex-math><?CDATA $n\geqslant
    1$?></jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:mi>n</mml:mi><mml:mo>⩾</mml:mo><mml:mn>1</mml:mn></mml:math><jats:inline-graphic
    xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="nonace22eieqn2.gif" xlink:type="simple"
    /></jats:inline-formula>and suitably regular initial data, where<jats:italic>φ</jats:italic>is
    assumed to reflect algebraic type cross-degeneracies by sharing essential features
    with<jats:inline-formula><jats:tex-math><?CDATA $0\leqslant \xi\mapsto \xi^\alpha$?></jats:tex-math><mml:math
    xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:mn>0</mml:mn><mml:mo>⩽</mml:mo><mml:mi>ξ</mml:mi><mml:mo
    stretchy="false">↦</mml:mo><mml:msup><mml:mi>ξ</mml:mi><mml:mi>α</mml:mi></mml:msup></mml:math><jats:inline-graphic
    xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="nonace22eieqn3.gif" xlink:type="simple"
    /></jats:inline-formula>for some<jats:inline-formula><jats:tex-math><?CDATA $\alpha\geqslant
    1$?></jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:mi>α</mml:mi><mml:mo>⩾</mml:mo><mml:mn>1</mml:mn></mml:math><jats:inline-graphic
    xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="nonace22eieqn4.gif" xlink:type="simple"
    /></jats:inline-formula>. Based on the discovery of a gradient structure acting
    at regularity levels mild enough to be consistent with degeneracy-driven limitations
    of smoothness information, in this general setting it is shown that with some
    measurable limit profile<jats:inline-formula><jats:tex-math><?CDATA $u_\infty$?></jats:tex-math><mml:math
    xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:msub><mml:mi>u</mml:mi><mml:mi
    mathvariant="normal">∞</mml:mi></mml:msub></mml:math><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink"
    xlink:href="nonace22eieqn5.gif" xlink:type="simple" /></jats:inline-formula>and
    some null set<jats:inline-formula><jats:tex-math><?CDATA $N_\star\subset (0,\infty)$?></jats:tex-math><mml:math
    xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:msub><mml:mi>N</mml:mi><mml:mo>⋆</mml:mo></mml:msub><mml:mo>⊂</mml:mo><mml:mo
    stretchy="false">(</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi mathvariant="normal">∞</mml:mi><mml:mo
    stretchy="false">)</mml:mo></mml:math><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink"
    xlink:href="nonace22eieqn6.gif" xlink:type="simple" /></jats:inline-formula>,
    a corresponding global generalized solution, known to exist according to recent
    literature, satisfies<jats:disp-formula id="nonace22eueqn2"><jats:tex-math><?CDATA
    \begin{align*} \rho(u(\cdot,t))\stackrel{\star}{\rightharpoonup} \rho(u_\infty)
    \quad \textrm{in } L^\infty(\Omega) \quad\;\; \textrm{ and } \quad\;\; v(\cdot,t)\to
    0 \quad \textrm{in } L^p(\Omega)\; \textrm{for all } p\geqslant 1 \end{align*}?></jats:tex-math><mml:math
    xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block" overflow="scroll"><mml:mtable
    columnalign="right left right left right left right left right left right left"
    columnspacing="0.2777777777777778em 2em 0.2777777777777778em 2em 0.2777777777777778em
    2em 0.2777777777777778em 2em 0.2777777777777778em 2em 0.2777777777777778em" rowspacing="3pt"><mml:mtr><mml:mtd><mml:mi>ρ</mml:mi><mml:mo
    stretchy="false">(</mml:mo><mml:mi>u</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mo>⋅</mml:mo><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo
    stretchy="false">)</mml:mo><mml:mo stretchy="false">)</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mo
    stretchy="false">⇀</mml:mo></mml:mrow><mml:mrow><mml:mo>⋆</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mi>ρ</mml:mi><mml:mo
    stretchy="false">(</mml:mo><mml:msub><mml:mi>u</mml:mi><mml:mi mathvariant="normal">∞</mml:mi></mml:msub><mml:mo
    stretchy="false">)</mml:mo><mml:mrow><mml:mtext>in </mml:mtext></mml:mrow><mml:msup><mml:mi>L</mml:mi><mml:mi
    mathvariant="normal">∞</mml:mi></mml:msup><mml:mo stretchy="false">(</mml:mo><mml:mi
    mathvariant="normal">Ω</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mrow><mml:mtext> and </mml:mtext></mml:mrow><mml:mi>v</mml:mi><mml:mo
    stretchy="false">(</mml:mo><mml:mo>⋅</mml:mo><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo
    stretchy="false">)</mml:mo><mml:mo stretchy="false">→</mml:mo><mml:mn>0</mml:mn><mml:mrow><mml:mtext>in </mml:mtext></mml:mrow><mml:msup><mml:mi>L</mml:mi><mml:mi>p</mml:mi></mml:msup><mml:mo
    stretchy="false">(</mml:mo><mml:mi mathvariant="normal">Ω</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mrow><mml:mtext>for
    all </mml:mtext></mml:mrow><mml:mi>p</mml:mi><mml:mo>⩾</mml:mo><mml:mn>1</mml:mn></mml:mtd></mml:mtr></mml:mtable></mml:math><jats:graphic
    xmlns:xlink="http://www.w3.org/1999/xlink" orientation="portrait" position="float"
    xlink:href="nonace22eueqn2.gif" xlink:type="simple" /></jats:disp-formula>as<jats:inline-formula><jats:tex-math><?CDATA
    $(0,\infty)\setminus N_\star \ni t\to \infty$?></jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"
    overflow="scroll"><mml:mo stretchy="false">(</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi
    mathvariant="normal">∞</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>∖</mml:mo><mml:msub><mml:mi>N</mml:mi><mml:mo>⋆</mml:mo></mml:msub><mml:mo>∋</mml:mo><mml:mi>t</mml:mi><mml:mo
    stretchy="false">→</mml:mo><mml:mi mathvariant="normal">∞</mml:mi></mml:math><jats:inline-graphic
    xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="nonace22eieqn7.gif" xlink:type="simple"
    /></jats:inline-formula>, where<jats:inline-formula><jats:tex-math><?CDATA $\rho(\xi):
    = \frac{\xi^2}{(\xi+1)^2}$?></jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"
    overflow="scroll"><mml:mi>ρ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ξ</mml:mi><mml:mo
    stretchy="false">)</mml:mo><mml:mo>:=</mml:mo><mml:mfrac><mml:msup><mml:mi>ξ</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mrow><mml:mo
    stretchy="false">(</mml:mo><mml:mi>ξ</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mrow><mml:msup><mml:mo
    stretchy="false">)</mml:mo><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:mrow></mml:mfrac></mml:math><jats:inline-graphic
    xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="nonace22eieqn8.gif" xlink:type="simple"
    /></jats:inline-formula>,<jats:inline-formula><jats:tex-math><?CDATA $\xi\geqslant
    0$?></jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:mi>ξ</mml:mi><mml:mo>⩾</mml:mo><mml:mn>0</mml:mn></mml:math><jats:inline-graphic
    xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="nonace22eieqn9.gif" xlink:type="simple"
    /></jats:inline-formula>. In the particular case when either<jats:inline-formula><jats:tex-math><?CDATA
    $n\leqslant 2$?></jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"
    overflow="scroll"><mml:mi>n</mml:mi><mml:mo>⩽</mml:mo><mml:mn>2</mml:mn></mml:math><jats:inline-graphic
    xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="nonace22eieqn10.gif" xlink:type="simple"
    /></jats:inline-formula>and<jats:inline-formula><jats:tex-math><?CDATA $\alpha\geqslant
    1$?></jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:mi>α</mml:mi><mml:mo>⩾</mml:mo><mml:mn>1</mml:mn></mml:math><jats:inline-graphic
    xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="nonace22eieqn11.gif" xlink:type="simple"
    /></jats:inline-formula>is arbitrary, or<jats:inline-formula><jats:tex-math><?CDATA
    $n\geqslant 1$?></jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"
    overflow="scroll"><mml:mi>n</mml:mi><mml:mo>⩾</mml:mo><mml:mn>1</mml:mn></mml:math><jats:inline-graphic
    xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="nonace22eieqn12.gif" xlink:type="simple"
    /></jats:inline-formula>and<jats:inline-formula><jats:tex-math><?CDATA $\alpha\in
    [1,2]$?></jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"
    overflow="scroll"><mml:mi>α</mml:mi><mml:mo>∈</mml:mo><mml:mo stretchy="false">[</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo
    stretchy="false">]</mml:mo></mml:math><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink"
    xlink:href="nonace22eieqn13.gif" xlink:type="simple" /></jats:inline-formula>,
    additional quantitative information on the deviation of trajectories from the
    initial data is derived. This is found to imply a lower estimate for the spatial
    oscillation of the respective first components throughout evolution, and moreover
    this is seen to entail that each of the uncountably many steady states<jats:inline-formula><jats:tex-math><?CDATA
    $(u_\star,0)$?></jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"
    overflow="scroll"><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>u</mml:mi><mml:mo>⋆</mml:mo></mml:msub><mml:mo>,</mml:mo><mml:mn>0</mml:mn><mml:mo
    stretchy="false">)</mml:mo></mml:math><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink"
    xlink:href="nonace22eieqn14.gif" xlink:type="simple" /></jats:inline-formula>of
    (<jats:inline-formula><jats:tex-math><?CDATA $\star$?></jats:tex-math><mml:math
    xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:mo>⋆</mml:mo></mml:math><jats:inline-graphic
    xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="nonace22eieqn15.gif" xlink:type="simple"
    /></jats:inline-formula>) is stable with respect to a suitably chosen norm topology.</jats:p>'
author:
- first_name: Michael
  full_name: Winkler, Michael
  last_name: Winkler
citation:
  ama: Winkler M. Stabilization despite pervasive strong cross-degeneracies in a nonlinear
    diffusion model for migration–consumption interaction. <i>Nonlinearity</i>. 2023;36(8):4438-4469.
    doi:<a href="https://doi.org/10.1088/1361-6544/ace22e">10.1088/1361-6544/ace22e</a>
  apa: Winkler, M. (2023). Stabilization despite pervasive strong cross-degeneracies
    in a nonlinear diffusion model for migration–consumption interaction. <i>Nonlinearity</i>,
    <i>36</i>(8), 4438–4469. <a href="https://doi.org/10.1088/1361-6544/ace22e">https://doi.org/10.1088/1361-6544/ace22e</a>
  bibtex: '@article{Winkler_2023, title={Stabilization despite pervasive strong cross-degeneracies
    in a nonlinear diffusion model for migration–consumption interaction}, volume={36},
    DOI={<a href="https://doi.org/10.1088/1361-6544/ace22e">10.1088/1361-6544/ace22e</a>},
    number={8}, journal={Nonlinearity}, publisher={IOP Publishing}, author={Winkler,
    Michael}, year={2023}, pages={4438–4469} }'
  chicago: 'Winkler, Michael. “Stabilization despite Pervasive Strong Cross-Degeneracies
    in a Nonlinear Diffusion Model for Migration–Consumption Interaction.” <i>Nonlinearity</i>
    36, no. 8 (2023): 4438–69. <a href="https://doi.org/10.1088/1361-6544/ace22e">https://doi.org/10.1088/1361-6544/ace22e</a>.'
  ieee: 'M. Winkler, “Stabilization despite pervasive strong cross-degeneracies in
    a nonlinear diffusion model for migration–consumption interaction,” <i>Nonlinearity</i>,
    vol. 36, no. 8, pp. 4438–4469, 2023, doi: <a href="https://doi.org/10.1088/1361-6544/ace22e">10.1088/1361-6544/ace22e</a>.'
  mla: Winkler, Michael. “Stabilization despite Pervasive Strong Cross-Degeneracies
    in a Nonlinear Diffusion Model for Migration–Consumption Interaction.” <i>Nonlinearity</i>,
    vol. 36, no. 8, IOP Publishing, 2023, pp. 4438–69, doi:<a href="https://doi.org/10.1088/1361-6544/ace22e">10.1088/1361-6544/ace22e</a>.
  short: M. Winkler, Nonlinearity 36 (2023) 4438–4469.
date_created: 2024-04-07T12:56:35Z
date_updated: 2024-04-07T12:56:40Z
doi: 10.1088/1361-6544/ace22e
intvolume: '        36'
issue: '8'
keyword:
- Applied Mathematics
- General Physics and Astronomy
- Mathematical Physics
- Statistical and Nonlinear Physics
language:
- iso: eng
page: 4438-4469
publication: Nonlinearity
publication_identifier:
  issn:
  - 0951-7715
  - 1361-6544
publication_status: published
publisher: IOP Publishing
status: public
title: Stabilization despite pervasive strong cross-degeneracies in a nonlinear diffusion
  model for migration–consumption interaction
type: journal_article
user_id: '31496'
volume: 36
year: '2023'
...
---
_id: '53341'
abstract:
- lang: eng
  text: "<jats:title>Abstract</jats:title><jats:p>The Cauchy problem in <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\mathbb
    {R}^n$$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\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:math></jats:alternatives></jats:inline-formula>
    is considered for the Keller–Segel system <jats:disp-formula><jats:alternatives><jats:tex-math>$$\\begin{aligned}
    \\left\\{ \\begin{array}{l}u_t = \\Delta u - \\nabla \\cdot (u\\nabla v), \\\\
    0 = \\Delta v + u, \\end{array} \\right. \\qquad \\qquad (\\star ) \\end{aligned}$$</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>Δ</mml:mi>\r\n                                        <mml:mi>u</mml:mi>\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:mi>u</mml:mi>\r\n
    \                                         <mml:mi>∇</mml:mi>\r\n                                          <mml:mi>v</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: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>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>with a focus
    on a detailed description of behavior in the presence of nonnegative radially
    symmetric initial data <jats:inline-formula><jats:alternatives><jats:tex-math>$$u_0$$</jats:tex-math><mml:math
    xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n                  <mml:msub>\r\n
    \                   <mml:mi>u</mml:mi>\r\n                    <mml:mn>0</mml:mn>\r\n
    \                 </mml:msub>\r\n                </mml:math></jats:alternatives></jats:inline-formula>
    with non-integrable behavior at spatial infinity. It is shown that if <jats:inline-formula><jats:alternatives><jats:tex-math>$$u_0$$</jats:tex-math><mml:math
    xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n                  <mml:msub>\r\n
    \                   <mml:mi>u</mml:mi>\r\n                    <mml:mn>0</mml:mn>\r\n
    \                 </mml:msub>\r\n                </mml:math></jats:alternatives></jats:inline-formula>
    is continuous and bounded, then (<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 a local-in-time classical solution, whereas if <jats:inline-formula><jats:alternatives><jats:tex-math>$$u_0(x)\\rightarrow
    +\\infty $$</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>u</mml:mi>\r\n
    \                     <mml:mn>0</mml:mn>\r\n                    </mml:msub>\r\n
    \                   <mml:mrow>\r\n                      <mml:mo>(</mml:mo>\r\n
    \                     <mml:mi>x</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: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>$$|x|\\rightarrow \\infty
    $$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n
    \                 <mml:mrow>\r\n                    <mml:mo>|</mml:mo>\r\n                    <mml:mi>x</mml:mi>\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>,
    then no such solution can be found. Furthermore, a collection of three sufficient
    criteria for either global existence or global nonexistence indicates that with
    respect to the occurrence of finite-time blow-up, spatial decay properties of
    an explicit singular steady state plays a critical role. In particular, this underlines
    that explosions 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>)
    need not be enforced by initially high concentrations near finite points, but
    can be exclusively due to large tails.</jats:p>"
author:
- first_name: Michael
  full_name: Winkler, Michael
  last_name: Winkler
citation:
  ama: Winkler M. Solutions to the Keller–Segel system with non-integrable behavior
    at spatial infinity. <i>Journal of Elliptic and Parabolic Equations</i>. 2023;9(2):919-959.
    doi:<a href="https://doi.org/10.1007/s41808-023-00230-y">10.1007/s41808-023-00230-y</a>
  apa: Winkler, M. (2023). Solutions to the Keller–Segel system with non-integrable
    behavior at spatial infinity. <i>Journal of Elliptic and Parabolic Equations</i>,
    <i>9</i>(2), 919–959. <a href="https://doi.org/10.1007/s41808-023-00230-y">https://doi.org/10.1007/s41808-023-00230-y</a>
  bibtex: '@article{Winkler_2023, title={Solutions to the Keller–Segel system with
    non-integrable behavior at spatial infinity}, volume={9}, DOI={<a href="https://doi.org/10.1007/s41808-023-00230-y">10.1007/s41808-023-00230-y</a>},
    number={2}, journal={Journal of Elliptic and Parabolic Equations}, publisher={Springer
    Science and Business Media LLC}, author={Winkler, Michael}, year={2023}, pages={919–959}
    }'
  chicago: 'Winkler, Michael. “Solutions to the Keller–Segel System with Non-Integrable
    Behavior at Spatial Infinity.” <i>Journal of Elliptic and Parabolic Equations</i>
    9, no. 2 (2023): 919–59. <a href="https://doi.org/10.1007/s41808-023-00230-y">https://doi.org/10.1007/s41808-023-00230-y</a>.'
  ieee: 'M. Winkler, “Solutions to the Keller–Segel system with non-integrable behavior
    at spatial infinity,” <i>Journal of Elliptic and Parabolic Equations</i>, vol.
    9, no. 2, pp. 919–959, 2023, doi: <a href="https://doi.org/10.1007/s41808-023-00230-y">10.1007/s41808-023-00230-y</a>.'
  mla: Winkler, Michael. “Solutions to the Keller–Segel System with Non-Integrable
    Behavior at Spatial Infinity.” <i>Journal of Elliptic and Parabolic Equations</i>,
    vol. 9, no. 2, Springer Science and Business Media LLC, 2023, pp. 919–59, doi:<a
    href="https://doi.org/10.1007/s41808-023-00230-y">10.1007/s41808-023-00230-y</a>.
  short: M. Winkler, Journal of Elliptic and Parabolic Equations 9 (2023) 919–959.
date_created: 2024-04-07T12:52:52Z
date_updated: 2024-04-07T12:52:55Z
doi: 10.1007/s41808-023-00230-y
intvolume: '         9'
issue: '2'
keyword:
- Applied Mathematics
- Numerical Analysis
- Analysis
language:
- iso: eng
page: 919-959
publication: Journal of Elliptic and Parabolic Equations
publication_identifier:
  issn:
  - 2296-9020
  - 2296-9039
publication_status: published
publisher: Springer Science and Business Media LLC
status: public
title: Solutions to the Keller–Segel system with non-integrable behavior at spatial
  infinity
type: journal_article
user_id: '31496'
volume: 9
year: '2023'
...
---
_id: '53339'
abstract:
- lang: eng
  text: "<jats:p>The chemotaxis‐Stokes system \r\n<jats:disp-formula>\r\n\r\n</jats:disp-formula>is
    considered along with homogeneous boundary conditions of no‐flux type for \r\n
    and \r\n, and of Dirichlet type for \r\n, in a smoothly bounded domain \r\n. Under
    the assumption that \r\n, that \r\n is bounded on each of the intervals \r\n with
    arbitrary \r\n, and that with some \r\n and \r\n, we have \r\n<jats:disp-formula>\r\n\r\n</jats:disp-formula>It
    is shown that for any suitably regular initial data, an associated initial‐boundary
    value problem admits a global very weak solution.</jats:p>"
author:
- first_name: Yu
  full_name: Tian, Yu
  last_name: Tian
- first_name: Michael
  full_name: Winkler, Michael
  last_name: Winkler
citation:
  ama: Tian Y, Winkler M. Keller–Segel–Stokes interaction involving signal‐dependent
    motilities. <i>Mathematical Methods in the Applied Sciences</i>. 2023;46(14):15667-15683.
    doi:<a href="https://doi.org/10.1002/mma.9419">10.1002/mma.9419</a>
  apa: Tian, Y., &#38; Winkler, M. (2023). Keller–Segel–Stokes interaction involving
    signal‐dependent motilities. <i>Mathematical Methods in the Applied Sciences</i>,
    <i>46</i>(14), 15667–15683. <a href="https://doi.org/10.1002/mma.9419">https://doi.org/10.1002/mma.9419</a>
  bibtex: '@article{Tian_Winkler_2023, title={Keller–Segel–Stokes interaction involving
    signal‐dependent motilities}, volume={46}, DOI={<a href="https://doi.org/10.1002/mma.9419">10.1002/mma.9419</a>},
    number={14}, journal={Mathematical Methods in the Applied Sciences}, publisher={Wiley},
    author={Tian, Yu and Winkler, Michael}, year={2023}, pages={15667–15683} }'
  chicago: 'Tian, Yu, and Michael Winkler. “Keller–Segel–Stokes Interaction Involving
    Signal‐dependent Motilities.” <i>Mathematical Methods in the Applied Sciences</i>
    46, no. 14 (2023): 15667–83. <a href="https://doi.org/10.1002/mma.9419">https://doi.org/10.1002/mma.9419</a>.'
  ieee: 'Y. Tian and M. Winkler, “Keller–Segel–Stokes interaction involving signal‐dependent
    motilities,” <i>Mathematical Methods in the Applied Sciences</i>, vol. 46, no.
    14, pp. 15667–15683, 2023, doi: <a href="https://doi.org/10.1002/mma.9419">10.1002/mma.9419</a>.'
  mla: Tian, Yu, and Michael Winkler. “Keller–Segel–Stokes Interaction Involving Signal‐dependent
    Motilities.” <i>Mathematical Methods in the Applied Sciences</i>, vol. 46, no.
    14, Wiley, 2023, pp. 15667–83, doi:<a href="https://doi.org/10.1002/mma.9419">10.1002/mma.9419</a>.
  short: Y. Tian, M. Winkler, Mathematical Methods in the Applied Sciences 46 (2023)
    15667–15683.
date_created: 2024-04-07T12:51:27Z
date_updated: 2024-04-07T12:51:31Z
doi: 10.1002/mma.9419
intvolume: '        46'
issue: '14'
keyword:
- General Engineering
- General Mathematics
language:
- iso: eng
page: 15667-15683
publication: Mathematical Methods in the Applied Sciences
publication_identifier:
  issn:
  - 0170-4214
  - 1099-1476
publication_status: published
publisher: Wiley
status: public
title: Keller–Segel–Stokes interaction involving signal‐dependent motilities
type: journal_article
user_id: '31496'
volume: 46
year: '2023'
...
---
_id: '53340'
author:
- first_name: Kevin J.
  full_name: Painter, Kevin J.
  last_name: Painter
- first_name: Michael
  full_name: Winkler, Michael
  last_name: Winkler
citation:
  ama: Painter KJ, Winkler M. Phenotype Switching in Chemotaxis Aggregation Models
    Controls the Spontaneous Emergence of Large Densities. <i>SIAM Journal on Applied
    Mathematics</i>. 2023;83(5):2096-2117. doi:<a href="https://doi.org/10.1137/22m1539393">10.1137/22m1539393</a>
  apa: Painter, K. J., &#38; Winkler, M. (2023). Phenotype Switching in Chemotaxis
    Aggregation Models Controls the Spontaneous Emergence of Large Densities. <i>SIAM
    Journal on Applied Mathematics</i>, <i>83</i>(5), 2096–2117. <a href="https://doi.org/10.1137/22m1539393">https://doi.org/10.1137/22m1539393</a>
  bibtex: '@article{Painter_Winkler_2023, title={Phenotype Switching in Chemotaxis
    Aggregation Models Controls the Spontaneous Emergence of Large Densities}, volume={83},
    DOI={<a href="https://doi.org/10.1137/22m1539393">10.1137/22m1539393</a>}, number={5},
    journal={SIAM Journal on Applied Mathematics}, publisher={Society for Industrial
    &#38; Applied Mathematics (SIAM)}, author={Painter, Kevin J. and Winkler, Michael},
    year={2023}, pages={2096–2117} }'
  chicago: 'Painter, Kevin J., and Michael Winkler. “Phenotype Switching in Chemotaxis
    Aggregation Models Controls the Spontaneous Emergence of Large Densities.” <i>SIAM
    Journal on Applied Mathematics</i> 83, no. 5 (2023): 2096–2117. <a href="https://doi.org/10.1137/22m1539393">https://doi.org/10.1137/22m1539393</a>.'
  ieee: 'K. J. Painter and M. Winkler, “Phenotype Switching in Chemotaxis Aggregation
    Models Controls the Spontaneous Emergence of Large Densities,” <i>SIAM Journal
    on Applied Mathematics</i>, vol. 83, no. 5, pp. 2096–2117, 2023, doi: <a href="https://doi.org/10.1137/22m1539393">10.1137/22m1539393</a>.'
  mla: Painter, Kevin J., and Michael Winkler. “Phenotype Switching in Chemotaxis
    Aggregation Models Controls the Spontaneous Emergence of Large Densities.” <i>SIAM
    Journal on Applied Mathematics</i>, vol. 83, no. 5, Society for Industrial &#38;
    Applied Mathematics (SIAM), 2023, pp. 2096–117, doi:<a href="https://doi.org/10.1137/22m1539393">10.1137/22m1539393</a>.
  short: K.J. Painter, M. Winkler, SIAM Journal on Applied Mathematics 83 (2023) 2096–2117.
date_created: 2024-04-07T12:52:03Z
date_updated: 2024-04-07T12:52:06Z
doi: 10.1137/22m1539393
intvolume: '        83'
issue: '5'
keyword:
- Applied Mathematics
language:
- iso: eng
page: 2096-2117
publication: SIAM Journal on Applied Mathematics
publication_identifier:
  issn:
  - 0036-1399
  - 1095-712X
publication_status: published
publisher: Society for Industrial & Applied Mathematics (SIAM)
status: public
title: Phenotype Switching in Chemotaxis Aggregation Models Controls the Spontaneous
  Emergence of Large Densities
type: journal_article
user_id: '31496'
volume: 83
year: '2023'
...
---
_id: '53342'
author:
- first_name: Michael
  full_name: Winkler, Michael
  last_name: Winkler
- first_name: Tomomi
  full_name: Yokota, Tomomi
  last_name: Yokota
citation:
  ama: Winkler M, Yokota T. Avoiding critical mass phenomena by arbitrarily mild saturation
    of cross-diffusive fluxes in two-dimensional Keller-Segel-Navier-Stokes systems.
    <i>Journal of Differential Equations</i>. 2023;374:1-28. doi:<a href="https://doi.org/10.1016/j.jde.2023.07.029">10.1016/j.jde.2023.07.029</a>
  apa: Winkler, M., &#38; Yokota, T. (2023). Avoiding critical mass phenomena by arbitrarily
    mild saturation of cross-diffusive fluxes in two-dimensional Keller-Segel-Navier-Stokes
    systems. <i>Journal of Differential Equations</i>, <i>374</i>, 1–28. <a href="https://doi.org/10.1016/j.jde.2023.07.029">https://doi.org/10.1016/j.jde.2023.07.029</a>
  bibtex: '@article{Winkler_Yokota_2023, title={Avoiding critical mass phenomena by
    arbitrarily mild saturation of cross-diffusive fluxes in two-dimensional Keller-Segel-Navier-Stokes
    systems}, volume={374}, DOI={<a href="https://doi.org/10.1016/j.jde.2023.07.029">10.1016/j.jde.2023.07.029</a>},
    journal={Journal of Differential Equations}, publisher={Elsevier BV}, author={Winkler,
    Michael and Yokota, Tomomi}, year={2023}, pages={1–28} }'
  chicago: 'Winkler, Michael, and Tomomi Yokota. “Avoiding Critical Mass Phenomena
    by Arbitrarily Mild Saturation of Cross-Diffusive Fluxes in Two-Dimensional Keller-Segel-Navier-Stokes
    Systems.” <i>Journal of Differential Equations</i> 374 (2023): 1–28. <a href="https://doi.org/10.1016/j.jde.2023.07.029">https://doi.org/10.1016/j.jde.2023.07.029</a>.'
  ieee: 'M. Winkler and T. Yokota, “Avoiding critical mass phenomena by arbitrarily
    mild saturation of cross-diffusive fluxes in two-dimensional Keller-Segel-Navier-Stokes
    systems,” <i>Journal of Differential Equations</i>, vol. 374, pp. 1–28, 2023,
    doi: <a href="https://doi.org/10.1016/j.jde.2023.07.029">10.1016/j.jde.2023.07.029</a>.'
  mla: Winkler, Michael, and Tomomi Yokota. “Avoiding Critical Mass Phenomena by Arbitrarily
    Mild Saturation of Cross-Diffusive Fluxes in Two-Dimensional Keller-Segel-Navier-Stokes
    Systems.” <i>Journal of Differential Equations</i>, vol. 374, Elsevier BV, 2023,
    pp. 1–28, doi:<a href="https://doi.org/10.1016/j.jde.2023.07.029">10.1016/j.jde.2023.07.029</a>.
  short: M. Winkler, T. Yokota, Journal of Differential Equations 374 (2023) 1–28.
date_created: 2024-04-07T12:53:32Z
date_updated: 2024-04-07T12:53:38Z
doi: 10.1016/j.jde.2023.07.029
intvolume: '       374'
keyword:
- Analysis
- Applied Mathematics
language:
- iso: eng
page: 1-28
publication: Journal of Differential Equations
publication_identifier:
  issn:
  - 0022-0396
publication_status: published
publisher: Elsevier BV
status: public
title: Avoiding critical mass phenomena by arbitrarily mild saturation of cross-diffusive
  fluxes in two-dimensional Keller-Segel-Navier-Stokes systems
type: journal_article
user_id: '31496'
volume: 374
year: '2023'
...
---
_id: '53346'
author:
- first_name: Michael
  full_name: Winkler, Michael
  last_name: Winkler
citation:
  ama: Winkler M. Absence of collapse into persistent Dirac-type singularities in
    a Keller-Segel-Navier-Stokes system involving local sensing. <i>Advances in Differential
    Equations</i>. 2023;28(11/12). doi:<a href="https://doi.org/10.57262/ade028-1112-921">10.57262/ade028-1112-921</a>
  apa: Winkler, M. (2023). Absence of collapse into persistent Dirac-type singularities
    in a Keller-Segel-Navier-Stokes system involving local sensing. <i>Advances in
    Differential Equations</i>, <i>28</i>(11/12). <a href="https://doi.org/10.57262/ade028-1112-921">https://doi.org/10.57262/ade028-1112-921</a>
  bibtex: '@article{Winkler_2023, title={Absence of collapse into persistent Dirac-type
    singularities in a Keller-Segel-Navier-Stokes system involving local sensing},
    volume={28}, DOI={<a href="https://doi.org/10.57262/ade028-1112-921">10.57262/ade028-1112-921</a>},
    number={11/12}, journal={Advances in Differential Equations}, publisher={Khayyam
    Publishing, Inc}, author={Winkler, Michael}, year={2023} }'
  chicago: Winkler, Michael. “Absence of Collapse into Persistent Dirac-Type Singularities
    in a Keller-Segel-Navier-Stokes System Involving Local Sensing.” <i>Advances in
    Differential Equations</i> 28, no. 11/12 (2023). <a href="https://doi.org/10.57262/ade028-1112-921">https://doi.org/10.57262/ade028-1112-921</a>.
  ieee: 'M. Winkler, “Absence of collapse into persistent Dirac-type singularities
    in a Keller-Segel-Navier-Stokes system involving local sensing,” <i>Advances in
    Differential Equations</i>, vol. 28, no. 11/12, 2023, doi: <a href="https://doi.org/10.57262/ade028-1112-921">10.57262/ade028-1112-921</a>.'
  mla: Winkler, Michael. “Absence of Collapse into Persistent Dirac-Type Singularities
    in a Keller-Segel-Navier-Stokes System Involving Local Sensing.” <i>Advances in
    Differential Equations</i>, vol. 28, no. 11/12, Khayyam Publishing, Inc, 2023,
    doi:<a href="https://doi.org/10.57262/ade028-1112-921">10.57262/ade028-1112-921</a>.
  short: M. Winkler, Advances in Differential Equations 28 (2023).
date_created: 2024-04-07T12:57:19Z
date_updated: 2024-04-07T12:57:23Z
doi: 10.57262/ade028-1112-921
intvolume: '        28'
issue: 11/12
keyword:
- Applied Mathematics
- Analysis
language:
- iso: eng
publication: Advances in Differential Equations
publication_identifier:
  issn:
  - 1079-9389
publication_status: published
publisher: Khayyam Publishing, Inc
status: public
title: Absence of collapse into persistent Dirac-type singularities in a Keller-Segel-Navier-Stokes
  system involving local sensing
type: journal_article
user_id: '31496'
volume: 28
year: '2023'
...
---
_id: '53229'
author:
- first_name: Francisco J.
  full_name: Santos-Arteaga, Francisco J.
  last_name: Santos-Arteaga
- first_name: Debora
  full_name: Di Caprio, Debora
  last_name: Di Caprio
- first_name: Madjid
  full_name: Tavana, Madjid
  id: '31858'
  last_name: Tavana
- first_name: Emilio Cerda
  full_name: Tena, Emilio Cerda
  last_name: Tena
citation:
  ama: Santos-Arteaga FJ, Di Caprio D, Tavana M, Tena EC. A Credibility and Strategic
    Behavior Approach in Hesitant Multiple Criteria Decision-Making With Application
    to Sustainable Transportation. <i>IEEE Transactions on Fuzzy Systems</i>. 2023;31(2):460-474.
    doi:<a href="https://doi.org/10.1109/tfuzz.2022.3188875">10.1109/tfuzz.2022.3188875</a>
  apa: Santos-Arteaga, F. J., Di Caprio, D., Tavana, M., &#38; Tena, E. C. (2023).
    A Credibility and Strategic Behavior Approach in Hesitant Multiple Criteria Decision-Making
    With Application to Sustainable Transportation. <i>IEEE Transactions on Fuzzy
    Systems</i>, <i>31</i>(2), 460–474. <a href="https://doi.org/10.1109/tfuzz.2022.3188875">https://doi.org/10.1109/tfuzz.2022.3188875</a>
  bibtex: '@article{Santos-Arteaga_Di Caprio_Tavana_Tena_2023, title={A Credibility
    and Strategic Behavior Approach in Hesitant Multiple Criteria Decision-Making
    With Application to Sustainable Transportation}, volume={31}, DOI={<a href="https://doi.org/10.1109/tfuzz.2022.3188875">10.1109/tfuzz.2022.3188875</a>},
    number={2}, journal={IEEE Transactions on Fuzzy Systems}, publisher={Institute
    of Electrical and Electronics Engineers (IEEE)}, author={Santos-Arteaga, Francisco
    J. and Di Caprio, Debora and Tavana, Madjid and Tena, Emilio Cerda}, year={2023},
    pages={460–474} }'
  chicago: 'Santos-Arteaga, Francisco J., Debora Di Caprio, Madjid Tavana, and Emilio
    Cerda Tena. “A Credibility and Strategic Behavior Approach in Hesitant Multiple
    Criteria Decision-Making With Application to Sustainable Transportation.” <i>IEEE
    Transactions on Fuzzy Systems</i> 31, no. 2 (2023): 460–74. <a href="https://doi.org/10.1109/tfuzz.2022.3188875">https://doi.org/10.1109/tfuzz.2022.3188875</a>.'
  ieee: 'F. J. Santos-Arteaga, D. Di Caprio, M. Tavana, and E. C. Tena, “A Credibility
    and Strategic Behavior Approach in Hesitant Multiple Criteria Decision-Making
    With Application to Sustainable Transportation,” <i>IEEE Transactions on Fuzzy
    Systems</i>, vol. 31, no. 2, pp. 460–474, 2023, doi: <a href="https://doi.org/10.1109/tfuzz.2022.3188875">10.1109/tfuzz.2022.3188875</a>.'
  mla: Santos-Arteaga, Francisco J., et al. “A Credibility and Strategic Behavior
    Approach in Hesitant Multiple Criteria Decision-Making With Application to Sustainable
    Transportation.” <i>IEEE Transactions on Fuzzy Systems</i>, vol. 31, no. 2, Institute
    of Electrical and Electronics Engineers (IEEE), 2023, pp. 460–74, doi:<a href="https://doi.org/10.1109/tfuzz.2022.3188875">10.1109/tfuzz.2022.3188875</a>.
  short: F.J. Santos-Arteaga, D. Di Caprio, M. Tavana, E.C. Tena, IEEE Transactions
    on Fuzzy Systems 31 (2023) 460–474.
date_created: 2024-04-04T14:00:53Z
date_updated: 2024-04-15T13:15:07Z
department:
- _id: '277'
doi: 10.1109/tfuzz.2022.3188875
intvolume: '        31'
issue: '2'
keyword:
- Applied Mathematics
- Artificial Intelligence
- Computational Theory and Mathematics
- Control and Systems Engineering
language:
- iso: eng
page: 460-474
publication: IEEE Transactions on Fuzzy Systems
publication_identifier:
  issn:
  - 1063-6706
  - 1941-0034
publication_status: published
publisher: Institute of Electrical and Electronics Engineers (IEEE)
status: public
title: A Credibility and Strategic Behavior Approach in Hesitant Multiple Criteria
  Decision-Making With Application to Sustainable Transportation
type: journal_article
user_id: '51811'
volume: 31
year: '2023'
...
---
_id: '53539'
abstract:
- lang: eng
  text: "<jats:title>Abstract</jats:title><jats:p>The infinite Brownian loop on a
    Riemannian manifold is the limit in distribution of the Brownian bridge of length
    <jats:italic>T</jats:italic> around a fixed origin when <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:mo>+</mml:mo>\r\n                  <mml:mi>∞</mml:mi>\r\n
    \               </mml:mrow>\r\n              </mml:math></jats:alternatives></jats:inline-formula>.
    The aim of this note is to study its long-time asymptotics on Riemannian symmetric
    spaces <jats:italic>G</jats:italic>/<jats:italic>K</jats:italic> of noncompact
    type and of general rank. This amounts to the behavior of solutions to the heat
    equation subject to the Doob transform induced by the ground spherical function.
    Unlike the standard Brownian motion, we observe in this case phenomena which are
    similar to the Euclidean setting, namely <jats:inline-formula><jats:alternatives><jats:tex-math>$$L^1$$</jats:tex-math><mml:math
    xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n                <mml:msup>\r\n
    \                 <mml:mi>L</mml:mi>\r\n                  <mml:mn>1</mml:mn>\r\n
    \               </mml:msup>\r\n              </mml:math></jats:alternatives></jats:inline-formula>
    asymptotic convergence without requiring bi-<jats:italic>K</jats:italic>-invariance
    for initial data, and strong <jats:inline-formula><jats:alternatives><jats:tex-math>$$L^{\\infty
    }$$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n
    \               <mml:msup>\r\n                  <mml:mi>L</mml:mi>\r\n                  <mml:mi>∞</mml:mi>\r\n
    \               </mml:msup>\r\n              </mml:math></jats:alternatives></jats:inline-formula>
    convergence.</jats:p>"
author:
- first_name: Efthymia
  full_name: Papageorgiou, Efthymia
  id: '100325'
  last_name: Papageorgiou
citation:
  ama: Papageorgiou E. Asymptotics for the infinite Brownian loop on noncompact symmetric
    spaces. <i>Journal of Elliptic and Parabolic Equations</i>. Published online 2023.
    doi:<a href="https://doi.org/10.1007/s41808-023-00250-8">10.1007/s41808-023-00250-8</a>
  apa: Papageorgiou, E. (2023). Asymptotics for the infinite Brownian loop on noncompact
    symmetric spaces. <i>Journal of Elliptic and Parabolic Equations</i>. <a href="https://doi.org/10.1007/s41808-023-00250-8">https://doi.org/10.1007/s41808-023-00250-8</a>
  bibtex: '@article{Papageorgiou_2023, title={Asymptotics for the infinite Brownian
    loop on noncompact symmetric spaces}, DOI={<a href="https://doi.org/10.1007/s41808-023-00250-8">10.1007/s41808-023-00250-8</a>},
    journal={Journal of Elliptic and Parabolic Equations}, publisher={Springer Science
    and Business Media LLC}, author={Papageorgiou, Efthymia}, year={2023} }'
  chicago: Papageorgiou, Efthymia. “Asymptotics for the Infinite Brownian Loop on
    Noncompact Symmetric Spaces.” <i>Journal of Elliptic and Parabolic Equations</i>,
    2023. <a href="https://doi.org/10.1007/s41808-023-00250-8">https://doi.org/10.1007/s41808-023-00250-8</a>.
  ieee: 'E. Papageorgiou, “Asymptotics for the infinite Brownian loop on noncompact
    symmetric spaces,” <i>Journal of Elliptic and Parabolic Equations</i>, 2023, doi:
    <a href="https://doi.org/10.1007/s41808-023-00250-8">10.1007/s41808-023-00250-8</a>.'
  mla: Papageorgiou, Efthymia. “Asymptotics for the Infinite Brownian Loop on Noncompact
    Symmetric Spaces.” <i>Journal of Elliptic and Parabolic Equations</i>, Springer
    Science and Business Media LLC, 2023, doi:<a href="https://doi.org/10.1007/s41808-023-00250-8">10.1007/s41808-023-00250-8</a>.
  short: E. Papageorgiou, Journal of Elliptic and Parabolic Equations (2023).
date_created: 2024-04-17T13:16:39Z
date_updated: 2024-04-17T13:17:10Z
department:
- _id: '555'
doi: 10.1007/s41808-023-00250-8
keyword:
- Applied Mathematics
- Numerical Analysis
- Analysis
language:
- iso: eng
publication: Journal of Elliptic and Parabolic Equations
publication_identifier:
  issn:
  - 2296-9020
  - 2296-9039
publication_status: published
publisher: Springer Science and Business Media LLC
status: public
title: Asymptotics for the infinite Brownian loop on noncompact symmetric spaces
type: journal_article
user_id: '100325'
year: '2023'
...
---
_id: '53538'
abstract:
- lang: eng
  text: <jats:title>Abstract</jats:title><jats:p>We study harmonic maps from a subset
    of the complex plane to a subset of the hyperbolic plane. In Fotiadis and Daskaloyannis
    (Nonlinear Anal 214, 112546, 2022), harmonic maps are related to the sinh-Gordon
    equation and a Bäcklund transformation is introduced, which connects solutions
    of the sinh-Gordon and sine-Gordon equation. We develop this machinery in order
    to construct new harmonic maps to the hyperbolic plane.</jats:p>
author:
- first_name: G.
  full_name: Polychrou, G.
  last_name: Polychrou
- first_name: Efthymia
  full_name: Papageorgiou, Efthymia
  id: '100325'
  last_name: Papageorgiou
- first_name: A.
  full_name: Fotiadis, A.
  last_name: Fotiadis
- first_name: C.
  full_name: Daskaloyannis, C.
  last_name: Daskaloyannis
citation:
  ama: Polychrou G, Papageorgiou E, Fotiadis A, Daskaloyannis C. New examples of harmonic
    maps to the hyperbolic plane via Bäcklund transformation. <i>Revista Matemática
    Complutense</i>. Published online 2023. doi:<a href="https://doi.org/10.1007/s13163-023-00476-z">10.1007/s13163-023-00476-z</a>
  apa: Polychrou, G., Papageorgiou, E., Fotiadis, A., &#38; Daskaloyannis, C. (2023).
    New examples of harmonic maps to the hyperbolic plane via Bäcklund transformation.
    <i>Revista Matemática Complutense</i>. <a href="https://doi.org/10.1007/s13163-023-00476-z">https://doi.org/10.1007/s13163-023-00476-z</a>
  bibtex: '@article{Polychrou_Papageorgiou_Fotiadis_Daskaloyannis_2023, title={New
    examples of harmonic maps to the hyperbolic plane via Bäcklund transformation},
    DOI={<a href="https://doi.org/10.1007/s13163-023-00476-z">10.1007/s13163-023-00476-z</a>},
    journal={Revista Matemática Complutense}, publisher={Springer Science and Business
    Media LLC}, author={Polychrou, G. and Papageorgiou, Efthymia and Fotiadis, A.
    and Daskaloyannis, C.}, year={2023} }'
  chicago: Polychrou, G., Efthymia Papageorgiou, A. Fotiadis, and C. Daskaloyannis.
    “New Examples of Harmonic Maps to the Hyperbolic Plane via Bäcklund Transformation.”
    <i>Revista Matemática Complutense</i>, 2023. <a href="https://doi.org/10.1007/s13163-023-00476-z">https://doi.org/10.1007/s13163-023-00476-z</a>.
  ieee: 'G. Polychrou, E. Papageorgiou, A. Fotiadis, and C. Daskaloyannis, “New examples
    of harmonic maps to the hyperbolic plane via Bäcklund transformation,” <i>Revista
    Matemática Complutense</i>, 2023, doi: <a href="https://doi.org/10.1007/s13163-023-00476-z">10.1007/s13163-023-00476-z</a>.'
  mla: Polychrou, G., et al. “New Examples of Harmonic Maps to the Hyperbolic Plane
    via Bäcklund Transformation.” <i>Revista Matemática Complutense</i>, Springer
    Science and Business Media LLC, 2023, doi:<a href="https://doi.org/10.1007/s13163-023-00476-z">10.1007/s13163-023-00476-z</a>.
  short: G. Polychrou, E. Papageorgiou, A. Fotiadis, C. Daskaloyannis, Revista Matemática
    Complutense (2023).
date_created: 2024-04-17T13:15:07Z
date_updated: 2024-04-17T13:15:51Z
department:
- _id: '555'
doi: 10.1007/s13163-023-00476-z
keyword:
- General Mathematics
language:
- iso: eng
publication: Revista Matemática Complutense
publication_identifier:
  issn:
  - 1139-1138
  - 1988-2807
publication_status: published
publisher: Springer Science and Business Media LLC
status: public
title: New examples of harmonic maps to the hyperbolic plane via Bäcklund transformation
type: journal_article
user_id: '100325'
year: '2023'
...
---
_id: '45786'
abstract:
- lang: eng
  text: Intending to counteract Klein’s second discontinuity in teacher education,
    we explored and applied the innovation of “interface ePortfolio” in the context
    of a geometry course for preservice teachers (PSTs). The tool offers the possibility
    of implementing the design principle of profession orientation. In the article,
    we theoretically clarify what we understand by this principle and locate our innovative
    concept against this theoretical background. We empirically investigate the extent
    to which counteraction against the second discontinuity is successful by analyzing
    reflection texts created in the interface ePortfolio, focusing on PSTs’ perspectives.
    Our qualitative content analysis shows that most of them perceive the innovation
    as helpful in the intended sense and indicates that the course concept, in general,
    and the interface ePortfolio, in particular, have helped establish relevant links
    between the course content and their later work as teachers.
article_type: original
author:
- first_name: Max
  full_name: Hoffmann, Max
  id: '32202'
  last_name: Hoffmann
  orcid: 0000-0002-6964-7123
- first_name: Rolf
  full_name: Biehler, Rolf
  id: '16274'
  last_name: Biehler
citation:
  ama: Hoffmann M, Biehler R. Implementing profession orientation as a design principle
    for overcoming Klein’s second discontinuity – preservice teacher’s perspectives
    on interface activities in the context of a geometry course. <i>ZDM – Mathematics
    Education</i>. Published online 2023. doi:<a href="https://doi.org/10.1007/s11858-023-01505-3">10.1007/s11858-023-01505-3</a>
  apa: Hoffmann, M., &#38; Biehler, R. (2023). Implementing profession orientation
    as a design principle for overcoming Klein’s second discontinuity – preservice
    teacher’s perspectives on interface activities in the context of a geometry course.
    <i>ZDM – Mathematics Education</i>. <a href="https://doi.org/10.1007/s11858-023-01505-3">https://doi.org/10.1007/s11858-023-01505-3</a>
  bibtex: '@article{Hoffmann_Biehler_2023, title={Implementing profession orientation
    as a design principle for overcoming Klein’s second discontinuity – preservice
    teacher’s perspectives on interface activities in the context of a geometry course},
    DOI={<a href="https://doi.org/10.1007/s11858-023-01505-3">10.1007/s11858-023-01505-3</a>},
    journal={ZDM – Mathematics Education}, publisher={Springer}, author={Hoffmann,
    Max and Biehler, Rolf}, year={2023} }'
  chicago: Hoffmann, Max, and Rolf Biehler. “Implementing Profession Orientation as
    a Design Principle for Overcoming Klein’s Second Discontinuity – Preservice Teacher’s
    Perspectives on Interface Activities in the Context of a Geometry Course.” <i>ZDM
    – Mathematics Education</i>, 2023. <a href="https://doi.org/10.1007/s11858-023-01505-3">https://doi.org/10.1007/s11858-023-01505-3</a>.
  ieee: 'M. Hoffmann and R. Biehler, “Implementing profession orientation as a design
    principle for overcoming Klein’s second discontinuity – preservice teacher’s perspectives
    on interface activities in the context of a geometry course,” <i>ZDM – Mathematics
    Education</i>, 2023, doi: <a href="https://doi.org/10.1007/s11858-023-01505-3">10.1007/s11858-023-01505-3</a>.'
  mla: Hoffmann, Max, and Rolf Biehler. “Implementing Profession Orientation as a
    Design Principle for Overcoming Klein’s Second Discontinuity – Preservice Teacher’s
    Perspectives on Interface Activities in the Context of a Geometry Course.” <i>ZDM
    – Mathematics Education</i>, Springer, 2023, doi:<a href="https://doi.org/10.1007/s11858-023-01505-3">10.1007/s11858-023-01505-3</a>.
  short: M. Hoffmann, R. Biehler, ZDM – Mathematics Education (2023).
date_created: 2023-06-27T11:45:25Z
date_updated: 2024-04-18T09:01:29Z
ddc:
- '510'
- '370'
department:
- _id: '643'
doi: 10.1007/s11858-023-01505-3
file:
- access_level: closed
  content_type: application/pdf
  creator: maxh
  date_created: 2023-07-13T09:40:02Z
  date_updated: 2023-07-13T09:40:02Z
  file_id: '46041'
  file_name: s11858-023-01505-3.pdf
  file_size: 1460246
  relation: main_file
  success: 1
file_date_updated: 2023-07-13T09:40:02Z
has_accepted_license: '1'
keyword:
- General Mathematics
- Education
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://link.springer.com/article/10.1007/s11858-023-01505-3
oa: '1'
publication: ZDM – Mathematics Education
publication_identifier:
  issn:
  - 1863-9690
  - 1863-9704
publication_status: published
publisher: Springer
quality_controlled: '1'
status: public
title: Implementing profession orientation as a design principle for overcoming Klein’s
  second discontinuity – preservice teacher’s perspectives on interface activities
  in the context of a geometry course
type: journal_article
user_id: '37888'
year: '2023'
...