---
_id: '63309'
abstract:
- lang: eng
  text: <jats:title>Abstract</jats:title><jats:p>This manuscript is concerned with
    the problem of efficiently estimating chemotactic gradients, as forming a ubiquitous
    issue of key importance in virtually any proof of boundedness features in Keller–Segel
    type systems. A strategy is proposed which at its core relies on bounds for such
    quantities, conditional in the sense of involving certain Lebesgue norms of solution
    components that explicitly influence the signal evolution.</jats:p><jats:p>Applications
    of this procedure firstly provide apparently novel boundedness results for two
    particular classes chemotaxis systems, and apart from that are shown to significantly
    condense proofs for basically well‐known statements on boundedness in two further
    Keller–Segel type problems.</jats:p>
author:
- first_name: Michael
  full_name: Winkler, Michael
  id: '31496'
  last_name: Winkler
citation:
  ama: Winkler M. A unifying approach toward boundedness in Keller–Segel type cross‐diffusion
    systems via conditional L∞$L^\infty$ estimates for taxis gradients. <i>Mathematische
    Nachrichten</i>. 2022;295(9):1840-1862. doi:<a href="https://doi.org/10.1002/mana.202000403">10.1002/mana.202000403</a>
  apa: Winkler, M. (2022). A unifying approach toward boundedness in Keller–Segel
    type cross‐diffusion systems via conditional L∞$L^\infty$ estimates for taxis
    gradients. <i>Mathematische Nachrichten</i>, <i>295</i>(9), 1840–1862. <a href="https://doi.org/10.1002/mana.202000403">https://doi.org/10.1002/mana.202000403</a>
  bibtex: '@article{Winkler_2022, title={A unifying approach toward boundedness in
    Keller–Segel type cross‐diffusion systems via conditional L∞$L^\infty$ estimates
    for taxis gradients}, volume={295}, DOI={<a href="https://doi.org/10.1002/mana.202000403">10.1002/mana.202000403</a>},
    number={9}, journal={Mathematische Nachrichten}, publisher={Wiley}, author={Winkler,
    Michael}, year={2022}, pages={1840–1862} }'
  chicago: 'Winkler, Michael. “A Unifying Approach toward Boundedness in Keller–Segel
    Type Cross‐diffusion Systems via Conditional L∞$L^\infty$ Estimates for Taxis
    Gradients.” <i>Mathematische Nachrichten</i> 295, no. 9 (2022): 1840–62. <a href="https://doi.org/10.1002/mana.202000403">https://doi.org/10.1002/mana.202000403</a>.'
  ieee: 'M. Winkler, “A unifying approach toward boundedness in Keller–Segel type
    cross‐diffusion systems via conditional L∞$L^\infty$ estimates for taxis gradients,”
    <i>Mathematische Nachrichten</i>, vol. 295, no. 9, pp. 1840–1862, 2022, doi: <a
    href="https://doi.org/10.1002/mana.202000403">10.1002/mana.202000403</a>.'
  mla: Winkler, Michael. “A Unifying Approach toward Boundedness in Keller–Segel Type
    Cross‐diffusion Systems via Conditional L∞$L^\infty$ Estimates for Taxis Gradients.”
    <i>Mathematische Nachrichten</i>, vol. 295, no. 9, Wiley, 2022, pp. 1840–62, doi:<a
    href="https://doi.org/10.1002/mana.202000403">10.1002/mana.202000403</a>.
  short: M. Winkler, Mathematische Nachrichten 295 (2022) 1840–1862.
date_created: 2025-12-18T19:28:46Z
date_updated: 2025-12-18T20:05:19Z
doi: 10.1002/mana.202000403
intvolume: '       295'
issue: '9'
language:
- iso: eng
page: 1840-1862
publication: Mathematische Nachrichten
publication_identifier:
  issn:
  - 0025-584X
  - 1522-2616
publication_status: published
publisher: Wiley
status: public
title: A unifying approach toward boundedness in Keller–Segel type cross‐diffusion
  systems via conditional L∞$L^\infty$ estimates for taxis gradients
type: journal_article
user_id: '31496'
volume: 295
year: '2022'
...
---
_id: '63306'
author:
- first_name: Michael
  full_name: Winkler, Michael
  id: '31496'
  last_name: Winkler
citation:
  ama: Winkler M. A critical blow-up exponent for flux limiation in a Keller-Segel
    system. <i>Indiana University Mathematics Journal</i>. 2022;71(4):1437-1465. doi:<a
    href="https://doi.org/10.1512/iumj.2022.71.9042">10.1512/iumj.2022.71.9042</a>
  apa: Winkler, M. (2022). A critical blow-up exponent for flux limiation in a Keller-Segel
    system. <i>Indiana University Mathematics Journal</i>, <i>71</i>(4), 1437–1465.
    <a href="https://doi.org/10.1512/iumj.2022.71.9042">https://doi.org/10.1512/iumj.2022.71.9042</a>
  bibtex: '@article{Winkler_2022, title={A critical blow-up exponent for flux limiation
    in a Keller-Segel system}, volume={71}, DOI={<a href="https://doi.org/10.1512/iumj.2022.71.9042">10.1512/iumj.2022.71.9042</a>},
    number={4}, journal={Indiana University Mathematics Journal}, publisher={Indiana
    University Mathematics Journal}, author={Winkler, Michael}, year={2022}, pages={1437–1465}
    }'
  chicago: 'Winkler, Michael. “A Critical Blow-up Exponent for Flux Limiation in a
    Keller-Segel System.” <i>Indiana University Mathematics Journal</i> 71, no. 4
    (2022): 1437–65. <a href="https://doi.org/10.1512/iumj.2022.71.9042">https://doi.org/10.1512/iumj.2022.71.9042</a>.'
  ieee: 'M. Winkler, “A critical blow-up exponent for flux limiation in a Keller-Segel
    system,” <i>Indiana University Mathematics Journal</i>, vol. 71, no. 4, pp. 1437–1465,
    2022, doi: <a href="https://doi.org/10.1512/iumj.2022.71.9042">10.1512/iumj.2022.71.9042</a>.'
  mla: Winkler, Michael. “A Critical Blow-up Exponent for Flux Limiation in a Keller-Segel
    System.” <i>Indiana University Mathematics Journal</i>, vol. 71, no. 4, Indiana
    University Mathematics Journal, 2022, pp. 1437–65, doi:<a href="https://doi.org/10.1512/iumj.2022.71.9042">10.1512/iumj.2022.71.9042</a>.
  short: M. Winkler, Indiana University Mathematics Journal 71 (2022) 1437–1465.
date_created: 2025-12-18T19:26:56Z
date_updated: 2025-12-18T20:04:53Z
doi: 10.1512/iumj.2022.71.9042
intvolume: '        71'
issue: '4'
language:
- iso: eng
page: 1437-1465
publication: Indiana University Mathematics Journal
publication_identifier:
  issn:
  - 0022-2518
publication_status: published
publisher: Indiana University Mathematics Journal
status: public
title: A critical blow-up exponent for flux limiation in a Keller-Segel system
type: journal_article
user_id: '31496'
volume: 71
year: '2022'
...
---
_id: '63284'
abstract:
- lang: eng
  text: '<jats:p> A no-flux initial-boundary value problem for the cross-diffusion
    system [Formula: see text] is considered in smoothly bounded domains [Formula:
    see text] with [Formula: see text]. It is shown that whenever [Formula: see text]
    is positive on [Formula: see text] and such that [Formula: see text] for some
    [Formula: see text], for all suitably regular positive initial data a global very
    weak solution, particularly preserving mass in its first component, can be constructed.
    This extends previous results which either concentrate on non-degenerate analogs,
    or are restricted to the special case [Formula: see text]. </jats:p><jats:p> To
    appropriately cope with the considerably stronger cross-degeneracies thus allowed
    through [Formula: see text] when [Formula: see text] is large, in its core part
    the analysis relies on the use of the Moser–Trudinger inequality in controlling
    the respective diffusion rates [Formula: see text] from below. </jats:p>'
article_number: '2250012'
author:
- first_name: Michael
  full_name: Winkler, Michael
  id: '31496'
  last_name: Winkler
citation:
  ama: Winkler M. Application of the Moser–Trudinger inequality in the construction
    of global solutions to a strongly degenerate migration model. <i>Bulletin of Mathematical
    Sciences</i>. 2022;13(02). doi:<a href="https://doi.org/10.1142/s1664360722500126">10.1142/s1664360722500126</a>
  apa: Winkler, M. (2022). Application of the Moser–Trudinger inequality in the construction
    of global solutions to a strongly degenerate migration model. <i>Bulletin of Mathematical
    Sciences</i>, <i>13</i>(02), Article 2250012. <a href="https://doi.org/10.1142/s1664360722500126">https://doi.org/10.1142/s1664360722500126</a>
  bibtex: '@article{Winkler_2022, title={Application of the Moser–Trudinger inequality
    in the construction of global solutions to a strongly degenerate migration model},
    volume={13}, DOI={<a href="https://doi.org/10.1142/s1664360722500126">10.1142/s1664360722500126</a>},
    number={022250012}, journal={Bulletin of Mathematical Sciences}, publisher={World
    Scientific Pub Co Pte Ltd}, author={Winkler, Michael}, year={2022} }'
  chicago: Winkler, Michael. “Application of the Moser–Trudinger Inequality in the
    Construction of Global Solutions to a Strongly Degenerate Migration Model.” <i>Bulletin
    of Mathematical Sciences</i> 13, no. 02 (2022). <a href="https://doi.org/10.1142/s1664360722500126">https://doi.org/10.1142/s1664360722500126</a>.
  ieee: 'M. Winkler, “Application of the Moser–Trudinger inequality in the construction
    of global solutions to a strongly degenerate migration model,” <i>Bulletin of
    Mathematical Sciences</i>, vol. 13, no. 02, Art. no. 2250012, 2022, doi: <a href="https://doi.org/10.1142/s1664360722500126">10.1142/s1664360722500126</a>.'
  mla: Winkler, Michael. “Application of the Moser–Trudinger Inequality in the Construction
    of Global Solutions to a Strongly Degenerate Migration Model.” <i>Bulletin of
    Mathematical Sciences</i>, vol. 13, no. 02, 2250012, World Scientific Pub Co Pte
    Ltd, 2022, doi:<a href="https://doi.org/10.1142/s1664360722500126">10.1142/s1664360722500126</a>.
  short: M. Winkler, Bulletin of Mathematical Sciences 13 (2022).
date_created: 2025-12-18T19:18:11Z
date_updated: 2025-12-18T20:07:05Z
doi: 10.1142/s1664360722500126
intvolume: '        13'
issue: '02'
language:
- iso: eng
publication: Bulletin of Mathematical Sciences
publication_identifier:
  issn:
  - 1664-3607
  - 1664-3615
publication_status: published
publisher: World Scientific Pub Co Pte Ltd
status: public
title: Application of the Moser–Trudinger inequality in the construction of global
  solutions to a strongly degenerate migration model
type: journal_article
user_id: '31496'
volume: 13
year: '2022'
...
---
_id: '63286'
abstract:
- lang: eng
  text: '<jats:p> The chemotaxis system [Formula: see text] is considered in a ball
    [Formula: see text]. </jats:p><jats:p> It is shown that if [Formula: see text]
    suitably generalizes the prototype given by [Formula: see text] with some [Formula:
    see text], and if diffusion is suitably weak in the sense that [Formula: see text]
    is such that there exist [Formula: see text] and [Formula: see text] fulfilling
    [Formula: see text] then for appropriate choices of sufficiently concentrated
    initial data, an associated no-flux initial-boundary value problem admits a global
    classical solution [Formula: see text] which blows up in infinite time and satisfies
    [Formula: see text] A major part of the proof is based on a comparison argument
    involving explicitly constructed subsolutions to a scalar parabolic problem satisfied
    by mass accumulation functions corresponding to solutions of ( ⋆ ). </jats:p>'
author:
- first_name: Michael
  full_name: Winkler, Michael
  id: '31496'
  last_name: Winkler
citation:
  ama: Winkler M. Exponential grow-up rates in a quasilinear Keller–Segel system.
    <i>Asymptotic Analysis</i>. 2022;131(1):33-57. doi:<a href="https://doi.org/10.3233/asy-221765">10.3233/asy-221765</a>
  apa: Winkler, M. (2022). Exponential grow-up rates in a quasilinear Keller–Segel
    system. <i>Asymptotic Analysis</i>, <i>131</i>(1), 33–57. <a href="https://doi.org/10.3233/asy-221765">https://doi.org/10.3233/asy-221765</a>
  bibtex: '@article{Winkler_2022, title={Exponential grow-up rates in a quasilinear
    Keller–Segel system}, volume={131}, DOI={<a href="https://doi.org/10.3233/asy-221765">10.3233/asy-221765</a>},
    number={1}, journal={Asymptotic Analysis}, publisher={SAGE Publications}, author={Winkler,
    Michael}, year={2022}, pages={33–57} }'
  chicago: 'Winkler, Michael. “Exponential Grow-up Rates in a Quasilinear Keller–Segel
    System.” <i>Asymptotic Analysis</i> 131, no. 1 (2022): 33–57. <a href="https://doi.org/10.3233/asy-221765">https://doi.org/10.3233/asy-221765</a>.'
  ieee: 'M. Winkler, “Exponential grow-up rates in a quasilinear Keller–Segel system,”
    <i>Asymptotic Analysis</i>, vol. 131, no. 1, pp. 33–57, 2022, doi: <a href="https://doi.org/10.3233/asy-221765">10.3233/asy-221765</a>.'
  mla: Winkler, Michael. “Exponential Grow-up Rates in a Quasilinear Keller–Segel
    System.” <i>Asymptotic Analysis</i>, vol. 131, no. 1, SAGE Publications, 2022,
    pp. 33–57, doi:<a href="https://doi.org/10.3233/asy-221765">10.3233/asy-221765</a>.
  short: M. Winkler, Asymptotic Analysis 131 (2022) 33–57.
date_created: 2025-12-18T19:18:51Z
date_updated: 2025-12-18T20:07:19Z
doi: 10.3233/asy-221765
intvolume: '       131'
issue: '1'
language:
- iso: eng
page: 33-57
publication: Asymptotic Analysis
publication_identifier:
  issn:
  - 0921-7134
  - 1875-8576
publication_status: published
publisher: SAGE Publications
status: public
title: Exponential grow-up rates in a quasilinear Keller–Segel system
type: journal_article
user_id: '31496'
volume: 131
year: '2022'
...
---
_id: '63293'
abstract:
- lang: eng
  text: "<jats:p xml:lang=\"fr\">&lt;p style='text-indent:20px;'&gt;The Cauchy problem
    in &lt;inline-formula&gt;&lt;tex-math id=\"M2\"&gt;\\begin{document}$  \\mathbb{R}^3
    $\\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; for the chemotaxis-Navier–Stokes
    system&lt;/p&gt;&lt;p style='text-indent:20px;'&gt;&lt;disp-formula&gt; &lt;label/&gt;
    &lt;tex-math id=\"FE1\"&gt; \\begin{document}$ \\begin{eqnarray*} \\left\\{ \\begin{array}{l}
    \     n_t + u\\cdot\\nabla n = \\Delta n - \\nabla \\cdot (n\\nabla c), \\\\\tc_t
    + u\\cdot\\nabla c = \\Delta c - nc, \\\\ \tu_t + (u\\cdot\\nabla) u = \\Delta
    u + \\nabla P + n\\nabla\\phi, \\qquad \\nabla \\cdot u = 0, \\ \t\\end{array}
    \\right. \\end{eqnarray*} $\\end{document} &lt;/tex-math&gt;&lt;/disp-formula&gt;&lt;/p&gt;&lt;p
    style='text-indent:20px;'&gt;is considered. Under suitable conditions on the initial
    data &lt;inline-formula&gt;&lt;tex-math id=\"M3\"&gt;\\begin{document}$ (n_0,
    c_0, u_0) $\\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;, with regard
    to the crucial first component requiring that &lt;inline-formula&gt;&lt;tex-math
    id=\"M4\"&gt;\\begin{document}$ n_0\\in L^1( \\mathbb{R}^3) $\\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;
    be nonnegative and such that &lt;inline-formula&gt;&lt;tex-math id=\"M5\"&gt;\\begin{document}$
    (n_0+1)\\ln (n_0+1) \\in L^1( \\mathbb{R}^3) $\\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;,
    a globally defined weak solution with &lt;inline-formula&gt;&lt;tex-math id=\"M6\"&gt;\\begin{document}$
    (n, c, u)|_{t = 0} = (n_0, c_0, u_0) $\\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;
    is constructed. Apart from that, assuming that moreover &lt;inline-formula&gt;&lt;tex-math
    id=\"M7\"&gt;\\begin{document}$ \\int_{ \\mathbb{R}^3} n_0(x) \\ln (1+|x|^2) dx
    $\\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; is finite, it is shown
    that a weak solution exists which enjoys further regularity features and preserves
    mass in an appropriate sense.&lt;/p&gt;</jats:p>"
article_number: '5201'
author:
- first_name: Kyungkeun
  full_name: Kang, Kyungkeun
  last_name: Kang
- first_name: Jihoon
  full_name: Lee, Jihoon
  last_name: Lee
- first_name: Michael
  full_name: Winkler, Michael
  id: '31496'
  last_name: Winkler
citation:
  ama: Kang K, Lee J, Winkler M. Global weak solutions to a chemotaxis-Navier-Stokes
    system in $  \mathbb{R}^3 $. <i>Discrete and Continuous Dynamical Systems</i>.
    2022;42(11). doi:<a href="https://doi.org/10.3934/dcds.2022091">10.3934/dcds.2022091</a>
  apa: Kang, K., Lee, J., &#38; Winkler, M. (2022). Global weak solutions to a chemotaxis-Navier-Stokes
    system in $  \mathbb{R}^3 $. <i>Discrete and Continuous Dynamical Systems</i>,
    <i>42</i>(11), Article 5201. <a href="https://doi.org/10.3934/dcds.2022091">https://doi.org/10.3934/dcds.2022091</a>
  bibtex: '@article{Kang_Lee_Winkler_2022, title={Global weak solutions to a chemotaxis-Navier-Stokes
    system in $  \mathbb{R}^3 $}, volume={42}, DOI={<a href="https://doi.org/10.3934/dcds.2022091">10.3934/dcds.2022091</a>},
    number={115201}, journal={Discrete and Continuous Dynamical Systems}, publisher={American
    Institute of Mathematical Sciences (AIMS)}, author={Kang, Kyungkeun and Lee, Jihoon
    and Winkler, Michael}, year={2022} }'
  chicago: Kang, Kyungkeun, Jihoon Lee, and Michael Winkler. “Global Weak Solutions
    to a Chemotaxis-Navier-Stokes System in $  \mathbb{R}^3 $.” <i>Discrete and Continuous
    Dynamical Systems</i> 42, no. 11 (2022). <a href="https://doi.org/10.3934/dcds.2022091">https://doi.org/10.3934/dcds.2022091</a>.
  ieee: 'K. Kang, J. Lee, and M. Winkler, “Global weak solutions to a chemotaxis-Navier-Stokes
    system in $  \mathbb{R}^3 $,” <i>Discrete and Continuous Dynamical Systems</i>,
    vol. 42, no. 11, Art. no. 5201, 2022, doi: <a href="https://doi.org/10.3934/dcds.2022091">10.3934/dcds.2022091</a>.'
  mla: Kang, Kyungkeun, et al. “Global Weak Solutions to a Chemotaxis-Navier-Stokes
    System in $  \mathbb{R}^3 $.” <i>Discrete and Continuous Dynamical Systems</i>,
    vol. 42, no. 11, 5201, American Institute of Mathematical Sciences (AIMS), 2022,
    doi:<a href="https://doi.org/10.3934/dcds.2022091">10.3934/dcds.2022091</a>.
  short: K. Kang, J. Lee, M. Winkler, Discrete and Continuous Dynamical Systems 42
    (2022).
date_created: 2025-12-18T19:22:04Z
date_updated: 2025-12-18T20:08:21Z
doi: 10.3934/dcds.2022091
intvolume: '        42'
issue: '11'
language:
- iso: eng
publication: Discrete and Continuous Dynamical Systems
publication_identifier:
  issn:
  - 1078-0947
  - 1553-5231
publication_status: published
publisher: American Institute of Mathematical Sciences (AIMS)
status: public
title: Global weak solutions to a chemotaxis-Navier-Stokes system in $  \mathbb{R}^3
  $
type: journal_article
user_id: '31496'
volume: 42
year: '2022'
...
---
_id: '63290'
abstract:
- lang: eng
  text: <jats:p> This paper proposes a review focused on exotic chemotaxis and cross-diffusion
    models in complex environments. The term exotic is used to denote the dynamics
    of models interacting with a time-evolving external system and, specifically,
    models derived with the aim of describing the dynamics of living systems. The
    presentation first, considers the derivation of phenomenological models of chemotaxis
    and cross-diffusion models with particular attention on nonlinear characteristics.
    Then, a variety of exotic models is presented with some hints toward the derivation
    of new models, by accounting for a critical analysis looking ahead to perspectives.
    The second part of the paper is devoted to a survey of analytical problems concerning
    the application of models to the study of real world dynamics. Finally, the focus
    shifts to research perspectives within the framework of a multiscale vision, where
    different paths are examined to move from the dynamics at the microscopic scale
    to collective behaviors at the macroscopic scale. </jats:p>
author:
- first_name: N.
  full_name: Bellomo, N.
  last_name: Bellomo
- first_name: N.
  full_name: Outada, N.
  last_name: Outada
- first_name: J.
  full_name: Soler, J.
  last_name: Soler
- first_name: Y.
  full_name: Tao, Y.
  last_name: Tao
- first_name: Michael
  full_name: Winkler, Michael
  id: '31496'
  last_name: Winkler
citation:
  ama: 'Bellomo N, Outada N, Soler J, Tao Y, Winkler M. Chemotaxis and cross-diffusion
    models in complex environments: Models and analytic problems toward a multiscale
    vision. <i>Mathematical Models and Methods in Applied Sciences</i>. 2022;32(04):713-792.
    doi:<a href="https://doi.org/10.1142/s0218202522500166">10.1142/s0218202522500166</a>'
  apa: 'Bellomo, N., Outada, N., Soler, J., Tao, Y., &#38; Winkler, M. (2022). Chemotaxis
    and cross-diffusion models in complex environments: Models and analytic problems
    toward a multiscale vision. <i>Mathematical Models and Methods in Applied Sciences</i>,
    <i>32</i>(04), 713–792. <a href="https://doi.org/10.1142/s0218202522500166">https://doi.org/10.1142/s0218202522500166</a>'
  bibtex: '@article{Bellomo_Outada_Soler_Tao_Winkler_2022, title={Chemotaxis and cross-diffusion
    models in complex environments: Models and analytic problems toward a multiscale
    vision}, volume={32}, DOI={<a href="https://doi.org/10.1142/s0218202522500166">10.1142/s0218202522500166</a>},
    number={04}, journal={Mathematical Models and Methods in Applied Sciences}, publisher={World
    Scientific Pub Co Pte Ltd}, author={Bellomo, N. and Outada, N. and Soler, J. and
    Tao, Y. and Winkler, Michael}, year={2022}, pages={713–792} }'
  chicago: 'Bellomo, N., N. Outada, J. Soler, Y. Tao, and Michael Winkler. “Chemotaxis
    and Cross-Diffusion Models in Complex Environments: Models and Analytic Problems
    toward a Multiscale Vision.” <i>Mathematical Models and Methods in Applied Sciences</i>
    32, no. 04 (2022): 713–92. <a href="https://doi.org/10.1142/s0218202522500166">https://doi.org/10.1142/s0218202522500166</a>.'
  ieee: 'N. Bellomo, N. Outada, J. Soler, Y. Tao, and M. Winkler, “Chemotaxis and
    cross-diffusion models in complex environments: Models and analytic problems toward
    a multiscale vision,” <i>Mathematical Models and Methods in Applied Sciences</i>,
    vol. 32, no. 04, pp. 713–792, 2022, doi: <a href="https://doi.org/10.1142/s0218202522500166">10.1142/s0218202522500166</a>.'
  mla: 'Bellomo, N., et al. “Chemotaxis and Cross-Diffusion Models in Complex Environments:
    Models and Analytic Problems toward a Multiscale Vision.” <i>Mathematical Models
    and Methods in Applied Sciences</i>, vol. 32, no. 04, World Scientific Pub Co
    Pte Ltd, 2022, pp. 713–92, doi:<a href="https://doi.org/10.1142/s0218202522500166">10.1142/s0218202522500166</a>.'
  short: N. Bellomo, N. Outada, J. Soler, Y. Tao, M. Winkler, Mathematical Models
    and Methods in Applied Sciences 32 (2022) 713–792.
date_created: 2025-12-18T19:20:25Z
date_updated: 2025-12-18T20:07:51Z
doi: 10.1142/s0218202522500166
intvolume: '        32'
issue: '04'
language:
- iso: eng
page: 713-792
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: 'Chemotaxis and cross-diffusion models in complex environments: Models and
  analytic problems toward a multiscale vision'
type: journal_article
user_id: '31496'
volume: 32
year: '2022'
...
---
_id: '63295'
abstract:
- lang: eng
  text: <jats:title>Abstract</jats:title><jats:p>We introduce a generalized concept
    of solutions for reaction–diffusion systems and prove their global existence.
    The only restriction on the reaction function beyond regularity, quasipositivity
    and mass control is special in that it merely controls the growth of cross-absorptive
    terms. The result covers nonlinear diffusion and does not rely on an entropy estimate.</jats:p>
article_number: '14'
author:
- first_name: Johannes
  full_name: Lankeit, Johannes
  last_name: Lankeit
- first_name: Michael
  full_name: Winkler, Michael
  id: '31496'
  last_name: Winkler
citation:
  ama: Lankeit J, Winkler M. Global existence in reaction–diffusion systems with mass
    control under relaxed assumptions merely referring to cross-absorptive effects.
    <i>Journal of Evolution Equations</i>. 2022;22(1). doi:<a href="https://doi.org/10.1007/s00028-022-00768-9">10.1007/s00028-022-00768-9</a>
  apa: Lankeit, J., &#38; Winkler, M. (2022). Global existence in reaction–diffusion
    systems with mass control under relaxed assumptions merely referring to cross-absorptive
    effects. <i>Journal of Evolution Equations</i>, <i>22</i>(1), Article 14. <a href="https://doi.org/10.1007/s00028-022-00768-9">https://doi.org/10.1007/s00028-022-00768-9</a>
  bibtex: '@article{Lankeit_Winkler_2022, title={Global existence in reaction–diffusion
    systems with mass control under relaxed assumptions merely referring to cross-absorptive
    effects}, volume={22}, DOI={<a href="https://doi.org/10.1007/s00028-022-00768-9">10.1007/s00028-022-00768-9</a>},
    number={114}, journal={Journal of Evolution Equations}, publisher={Springer Science
    and Business Media LLC}, author={Lankeit, Johannes and Winkler, Michael}, year={2022}
    }'
  chicago: Lankeit, Johannes, and Michael Winkler. “Global Existence in Reaction–Diffusion
    Systems with Mass Control under Relaxed Assumptions Merely Referring to Cross-Absorptive
    Effects.” <i>Journal of Evolution Equations</i> 22, no. 1 (2022). <a href="https://doi.org/10.1007/s00028-022-00768-9">https://doi.org/10.1007/s00028-022-00768-9</a>.
  ieee: 'J. Lankeit and M. Winkler, “Global existence in reaction–diffusion systems
    with mass control under relaxed assumptions merely referring to cross-absorptive
    effects,” <i>Journal of Evolution Equations</i>, vol. 22, no. 1, Art. no. 14,
    2022, doi: <a href="https://doi.org/10.1007/s00028-022-00768-9">10.1007/s00028-022-00768-9</a>.'
  mla: Lankeit, Johannes, and Michael Winkler. “Global Existence in Reaction–Diffusion
    Systems with Mass Control under Relaxed Assumptions Merely Referring to Cross-Absorptive
    Effects.” <i>Journal of Evolution Equations</i>, vol. 22, no. 1, 14, Springer
    Science and Business Media LLC, 2022, doi:<a href="https://doi.org/10.1007/s00028-022-00768-9">10.1007/s00028-022-00768-9</a>.
  short: J. Lankeit, M. Winkler, Journal of Evolution Equations 22 (2022).
date_created: 2025-12-18T19:22:46Z
date_updated: 2025-12-18T20:08:35Z
doi: 10.1007/s00028-022-00768-9
intvolume: '        22'
issue: '1'
language:
- iso: eng
publication: Journal of Evolution Equations
publication_identifier:
  issn:
  - 1424-3199
  - 1424-3202
publication_status: published
publisher: Springer Science and Business Media LLC
status: public
title: Global existence in reaction–diffusion systems with mass control under relaxed
  assumptions merely referring to cross-absorptive effects
type: journal_article
user_id: '31496'
volume: 22
year: '2022'
...
---
_id: '63299'
author:
- first_name: Youshan
  full_name: Tao, Youshan
  last_name: Tao
- first_name: Michael
  full_name: Winkler, Michael
  id: '31496'
  last_name: Winkler
citation:
  ama: Tao Y, Winkler M. Existence Theory and Qualitative Analysis for a Fully Cross-Diffusive
    Predator-Prey System. <i>SIAM Journal on Mathematical Analysis</i>. 2022;54(4):4806-4864.
    doi:<a href="https://doi.org/10.1137/21m1449841">10.1137/21m1449841</a>
  apa: Tao, Y., &#38; Winkler, M. (2022). Existence Theory and Qualitative Analysis
    for a Fully Cross-Diffusive Predator-Prey System. <i>SIAM Journal on Mathematical
    Analysis</i>, <i>54</i>(4), 4806–4864. <a href="https://doi.org/10.1137/21m1449841">https://doi.org/10.1137/21m1449841</a>
  bibtex: '@article{Tao_Winkler_2022, title={Existence Theory and Qualitative Analysis
    for a Fully Cross-Diffusive Predator-Prey System}, volume={54}, DOI={<a href="https://doi.org/10.1137/21m1449841">10.1137/21m1449841</a>},
    number={4}, journal={SIAM Journal on Mathematical Analysis}, publisher={Society
    for Industrial &#38; Applied Mathematics (SIAM)}, author={Tao, Youshan and Winkler,
    Michael}, year={2022}, pages={4806–4864} }'
  chicago: 'Tao, Youshan, and Michael Winkler. “Existence Theory and Qualitative Analysis
    for a Fully Cross-Diffusive Predator-Prey System.” <i>SIAM Journal on Mathematical
    Analysis</i> 54, no. 4 (2022): 4806–64. <a href="https://doi.org/10.1137/21m1449841">https://doi.org/10.1137/21m1449841</a>.'
  ieee: 'Y. Tao and M. Winkler, “Existence Theory and Qualitative Analysis for a Fully
    Cross-Diffusive Predator-Prey System,” <i>SIAM Journal on Mathematical Analysis</i>,
    vol. 54, no. 4, pp. 4806–4864, 2022, doi: <a href="https://doi.org/10.1137/21m1449841">10.1137/21m1449841</a>.'
  mla: Tao, Youshan, and Michael Winkler. “Existence Theory and Qualitative Analysis
    for a Fully Cross-Diffusive Predator-Prey System.” <i>SIAM Journal on Mathematical
    Analysis</i>, vol. 54, no. 4, Society for Industrial &#38; Applied Mathematics
    (SIAM), 2022, pp. 4806–64, doi:<a href="https://doi.org/10.1137/21m1449841">10.1137/21m1449841</a>.
  short: Y. Tao, M. Winkler, SIAM Journal on Mathematical Analysis 54 (2022) 4806–4864.
date_created: 2025-12-18T19:24:16Z
date_updated: 2025-12-18T20:09:05Z
doi: 10.1137/21m1449841
intvolume: '        54'
issue: '4'
language:
- iso: eng
page: 4806-4864
publication: SIAM Journal on Mathematical Analysis
publication_identifier:
  issn:
  - 0036-1410
  - 1095-7154
publication_status: published
publisher: Society for Industrial & Applied Mathematics (SIAM)
status: public
title: Existence Theory and Qualitative Analysis for a Fully Cross-Diffusive Predator-Prey
  System
type: journal_article
user_id: '31496'
volume: 54
year: '2022'
...
---
_id: '63298'
author:
- first_name: Angela
  full_name: Stevens, Angela
  last_name: Stevens
- first_name: Michael
  full_name: Winkler, Michael
  id: '31496'
  last_name: Winkler
citation:
  ama: Stevens A, Winkler M. Taxis-driven persistent localization in a degenerate
    Keller-Segel system. <i>Communications in Partial Differential Equations</i>.
    2022;47(12):2341-2362. doi:<a href="https://doi.org/10.1080/03605302.2022.2122836">10.1080/03605302.2022.2122836</a>
  apa: Stevens, A., &#38; Winkler, M. (2022). Taxis-driven persistent localization
    in a degenerate Keller-Segel system. <i>Communications in Partial Differential
    Equations</i>, <i>47</i>(12), 2341–2362. <a href="https://doi.org/10.1080/03605302.2022.2122836">https://doi.org/10.1080/03605302.2022.2122836</a>
  bibtex: '@article{Stevens_Winkler_2022, title={Taxis-driven persistent localization
    in a degenerate Keller-Segel system}, volume={47}, DOI={<a href="https://doi.org/10.1080/03605302.2022.2122836">10.1080/03605302.2022.2122836</a>},
    number={12}, journal={Communications in Partial Differential Equations}, publisher={Informa
    UK Limited}, author={Stevens, Angela and Winkler, Michael}, year={2022}, pages={2341–2362}
    }'
  chicago: 'Stevens, Angela, and Michael Winkler. “Taxis-Driven Persistent Localization
    in a Degenerate Keller-Segel System.” <i>Communications in Partial Differential
    Equations</i> 47, no. 12 (2022): 2341–62. <a href="https://doi.org/10.1080/03605302.2022.2122836">https://doi.org/10.1080/03605302.2022.2122836</a>.'
  ieee: 'A. Stevens and M. Winkler, “Taxis-driven persistent localization in a degenerate
    Keller-Segel system,” <i>Communications in Partial Differential Equations</i>,
    vol. 47, no. 12, pp. 2341–2362, 2022, doi: <a href="https://doi.org/10.1080/03605302.2022.2122836">10.1080/03605302.2022.2122836</a>.'
  mla: Stevens, Angela, and Michael Winkler. “Taxis-Driven Persistent Localization
    in a Degenerate Keller-Segel System.” <i>Communications in Partial Differential
    Equations</i>, vol. 47, no. 12, Informa UK Limited, 2022, pp. 2341–62, doi:<a
    href="https://doi.org/10.1080/03605302.2022.2122836">10.1080/03605302.2022.2122836</a>.
  short: A. Stevens, M. Winkler, Communications in Partial Differential Equations
    47 (2022) 2341–2362.
date_created: 2025-12-18T19:23:52Z
date_updated: 2025-12-18T20:08:58Z
doi: 10.1080/03605302.2022.2122836
intvolume: '        47'
issue: '12'
language:
- iso: eng
page: 2341-2362
publication: Communications in Partial Differential Equations
publication_identifier:
  issn:
  - 0360-5302
  - 1532-4133
publication_status: published
publisher: Informa UK Limited
status: public
title: Taxis-driven persistent localization in a degenerate Keller-Segel system
type: journal_article
user_id: '31496'
volume: 47
year: '2022'
...
---
_id: '63266'
abstract:
- lang: eng
  text: "<jats:title>Abstract</jats:title><jats:p>In a ball <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\Omega
    =B_R(0)\\subset \\mathbb {R}^n$$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n
    \                 <mml:mrow>\r\n                    <mml:mi>Ω</mml:mi>\r\n                    <mml:mo>=</mml:mo>\r\n
    \                   <mml:msub>\r\n                      <mml:mi>B</mml:mi>\r\n
    \                     <mml:mi>R</mml:mi>\r\n                    </mml:msub>\r\n
    \                   <mml:mrow>\r\n                      <mml:mo>(</mml:mo>\r\n
    \                     <mml:mn>0</mml:mn>\r\n                      <mml:mo>)</mml:mo>\r\n
    \                   </mml:mrow>\r\n                    <mml:mo>⊂</mml:mo>\r\n
    \                   <mml:msup>\r\n                      <mml:mrow>\r\n                        <mml:mi>R</mml:mi>\r\n
    \                     </mml:mrow>\r\n                      <mml:mi>n</mml:mi>\r\n
    \                   </mml:msup>\r\n                  </mml:mrow>\r\n                </mml:math></jats:alternatives></jats:inline-formula>,
    <jats:inline-formula><jats:alternatives><jats:tex-math>$$n\\ge 2$$</jats:tex-math><mml:math
    xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n                  <mml:mrow>\r\n
    \                   <mml:mi>n</mml:mi>\r\n                    <mml:mo>≥</mml:mo>\r\n
    \                   <mml:mn>2</mml:mn>\r\n                  </mml:mrow>\r\n                </mml:math></jats:alternatives></jats:inline-formula>,
    the chemotaxis system <jats:disp-formula><jats:alternatives><jats:tex-math>$$\\begin{aligned}
    \\left\\{ \\begin{array}{l}u_t = \\nabla \\cdot \\big ( D(u) \\nabla u \\big )
    - \\nabla \\cdot \\big ( uS(u)\\nabla v\\big ), \\\\ 0 = \\Delta v - \\mu + u,
    \\qquad \\mu =\\frac{1}{|\\Omega |} \\int _\\Omega u, \\end{array} \\right. \\qquad
    \\qquad (\\star ) \\end{aligned}$$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n
    \                 <mml:mrow>\r\n                    <mml:mtable>\r\n                      <mml:mtr>\r\n
    \                       <mml:mtd>\r\n                          <mml:mrow>\r\n
    \                           <mml:mfenced>\r\n                              <mml:mrow>\r\n
    \                               <mml:mtable>\r\n                                  <mml:mtr>\r\n
    \                                   <mml:mtd>\r\n                                      <mml:mrow>\r\n
    \                                       <mml:msub>\r\n                                          <mml:mi>u</mml:mi>\r\n
    \                                         <mml:mi>t</mml:mi>\r\n                                        </mml:msub>\r\n
    \                                       <mml:mo>=</mml:mo>\r\n                                        <mml:mi>∇</mml:mi>\r\n
    \                                       <mml:mo>·</mml:mo>\r\n                                        <mml:mrow>\r\n
    \                                         <mml:mo>(</mml:mo>\r\n                                        </mml:mrow>\r\n
    \                                       <mml:mi>D</mml:mi>\r\n                                        <mml:mrow>\r\n
    \                                         <mml:mo>(</mml:mo>\r\n                                          <mml:mi>u</mml:mi>\r\n
    \                                         <mml:mo>)</mml:mo>\r\n                                        </mml:mrow>\r\n
    \                                       <mml:mi>∇</mml:mi>\r\n                                        <mml:mi>u</mml:mi>\r\n
    \                                       <mml:mrow>\r\n                                          <mml:mo>)</mml:mo>\r\n
    \                                       </mml:mrow>\r\n                                        <mml:mo>-</mml:mo>\r\n
    \                                       <mml:mi>∇</mml:mi>\r\n                                        <mml:mo>·</mml:mo>\r\n
    \                                       <mml:mrow>\r\n                                          <mml:mo>(</mml:mo>\r\n
    \                                       </mml:mrow>\r\n                                        <mml:mi>u</mml:mi>\r\n
    \                                       <mml:mi>S</mml:mi>\r\n                                        <mml:mrow>\r\n
    \                                         <mml:mo>(</mml:mo>\r\n                                          <mml:mi>u</mml:mi>\r\n
    \                                         <mml:mo>)</mml:mo>\r\n                                        </mml:mrow>\r\n
    \                                       <mml:mi>∇</mml:mi>\r\n                                        <mml:mi>v</mml:mi>\r\n
    \                                       <mml:mrow>\r\n                                          <mml:mo>)</mml:mo>\r\n
    \                                       </mml:mrow>\r\n                                        <mml:mo>,</mml:mo>\r\n
    \                                     </mml:mrow>\r\n                                    </mml:mtd>\r\n
    \                                 </mml:mtr>\r\n                                  <mml:mtr>\r\n
    \                                   <mml:mtd>\r\n                                      <mml:mrow>\r\n
    \                                       <mml:mrow/>\r\n                                        <mml:mn>0</mml:mn>\r\n
    \                                       <mml:mo>=</mml:mo>\r\n                                        <mml:mi>Δ</mml:mi>\r\n
    \                                       <mml:mi>v</mml:mi>\r\n                                        <mml:mo>-</mml:mo>\r\n
    \                                       <mml:mi>μ</mml:mi>\r\n                                        <mml:mo>+</mml:mo>\r\n
    \                                       <mml:mi>u</mml:mi>\r\n                                        <mml:mo>,</mml:mo>\r\n
    \                                       <mml:mspace/>\r\n                                        <mml:mi>μ</mml:mi>\r\n
    \                                       <mml:mo>=</mml:mo>\r\n                                        <mml:mfrac>\r\n
    \                                         <mml:mn>1</mml:mn>\r\n                                          <mml:mrow>\r\n
    \                                           <mml:mo>|</mml:mo>\r\n                                            <mml:mi>Ω</mml:mi>\r\n
    \                                           <mml:mo>|</mml:mo>\r\n                                          </mml:mrow>\r\n
    \                                       </mml:mfrac>\r\n                                        <mml:msub>\r\n
    \                                         <mml:mo>∫</mml:mo>\r\n                                          <mml:mi>Ω</mml:mi>\r\n
    \                                       </mml:msub>\r\n                                        <mml:mi>u</mml:mi>\r\n
    \                                       <mml:mo>,</mml:mo>\r\n                                      </mml:mrow>\r\n
    \                                   </mml:mtd>\r\n                                  </mml:mtr>\r\n
    \                               </mml:mtable>\r\n                              </mml:mrow>\r\n
    \                           </mml:mfenced>\r\n                            <mml:mspace/>\r\n
    \                           <mml:mspace/>\r\n                            <mml:mrow>\r\n
    \                             <mml:mo>(</mml:mo>\r\n                              <mml:mo>⋆</mml:mo>\r\n
    \                             <mml:mo>)</mml:mo>\r\n                            </mml:mrow>\r\n
    \                         </mml:mrow>\r\n                        </mml:mtd>\r\n
    \                     </mml:mtr>\r\n                    </mml:mtable>\r\n                  </mml:mrow>\r\n
    \               </mml:math></jats:alternatives></jats:disp-formula>is considered
    under no-flux boundary conditions, with a focus on nonlinearities <jats:inline-formula><jats:alternatives><jats:tex-math>$$S\\in
    C^2([0,\\infty ))$$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n
    \                 <mml:mrow>\r\n                    <mml:mi>S</mml:mi>\r\n                    <mml:mo>∈</mml:mo>\r\n
    \                   <mml:msup>\r\n                      <mml:mi>C</mml:mi>\r\n
    \                     <mml:mn>2</mml:mn>\r\n                    </mml:msup>\r\n
    \                   <mml:mrow>\r\n                      <mml:mo>(</mml:mo>\r\n
    \                     <mml:mrow>\r\n                        <mml:mo>[</mml:mo>\r\n
    \                       <mml:mn>0</mml:mn>\r\n                        <mml:mo>,</mml:mo>\r\n
    \                       <mml:mi>∞</mml:mi>\r\n                        <mml:mo>)</mml:mo>\r\n
    \                     </mml:mrow>\r\n                      <mml:mo>)</mml:mo>\r\n
    \                   </mml:mrow>\r\n                  </mml:mrow>\r\n                </mml:math></jats:alternatives></jats:inline-formula>
    which exhibit super-algebraically fast decay in the sense that with some <jats:inline-formula><jats:alternatives><jats:tex-math>$$K_S&gt;0,
    \\beta \\in [0,1)$$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n
    \                 <mml:mrow>\r\n                    <mml:msub>\r\n                      <mml:mi>K</mml:mi>\r\n
    \                     <mml:mi>S</mml:mi>\r\n                    </mml:msub>\r\n
    \                   <mml:mo>&gt;</mml:mo>\r\n                    <mml:mn>0</mml:mn>\r\n
    \                   <mml:mo>,</mml:mo>\r\n                    <mml:mi>β</mml:mi>\r\n
    \                   <mml:mo>∈</mml:mo>\r\n                    <mml:mrow>\r\n                      <mml:mo>[</mml:mo>\r\n
    \                     <mml:mn>0</mml:mn>\r\n                      <mml:mo>,</mml:mo>\r\n
    \                     <mml:mn>1</mml:mn>\r\n                      <mml:mo>)</mml:mo>\r\n
    \                   </mml:mrow>\r\n                  </mml:mrow>\r\n                </mml:math></jats:alternatives></jats:inline-formula>
    and <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\xi _0&gt;0$$</jats:tex-math><mml:math
    xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n                  <mml:mrow>\r\n
    \                   <mml:msub>\r\n                      <mml:mi>ξ</mml:mi>\r\n
    \                     <mml:mn>0</mml:mn>\r\n                    </mml:msub>\r\n
    \                   <mml:mo>&gt;</mml:mo>\r\n                    <mml:mn>0</mml:mn>\r\n
    \                 </mml:mrow>\r\n                </mml:math></jats:alternatives></jats:inline-formula>,
    <jats:disp-formula><jats:alternatives><jats:tex-math>$$\\begin{aligned} S(\\xi
    )&gt;0 \\quad \\text{ and } \\quad S'(\\xi ) \\le -K_S\\xi ^{-\\beta } S(\\xi
    ) \\qquad \\text{ for } \\text{ all } \\xi \\ge \\xi _0. \\end{aligned}$$</jats:tex-math><mml:math
    xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n                  <mml:mrow>\r\n
    \                   <mml:mtable>\r\n                      <mml:mtr>\r\n                        <mml:mtd>\r\n
    \                         <mml:mrow>\r\n                            <mml:mi>S</mml:mi>\r\n
    \                           <mml:mrow>\r\n                              <mml:mo>(</mml:mo>\r\n
    \                             <mml:mi>ξ</mml:mi>\r\n                              <mml:mo>)</mml:mo>\r\n
    \                           </mml:mrow>\r\n                            <mml:mo>&gt;</mml:mo>\r\n
    \                           <mml:mn>0</mml:mn>\r\n                            <mml:mspace/>\r\n
    \                           <mml:mspace/>\r\n                            <mml:mtext>and</mml:mtext>\r\n
    \                           <mml:mspace/>\r\n                            <mml:mspace/>\r\n
    \                           <mml:msup>\r\n                              <mml:mi>S</mml:mi>\r\n
    \                             <mml:mo>′</mml:mo>\r\n                            </mml:msup>\r\n
    \                           <mml:mrow>\r\n                              <mml:mo>(</mml:mo>\r\n
    \                             <mml:mi>ξ</mml:mi>\r\n                              <mml:mo>)</mml:mo>\r\n
    \                           </mml:mrow>\r\n                            <mml:mo>≤</mml:mo>\r\n
    \                           <mml:mo>-</mml:mo>\r\n                            <mml:msub>\r\n
    \                             <mml:mi>K</mml:mi>\r\n                              <mml:mi>S</mml:mi>\r\n
    \                           </mml:msub>\r\n                            <mml:msup>\r\n
    \                             <mml:mi>ξ</mml:mi>\r\n                              <mml:mrow>\r\n
    \                               <mml:mo>-</mml:mo>\r\n                                <mml:mi>β</mml:mi>\r\n
    \                             </mml:mrow>\r\n                            </mml:msup>\r\n
    \                           <mml:mi>S</mml:mi>\r\n                            <mml:mrow>\r\n
    \                             <mml:mo>(</mml:mo>\r\n                              <mml:mi>ξ</mml:mi>\r\n
    \                             <mml:mo>)</mml:mo>\r\n                            </mml:mrow>\r\n
    \                           <mml:mspace/>\r\n                            <mml:mspace/>\r\n
    \                           <mml:mtext>for</mml:mtext>\r\n                            <mml:mspace/>\r\n
    \                           <mml:mspace/>\r\n                            <mml:mtext>all</mml:mtext>\r\n
    \                           <mml:mspace/>\r\n                            <mml:mi>ξ</mml:mi>\r\n
    \                           <mml:mo>≥</mml:mo>\r\n                            <mml:msub>\r\n
    \                             <mml:mi>ξ</mml:mi>\r\n                              <mml:mn>0</mml:mn>\r\n
    \                           </mml:msub>\r\n                            <mml:mo>.</mml:mo>\r\n
    \                         </mml:mrow>\r\n                        </mml:mtd>\r\n
    \                     </mml:mtr>\r\n                    </mml:mtable>\r\n                  </mml:mrow>\r\n
    \               </mml:math></jats:alternatives></jats:disp-formula>It is, inter
    alia, shown that if furthermore <jats:inline-formula><jats:alternatives><jats:tex-math>$$D\\in
    C^2((0,\\infty ))$$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n
    \                 <mml:mrow>\r\n                    <mml:mi>D</mml:mi>\r\n                    <mml:mo>∈</mml:mo>\r\n
    \                   <mml:msup>\r\n                      <mml:mi>C</mml:mi>\r\n
    \                     <mml:mn>2</mml:mn>\r\n                    </mml:msup>\r\n
    \                   <mml:mrow>\r\n                      <mml:mo>(</mml:mo>\r\n
    \                     <mml:mrow>\r\n                        <mml:mo>(</mml:mo>\r\n
    \                       <mml:mn>0</mml:mn>\r\n                        <mml:mo>,</mml:mo>\r\n
    \                       <mml:mi>∞</mml:mi>\r\n                        <mml:mo>)</mml:mo>\r\n
    \                     </mml:mrow>\r\n                      <mml:mo>)</mml:mo>\r\n
    \                   </mml:mrow>\r\n                  </mml:mrow>\r\n                </mml:math></jats:alternatives></jats:inline-formula>
    is positive and suitably small in relation to <jats:italic>S</jats:italic> by
    satisfying <jats:disp-formula><jats:alternatives><jats:tex-math>$$\\begin{aligned}
    \\frac{\\xi S(\\xi )}{D(\\xi )} \\ge K_{SD}\\xi ^\\lambda \\qquad \\text{ for
    } \\text{ all } \\xi \\ge \\xi _0 \\end{aligned}$$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n
    \                 <mml:mrow>\r\n                    <mml:mtable>\r\n                      <mml:mtr>\r\n
    \                       <mml:mtd>\r\n                          <mml:mrow>\r\n
    \                           <mml:mfrac>\r\n                              <mml:mrow>\r\n
    \                               <mml:mi>ξ</mml:mi>\r\n                                <mml:mi>S</mml:mi>\r\n
    \                               <mml:mo>(</mml:mo>\r\n                                <mml:mi>ξ</mml:mi>\r\n
    \                               <mml:mo>)</mml:mo>\r\n                              </mml:mrow>\r\n
    \                             <mml:mrow>\r\n                                <mml:mi>D</mml:mi>\r\n
    \                               <mml:mo>(</mml:mo>\r\n                                <mml:mi>ξ</mml:mi>\r\n
    \                               <mml:mo>)</mml:mo>\r\n                              </mml:mrow>\r\n
    \                           </mml:mfrac>\r\n                            <mml:mo>≥</mml:mo>\r\n
    \                           <mml:msub>\r\n                              <mml:mi>K</mml:mi>\r\n
    \                             <mml:mrow>\r\n                                <mml:mi>SD</mml:mi>\r\n
    \                             </mml:mrow>\r\n                            </mml:msub>\r\n
    \                           <mml:msup>\r\n                              <mml:mi>ξ</mml:mi>\r\n
    \                             <mml:mi>λ</mml:mi>\r\n                            </mml:msup>\r\n
    \                           <mml:mspace/>\r\n                            <mml:mspace/>\r\n
    \                           <mml:mtext>for</mml:mtext>\r\n                            <mml:mspace/>\r\n
    \                           <mml:mspace/>\r\n                            <mml:mtext>all</mml:mtext>\r\n
    \                           <mml:mspace/>\r\n                            <mml:mi>ξ</mml:mi>\r\n
    \                           <mml:mo>≥</mml:mo>\r\n                            <mml:msub>\r\n
    \                             <mml:mi>ξ</mml:mi>\r\n                              <mml:mn>0</mml:mn>\r\n
    \                           </mml:msub>\r\n                          </mml:mrow>\r\n
    \                       </mml:mtd>\r\n                      </mml:mtr>\r\n                    </mml:mtable>\r\n
    \                 </mml:mrow>\r\n                </mml:math></jats:alternatives></jats:disp-formula>with
    some <jats:inline-formula><jats:alternatives><jats:tex-math>$$K_{SD}&gt;0$$</jats:tex-math><mml:math
    xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n                  <mml:mrow>\r\n
    \                   <mml:msub>\r\n                      <mml:mi>K</mml:mi>\r\n
    \                     <mml:mrow>\r\n                        <mml:mi>SD</mml:mi>\r\n
    \                     </mml:mrow>\r\n                    </mml:msub>\r\n                    <mml:mo>&gt;</mml:mo>\r\n
    \                   <mml:mn>0</mml:mn>\r\n                  </mml:mrow>\r\n                </mml:math></jats:alternatives></jats:inline-formula>
    and <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\lambda &gt;\\frac{2}{n}$$</jats:tex-math><mml:math
    xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n                  <mml:mrow>\r\n
    \                   <mml:mi>λ</mml:mi>\r\n                    <mml:mo>&gt;</mml:mo>\r\n
    \                   <mml:mfrac>\r\n                      <mml:mn>2</mml:mn>\r\n
    \                     <mml:mi>n</mml:mi>\r\n                    </mml:mfrac>\r\n
    \                 </mml:mrow>\r\n                </mml:math></jats:alternatives></jats:inline-formula>,
    then throughout a considerably large set of initial data, (<jats:inline-formula><jats:alternatives><jats:tex-math>$$\\star
    $$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n
    \                 <mml:mo>⋆</mml:mo>\r\n                </mml:math></jats:alternatives></jats:inline-formula>)
    admits global classical solutions (<jats:italic>u</jats:italic>, <jats:italic>v</jats:italic>)
    fulfilling <jats:disp-formula><jats:alternatives><jats:tex-math>$$\\begin{aligned}
    \\frac{z(t)}{C} \\le \\Vert u(\\cdot ,t)\\Vert _{L^\\infty (\\Omega )} \\le Cz(t)
    \\qquad \\text{ for } \\text{ all } t&gt;0, \\end{aligned}$$</jats:tex-math><mml:math
    xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n                  <mml:mrow>\r\n
    \                   <mml:mtable>\r\n                      <mml:mtr>\r\n                        <mml:mtd>\r\n
    \                         <mml:mrow>\r\n                            <mml:mfrac>\r\n
    \                             <mml:mrow>\r\n                                <mml:mi>z</mml:mi>\r\n
    \                               <mml:mo>(</mml:mo>\r\n                                <mml:mi>t</mml:mi>\r\n
    \                               <mml:mo>)</mml:mo>\r\n                              </mml:mrow>\r\n
    \                             <mml:mi>C</mml:mi>\r\n                            </mml:mfrac>\r\n
    \                           <mml:mo>≤</mml:mo>\r\n                            <mml:msub>\r\n
    \                             <mml:mrow>\r\n                                <mml:mo>‖</mml:mo>\r\n
    \                               <mml:mi>u</mml:mi>\r\n                                <mml:mrow>\r\n
    \                                 <mml:mo>(</mml:mo>\r\n                                  <mml:mo>·</mml:mo>\r\n
    \                                 <mml:mo>,</mml:mo>\r\n                                  <mml:mi>t</mml:mi>\r\n
    \                                 <mml:mo>)</mml:mo>\r\n                                </mml:mrow>\r\n
    \                               <mml:mo>‖</mml:mo>\r\n                              </mml:mrow>\r\n
    \                             <mml:mrow>\r\n                                <mml:msup>\r\n
    \                                 <mml:mi>L</mml:mi>\r\n                                  <mml:mi>∞</mml:mi>\r\n
    \                               </mml:msup>\r\n                                <mml:mrow>\r\n
    \                                 <mml:mo>(</mml:mo>\r\n                                  <mml:mi>Ω</mml:mi>\r\n
    \                                 <mml:mo>)</mml:mo>\r\n                                </mml:mrow>\r\n
    \                             </mml:mrow>\r\n                            </mml:msub>\r\n
    \                           <mml:mo>≤</mml:mo>\r\n                            <mml:mi>C</mml:mi>\r\n
    \                           <mml:mi>z</mml:mi>\r\n                            <mml:mrow>\r\n
    \                             <mml:mo>(</mml:mo>\r\n                              <mml:mi>t</mml:mi>\r\n
    \                             <mml:mo>)</mml:mo>\r\n                            </mml:mrow>\r\n
    \                           <mml:mspace/>\r\n                            <mml:mspace/>\r\n
    \                           <mml:mtext>for</mml:mtext>\r\n                            <mml:mspace/>\r\n
    \                           <mml:mspace/>\r\n                            <mml:mtext>all</mml:mtext>\r\n
    \                           <mml:mspace/>\r\n                            <mml:mi>t</mml:mi>\r\n
    \                           <mml:mo>&gt;</mml:mo>\r\n                            <mml:mn>0</mml:mn>\r\n
    \                           <mml:mo>,</mml:mo>\r\n                          </mml:mrow>\r\n
    \                       </mml:mtd>\r\n                      </mml:mtr>\r\n                    </mml:mtable>\r\n
    \                 </mml:mrow>\r\n                </mml:math></jats:alternatives></jats:disp-formula>with
    some <jats:inline-formula><jats:alternatives><jats:tex-math>$$C=C^{(u,v)}\\ge
    1$$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n
    \                 <mml:mrow>\r\n                    <mml:mi>C</mml:mi>\r\n                    <mml:mo>=</mml:mo>\r\n
    \                   <mml:msup>\r\n                      <mml:mi>C</mml:mi>\r\n
    \                     <mml:mrow>\r\n                        <mml:mo>(</mml:mo>\r\n
    \                       <mml:mi>u</mml:mi>\r\n                        <mml:mo>,</mml:mo>\r\n
    \                       <mml:mi>v</mml:mi>\r\n                        <mml:mo>)</mml:mo>\r\n
    \                     </mml:mrow>\r\n                    </mml:msup>\r\n                    <mml:mo>≥</mml:mo>\r\n
    \                   <mml:mn>1</mml:mn>\r\n                  </mml:mrow>\r\n                </mml:math></jats:alternatives></jats:inline-formula>,
    where <jats:italic>z</jats:italic> denotes the solution of <jats:disp-formula><jats:alternatives><jats:tex-math>$$\\begin{aligned}
    \\left\\{ \\begin{array}{l}z'(t) = z^2(t) \\cdot S\\big ( z(t)\\big ), \\qquad
    t&gt;0, \\\\ z(0)=\\xi _0, \\end{array} \\right. \\end{aligned}$$</jats:tex-math><mml:math
    xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n                  <mml:mrow>\r\n
    \                   <mml:mtable>\r\n                      <mml:mtr>\r\n                        <mml:mtd>\r\n
    \                         <mml:mfenced>\r\n                            <mml:mrow>\r\n
    \                             <mml:mtable>\r\n                                <mml:mtr>\r\n
    \                                 <mml:mtd>\r\n                                    <mml:mrow>\r\n
    \                                     <mml:msup>\r\n                                        <mml:mi>z</mml:mi>\r\n
    \                                       <mml:mo>′</mml:mo>\r\n                                      </mml:msup>\r\n
    \                                     <mml:mrow>\r\n                                        <mml:mo>(</mml:mo>\r\n
    \                                       <mml:mi>t</mml:mi>\r\n                                        <mml:mo>)</mml:mo>\r\n
    \                                     </mml:mrow>\r\n                                      <mml:mo>=</mml:mo>\r\n
    \                                     <mml:msup>\r\n                                        <mml:mi>z</mml:mi>\r\n
    \                                       <mml:mn>2</mml:mn>\r\n                                      </mml:msup>\r\n
    \                                     <mml:mrow>\r\n                                        <mml:mo>(</mml:mo>\r\n
    \                                       <mml:mi>t</mml:mi>\r\n                                        <mml:mo>)</mml:mo>\r\n
    \                                     </mml:mrow>\r\n                                      <mml:mo>·</mml:mo>\r\n
    \                                     <mml:mi>S</mml:mi>\r\n                                      <mml:mrow>\r\n
    \                                       <mml:mo>(</mml:mo>\r\n                                      </mml:mrow>\r\n
    \                                     <mml:mi>z</mml:mi>\r\n                                      <mml:mrow>\r\n
    \                                       <mml:mo>(</mml:mo>\r\n                                        <mml:mi>t</mml:mi>\r\n
    \                                       <mml:mo>)</mml:mo>\r\n                                      </mml:mrow>\r\n
    \                                     <mml:mrow>\r\n                                        <mml:mo>)</mml:mo>\r\n
    \                                     </mml:mrow>\r\n                                      <mml:mo>,</mml:mo>\r\n
    \                                     <mml:mspace/>\r\n                                      <mml:mi>t</mml:mi>\r\n
    \                                     <mml:mo>&gt;</mml:mo>\r\n                                      <mml:mn>0</mml:mn>\r\n
    \                                     <mml:mo>,</mml:mo>\r\n                                    </mml:mrow>\r\n
    \                                 </mml:mtd>\r\n                                </mml:mtr>\r\n
    \                               <mml:mtr>\r\n                                  <mml:mtd>\r\n
    \                                   <mml:mrow>\r\n                                      <mml:mrow/>\r\n
    \                                     <mml:mi>z</mml:mi>\r\n                                      <mml:mrow>\r\n
    \                                       <mml:mo>(</mml:mo>\r\n                                        <mml:mn>0</mml:mn>\r\n
    \                                       <mml:mo>)</mml:mo>\r\n                                      </mml:mrow>\r\n
    \                                     <mml:mo>=</mml:mo>\r\n                                      <mml:msub>\r\n
    \                                       <mml:mi>ξ</mml:mi>\r\n                                        <mml:mn>0</mml:mn>\r\n
    \                                     </mml:msub>\r\n                                      <mml:mo>,</mml:mo>\r\n
    \                                   </mml:mrow>\r\n                                  </mml:mtd>\r\n
    \                               </mml:mtr>\r\n                              </mml:mtable>\r\n
    \                           </mml:mrow>\r\n                          </mml:mfenced>\r\n
    \                       </mml:mtd>\r\n                      </mml:mtr>\r\n                    </mml:mtable>\r\n
    \                 </mml:mrow>\r\n                </mml:math></jats:alternatives></jats:disp-formula>which
    is seen to exist globally, and to satisfy <jats:inline-formula><jats:alternatives><jats:tex-math>$$z(t)\\rightarrow
    +\\infty $$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n
    \                 <mml:mrow>\r\n                    <mml:mi>z</mml:mi>\r\n                    <mml:mo>(</mml:mo>\r\n
    \                   <mml:mi>t</mml:mi>\r\n                    <mml:mo>)</mml:mo>\r\n
    \                   <mml:mo>→</mml:mo>\r\n                    <mml:mo>+</mml:mo>\r\n
    \                   <mml:mi>∞</mml:mi>\r\n                  </mml:mrow>\r\n                </mml:math></jats:alternatives></jats:inline-formula>
    as <jats:inline-formula><jats:alternatives><jats:tex-math>$$t\\rightarrow \\infty
    $$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n
    \                 <mml:mrow>\r\n                    <mml:mi>t</mml:mi>\r\n                    <mml:mo>→</mml:mo>\r\n
    \                   <mml:mi>∞</mml:mi>\r\n                  </mml:mrow>\r\n                </mml:math></jats:alternatives></jats:inline-formula>.
    As particular examples, exponentially and doubly exponentially decaying <jats:italic>S</jats:italic>
    are found to imply corresponding infinite-time blow-up properties in (<jats:inline-formula><jats:alternatives><jats:tex-math>$$\\star
    $$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n
    \                 <mml:mo>⋆</mml:mo>\r\n                </mml:math></jats:alternatives></jats:inline-formula>)
    at logarithmic and doubly logarithmic rates, respectively.</jats:p>"
author:
- first_name: Michael
  full_name: Winkler, Michael
  id: '31496'
  last_name: Winkler
citation:
  ama: Winkler M. Slow Grow-up in a Quasilinear Keller–Segel System. <i>Journal of
    Dynamics and Differential Equations</i>. 2022;36(2):1677-1702. doi:<a href="https://doi.org/10.1007/s10884-022-10167-w">10.1007/s10884-022-10167-w</a>
  apa: Winkler, M. (2022). Slow Grow-up in a Quasilinear Keller–Segel System. <i>Journal
    of Dynamics and Differential Equations</i>, <i>36</i>(2), 1677–1702. <a href="https://doi.org/10.1007/s10884-022-10167-w">https://doi.org/10.1007/s10884-022-10167-w</a>
  bibtex: '@article{Winkler_2022, title={Slow Grow-up in a Quasilinear Keller–Segel
    System}, volume={36}, DOI={<a href="https://doi.org/10.1007/s10884-022-10167-w">10.1007/s10884-022-10167-w</a>},
    number={2}, journal={Journal of Dynamics and Differential Equations}, publisher={Springer
    Science and Business Media LLC}, author={Winkler, Michael}, year={2022}, pages={1677–1702}
    }'
  chicago: 'Winkler, Michael. “Slow Grow-up in a Quasilinear Keller–Segel System.”
    <i>Journal of Dynamics and Differential Equations</i> 36, no. 2 (2022): 1677–1702.
    <a href="https://doi.org/10.1007/s10884-022-10167-w">https://doi.org/10.1007/s10884-022-10167-w</a>.'
  ieee: 'M. Winkler, “Slow Grow-up in a Quasilinear Keller–Segel System,” <i>Journal
    of Dynamics and Differential Equations</i>, vol. 36, no. 2, pp. 1677–1702, 2022,
    doi: <a href="https://doi.org/10.1007/s10884-022-10167-w">10.1007/s10884-022-10167-w</a>.'
  mla: Winkler, Michael. “Slow Grow-up in a Quasilinear Keller–Segel System.” <i>Journal
    of Dynamics and Differential Equations</i>, vol. 36, no. 2, Springer Science and
    Business Media LLC, 2022, pp. 1677–702, doi:<a href="https://doi.org/10.1007/s10884-022-10167-w">10.1007/s10884-022-10167-w</a>.
  short: M. Winkler, Journal of Dynamics and Differential Equations 36 (2022) 1677–1702.
date_created: 2025-12-18T19:10:32Z
date_updated: 2025-12-18T20:10:14Z
doi: 10.1007/s10884-022-10167-w
intvolume: '        36'
issue: '2'
language:
- iso: eng
page: 1677-1702
publication: Journal of Dynamics and Differential Equations
publication_identifier:
  issn:
  - 1040-7294
  - 1572-9222
publication_status: published
publisher: Springer Science and Business Media LLC
status: public
title: Slow Grow-up in a Quasilinear Keller–Segel System
type: journal_article
user_id: '31496'
volume: 36
year: '2022'
...
---
_id: '63272'
author:
- first_name: Youshan
  full_name: Tao, Youshan
  last_name: Tao
- first_name: Michael
  full_name: Winkler, Michael
  id: '31496'
  last_name: Winkler
citation:
  ama: Tao Y, Winkler M. Global solutions to a Keller-Segel-consumption system involving
    singularly signal-dependent motilities in domains of arbitrary dimension. <i>Journal
    of Differential Equations</i>. 2022;343:390-418. doi:<a href="https://doi.org/10.1016/j.jde.2022.10.022">10.1016/j.jde.2022.10.022</a>
  apa: Tao, Y., &#38; Winkler, M. (2022). Global solutions to a Keller-Segel-consumption
    system involving singularly signal-dependent motilities in domains of arbitrary
    dimension. <i>Journal of Differential Equations</i>, <i>343</i>, 390–418. <a href="https://doi.org/10.1016/j.jde.2022.10.022">https://doi.org/10.1016/j.jde.2022.10.022</a>
  bibtex: '@article{Tao_Winkler_2022, title={Global solutions to a Keller-Segel-consumption
    system involving singularly signal-dependent motilities in domains of arbitrary
    dimension}, volume={343}, DOI={<a href="https://doi.org/10.1016/j.jde.2022.10.022">10.1016/j.jde.2022.10.022</a>},
    journal={Journal of Differential Equations}, publisher={Elsevier BV}, author={Tao,
    Youshan and Winkler, Michael}, year={2022}, pages={390–418} }'
  chicago: 'Tao, Youshan, and Michael Winkler. “Global Solutions to a Keller-Segel-Consumption
    System Involving Singularly Signal-Dependent Motilities in Domains of Arbitrary
    Dimension.” <i>Journal of Differential Equations</i> 343 (2022): 390–418. <a href="https://doi.org/10.1016/j.jde.2022.10.022">https://doi.org/10.1016/j.jde.2022.10.022</a>.'
  ieee: 'Y. Tao and M. Winkler, “Global solutions to a Keller-Segel-consumption system
    involving singularly signal-dependent motilities in domains of arbitrary dimension,”
    <i>Journal of Differential Equations</i>, vol. 343, pp. 390–418, 2022, doi: <a
    href="https://doi.org/10.1016/j.jde.2022.10.022">10.1016/j.jde.2022.10.022</a>.'
  mla: Tao, Youshan, and Michael Winkler. “Global Solutions to a Keller-Segel-Consumption
    System Involving Singularly Signal-Dependent Motilities in Domains of Arbitrary
    Dimension.” <i>Journal of Differential Equations</i>, vol. 343, Elsevier BV, 2022,
    pp. 390–418, doi:<a href="https://doi.org/10.1016/j.jde.2022.10.022">10.1016/j.jde.2022.10.022</a>.
  short: Y. Tao, M. Winkler, Journal of Differential Equations 343 (2022) 390–418.
date_created: 2025-12-18T19:13:04Z
date_updated: 2025-12-18T20:11:02Z
doi: 10.1016/j.jde.2022.10.022
intvolume: '       343'
language:
- iso: eng
page: 390-418
publication: Journal of Differential Equations
publication_identifier:
  issn:
  - 0022-0396
publication_status: published
publisher: Elsevier BV
status: public
title: Global solutions to a Keller-Segel-consumption system involving singularly
  signal-dependent motilities in domains of arbitrary dimension
type: journal_article
user_id: '31496'
volume: 343
year: '2022'
...
---
_id: '63268'
article_number: '113153'
author:
- first_name: Laurent
  full_name: Desvillettes, Laurent
  last_name: Desvillettes
- first_name: Philippe
  full_name: Laurençot, Philippe
  last_name: Laurençot
- first_name: Ariane
  full_name: Trescases, Ariane
  last_name: Trescases
- first_name: Michael
  full_name: Winkler, Michael
  id: '31496'
  last_name: Winkler
citation:
  ama: Desvillettes L, Laurençot P, Trescases A, Winkler M. Weak solutions to triangular
    cross diffusion systems modeling chemotaxis with local sensing. <i>Nonlinear Analysis</i>.
    2022;226. doi:<a href="https://doi.org/10.1016/j.na.2022.113153">10.1016/j.na.2022.113153</a>
  apa: Desvillettes, L., Laurençot, P., Trescases, A., &#38; Winkler, M. (2022). Weak
    solutions to triangular cross diffusion systems modeling chemotaxis with local
    sensing. <i>Nonlinear Analysis</i>, <i>226</i>, Article 113153. <a href="https://doi.org/10.1016/j.na.2022.113153">https://doi.org/10.1016/j.na.2022.113153</a>
  bibtex: '@article{Desvillettes_Laurençot_Trescases_Winkler_2022, title={Weak solutions
    to triangular cross diffusion systems modeling chemotaxis with local sensing},
    volume={226}, DOI={<a href="https://doi.org/10.1016/j.na.2022.113153">10.1016/j.na.2022.113153</a>},
    number={113153}, journal={Nonlinear Analysis}, publisher={Elsevier BV}, author={Desvillettes,
    Laurent and Laurençot, Philippe and Trescases, Ariane and Winkler, Michael}, year={2022}
    }'
  chicago: Desvillettes, Laurent, Philippe Laurençot, Ariane Trescases, and Michael
    Winkler. “Weak Solutions to Triangular Cross Diffusion Systems Modeling Chemotaxis
    with Local Sensing.” <i>Nonlinear Analysis</i> 226 (2022). <a href="https://doi.org/10.1016/j.na.2022.113153">https://doi.org/10.1016/j.na.2022.113153</a>.
  ieee: 'L. Desvillettes, P. Laurençot, A. Trescases, and M. Winkler, “Weak solutions
    to triangular cross diffusion systems modeling chemotaxis with local sensing,”
    <i>Nonlinear Analysis</i>, vol. 226, Art. no. 113153, 2022, doi: <a href="https://doi.org/10.1016/j.na.2022.113153">10.1016/j.na.2022.113153</a>.'
  mla: Desvillettes, Laurent, et al. “Weak Solutions to Triangular Cross Diffusion
    Systems Modeling Chemotaxis with Local Sensing.” <i>Nonlinear Analysis</i>, vol.
    226, 113153, Elsevier BV, 2022, doi:<a href="https://doi.org/10.1016/j.na.2022.113153">10.1016/j.na.2022.113153</a>.
  short: L. Desvillettes, P. Laurençot, A. Trescases, M. Winkler, Nonlinear Analysis
    226 (2022).
date_created: 2025-12-18T19:11:16Z
date_updated: 2025-12-18T20:10:32Z
doi: 10.1016/j.na.2022.113153
intvolume: '       226'
language:
- iso: eng
publication: Nonlinear Analysis
publication_identifier:
  issn:
  - 0362-546X
publication_status: published
publisher: Elsevier BV
status: public
title: Weak solutions to triangular cross diffusion systems modeling chemotaxis with
  local sensing
type: journal_article
user_id: '31496'
volume: 226
year: '2022'
...
---
_id: '63278'
abstract:
- lang: eng
  text: "<jats:title>Abstract</jats:title>\r\n               <jats:p>The Neumann problem
    for (0.1)$$ \\begin{align}&amp; V_t = \\Delta V-aV+f(x,t) \\end{align}$$is considered
    in bounded domains $\\Omega \\subset {\\mathbb {R}}^n$ with smooth boundary, where
    $n\\ge 1$ and $a\\in {\\mathbb {R}}$. By means of a variational approach, a statement
    on boundedness of the quantities $$ \\begin{eqnarray*} \\sup_{t\\in (0,T)} \\int_\\Omega
    \\big|\\nabla V(\\cdot,t)\\big|^p L^{\\frac{n+p}{n+2}} \\Big( \\big|\\nabla V(\\cdot,t)\\big|
    \\Big) \\end{eqnarray*}$$in dependence on the expressions (0.2)$$ \\begin{align}&amp;
    \\sup_{t\\in (0,T-\\tau)} \\int_t^{t+\\tau} \\int_\\Omega |f|^{\\frac{(n+2)p}{n+p}}
    L\\big( |f|\\big) \\end{align}$$is derived for $p\\ge 2$, $\\tau&amp;gt;0$, and
    $T\\ge 2\\tau $, provided that $L\\in C^0([0,\\infty ))$ is positive, strictly
    increasing, unbounded, and slowly growing in the sense that $\\limsup _{s\\to
    \\infty } \\frac {L(s^{\\lambda _0})}{L(s)} &amp;lt;\\infty $ for some $\\lambda
    _0&amp;gt;1$. In the particular case when $p=n\\ge 2$, an additional condition
    on growth of $L$, particularly satisfied by $L(\\xi ):=\\ln ^\\alpha (\\xi +b)$
    whenever $b&amp;gt;0$ and $\\alpha&amp;gt;\\frac {(n+2)(n-1)}{2n}$, is identified
    as sufficient to ensure that as a consequence of the above, bounds for theintegrals
    in (0.2) even imply estimates for the spatio-temporal modulus of continuity of
    solutions to (0.1). A subsequent application to the Keller–Segel system $$ \\begin{eqnarray*}
    \\left\\{ \\begin{array}{l} u_t = \\nabla \\cdot \\big( D(v)\\nabla u\\big) -
    \\nabla \\cdot \\big( uS(v)\\nabla v\\big) + ru - \\mu u^2, \\\\[1mm] v_t = \\Delta
    v-v+u, \\end{array} \\right. \\end{eqnarray*}$$shows that when $n=2$, $r\\in {\\mathbb
    {R}}$, $0&amp;lt;D\\in C^2([0,\\infty ))$, and $S\\in C^2([0,\\infty )) \\cap
    W^{1,\\infty }((0,\\infty ))$ and thus especially in the presence of arbitrarily
    strong diffusion degeneracies implied by rapid decay of $D$, any choice of $\\mu&amp;gt;0$
    excludes blowup in the sense that for all suitably regular nonnegative initial
    data, an associated initial-boundary value problem admits a global bounded classical
    solution.</jats:p>"
author:
- first_name: Michael
  full_name: Winkler, Michael
  id: '31496'
  last_name: Winkler
citation:
  ama: Winkler M. A Result on Parabolic Gradient Regularity in Orlicz Spaces and Application
    to Absorption-Induced Blow-Up Prevention in a Keller–Segel-Type Cross-Diffusion
    System. <i>International Mathematics Research Notices</i>. 2022;2023(19):16336-16393.
    doi:<a href="https://doi.org/10.1093/imrn/rnac286">10.1093/imrn/rnac286</a>
  apa: Winkler, M. (2022). A Result on Parabolic Gradient Regularity in Orlicz Spaces
    and Application to Absorption-Induced Blow-Up Prevention in a Keller–Segel-Type
    Cross-Diffusion System. <i>International Mathematics Research Notices</i>, <i>2023</i>(19),
    16336–16393. <a href="https://doi.org/10.1093/imrn/rnac286">https://doi.org/10.1093/imrn/rnac286</a>
  bibtex: '@article{Winkler_2022, title={A Result on Parabolic Gradient Regularity
    in Orlicz Spaces and Application to Absorption-Induced Blow-Up Prevention in a
    Keller–Segel-Type Cross-Diffusion System}, volume={2023}, DOI={<a href="https://doi.org/10.1093/imrn/rnac286">10.1093/imrn/rnac286</a>},
    number={19}, journal={International Mathematics Research Notices}, publisher={Oxford
    University Press (OUP)}, author={Winkler, Michael}, year={2022}, pages={16336–16393}
    }'
  chicago: 'Winkler, Michael. “A Result on Parabolic Gradient Regularity in Orlicz
    Spaces and Application to Absorption-Induced Blow-Up Prevention in a Keller–Segel-Type
    Cross-Diffusion System.” <i>International Mathematics Research Notices</i> 2023,
    no. 19 (2022): 16336–93. <a href="https://doi.org/10.1093/imrn/rnac286">https://doi.org/10.1093/imrn/rnac286</a>.'
  ieee: 'M. Winkler, “A Result on Parabolic Gradient Regularity in Orlicz Spaces and
    Application to Absorption-Induced Blow-Up Prevention in a Keller–Segel-Type Cross-Diffusion
    System,” <i>International Mathematics Research Notices</i>, vol. 2023, no. 19,
    pp. 16336–16393, 2022, doi: <a href="https://doi.org/10.1093/imrn/rnac286">10.1093/imrn/rnac286</a>.'
  mla: Winkler, Michael. “A Result on Parabolic Gradient Regularity in Orlicz Spaces
    and Application to Absorption-Induced Blow-Up Prevention in a Keller–Segel-Type
    Cross-Diffusion System.” <i>International Mathematics Research Notices</i>, vol.
    2023, no. 19, Oxford University Press (OUP), 2022, pp. 16336–93, doi:<a href="https://doi.org/10.1093/imrn/rnac286">10.1093/imrn/rnac286</a>.
  short: M. Winkler, International Mathematics Research Notices 2023 (2022) 16336–16393.
date_created: 2025-12-18T19:15:52Z
date_updated: 2025-12-18T20:11:43Z
doi: 10.1093/imrn/rnac286
intvolume: '      2023'
issue: '19'
language:
- iso: eng
page: 16336-16393
publication: International Mathematics Research Notices
publication_identifier:
  issn:
  - 1073-7928
  - 1687-0247
publication_status: published
publisher: Oxford University Press (OUP)
status: public
title: A Result on Parabolic Gradient Regularity in Orlicz Spaces and Application
  to Absorption-Induced Blow-Up Prevention in a Keller–Segel-Type Cross-Diffusion
  System
type: journal_article
user_id: '31496'
volume: 2023
year: '2022'
...
---
_id: '63279'
abstract:
- lang: eng
  text: "<jats:p>\r\n                    In a smoothly bounded convex domain\r\n                    <jats:inline-formula>\r\n
    \                     <jats:tex-math>\\Omega \\subset \\mathbb{R}^3</jats:tex-math>\r\n
    \                   </jats:inline-formula>\r\n                    , we consider
    the chemotaxis-Navier–Stokes model\r\n                  </jats:p>\r\n                  <jats:p>\r\n
    \                   <jats:disp-formula>\r\n                      <jats:tex-math>\\begin{cases}
    n_t + u\\cdot\\nabla n = \\Delta n - \\nabla \\cdot (n\\nabla c), &amp; x\\in
    \\Omega, \\, t&gt;0, \\\\ c_t + u\\cdot\\nabla c = \\Delta c -nc, &amp; x\\in
    \\Omega, \\, t&gt;0, \\\\ u_t + (u\\cdot\\nabla) u = \\Delta u + \\nabla P + n\\nabla\\Phi,
    \\quad \\nabla\\cdot u=0, &amp; x\\in \\Omega, \\, t&gt;0, \\end{cases} \\quad
    (\\star)</jats:tex-math>\r\n                    </jats:disp-formula>\r\n                  </jats:p>\r\n
    \                 <jats:p>\r\n                    proposed by Goldstein et al.
    to describe pattern formation in populations of aerobic bacteria interacting with
    their liquid environment via transport and buoyancy. Known results have asserted
    that under appropriate regularity assumptions on\r\n                    <jats:inline-formula>\r\n
    \                     <jats:tex-math>\\Phi</jats:tex-math>\r\n                    </jats:inline-formula>\r\n
    \                   and the initial data, a corresponding no-flux/no-flux/Dirichlet
    initial-boundary value problem is globally solvable in a framework of so-called
    weak energy solutions, and that any such solution eventually becomes smooth and
    classical.\r\n                  </jats:p>\r\n                  <jats:p>\r\n                    Going
    beyond this, the present work focuses on the possible extent of unboundedness
    phenomena also on short timescales, and hence investigates in more detail the
    set of times in\r\n                    <jats:inline-formula>\r\n                      <jats:tex-math>(0,\\infty)</jats:tex-math>\r\n
    \                   </jats:inline-formula>\r\n                    at which solutions
    may develop singularities. The main results in this direction reveal the existence
    of a global weak energy solution which coincides with a smooth function throughout\r\n
    \                   <jats:inline-formula>\r\n                      <jats:tex-math>\\overline{\\Omega}\\times
    E</jats:tex-math>\r\n                    </jats:inline-formula>\r\n                    ,
    where\r\n                    <jats:inline-formula>\r\n                      <jats:tex-math>E</jats:tex-math>\r\n
    \                   </jats:inline-formula>\r\n                    denotes a countable
    union of open intervals which is such that\r\n                    <jats:inline-formula>\r\n
    \                     <jats:tex-math>|(0,\\infty)\\setminus E|=0</jats:tex-math>\r\n
    \                   </jats:inline-formula>\r\n                    . In particular,
    this indicates that a similar feature of the unperturbed Navie–Stokes equations,
    known as Leray’s structure theorem, persists even in the presence of the coupling
    to the attractive and hence potentially destabilizing cross-diffusive mechanism
    in the full system (\r\n                    <jats:inline-formula>\r\n                      <jats:tex-math>\\star</jats:tex-math>\r\n
    \                   </jats:inline-formula>\r\n                    ).\r\n                  </jats:p>"
author:
- first_name: Michael
  full_name: Winkler, Michael
  id: '31496'
  last_name: Winkler
citation:
  ama: Winkler M. Does Leray’s structure theorem withstand buoyancy-driven chemotaxis-fluid
    interaction? <i>Journal of the European Mathematical Society</i>. 2022;25(4):1423-1456.
    doi:<a href="https://doi.org/10.4171/jems/1226">10.4171/jems/1226</a>
  apa: Winkler, M. (2022). Does Leray’s structure theorem withstand buoyancy-driven
    chemotaxis-fluid interaction? <i>Journal of the European Mathematical Society</i>,
    <i>25</i>(4), 1423–1456. <a href="https://doi.org/10.4171/jems/1226">https://doi.org/10.4171/jems/1226</a>
  bibtex: '@article{Winkler_2022, title={Does Leray’s structure theorem withstand
    buoyancy-driven chemotaxis-fluid interaction?}, volume={25}, DOI={<a href="https://doi.org/10.4171/jems/1226">10.4171/jems/1226</a>},
    number={4}, journal={Journal of the European Mathematical Society}, publisher={European
    Mathematical Society - EMS - Publishing House GmbH}, author={Winkler, Michael},
    year={2022}, pages={1423–1456} }'
  chicago: 'Winkler, Michael. “Does Leray’s Structure Theorem Withstand Buoyancy-Driven
    Chemotaxis-Fluid Interaction?” <i>Journal of the European Mathematical Society</i>
    25, no. 4 (2022): 1423–56. <a href="https://doi.org/10.4171/jems/1226">https://doi.org/10.4171/jems/1226</a>.'
  ieee: 'M. Winkler, “Does Leray’s structure theorem withstand buoyancy-driven chemotaxis-fluid
    interaction?,” <i>Journal of the European Mathematical Society</i>, vol. 25, no.
    4, pp. 1423–1456, 2022, doi: <a href="https://doi.org/10.4171/jems/1226">10.4171/jems/1226</a>.'
  mla: Winkler, Michael. “Does Leray’s Structure Theorem Withstand Buoyancy-Driven
    Chemotaxis-Fluid Interaction?” <i>Journal of the European Mathematical Society</i>,
    vol. 25, no. 4, European Mathematical Society - EMS - Publishing House GmbH, 2022,
    pp. 1423–56, doi:<a href="https://doi.org/10.4171/jems/1226">10.4171/jems/1226</a>.
  short: M. Winkler, Journal of the European Mathematical Society 25 (2022) 1423–1456.
date_created: 2025-12-18T19:16:13Z
date_updated: 2025-12-18T20:11:51Z
doi: 10.4171/jems/1226
intvolume: '        25'
issue: '4'
language:
- iso: eng
page: 1423-1456
publication: Journal of the European Mathematical Society
publication_identifier:
  issn:
  - 1435-9855
  - 1435-9863
publication_status: published
publisher: European Mathematical Society - EMS - Publishing House GmbH
status: public
title: Does Leray’s structure theorem withstand buoyancy-driven chemotaxis-fluid interaction?
type: journal_article
user_id: '31496'
volume: 25
year: '2022'
...
---
_id: '63274'
abstract:
- lang: eng
  text: "<jats:p>In a ball <jats:inline-formula><jats:alternatives><jats:tex-math>$\\Omega
    \\subset \\mathbb {R}^{n}$</jats:tex-math><jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\"
    mime-subtype=\"png\" xlink:href=\"S0308210522000397_inline1.png\" /></jats:alternatives></jats:inline-formula>
    with <jats:inline-formula><jats:alternatives><jats:tex-math>$n\\ge 2$</jats:tex-math><jats:inline-graphic
    xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210522000397_inline2.png\"
    /></jats:alternatives></jats:inline-formula>, the chemotaxis system\r\n<jats:disp-formula><jats:alternatives><jats:tex-math>\\[
    \\left\\{ \\begin{array}{@{}l} u_t = \\nabla \\cdot \\big( D(u)\\nabla u\\big)
    + \\nabla\\cdot \\big(\\dfrac{u}{v} \\nabla v\\big), \\\\ 0=\\Delta v - uv \\end{array}
    \\right. \\]</jats:tex-math><jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\"
    mime-subtype=\"png\" mimetype=\"image\" position=\"float\" xlink:href=\"S0308210522000397_eqnU1.png\"
    /></jats:alternatives></jats:disp-formula>is considered along with no-flux boundary
    conditions for <jats:inline-formula><jats:alternatives><jats:tex-math>$u$</jats:tex-math><jats:inline-graphic
    xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210522000397_inline3.png\"
    /></jats:alternatives></jats:inline-formula> and with prescribed constant positive
    Dirichlet boundary data for <jats:inline-formula><jats:alternatives><jats:tex-math>$v$</jats:tex-math><jats:inline-graphic
    xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210522000397_inline4.png\"
    /></jats:alternatives></jats:inline-formula>. It is shown that if <jats:inline-formula><jats:alternatives><jats:tex-math>$D\\in
    C^{3}([0,\\infty ))$</jats:tex-math><jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\"
    mime-subtype=\"png\" xlink:href=\"S0308210522000397_inline5.png\" /></jats:alternatives></jats:inline-formula>
    is such that <jats:inline-formula><jats:alternatives><jats:tex-math>$0&lt; D(\\xi
    ) \\le {K_D} (\\xi +1)^{-\\alpha }$</jats:tex-math><jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\"
    mime-subtype=\"png\" xlink:href=\"S0308210522000397_inline6.png\" /></jats:alternatives></jats:inline-formula>
    for all <jats:inline-formula><jats:alternatives><jats:tex-math>$\\xi &gt;0$</jats:tex-math><jats:inline-graphic
    xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210522000397_inline7.png\"
    /></jats:alternatives></jats:inline-formula> with some <jats:inline-formula><jats:alternatives><jats:tex-math>${K_D}&gt;0$</jats:tex-math><jats:inline-graphic
    xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210522000397_inline8.png\"
    /></jats:alternatives></jats:inline-formula> and <jats:inline-formula><jats:alternatives><jats:tex-math>$\\alpha
    &gt;0$</jats:tex-math><jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\"
    mime-subtype=\"png\" xlink:href=\"S0308210522000397_inline9.png\" /></jats:alternatives></jats:inline-formula>,
    then for all initial data from a considerably large set of radial functions on
    <jats:inline-formula><jats:alternatives><jats:tex-math>$\\Omega$</jats:tex-math><jats:inline-graphic
    xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210522000397_inline10.png\"
    /></jats:alternatives></jats:inline-formula>, the corresponding initial-boundary
    value problem admits a solution blowing up in finite time.</jats:p>"
author:
- first_name: Yulan
  full_name: Wang, Yulan
  last_name: Wang
- first_name: Michael
  full_name: Winkler, Michael
  id: '31496'
  last_name: Winkler
citation:
  ama: 'Wang Y, Winkler M. Finite-time blow-up in a repulsive chemotaxis-consumption
    system. <i>Proceedings of the Royal Society of Edinburgh: Section A Mathematics</i>.
    2022;153(4):1150-1166. doi:<a href="https://doi.org/10.1017/prm.2022.39">10.1017/prm.2022.39</a>'
  apa: 'Wang, Y., &#38; Winkler, M. (2022). Finite-time blow-up in a repulsive chemotaxis-consumption
    system. <i>Proceedings of the Royal Society of Edinburgh: Section A Mathematics</i>,
    <i>153</i>(4), 1150–1166. <a href="https://doi.org/10.1017/prm.2022.39">https://doi.org/10.1017/prm.2022.39</a>'
  bibtex: '@article{Wang_Winkler_2022, title={Finite-time blow-up in a repulsive chemotaxis-consumption
    system}, volume={153}, DOI={<a href="https://doi.org/10.1017/prm.2022.39">10.1017/prm.2022.39</a>},
    number={4}, journal={Proceedings of the Royal Society of Edinburgh: Section A
    Mathematics}, publisher={Cambridge University Press (CUP)}, author={Wang, Yulan
    and Winkler, Michael}, year={2022}, pages={1150–1166} }'
  chicago: 'Wang, Yulan, and Michael Winkler. “Finite-Time Blow-up in a Repulsive
    Chemotaxis-Consumption System.” <i>Proceedings of the Royal Society of Edinburgh:
    Section A Mathematics</i> 153, no. 4 (2022): 1150–66. <a href="https://doi.org/10.1017/prm.2022.39">https://doi.org/10.1017/prm.2022.39</a>.'
  ieee: 'Y. Wang and M. Winkler, “Finite-time blow-up in a repulsive chemotaxis-consumption
    system,” <i>Proceedings of the Royal Society of Edinburgh: Section A Mathematics</i>,
    vol. 153, no. 4, pp. 1150–1166, 2022, doi: <a href="https://doi.org/10.1017/prm.2022.39">10.1017/prm.2022.39</a>.'
  mla: 'Wang, Yulan, and Michael Winkler. “Finite-Time Blow-up in a Repulsive Chemotaxis-Consumption
    System.” <i>Proceedings of the Royal Society of Edinburgh: Section A Mathematics</i>,
    vol. 153, no. 4, Cambridge University Press (CUP), 2022, pp. 1150–66, doi:<a href="https://doi.org/10.1017/prm.2022.39">10.1017/prm.2022.39</a>.'
  short: 'Y. Wang, M. Winkler, Proceedings of the Royal Society of Edinburgh: Section
    A Mathematics 153 (2022) 1150–1166.'
date_created: 2025-12-18T19:14:20Z
date_updated: 2025-12-18T20:11:15Z
doi: 10.1017/prm.2022.39
intvolume: '       153'
issue: '4'
language:
- iso: eng
page: 1150-1166
publication: 'Proceedings of the Royal Society of Edinburgh: Section A Mathematics'
publication_identifier:
  issn:
  - 0308-2105
  - 1473-7124
publication_status: published
publisher: Cambridge University Press (CUP)
status: public
title: Finite-time blow-up in a repulsive chemotaxis-consumption system
type: journal_article
user_id: '31496'
volume: 153
year: '2022'
...
---
_id: '63282'
abstract:
- lang: eng
  text: '<jats:p> The chemotaxis system [Formula: see text] is considered in a ball
    [Formula: see text], [Formula: see text], where the positive function [Formula:
    see text] reflects suitably weak diffusion by satisfying [Formula: see text] for
    some [Formula: see text]. It is shown that whenever [Formula: see text] is positive
    and satisfies [Formula: see text] as [Formula: see text], one can find a suitably
    regular nonlinearity [Formula: see text] with the property that at each sufficiently
    large mass level [Formula: see text] there exists a globally defined radially
    symmetric classical solution to a Neumann-type boundary value problem for (⋆)
    which satisfies [Formula: see text] </jats:p>'
article_number: '2250062'
author:
- first_name: Michael
  full_name: Winkler, Michael
  id: '31496'
  last_name: Winkler
citation:
  ama: Winkler M. Arbitrarily fast grow-up rates in quasilinear Keller–Segel systems.
    <i>Communications in Contemporary Mathematics</i>. 2022;25(10). doi:<a href="https://doi.org/10.1142/s0219199722500626">10.1142/s0219199722500626</a>
  apa: Winkler, M. (2022). Arbitrarily fast grow-up rates in quasilinear Keller–Segel
    systems. <i>Communications in Contemporary Mathematics</i>, <i>25</i>(10), Article
    2250062. <a href="https://doi.org/10.1142/s0219199722500626">https://doi.org/10.1142/s0219199722500626</a>
  bibtex: '@article{Winkler_2022, title={Arbitrarily fast grow-up rates in quasilinear
    Keller–Segel systems}, volume={25}, DOI={<a href="https://doi.org/10.1142/s0219199722500626">10.1142/s0219199722500626</a>},
    number={102250062}, journal={Communications in Contemporary Mathematics}, publisher={World
    Scientific Pub Co Pte Ltd}, author={Winkler, Michael}, year={2022} }'
  chicago: Winkler, Michael. “Arbitrarily Fast Grow-up Rates in Quasilinear Keller–Segel
    Systems.” <i>Communications in Contemporary Mathematics</i> 25, no. 10 (2022).
    <a href="https://doi.org/10.1142/s0219199722500626">https://doi.org/10.1142/s0219199722500626</a>.
  ieee: 'M. Winkler, “Arbitrarily fast grow-up rates in quasilinear Keller–Segel systems,”
    <i>Communications in Contemporary Mathematics</i>, vol. 25, no. 10, Art. no. 2250062,
    2022, doi: <a href="https://doi.org/10.1142/s0219199722500626">10.1142/s0219199722500626</a>.'
  mla: Winkler, Michael. “Arbitrarily Fast Grow-up Rates in Quasilinear Keller–Segel
    Systems.” <i>Communications in Contemporary Mathematics</i>, vol. 25, no. 10,
    2250062, World Scientific Pub Co Pte Ltd, 2022, doi:<a href="https://doi.org/10.1142/s0219199722500626">10.1142/s0219199722500626</a>.
  short: M. Winkler, Communications in Contemporary Mathematics 25 (2022).
date_created: 2025-12-18T19:17:23Z
date_updated: 2025-12-18T20:12:13Z
doi: 10.1142/s0219199722500626
intvolume: '        25'
issue: '10'
language:
- iso: eng
publication: Communications in Contemporary Mathematics
publication_identifier:
  issn:
  - 0219-1997
  - 1793-6683
publication_status: published
publisher: World Scientific Pub Co Pte Ltd
status: public
title: Arbitrarily fast grow-up rates in quasilinear Keller–Segel systems
type: journal_article
user_id: '31496'
volume: 25
year: '2022'
...
---
_id: '40212'
author:
- first_name: Barnabas
  full_name: Haucke-Korber, Barnabas
  id: '93461'
  last_name: Haucke-Korber
  orcid: 0000-0003-0862-2069
- first_name: Maximilian
  full_name: Schenke, Maximilian
  id: '52638'
  last_name: Schenke
  orcid: 0000-0001-5427-9527
- first_name: Oliver
  full_name: Wallscheid, Oliver
  id: '11291'
  last_name: Wallscheid
  orcid: https://orcid.org/0000-0001-9362-8777
citation:
  ama: 'Haucke-Korber B, Schenke M, Wallscheid O. Reinforcement Learning-Based Deep
    Q Direct Torque Control with Adaptable Switching Frequency Towards Six-Step Operation
    of Permanent Magnet Synchronous Motors. In: <i>IKMT 2022; 13. GMM/ETG-Symposium</i>.
    ; 2022:1-6.'
  apa: Haucke-Korber, B., Schenke, M., &#38; Wallscheid, O. (2022). Reinforcement
    Learning-Based Deep Q Direct Torque Control with Adaptable Switching Frequency
    Towards Six-Step Operation of Permanent Magnet Synchronous Motors. <i>IKMT 2022;
    13. GMM/ETG-Symposium</i>, 1–6.
  bibtex: '@inproceedings{Haucke-Korber_Schenke_Wallscheid_2022, title={Reinforcement
    Learning-Based Deep Q Direct Torque Control with Adaptable Switching Frequency
    Towards Six-Step Operation of Permanent Magnet Synchronous Motors}, booktitle={IKMT
    2022; 13. GMM/ETG-Symposium}, author={Haucke-Korber, Barnabas and Schenke, Maximilian
    and Wallscheid, Oliver}, year={2022}, pages={1–6} }'
  chicago: Haucke-Korber, Barnabas, Maximilian Schenke, and Oliver Wallscheid. “Reinforcement
    Learning-Based Deep Q Direct Torque Control with Adaptable Switching Frequency
    Towards Six-Step Operation of Permanent Magnet Synchronous Motors.” In <i>IKMT
    2022; 13. GMM/ETG-Symposium</i>, 1–6, 2022.
  ieee: B. Haucke-Korber, M. Schenke, and O. Wallscheid, “Reinforcement Learning-Based
    Deep Q Direct Torque Control with Adaptable Switching Frequency Towards Six-Step
    Operation of Permanent Magnet Synchronous Motors,” in <i>IKMT 2022; 13. GMM/ETG-Symposium</i>,
    2022, pp. 1–6.
  mla: Haucke-Korber, Barnabas, et al. “Reinforcement Learning-Based Deep Q Direct
    Torque Control with Adaptable Switching Frequency Towards Six-Step Operation of
    Permanent Magnet Synchronous Motors.” <i>IKMT 2022; 13. GMM/ETG-Symposium</i>,
    2022, pp. 1–6.
  short: 'B. Haucke-Korber, M. Schenke, O. Wallscheid, in: IKMT 2022; 13. GMM/ETG-Symposium,
    2022, pp. 1–6.'
date_created: 2023-01-26T08:51:40Z
date_updated: 2025-12-19T12:44:31Z
department:
- _id: '52'
language:
- iso: eng
page: 1-6
publication: IKMT 2022; 13. GMM/ETG-Symposium
status: public
title: Reinforcement Learning-Based Deep Q Direct Torque Control with Adaptable Switching
  Frequency Towards Six-Step Operation of Permanent Magnet Synchronous Motors
type: conference
user_id: '93461'
year: '2022'
...
---
_id: '63428'
author:
- first_name: Claudia
  full_name: Kirschtein, Claudia
  id: '28261'
  last_name: Kirschtein
citation:
  ama: 'Kirschtein C. Mediendidaktische Konzeption mit Emotion. In: Pauschenwein J,
    Hernády B, Michelitsch L, eds. <i>Wie beeinflussen Gefühle und Sprache den (Online-)
    Lernprozess? Tagungsband zum 21. E-Learning Tag der FH JOANNEUM am 21.09.2022</i>.
    FH JOANNEUM Gesellschaft; 2022:73-84.'
  apa: Kirschtein, C. (2022). Mediendidaktische Konzeption mit Emotion. In J. Pauschenwein,
    B. Hernády, &#38; L. Michelitsch (Eds.), <i>Wie beeinflussen Gefühle und Sprache
    den (Online-) Lernprozess? Tagungsband zum 21. E-Learning Tag der FH JOANNEUM
    am 21.09.2022</i> (pp. 73–84). FH JOANNEUM Gesellschaft.
  bibtex: '@inbook{Kirschtein_2022, place={Graz}, title={Mediendidaktische Konzeption
    mit Emotion}, booktitle={Wie beeinflussen Gefühle und Sprache den (Online-) Lernprozess?
    Tagungsband zum 21. E-Learning Tag der FH JOANNEUM am 21.09.2022}, publisher={FH
    JOANNEUM Gesellschaft}, author={Kirschtein, Claudia}, editor={Pauschenwein, Jutta
    and Hernády, Birgit and Michelitsch, Linda}, year={2022}, pages={73–84} }'
  chicago: 'Kirschtein, Claudia. “Mediendidaktische Konzeption mit Emotion.” In <i>Wie
    beeinflussen Gefühle und Sprache den (Online-) Lernprozess? Tagungsband zum 21.
    E-Learning Tag der FH JOANNEUM am 21.09.2022</i>, edited by Jutta Pauschenwein,
    Birgit Hernády, and Linda Michelitsch, 73–84. Graz: FH JOANNEUM Gesellschaft,
    2022.'
  ieee: 'C. Kirschtein, “Mediendidaktische Konzeption mit Emotion,” in <i>Wie beeinflussen
    Gefühle und Sprache den (Online-) Lernprozess? Tagungsband zum 21. E-Learning
    Tag der FH JOANNEUM am 21.09.2022</i>, J. Pauschenwein, B. Hernády, and L. Michelitsch,
    Eds. Graz: FH JOANNEUM Gesellschaft, 2022, pp. 73–84.'
  mla: Kirschtein, Claudia. “Mediendidaktische Konzeption mit Emotion.” <i>Wie beeinflussen
    Gefühle und Sprache den (Online-) Lernprozess? Tagungsband zum 21. E-Learning
    Tag der FH JOANNEUM am 21.09.2022</i>, edited by Jutta Pauschenwein et al., FH
    JOANNEUM Gesellschaft, 2022, pp. 73–84.
  short: 'C. Kirschtein, in: J. Pauschenwein, B. Hernády, L. Michelitsch (Eds.), Wie
    beeinflussen Gefühle und Sprache den (Online-) Lernprozess? Tagungsband zum 21.
    E-Learning Tag der FH JOANNEUM am 21.09.2022, FH JOANNEUM Gesellschaft, Graz,
    2022, pp. 73–84.'
date_created: 2025-12-30T13:52:59Z
date_updated: 2025-12-30T13:54:08Z
editor:
- first_name: Jutta
  full_name: Pauschenwein, Jutta
  last_name: Pauschenwein
- first_name: Birgit
  full_name: Hernády, Birgit
  last_name: Hernády
- first_name: Linda
  full_name: Michelitsch, Linda
  last_name: Michelitsch
language:
- iso: ger
page: 73-84
place: Graz
publication: Wie beeinflussen Gefühle und Sprache den (Online-) Lernprozess? Tagungsband
  zum 21. E-Learning Tag der FH JOANNEUM am 21.09.2022
publication_identifier:
  unknown:
  - 978-3-903318-07-6
publication_status: published
publisher: FH JOANNEUM Gesellschaft
status: public
title: Mediendidaktische Konzeption mit Emotion
type: book_chapter
user_id: '28261'
year: '2022'
...
---
_id: '6553'
author:
- first_name: Leander
  full_name: Claes, Leander
  id: '11829'
  last_name: Claes
  orcid: 0000-0002-4393-268X
- first_name: Nadine
  full_name: Feldmann, Nadine
  id: '23082'
  last_name: Feldmann
- first_name: Veronika
  full_name: Schulze, Veronika
  last_name: Schulze
- first_name: Benjamin
  full_name: Jurgelucks, Benjamin
  last_name: Jurgelucks
- first_name: Andrea
  full_name: Walther, Andrea
  last_name: Walther
- first_name: Bernd
  full_name: Henning, Bernd
  id: '213'
  last_name: Henning
citation:
  ama: 'Claes L, Feldmann N, Schulze V, Jurgelucks B, Walther A, Henning B. Identification
    of piezoelectric material parameters using optimised multi-electrode specimens.
    In: <i>Fortschritte Der Akustik - DAGA 2022</i>. ; 2022:1326-1329.'
  apa: Claes, L., Feldmann, N., Schulze, V., Jurgelucks, B., Walther, A., &#38; Henning,
    B. (2022). Identification of piezoelectric material parameters using optimised
    multi-electrode specimens. <i>Fortschritte Der Akustik - DAGA 2022</i>, 1326–1329.
  bibtex: '@inproceedings{Claes_Feldmann_Schulze_Jurgelucks_Walther_Henning_2022,
    title={Identification of piezoelectric material parameters using optimised multi-electrode
    specimens}, booktitle={Fortschritte der Akustik - DAGA 2022}, author={Claes, Leander
    and Feldmann, Nadine and Schulze, Veronika and Jurgelucks, Benjamin and Walther,
    Andrea and Henning, Bernd}, year={2022}, pages={1326–1329} }'
  chicago: Claes, Leander, Nadine Feldmann, Veronika Schulze, Benjamin Jurgelucks,
    Andrea Walther, and Bernd Henning. “Identification of Piezoelectric Material Parameters
    Using Optimised Multi-Electrode Specimens.” In <i>Fortschritte Der Akustik - DAGA
    2022</i>, 1326–29, 2022.
  ieee: L. Claes, N. Feldmann, V. Schulze, B. Jurgelucks, A. Walther, and B. Henning,
    “Identification of piezoelectric material parameters using optimised multi-electrode
    specimens,” in <i>Fortschritte der Akustik - DAGA 2022</i>, Stuttgart, 2022, pp.
    1326–1329.
  mla: Claes, Leander, et al. “Identification of Piezoelectric Material Parameters
    Using Optimised Multi-Electrode Specimens.” <i>Fortschritte Der Akustik - DAGA
    2022</i>, 2022, pp. 1326–29.
  short: 'L. Claes, N. Feldmann, V. Schulze, B. Jurgelucks, A. Walther, B. Henning,
    in: Fortschritte Der Akustik - DAGA 2022, 2022, pp. 1326–1329.'
conference:
  end_date: 2022-03-24
  location: Stuttgart
  name: DAGA 2022 - 48. Jahrestagung für Akustik
  start_date: 2022-03-21
date_created: 2019-01-09T14:36:59Z
date_updated: 2026-01-05T07:52:51Z
ddc:
- '620'
department:
- _id: '49'
file:
- access_level: closed
  content_type: application/pdf
  creator: leanderc
  date_created: 2022-03-28T14:21:06Z
  date_updated: 2022-03-28T14:21:06Z
  file_id: '30666'
  file_name: daga2022 cla.pdf
  file_size: 1143917
  relation: main_file
  success: 1
file_date_updated: 2022-03-28T14:21:06Z
has_accepted_license: '1'
language:
- iso: eng
page: 1326-1329
project:
- _id: '90'
  name: 'ChaMP: Ein modellbasiertes Messverfahren zur Charakterisierung der frequenzabhängigen
    Materialeigenschaften von Piezokeramiken unter Verwendung eines einzelnen Probekörperindividuums'
- _id: '245'
  name: 'FOR 5208: Modellbasierte Bestimmung nichtlinearer Eigenschaften von Piezokeramiken
    für Leistungsschallanwendungen (NEPTUN)'
publication: Fortschritte der Akustik - DAGA 2022
status: public
title: Identification of piezoelectric material parameters using optimised multi-electrode
  specimens
type: conference
user_id: '11829'
year: '2022'
...
---
_id: '6558'
author:
- first_name: Olga
  full_name: Friesen, Olga
  id: '44026'
  last_name: Friesen
- first_name: Leander
  full_name: Claes, Leander
  id: '11829'
  last_name: Claes
  orcid: 0000-0002-4393-268X
- first_name: Nadine
  full_name: Feldmann, Nadine
  id: '23082'
  last_name: Feldmann
- first_name: Bernd
  full_name: Henning, Bernd
  id: '213'
  last_name: Henning
citation:
  ama: Friesen O, Claes L, Feldmann N, Henning B. <i>Estimation of Piezoelectric Material
    Parameters of Ring-Shaped Specimens</i>.; 2022.
  apa: Friesen, O., Claes, L., Feldmann, N., &#38; Henning, B. (2022). <i>Estimation
    of piezoelectric material parameters of ring-shaped specimens</i>.
  bibtex: '@book{Friesen_Claes_Feldmann_Henning_2022, place={International Workshop
    on Piezoelectric Materials and Applications in Actuators (IWPMA)}, title={Estimation
    of piezoelectric material parameters of ring-shaped specimens}, author={Friesen,
    Olga and Claes, Leander and Feldmann, Nadine and Henning, Bernd}, year={2022}
    }'
  chicago: Friesen, Olga, Leander Claes, Nadine Feldmann, and Bernd Henning. <i>Estimation
    of Piezoelectric Material Parameters of Ring-Shaped Specimens</i>. International
    Workshop on Piezoelectric Materials and Applications in Actuators (IWPMA), 2022.
  ieee: O. Friesen, L. Claes, N. Feldmann, and B. Henning, <i>Estimation of piezoelectric
    material parameters of ring-shaped specimens</i>. International Workshop on Piezoelectric
    Materials and Applications in Actuators (IWPMA), 2022.
  mla: Friesen, Olga, et al. <i>Estimation of Piezoelectric Material Parameters of
    Ring-Shaped Specimens</i>. 2022.
  short: O. Friesen, L. Claes, N. Feldmann, B. Henning, Estimation of Piezoelectric
    Material Parameters of Ring-Shaped Specimens, International Workshop on Piezoelectric
    Materials and Applications in Actuators (IWPMA), 2022.
date_created: 2019-01-09T14:37:05Z
date_updated: 2026-01-05T08:02:08Z
department:
- _id: '49'
language:
- iso: eng
place: International Workshop on Piezoelectric Materials and Applications in Actuators
  (IWPMA)
project:
- _id: '90'
  name: 'ChaMP: Ein modellbasiertes Messverfahren zur Charakterisierung der frequenzabhängigen
    Materialeigenschaften von Piezokeramiken unter Verwendung eines einzelnen Probekörperindividuums'
- _id: '245'
  name: 'FOR 5208: Modellbasierte Bestimmung nichtlinearer Eigenschaften von Piezokeramiken
    für Leistungsschallanwendungen (NEPTUN)'
status: public
title: Estimation of piezoelectric material parameters of ring-shaped specimens
type: misc
user_id: '11829'
year: '2022'
...