---
_id: '63313'
article_number: '47'
author:
- first_name: Mengyao
  full_name: Ding, Mengyao
  last_name: Ding
- first_name: Michael
  full_name: Winkler, Michael
  id: '31496'
  last_name: Winkler
citation:
  ama: Ding M, Winkler M. Small-density solutions in Keller–Segel systems involving
    rapidly decaying diffusivities. <i>Nonlinear Differential Equations and Applications
    NoDEA</i>. 2021;28(5). doi:<a href="https://doi.org/10.1007/s00030-021-00709-4">10.1007/s00030-021-00709-4</a>
  apa: Ding, M., &#38; Winkler, M. (2021). Small-density solutions in Keller–Segel
    systems involving rapidly decaying diffusivities. <i>Nonlinear Differential Equations
    and Applications NoDEA</i>, <i>28</i>(5), Article 47. <a href="https://doi.org/10.1007/s00030-021-00709-4">https://doi.org/10.1007/s00030-021-00709-4</a>
  bibtex: '@article{Ding_Winkler_2021, title={Small-density solutions in Keller–Segel
    systems involving rapidly decaying diffusivities}, volume={28}, DOI={<a href="https://doi.org/10.1007/s00030-021-00709-4">10.1007/s00030-021-00709-4</a>},
    number={547}, journal={Nonlinear Differential Equations and Applications NoDEA},
    publisher={Springer Science and Business Media LLC}, author={Ding, Mengyao and
    Winkler, Michael}, year={2021} }'
  chicago: Ding, Mengyao, and Michael Winkler. “Small-Density Solutions in Keller–Segel
    Systems Involving Rapidly Decaying Diffusivities.” <i>Nonlinear Differential Equations
    and Applications NoDEA</i> 28, no. 5 (2021). <a href="https://doi.org/10.1007/s00030-021-00709-4">https://doi.org/10.1007/s00030-021-00709-4</a>.
  ieee: 'M. Ding and M. Winkler, “Small-density solutions in Keller–Segel systems
    involving rapidly decaying diffusivities,” <i>Nonlinear Differential Equations
    and Applications NoDEA</i>, vol. 28, no. 5, Art. no. 47, 2021, doi: <a href="https://doi.org/10.1007/s00030-021-00709-4">10.1007/s00030-021-00709-4</a>.'
  mla: Ding, Mengyao, and Michael Winkler. “Small-Density Solutions in Keller–Segel
    Systems Involving Rapidly Decaying Diffusivities.” <i>Nonlinear Differential Equations
    and Applications NoDEA</i>, vol. 28, no. 5, 47, Springer Science and Business
    Media LLC, 2021, doi:<a href="https://doi.org/10.1007/s00030-021-00709-4">10.1007/s00030-021-00709-4</a>.
  short: M. Ding, M. Winkler, Nonlinear Differential Equations and Applications NoDEA
    28 (2021).
date_created: 2025-12-18T19:30:53Z
date_updated: 2025-12-18T20:05:56Z
doi: 10.1007/s00030-021-00709-4
intvolume: '        28'
issue: '5'
language:
- iso: eng
publication: Nonlinear Differential Equations and Applications NoDEA
publication_identifier:
  issn:
  - 1021-9722
  - 1420-9004
publication_status: published
publisher: Springer Science and Business Media LLC
status: public
title: Small-density solutions in Keller–Segel systems involving rapidly decaying
  diffusivities
type: journal_article
user_id: '31496'
volume: 28
year: '2021'
...
---
_id: '63308'
article_number: '103407'
author:
- first_name: Michael
  full_name: Winkler, Michael
  id: '31496'
  last_name: Winkler
citation:
  ama: 'Winkler M. Small-signal solutions of a two-dimensional doubly degenerate taxis
    system modeling bacterial motion in nutrient-poor environments. <i>Nonlinear Analysis:
    Real World Applications</i>. 2021;63. doi:<a href="https://doi.org/10.1016/j.nonrwa.2021.103407">10.1016/j.nonrwa.2021.103407</a>'
  apa: 'Winkler, M. (2021). Small-signal solutions of a two-dimensional doubly degenerate
    taxis system modeling bacterial motion in nutrient-poor environments. <i>Nonlinear
    Analysis: Real World Applications</i>, <i>63</i>, Article 103407. <a href="https://doi.org/10.1016/j.nonrwa.2021.103407">https://doi.org/10.1016/j.nonrwa.2021.103407</a>'
  bibtex: '@article{Winkler_2021, title={Small-signal solutions of a two-dimensional
    doubly degenerate taxis system modeling bacterial motion in nutrient-poor environments},
    volume={63}, DOI={<a href="https://doi.org/10.1016/j.nonrwa.2021.103407">10.1016/j.nonrwa.2021.103407</a>},
    number={103407}, journal={Nonlinear Analysis: Real World Applications}, publisher={Elsevier
    BV}, author={Winkler, Michael}, year={2021} }'
  chicago: 'Winkler, Michael. “Small-Signal Solutions of a Two-Dimensional Doubly
    Degenerate Taxis System Modeling Bacterial Motion in Nutrient-Poor Environments.”
    <i>Nonlinear Analysis: Real World Applications</i> 63 (2021). <a href="https://doi.org/10.1016/j.nonrwa.2021.103407">https://doi.org/10.1016/j.nonrwa.2021.103407</a>.'
  ieee: 'M. Winkler, “Small-signal solutions of a two-dimensional doubly degenerate
    taxis system modeling bacterial motion in nutrient-poor environments,” <i>Nonlinear
    Analysis: Real World Applications</i>, vol. 63, Art. no. 103407, 2021, doi: <a
    href="https://doi.org/10.1016/j.nonrwa.2021.103407">10.1016/j.nonrwa.2021.103407</a>.'
  mla: 'Winkler, Michael. “Small-Signal Solutions of a Two-Dimensional Doubly Degenerate
    Taxis System Modeling Bacterial Motion in Nutrient-Poor Environments.” <i>Nonlinear
    Analysis: Real World Applications</i>, vol. 63, 103407, Elsevier BV, 2021, doi:<a
    href="https://doi.org/10.1016/j.nonrwa.2021.103407">10.1016/j.nonrwa.2021.103407</a>.'
  short: 'M. Winkler, Nonlinear Analysis: Real World Applications 63 (2021).'
date_created: 2025-12-18T19:27:47Z
date_updated: 2025-12-18T20:05:10Z
doi: 10.1016/j.nonrwa.2021.103407
intvolume: '        63'
language:
- iso: eng
publication: 'Nonlinear Analysis: Real World Applications'
publication_identifier:
  issn:
  - 1468-1218
publication_status: published
publisher: Elsevier BV
status: public
title: Small-signal solutions of a two-dimensional doubly degenerate taxis system
  modeling bacterial motion in nutrient-poor environments
type: journal_article
user_id: '31496'
volume: 63
year: '2021'
...
---
_id: '63307'
author:
- first_name: Michael
  full_name: Winkler, Michael
  id: '31496'
  last_name: Winkler
citation:
  ama: Winkler M. Unlimited growth in logarithmic Keller-Segel systems. <i>Journal
    of Differential Equations</i>. 2021;309:74-97. doi:<a href="https://doi.org/10.1016/j.jde.2021.11.026">10.1016/j.jde.2021.11.026</a>
  apa: Winkler, M. (2021). Unlimited growth in logarithmic Keller-Segel systems. <i>Journal
    of Differential Equations</i>, <i>309</i>, 74–97. <a href="https://doi.org/10.1016/j.jde.2021.11.026">https://doi.org/10.1016/j.jde.2021.11.026</a>
  bibtex: '@article{Winkler_2021, title={Unlimited growth in logarithmic Keller-Segel
    systems}, volume={309}, DOI={<a href="https://doi.org/10.1016/j.jde.2021.11.026">10.1016/j.jde.2021.11.026</a>},
    journal={Journal of Differential Equations}, publisher={Elsevier BV}, author={Winkler,
    Michael}, year={2021}, pages={74–97} }'
  chicago: 'Winkler, Michael. “Unlimited Growth in Logarithmic Keller-Segel Systems.”
    <i>Journal of Differential Equations</i> 309 (2021): 74–97. <a href="https://doi.org/10.1016/j.jde.2021.11.026">https://doi.org/10.1016/j.jde.2021.11.026</a>.'
  ieee: 'M. Winkler, “Unlimited growth in logarithmic Keller-Segel systems,” <i>Journal
    of Differential Equations</i>, vol. 309, pp. 74–97, 2021, doi: <a href="https://doi.org/10.1016/j.jde.2021.11.026">10.1016/j.jde.2021.11.026</a>.'
  mla: Winkler, Michael. “Unlimited Growth in Logarithmic Keller-Segel Systems.” <i>Journal
    of Differential Equations</i>, vol. 309, Elsevier BV, 2021, pp. 74–97, doi:<a
    href="https://doi.org/10.1016/j.jde.2021.11.026">10.1016/j.jde.2021.11.026</a>.
  short: M. Winkler, Journal of Differential Equations 309 (2021) 74–97.
date_created: 2025-12-18T19:27:16Z
date_updated: 2025-12-18T20:05:02Z
doi: 10.1016/j.jde.2021.11.026
intvolume: '       309'
language:
- iso: eng
page: 74-97
publication: Journal of Differential Equations
publication_identifier:
  issn:
  - 0022-0396
publication_status: published
publisher: Elsevier BV
status: public
title: Unlimited growth in logarithmic Keller-Segel systems
type: journal_article
user_id: '31496'
volume: 309
year: '2021'
...
---
_id: '63316'
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. Taxis-driven Formation of Singular Hotspots in a May--Nowak
    Type Model for Virus Infection. <i>SIAM Journal on Mathematical Analysis</i>.
    2021;53(2):1411-1433. doi:<a href="https://doi.org/10.1137/20m1362851">10.1137/20m1362851</a>
  apa: Tao, Y., &#38; Winkler, M. (2021). Taxis-driven Formation of Singular Hotspots
    in a May--Nowak Type Model for Virus Infection. <i>SIAM Journal on Mathematical
    Analysis</i>, <i>53</i>(2), 1411–1433. <a href="https://doi.org/10.1137/20m1362851">https://doi.org/10.1137/20m1362851</a>
  bibtex: '@article{Tao_Winkler_2021, title={Taxis-driven Formation of Singular Hotspots
    in a May--Nowak Type Model for Virus Infection}, volume={53}, DOI={<a href="https://doi.org/10.1137/20m1362851">10.1137/20m1362851</a>},
    number={2}, journal={SIAM Journal on Mathematical Analysis}, publisher={Society
    for Industrial &#38; Applied Mathematics (SIAM)}, author={Tao, Youshan and Winkler,
    Michael}, year={2021}, pages={1411–1433} }'
  chicago: 'Tao, Youshan, and Michael Winkler. “Taxis-Driven Formation of Singular
    Hotspots in a May--Nowak Type Model for Virus Infection.” <i>SIAM Journal on Mathematical
    Analysis</i> 53, no. 2 (2021): 1411–33. <a href="https://doi.org/10.1137/20m1362851">https://doi.org/10.1137/20m1362851</a>.'
  ieee: 'Y. Tao and M. Winkler, “Taxis-driven Formation of Singular Hotspots in a
    May--Nowak Type Model for Virus Infection,” <i>SIAM Journal on Mathematical Analysis</i>,
    vol. 53, no. 2, pp. 1411–1433, 2021, doi: <a href="https://doi.org/10.1137/20m1362851">10.1137/20m1362851</a>.'
  mla: Tao, Youshan, and Michael Winkler. “Taxis-Driven Formation of Singular Hotspots
    in a May--Nowak Type Model for Virus Infection.” <i>SIAM Journal on Mathematical
    Analysis</i>, vol. 53, no. 2, Society for Industrial &#38; Applied Mathematics
    (SIAM), 2021, pp. 1411–33, doi:<a href="https://doi.org/10.1137/20m1362851">10.1137/20m1362851</a>.
  short: Y. Tao, M. Winkler, SIAM Journal on Mathematical Analysis 53 (2021) 1411–1433.
date_created: 2025-12-18T19:32:18Z
date_updated: 2025-12-18T20:06:20Z
doi: 10.1137/20m1362851
intvolume: '        53'
issue: '2'
language:
- iso: eng
page: 1411-1433
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: Taxis-driven Formation of Singular Hotspots in a May--Nowak Type Model for
  Virus Infection
type: journal_article
user_id: '31496'
volume: 53
year: '2021'
...
---
_id: '63319'
article_number: '112324'
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. The dampening role of large repulsive convection in a chemotaxis
    system modeling tumor angiogenesis. <i>Nonlinear Analysis</i>. 2021;208. doi:<a
    href="https://doi.org/10.1016/j.na.2021.112324">10.1016/j.na.2021.112324</a>
  apa: Tao, Y., &#38; Winkler, M. (2021). The dampening role of large repulsive convection
    in a chemotaxis system modeling tumor angiogenesis. <i>Nonlinear Analysis</i>,
    <i>208</i>, Article 112324. <a href="https://doi.org/10.1016/j.na.2021.112324">https://doi.org/10.1016/j.na.2021.112324</a>
  bibtex: '@article{Tao_Winkler_2021, title={The dampening role of large repulsive
    convection in a chemotaxis system modeling tumor angiogenesis}, volume={208},
    DOI={<a href="https://doi.org/10.1016/j.na.2021.112324">10.1016/j.na.2021.112324</a>},
    number={112324}, journal={Nonlinear Analysis}, publisher={Elsevier BV}, author={Tao,
    Youshan and Winkler, Michael}, year={2021} }'
  chicago: Tao, Youshan, and Michael Winkler. “The Dampening Role of Large Repulsive
    Convection in a Chemotaxis System Modeling Tumor Angiogenesis.” <i>Nonlinear Analysis</i>
    208 (2021). <a href="https://doi.org/10.1016/j.na.2021.112324">https://doi.org/10.1016/j.na.2021.112324</a>.
  ieee: 'Y. Tao and M. Winkler, “The dampening role of large repulsive convection
    in a chemotaxis system modeling tumor angiogenesis,” <i>Nonlinear Analysis</i>,
    vol. 208, Art. no. 112324, 2021, doi: <a href="https://doi.org/10.1016/j.na.2021.112324">10.1016/j.na.2021.112324</a>.'
  mla: Tao, Youshan, and Michael Winkler. “The Dampening Role of Large Repulsive Convection
    in a Chemotaxis System Modeling Tumor Angiogenesis.” <i>Nonlinear Analysis</i>,
    vol. 208, 112324, Elsevier BV, 2021, doi:<a href="https://doi.org/10.1016/j.na.2021.112324">10.1016/j.na.2021.112324</a>.
  short: Y. Tao, M. Winkler, Nonlinear Analysis 208 (2021).
date_created: 2025-12-18T19:33:25Z
date_updated: 2025-12-18T20:06:43Z
doi: 10.1016/j.na.2021.112324
intvolume: '       208'
language:
- iso: eng
publication: Nonlinear Analysis
publication_identifier:
  issn:
  - 0362-546X
publication_status: published
publisher: Elsevier BV
status: public
title: The dampening role of large repulsive convection in a chemotaxis system modeling
  tumor angiogenesis
type: journal_article
user_id: '31496'
volume: 208
year: '2021'
...
---
_id: '63291'
abstract:
- lang: eng
  text: <jats:p> An initial-boundary value problem for a coupled chemotaxis-Navier–Stokes
    model with porous medium type diffusion is considered. Previous related literature
    has provided profound knowledge in cases when the system is augmented with no-flux/no-flux/no-slip
    boundary conditions for the density of cells, the chemical concentration and the
    fluid velocity field, respectively; in particular, available qualitative results
    strongly indicate that only trivial solution behavior can be expected on large
    time scales. In line with refined modeling approaches to oxygen evolution near
    fluid-air interfaces, this study now focuses on situations involving a fixed chemoattractant
    concentration on the boundary. Despite an apparent loss of mathematically favorable
    energy structures thereby induced, by means of an alternative variational approach
    a basic theory of global existence is developed in a natural framework of weak
    solvability. Beyond this, some additional qualitative information on the large
    time behavior of these solutions is derived by identifying a certain global relaxation
    property. Specifically, a second result asserts, within a suitable topological
    setting, the existence of a bounded set which eventually absorbs each individual
    of the obtained trajectories, and the diameter of which is bounded only by the
    physically relevant quantities of total population size and prescribed boundary
    concentration of the chemical signal. </jats:p>
author:
- first_name: Tobias
  full_name: Black, Tobias
  id: '23686'
  last_name: Black
  orcid: 0000-0001-9963-0800
- first_name: Michael
  full_name: Winkler, Michael
  id: '31496'
  last_name: Winkler
citation:
  ama: Black T, Winkler M. Global weak solutions and absorbing sets in a chemotaxis-Navier–Stokes
    system with prescribed signal concentration on the boundary. <i>Mathematical Models
    and Methods in Applied Sciences</i>. 2021;32(01):137-173. doi:<a href="https://doi.org/10.1142/s021820252250004x">10.1142/s021820252250004x</a>
  apa: Black, T., &#38; Winkler, M. (2021). Global weak solutions and absorbing sets
    in a chemotaxis-Navier–Stokes system with prescribed signal concentration on the
    boundary. <i>Mathematical Models and Methods in Applied Sciences</i>, <i>32</i>(01),
    137–173. <a href="https://doi.org/10.1142/s021820252250004x">https://doi.org/10.1142/s021820252250004x</a>
  bibtex: '@article{Black_Winkler_2021, title={Global weak solutions and absorbing
    sets in a chemotaxis-Navier–Stokes system with prescribed signal concentration
    on the boundary}, volume={32}, DOI={<a href="https://doi.org/10.1142/s021820252250004x">10.1142/s021820252250004x</a>},
    number={01}, journal={Mathematical Models and Methods in Applied Sciences}, publisher={World
    Scientific Pub Co Pte Ltd}, author={Black, Tobias and Winkler, Michael}, year={2021},
    pages={137–173} }'
  chicago: 'Black, Tobias, and Michael Winkler. “Global Weak Solutions and Absorbing
    Sets in a Chemotaxis-Navier–Stokes System with Prescribed Signal Concentration
    on the Boundary.” <i>Mathematical Models and Methods in Applied Sciences</i> 32,
    no. 01 (2021): 137–73. <a href="https://doi.org/10.1142/s021820252250004x">https://doi.org/10.1142/s021820252250004x</a>.'
  ieee: 'T. Black and M. Winkler, “Global weak solutions and absorbing sets in a chemotaxis-Navier–Stokes
    system with prescribed signal concentration on the boundary,” <i>Mathematical
    Models and Methods in Applied Sciences</i>, vol. 32, no. 01, pp. 137–173, 2021,
    doi: <a href="https://doi.org/10.1142/s021820252250004x">10.1142/s021820252250004x</a>.'
  mla: Black, Tobias, and Michael Winkler. “Global Weak Solutions and Absorbing Sets
    in a Chemotaxis-Navier–Stokes System with Prescribed Signal Concentration on the
    Boundary.” <i>Mathematical Models and Methods in Applied Sciences</i>, vol. 32,
    no. 01, World Scientific Pub Co Pte Ltd, 2021, pp. 137–73, doi:<a href="https://doi.org/10.1142/s021820252250004x">10.1142/s021820252250004x</a>.
  short: T. Black, M. Winkler, Mathematical Models and Methods in Applied Sciences
    32 (2021) 137–173.
date_created: 2025-12-18T19:20:48Z
date_updated: 2025-12-18T20:08:01Z
doi: 10.1142/s021820252250004x
intvolume: '        32'
issue: '01'
language:
- iso: eng
page: 137-173
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: Global weak solutions and absorbing sets in a chemotaxis-Navier–Stokes system
  with prescribed signal concentration on the boundary
type: journal_article
user_id: '31496'
volume: 32
year: '2021'
...
---
_id: '63297'
abstract:
- lang: eng
  text: "<jats:p>We consider the no-flux initial-boundary value problem for the cross-diffusive
    evolution system:<jats:disp-formula><jats:alternatives><jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\"
    mime-subtype=\"png\" mimetype=\"image\" xlink:href=\"S0956792521000279_eqnU1.png\"/><jats:tex-math>\r\n\\begin{eqnarray*}
    \       \\left\\{ \\begin{array}{ll}        u_t = u_{xx} - \\chi \\big(\\frac{u}{v}
    \\partial_x v \\big)_x - uv +B_1(x,t),        \\qquad &amp; x\\in \\Omega, \\
    t&gt;0, \\\\[1mm]        v_t = v_{xx} +uv - v + B_2(x,t),        \\qquad &amp;
    x\\in \\Omega, \\ t&gt;0,        \\end{array} \\right.  \\end{eqnarray*}\r\n</jats:tex-math></jats:alternatives></jats:disp-formula>which
    was introduced by Short <jats:italic>et al.</jats:italic> in [40] with <jats:inline-formula><jats:alternatives><jats:inline-graphic
    xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0956792521000279_inline1.png\"/><jats:tex-math>\r\n$\\chi=2$\r\n</jats:tex-math></jats:alternatives></jats:inline-formula>
    to describe the dynamics of urban crime.</jats:p><jats:p>In bounded intervals
    <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\"
    mime-subtype=\"png\" xlink:href=\"S0956792521000279_inline2.png\"/><jats:tex-math>\r\n$\\Omega\\subset\\mathbb{R}$\r\n</jats:tex-math></jats:alternatives></jats:inline-formula>
    and with prescribed suitably regular non-negative functions <jats:inline-formula><jats:alternatives><jats:inline-graphic
    xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0956792521000279_inline3.png\"/><jats:tex-math>\r\n$B_1$\r\n</jats:tex-math></jats:alternatives></jats:inline-formula>
    and <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\"
    mime-subtype=\"png\" xlink:href=\"S0956792521000279_inline4.png\"/><jats:tex-math>\r\n$B_2$\r\n</jats:tex-math></jats:alternatives></jats:inline-formula>,
    we first prove the existence of global classical solutions for any choice of <jats:inline-formula><jats:alternatives><jats:inline-graphic
    xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0956792521000279_inline5.png\"/><jats:tex-math>\r\n$\\chi&gt;0$\r\n</jats:tex-math></jats:alternatives></jats:inline-formula>
    and all reasonably regular non-negative initial data.</jats:p><jats:p>We next
    address the issue of determining the qualitative behaviour of solutions under
    appropriate assumptions on the asymptotic properties of <jats:inline-formula><jats:alternatives><jats:inline-graphic
    xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0956792521000279_inline6.png\"/><jats:tex-math>\r\n$B_1$\r\n</jats:tex-math></jats:alternatives></jats:inline-formula>
    and <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\"
    mime-subtype=\"png\" xlink:href=\"S0956792521000279_inline7.png\"/><jats:tex-math>\r\n$B_2$\r\n</jats:tex-math></jats:alternatives></jats:inline-formula>.
    Indeed, for arbitrary <jats:inline-formula><jats:alternatives><jats:inline-graphic
    xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0956792521000279_inline8.png\"/><jats:tex-math>\r\n$\\chi&gt;0$\r\n</jats:tex-math></jats:alternatives></jats:inline-formula>,
    we obtain boundedness of the solutions given strict positivity of the average
    of <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\"
    mime-subtype=\"png\" xlink:href=\"S0956792521000279_inline9.png\"/><jats:tex-math>\r\n$B_2$\r\n</jats:tex-math></jats:alternatives></jats:inline-formula>
    over the domain; moreover, it is seen that imposing a mild decay assumption on
    <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\"
    mime-subtype=\"png\" xlink:href=\"S0956792521000279_inline10.png\"/><jats:tex-math>\r\n$B_1$\r\n</jats:tex-math></jats:alternatives></jats:inline-formula>
    implies that <jats:italic>u</jats:italic> must decay to zero in the long-term
    limit. Our final result, valid for all <jats:inline-formula><jats:alternatives><jats:inline-graphic
    xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0956792521000279_inline11.png\"/><jats:tex-math>\r\n$\\chi\\in\\left(0,\\frac{\\sqrt{6\\sqrt{3}+9}}{2}\\right),$\r\n</jats:tex-math></jats:alternatives></jats:inline-formula>
    which contains the relevant value <jats:inline-formula><jats:alternatives><jats:inline-graphic
    xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0956792521000279_inline12.png\"/><jats:tex-math>\r\n$\\chi=2$\r\n</jats:tex-math></jats:alternatives></jats:inline-formula>,
    states that under the above decay assumption on <jats:inline-formula><jats:alternatives><jats:inline-graphic
    xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0956792521000279_inline13.png\"/><jats:tex-math>\r\n$B_1$\r\n</jats:tex-math></jats:alternatives></jats:inline-formula>,
    if furthermore <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\"
    mime-subtype=\"png\" xlink:href=\"S0956792521000279_inline14.png\"/><jats:tex-math>\r\n$B_2$\r\n</jats:tex-math></jats:alternatives></jats:inline-formula>
    appropriately stabilises to a non-trivial function <jats:inline-formula><jats:alternatives><jats:inline-graphic
    xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0956792521000279_inline15.png\"/><jats:tex-math>\r\n$B_{2,\\infty}$\r\n</jats:tex-math></jats:alternatives></jats:inline-formula>,
    then (<jats:italic>u</jats:italic>,<jats:italic>v</jats:italic>) approaches the
    limit <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\"
    mime-subtype=\"png\" xlink:href=\"S0956792521000279_inline16.png\"/><jats:tex-math>\r\n$(0,v_\\infty)$\r\n</jats:tex-math></jats:alternatives></jats:inline-formula>,
    where <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\"
    mime-subtype=\"png\" xlink:href=\"S0956792521000279_inline17.png\"/><jats:tex-math>\r\n$v_\\infty$\r\n</jats:tex-math></jats:alternatives></jats:inline-formula>
    denotes the solution of <jats:disp-formula><jats:alternatives><jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\"
    mime-subtype=\"png\" mimetype=\"image\" xlink:href=\"S0956792521000279_eqnU2.png\"/><jats:tex-math>\r\n\\begin{eqnarray*}
    \       \\left\\{ \\begin{array}{l}        -\\partial_{xx}v_\\infty + v_\\infty
    = B_{2,\\infty},        \\qquad x\\in \\Omega, \\\\[1mm]        \\partial_x v_{\\infty}=0,
    \       \\qquad x\\in\\partial\\Omega.        \\end{array} \\right.  \\end{eqnarray*}\r\n</jats:tex-math></jats:alternatives></jats:disp-formula>We
    conclude with some numerical simulations exploring possible effects that may arise
    when considering large values of <jats:inline-formula><jats:alternatives><jats:inline-graphic
    xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0956792521000279_inline18.png\"/><jats:tex-math>\r\n$\\chi$\r\n</jats:tex-math></jats:alternatives></jats:inline-formula>
    not covered by our qualitative analysis. We observe that when <jats:inline-formula><jats:alternatives><jats:inline-graphic
    xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0956792521000279_inline19.png\"/><jats:tex-math>\r\n$\\chi$\r\n</jats:tex-math></jats:alternatives></jats:inline-formula>
    increases, solutions may grow substantially on short time intervals, whereas only
    on large timescales diffusion will dominate and enforce equilibration.</jats:p>"
author:
- first_name: NANCY
  full_name: RODRIGUEZ, NANCY
  last_name: RODRIGUEZ
- first_name: Michael
  full_name: Winkler, Michael
  id: '31496'
  last_name: Winkler
citation:
  ama: RODRIGUEZ N, Winkler M. On the global existence and qualitative behaviour of
    one-dimensional solutions to a model for urban crime. <i>European Journal of Applied
    Mathematics</i>. 2021;33(5):919-959. doi:<a href="https://doi.org/10.1017/s0956792521000279">10.1017/s0956792521000279</a>
  apa: RODRIGUEZ, N., &#38; Winkler, M. (2021). On the global existence and qualitative
    behaviour of one-dimensional solutions to a model for urban crime. <i>European
    Journal of Applied Mathematics</i>, <i>33</i>(5), 919–959. <a href="https://doi.org/10.1017/s0956792521000279">https://doi.org/10.1017/s0956792521000279</a>
  bibtex: '@article{RODRIGUEZ_Winkler_2021, title={On the global existence and qualitative
    behaviour of one-dimensional solutions to a model for urban crime}, volume={33},
    DOI={<a href="https://doi.org/10.1017/s0956792521000279">10.1017/s0956792521000279</a>},
    number={5}, journal={European Journal of Applied Mathematics}, publisher={Cambridge
    University Press (CUP)}, author={RODRIGUEZ, NANCY and Winkler, Michael}, year={2021},
    pages={919–959} }'
  chicago: 'RODRIGUEZ, NANCY, and Michael Winkler. “On the Global Existence and Qualitative
    Behaviour of One-Dimensional Solutions to a Model for Urban Crime.” <i>European
    Journal of Applied Mathematics</i> 33, no. 5 (2021): 919–59. <a href="https://doi.org/10.1017/s0956792521000279">https://doi.org/10.1017/s0956792521000279</a>.'
  ieee: 'N. RODRIGUEZ and M. Winkler, “On the global existence and qualitative behaviour
    of one-dimensional solutions to a model for urban crime,” <i>European Journal
    of Applied Mathematics</i>, vol. 33, no. 5, pp. 919–959, 2021, doi: <a href="https://doi.org/10.1017/s0956792521000279">10.1017/s0956792521000279</a>.'
  mla: RODRIGUEZ, NANCY, and Michael Winkler. “On the Global Existence and Qualitative
    Behaviour of One-Dimensional Solutions to a Model for Urban Crime.” <i>European
    Journal of Applied Mathematics</i>, vol. 33, no. 5, Cambridge University Press
    (CUP), 2021, pp. 919–59, doi:<a href="https://doi.org/10.1017/s0956792521000279">10.1017/s0956792521000279</a>.
  short: N. RODRIGUEZ, M. Winkler, European Journal of Applied Mathematics 33 (2021)
    919–959.
date_created: 2025-12-18T19:23:28Z
date_updated: 2025-12-18T20:08:49Z
doi: 10.1017/s0956792521000279
intvolume: '        33'
issue: '5'
language:
- iso: eng
page: 919-959
publication: European Journal of Applied Mathematics
publication_identifier:
  issn:
  - 0956-7925
  - 1469-4425
publication_status: published
publisher: Cambridge University Press (CUP)
status: public
title: On the global existence and qualitative behaviour of one-dimensional solutions
  to a model for urban crime
type: journal_article
user_id: '31496'
volume: 33
year: '2021'
...
---
_id: '53333'
author:
- first_name: Michael
  full_name: Winkler, Michael
  id: '31496'
  last_name: Winkler
citation:
  ama: Winkler M. $L^1$ solutions to parabolic Keller-Segel systems involving arbitrary
    superlinear degradation. <i>ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE</i>.
    Published online 2021:141-172. doi:<a href="https://doi.org/10.2422/2036-2145.202005_016">10.2422/2036-2145.202005_016</a>
  apa: Winkler, M. (2021). $L^1$ solutions to parabolic Keller-Segel systems involving
    arbitrary superlinear degradation. <i>ANNALI SCUOLA NORMALE SUPERIORE - CLASSE
    DI SCIENZE</i>, 141–172. <a href="https://doi.org/10.2422/2036-2145.202005_016">https://doi.org/10.2422/2036-2145.202005_016</a>
  bibtex: '@article{Winkler_2021, title={$L^1$ solutions to parabolic Keller-Segel
    systems involving arbitrary superlinear degradation}, DOI={<a href="https://doi.org/10.2422/2036-2145.202005_016">10.2422/2036-2145.202005_016</a>},
    journal={ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE}, publisher={Scuola
    Normale Superiore - Edizioni della Normale}, author={Winkler, Michael}, year={2021},
    pages={141–172} }'
  chicago: Winkler, Michael. “$L^1$ Solutions to Parabolic Keller-Segel Systems Involving
    Arbitrary Superlinear Degradation.” <i>ANNALI SCUOLA NORMALE SUPERIORE - CLASSE
    DI SCIENZE</i>, 2021, 141–72. <a href="https://doi.org/10.2422/2036-2145.202005_016">https://doi.org/10.2422/2036-2145.202005_016</a>.
  ieee: 'M. Winkler, “$L^1$ solutions to parabolic Keller-Segel systems involving
    arbitrary superlinear degradation,” <i>ANNALI SCUOLA NORMALE SUPERIORE - CLASSE
    DI SCIENZE</i>, pp. 141–172, 2021, doi: <a href="https://doi.org/10.2422/2036-2145.202005_016">10.2422/2036-2145.202005_016</a>.'
  mla: Winkler, Michael. “$L^1$ Solutions to Parabolic Keller-Segel Systems Involving
    Arbitrary Superlinear Degradation.” <i>ANNALI SCUOLA NORMALE SUPERIORE - CLASSE
    DI SCIENZE</i>, Scuola Normale Superiore - Edizioni della Normale, 2021, pp. 141–72,
    doi:<a href="https://doi.org/10.2422/2036-2145.202005_016">10.2422/2036-2145.202005_016</a>.
  short: M. Winkler, ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE (2021) 141–172.
date_created: 2024-04-07T12:45:49Z
date_updated: 2025-12-18T20:15:27Z
doi: 10.2422/2036-2145.202005_016
keyword:
- Mathematics (miscellaneous)
- Theoretical Computer Science
language:
- iso: eng
page: 141-172
publication: ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE
publication_identifier:
  issn:
  - 2036-2145
  - 0391-173X
publication_status: published
publisher: Scuola Normale Superiore - Edizioni della Normale
status: public
title: $L^1$ solutions to parabolic Keller-Segel systems involving arbitrary superlinear
  degradation
type: journal_article
user_id: '31496'
year: '2021'
...
---
_id: '63373'
article_number: 421-466
author:
- first_name: Yulan
  full_name: Wang, Yulan
  last_name: Wang
- first_name: Michael
  full_name: Winkler, Michael
  id: '31496'
  last_name: Winkler
- first_name: Zhaoyin
  full_name: Xiang, Zhaoyin
  last_name: Xiang
citation:
  ama: Wang Y, Winkler M, Xiang Z. Global classical solutions in a two-dimensional
    chemotaxis-Navier-Stokes system with subcritical sensitivity. <i>ANNALI SCUOLA
    NORMALE SUPERIORE - CLASSE DI SCIENZE</i>. Published online 2021. doi:<a href="https://doi.org/10.2422/2036-2145.201603_004">10.2422/2036-2145.201603_004</a>
  apa: Wang, Y., Winkler, M., &#38; Xiang, Z. (2021). Global classical solutions in
    a two-dimensional chemotaxis-Navier-Stokes system with subcritical sensitivity.
    <i>ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE</i>, Article 421–466. <a
    href="https://doi.org/10.2422/2036-2145.201603_004">https://doi.org/10.2422/2036-2145.201603_004</a>
  bibtex: '@article{Wang_Winkler_Xiang_2021, title={Global classical solutions in
    a two-dimensional chemotaxis-Navier-Stokes system with subcritical sensitivity},
    DOI={<a href="https://doi.org/10.2422/2036-2145.201603_004">10.2422/2036-2145.201603_004</a>},
    number={421–466}, journal={ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE},
    publisher={Scuola Normale Superiore - Edizioni della Normale}, author={Wang, Yulan
    and Winkler, Michael and Xiang, Zhaoyin}, year={2021} }'
  chicago: Wang, Yulan, Michael Winkler, and Zhaoyin Xiang. “Global Classical Solutions
    in a Two-Dimensional Chemotaxis-Navier-Stokes System with Subcritical Sensitivity.”
    <i>ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE</i>, 2021. <a href="https://doi.org/10.2422/2036-2145.201603_004">https://doi.org/10.2422/2036-2145.201603_004</a>.
  ieee: 'Y. Wang, M. Winkler, and Z. Xiang, “Global classical solutions in a two-dimensional
    chemotaxis-Navier-Stokes system with subcritical sensitivity,” <i>ANNALI SCUOLA
    NORMALE SUPERIORE - CLASSE DI SCIENZE</i>, Art. no. 421–466, 2021, doi: <a href="https://doi.org/10.2422/2036-2145.201603_004">10.2422/2036-2145.201603_004</a>.'
  mla: Wang, Yulan, et al. “Global Classical Solutions in a Two-Dimensional Chemotaxis-Navier-Stokes
    System with Subcritical Sensitivity.” <i>ANNALI SCUOLA NORMALE SUPERIORE - CLASSE
    DI SCIENZE</i>, 421–466, Scuola Normale Superiore - Edizioni della Normale, 2021,
    doi:<a href="https://doi.org/10.2422/2036-2145.201603_004">10.2422/2036-2145.201603_004</a>.
  short: Y. Wang, M. Winkler, Z. Xiang, ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI
    SCIENZE (2021).
date_created: 2025-12-19T11:05:07Z
date_updated: 2025-12-19T11:05:13Z
doi: 10.2422/2036-2145.201603_004
language:
- iso: eng
publication: ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE
publication_identifier:
  issn:
  - 2036-2145
  - 0391-173X
publication_status: published
publisher: Scuola Normale Superiore - Edizioni della Normale
status: public
title: Global classical solutions in a two-dimensional chemotaxis-Navier-Stokes system
  with subcritical sensitivity
type: journal_article
user_id: '31496'
year: '2021'
...
---
_id: '37373'
author:
- first_name: Michael
  full_name: Winkler, Michael
  id: '31496'
  last_name: Winkler
citation:
  ama: Winkler M. Single-point blow-up in the Cauchy problem for the higher-dimensional
    Keller-Segel system. <i>Nonlinearity</i>. 2020;33:5007-5048.
  apa: Winkler, M. (2020). Single-point blow-up in the Cauchy problem for the higher-dimensional
    Keller-Segel system. <i>Nonlinearity</i>, <i>33</i>, 5007–5048.
  bibtex: '@article{Winkler_2020, title={Single-point blow-up in the Cauchy problem
    for the higher-dimensional Keller-Segel system}, volume={33}, journal={Nonlinearity},
    author={Winkler, Michael}, year={2020}, pages={5007–5048} }'
  chicago: 'Winkler, Michael. “Single-Point Blow-up in the Cauchy Problem for the
    Higher-Dimensional Keller-Segel System.” <i>Nonlinearity</i> 33 (2020): 5007–48.'
  ieee: M. Winkler, “Single-point blow-up in the Cauchy problem for the higher-dimensional
    Keller-Segel system,” <i>Nonlinearity</i>, vol. 33, pp. 5007–5048, 2020.
  mla: Winkler, Michael. “Single-Point Blow-up in the Cauchy Problem for the Higher-Dimensional
    Keller-Segel System.” <i>Nonlinearity</i>, vol. 33, 2020, pp. 5007–48.
  short: M. Winkler, Nonlinearity 33 (2020) 5007–5048.
date_created: 2023-01-18T12:46:13Z
date_updated: 2023-01-20T13:12:29Z
department:
- _id: '34'
- _id: '10'
- _id: '90'
intvolume: '        33'
language:
- iso: eng
page: 5007-5048
publication: Nonlinearity
status: public
title: Single-point blow-up in the Cauchy problem for the higher-dimensional Keller-Segel
  system
type: journal_article
user_id: '15645'
volume: 33
year: '2020'
...
---
_id: '37380'
article_number: '10'
author:
- first_name: Michael
  full_name: Winkler, Michael
  id: '31496'
  last_name: Winkler
citation:
  ama: Winkler M. Boundedness in a two-dimensional Keller-Segel-Navier-Stokes system
    involving a rapidly diffusing repulsive signal. <i>Zeitschrift für angewandte
    Mathematik und Physik</i>. 2020;71.
  apa: Winkler, M. (2020). Boundedness in a two-dimensional Keller-Segel-Navier-Stokes
    system involving a rapidly diffusing repulsive signal. <i>Zeitschrift Für Angewandte
    Mathematik Und Physik</i>, <i>71</i>, Article 10.
  bibtex: '@article{Winkler_2020, title={Boundedness in a two-dimensional Keller-Segel-Navier-Stokes
    system involving a rapidly diffusing repulsive signal.}, volume={71}, number={10},
    journal={Zeitschrift für angewandte Mathematik und Physik}, author={Winkler, Michael},
    year={2020} }'
  chicago: Winkler, Michael. “Boundedness in a Two-Dimensional Keller-Segel-Navier-Stokes
    System Involving a Rapidly Diffusing Repulsive Signal.” <i>Zeitschrift Für Angewandte
    Mathematik Und Physik</i> 71 (2020).
  ieee: M. Winkler, “Boundedness in a two-dimensional Keller-Segel-Navier-Stokes system
    involving a rapidly diffusing repulsive signal.,” <i>Zeitschrift für angewandte
    Mathematik und Physik</i>, vol. 71, Art. no. 10, 2020.
  mla: Winkler, Michael. “Boundedness in a Two-Dimensional Keller-Segel-Navier-Stokes
    System Involving a Rapidly Diffusing Repulsive Signal.” <i>Zeitschrift Für Angewandte
    Mathematik Und Physik</i>, vol. 71, 10, 2020.
  short: M. Winkler, Zeitschrift Für Angewandte Mathematik Und Physik 71 (2020).
date_created: 2023-01-18T12:55:58Z
date_updated: 2023-01-20T13:13:00Z
department:
- _id: '34'
- _id: '10'
- _id: '90'
intvolume: '        71'
language:
- iso: eng
publication: Zeitschrift für angewandte Mathematik und Physik
status: public
title: Boundedness in a two-dimensional Keller-Segel-Navier-Stokes system involving
  a rapidly diffusing repulsive signal.
type: journal_article
user_id: '15645'
volume: 71
year: '2020'
...
---
_id: '37367'
author:
- first_name: Michael
  full_name: Winkler, Michael
  id: '31496'
  last_name: Winkler
citation:
  ama: Winkler M. Can simultaneous density-determined enhacement of diffusion and
    cross-diffusion foster boundedness in Keller-Segel type systems involving signal-dependent
    motilities? <i>Nonlinearity</i>. 2020;33:6590-6623.
  apa: Winkler, M. (2020). Can simultaneous density-determined enhacement of diffusion
    and cross-diffusion foster boundedness in Keller-Segel type systems involving
    signal-dependent motilities? <i>Nonlinearity</i>, <i>33</i>, 6590–6623.
  bibtex: '@article{Winkler_2020, title={Can simultaneous density-determined enhacement
    of diffusion and cross-diffusion foster boundedness in Keller-Segel type systems
    involving signal-dependent motilities?}, volume={33}, journal={Nonlinearity},
    author={Winkler, Michael}, year={2020}, pages={6590–6623} }'
  chicago: 'Winkler, Michael. “Can Simultaneous Density-Determined Enhacement of Diffusion
    and Cross-Diffusion Foster Boundedness in Keller-Segel Type Systems Involving
    Signal-Dependent Motilities?” <i>Nonlinearity</i> 33 (2020): 6590–6623.'
  ieee: M. Winkler, “Can simultaneous density-determined enhacement of diffusion and
    cross-diffusion foster boundedness in Keller-Segel type systems involving signal-dependent
    motilities?,” <i>Nonlinearity</i>, vol. 33, pp. 6590–6623, 2020.
  mla: Winkler, Michael. “Can Simultaneous Density-Determined Enhacement of Diffusion
    and Cross-Diffusion Foster Boundedness in Keller-Segel Type Systems Involving
    Signal-Dependent Motilities?” <i>Nonlinearity</i>, vol. 33, 2020, pp. 6590–623.
  short: M. Winkler, Nonlinearity 33 (2020) 6590–6623.
date_created: 2023-01-18T12:42:55Z
date_updated: 2023-01-20T13:13:34Z
department:
- _id: '34'
- _id: '10'
- _id: '90'
intvolume: '        33'
language:
- iso: eng
page: 6590-6623
publication: Nonlinearity
status: public
title: Can simultaneous density-determined enhacement of diffusion and cross-diffusion
  foster boundedness in Keller-Segel type systems involving signal-dependent motilities?
type: journal_article
user_id: '15645'
volume: 33
year: '2020'
...
---
_id: '37375'
author:
- first_name: Michael
  full_name: Winkler, Michael
  id: '31496'
  last_name: Winkler
citation:
  ama: Winkler M. Attractiveness  of constant states in logistic-type Keller-Segel
    systems involving subquadratic growth restrictions. <i>Advanced Nonlinear Studies</i>.
    2020;20:795-817.
  apa: Winkler, M. (2020). Attractiveness  of constant states in logistic-type Keller-Segel
    systems involving subquadratic growth restrictions. <i>Advanced Nonlinear Studies</i>,
    <i>20</i>, 795–817.
  bibtex: '@article{Winkler_2020, title={Attractiveness  of constant states in logistic-type
    Keller-Segel systems involving subquadratic growth restrictions.}, volume={20},
    journal={Advanced Nonlinear Studies}, author={Winkler, Michael}, year={2020},
    pages={795–817} }'
  chicago: 'Winkler, Michael. “Attractiveness  of Constant States in Logistic-Type
    Keller-Segel Systems Involving Subquadratic Growth Restrictions.” <i>Advanced
    Nonlinear Studies</i> 20 (2020): 795–817.'
  ieee: M. Winkler, “Attractiveness  of constant states in logistic-type Keller-Segel
    systems involving subquadratic growth restrictions.,” <i>Advanced Nonlinear Studies</i>,
    vol. 20, pp. 795–817, 2020.
  mla: Winkler, Michael. “Attractiveness  of Constant States in Logistic-Type Keller-Segel
    Systems Involving Subquadratic Growth Restrictions.” <i>Advanced Nonlinear Studies</i>,
    vol. 20, 2020, pp. 795–817.
  short: M. Winkler, Advanced Nonlinear Studies 20 (2020) 795–817.
date_created: 2023-01-18T12:49:46Z
date_updated: 2023-01-20T13:13:27Z
department:
- _id: '34'
- _id: '10'
- _id: '90'
intvolume: '        20'
language:
- iso: eng
page: 795-817
publication: Advanced Nonlinear Studies
status: public
title: Attractiveness  of constant states in logistic-type Keller-Segel systems involving
  subquadratic growth restrictions.
type: journal_article
user_id: '15645'
volume: 20
year: '2020'
...
---
_id: '37347'
author:
- first_name: Nancy
  full_name: Rodriguez, Nancy
  last_name: Rodriguez
- first_name: Michael
  full_name: Winkler, Michael
  id: '31496'
  last_name: Winkler
citation:
  ama: Rodriguez N, Winkler M. Relaxation by nonlinear diffusion enhancement in a
    two-dimensional cross-diffusion model for urban crime propagation. <i>Mathematical
    Models &#38; Methods in Applied Sciences</i>. 2020;30:2105-2137.
  apa: Rodriguez, N., &#38; Winkler, M. (2020). Relaxation by nonlinear diffusion
    enhancement in a two-dimensional cross-diffusion model for urban crime propagation.
    <i>Mathematical Models &#38; Methods in Applied Sciences</i>, <i>30</i>, 2105–2137.
  bibtex: '@article{Rodriguez_Winkler_2020, title={Relaxation by nonlinear diffusion
    enhancement in a two-dimensional cross-diffusion model for urban crime propagation.},
    volume={30}, journal={Mathematical Models &#38; Methods in Applied Sciences},
    author={Rodriguez, Nancy and Winkler, Michael}, year={2020}, pages={2105–2137}
    }'
  chicago: 'Rodriguez, Nancy, and Michael Winkler. “Relaxation by Nonlinear Diffusion
    Enhancement in a Two-Dimensional Cross-Diffusion Model for Urban Crime Propagation.”
    <i>Mathematical Models &#38; Methods in Applied Sciences</i> 30 (2020): 2105–37.'
  ieee: N. Rodriguez and M. Winkler, “Relaxation by nonlinear diffusion enhancement
    in a two-dimensional cross-diffusion model for urban crime propagation.,” <i>Mathematical
    Models &#38; Methods in Applied Sciences</i>, vol. 30, pp. 2105–2137, 2020.
  mla: Rodriguez, Nancy, and Michael Winkler. “Relaxation by Nonlinear Diffusion Enhancement
    in a Two-Dimensional Cross-Diffusion Model for Urban Crime Propagation.” <i>Mathematical
    Models &#38; Methods in Applied Sciences</i>, vol. 30, 2020, pp. 2105–37.
  short: N. Rodriguez, M. Winkler, Mathematical Models &#38; Methods in Applied Sciences
    30 (2020) 2105–2137.
date_created: 2023-01-18T12:10:04Z
date_updated: 2023-01-20T13:14:37Z
department:
- _id: '34'
- _id: '10'
- _id: '90'
intvolume: '        30'
language:
- iso: eng
page: 2105-2137
publication: Mathematical Models & Methods in Applied Sciences
status: public
title: Relaxation by nonlinear diffusion enhancement in a two-dimensional cross-diffusion
  model for urban crime propagation.
type: journal_article
user_id: '15645'
volume: 30
year: '2020'
...
---
_id: '37349'
author:
- first_name: Christian
  full_name: Stinner, Christian
  last_name: Stinner
- first_name: Michael
  full_name: Winkler, Michael
  id: '31496'
  last_name: Winkler
citation:
  ama: Stinner C, Winkler M. Refined regularity and stabilization properties in a
    degenerate haptotaxis system. <i>Discrete and Continuous Dynamical Systems --A</i>.
    2020;40:4039-4058.
  apa: Stinner, C., &#38; Winkler, M. (2020). Refined regularity and stabilization
    properties in a degenerate haptotaxis system. <i>Discrete and Continuous Dynamical
    Systems --A</i>, <i>40</i>, 4039–4058.
  bibtex: '@article{Stinner_Winkler_2020, title={Refined regularity and stabilization
    properties in a degenerate haptotaxis system}, volume={40}, journal={Discrete
    and Continuous Dynamical Systems --A}, author={Stinner, Christian and Winkler,
    Michael}, year={2020}, pages={4039–4058} }'
  chicago: 'Stinner, Christian, and Michael Winkler. “Refined Regularity and Stabilization
    Properties in a Degenerate Haptotaxis System.” <i>Discrete and Continuous Dynamical
    Systems --A</i> 40 (2020): 4039–58.'
  ieee: C. Stinner and M. Winkler, “Refined regularity and stabilization properties
    in a degenerate haptotaxis system,” <i>Discrete and Continuous Dynamical Systems
    --A</i>, vol. 40, pp. 4039–4058, 2020.
  mla: Stinner, Christian, and Michael Winkler. “Refined Regularity and Stabilization
    Properties in a Degenerate Haptotaxis System.” <i>Discrete and Continuous Dynamical
    Systems --A</i>, vol. 40, 2020, pp. 4039–58.
  short: C. Stinner, M. Winkler, Discrete and Continuous Dynamical Systems --A 40
    (2020) 4039–4058.
date_created: 2023-01-18T12:17:21Z
date_updated: 2023-01-20T13:14:30Z
department:
- _id: '34'
- _id: '10'
- _id: '90'
intvolume: '        40'
language:
- iso: eng
page: 4039-4058
publication: Discrete and Continuous Dynamical Systems --A
status: public
title: Refined regularity and stabilization properties in a degenerate haptotaxis
  system
type: journal_article
user_id: '15645'
volume: 40
year: '2020'
...
---
_id: '63340'
abstract:
- lang: eng
  text: "<jats:title>Abstract</jats:title>\r\n               <jats:p>The chemotaxis-growth
    system</jats:p>\r\n               <jats:p>\r\n                  <jats:disp-formula
    id=\"j_ans-2020-2107_eq_0001\">\r\n                     <jats:label>($\\star$)</jats:label>\r\n
    \                    <jats:alternatives>\r\n                        <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\r\n
    \                          <m:mrow>\r\n                              <m:mo>{</m:mo>\r\n
    \                             <m:mtable columnspacing=\"0pt\" displaystyle=\"true\"
    rowspacing=\"0pt\">\r\n                                 <m:mtr>\r\n                                    <m:mtd
    columnalign=\"right\">\r\n                                       <m:msub>\r\n
    \                                         <m:mi>u</m:mi>\r\n                                          <m:mi>t</m:mi>\r\n
    \                                      </m:msub>\r\n                                    </m:mtd>\r\n
    \                                   <m:mtd columnalign=\"left\">\r\n                                       <m:mrow>\r\n
    \                                         <m:mrow>\r\n                                             <m:mi
    />\r\n                                             <m:mo>=</m:mo>\r\n                                             <m:mrow>\r\n
    \                                               <m:mrow>\r\n                                                   <m:mrow>\r\n
    \                                                     <m:mrow>\r\n                                                         <m:mi>D</m:mi>\r\n
    \                                                        <m:mo>⁢</m:mo>\r\n                                                         <m:mi
    mathvariant=\"normal\">Δ</m:mi>\r\n                                                         <m:mo>⁢</m:mo>\r\n
    \                                                        <m:mi>u</m:mi>\r\n                                                      </m:mrow>\r\n
    \                                                     <m:mo>-</m:mo>\r\n                                                      <m:mrow>\r\n
    \                                                        <m:mrow>\r\n                                                            <m:mi>χ</m:mi>\r\n
    \                                                           <m:mo>⁢</m:mo>\r\n
    \                                                           <m:mo>∇</m:mo>\r\n
    \                                                        </m:mrow>\r\n                                                         <m:mo>⋅</m:mo>\r\n
    \                                                        <m:mrow>\r\n                                                            <m:mo
    stretchy=\"false\">(</m:mo>\r\n                                                            <m:mrow>\r\n
    \                                                              <m:mi>u</m:mi>\r\n
    \                                                              <m:mo>⁢</m:mo>\r\n
    \                                                              <m:mrow>\r\n                                                                  <m:mo>∇</m:mo>\r\n
    \                                                                 <m:mo>⁡</m:mo>\r\n
    \                                                                 <m:mi>v</m:mi>\r\n
    \                                                              </m:mrow>\r\n                                                            </m:mrow>\r\n
    \                                                           <m:mo stretchy=\"false\">)</m:mo>\r\n
    \                                                        </m:mrow>\r\n                                                      </m:mrow>\r\n
    \                                                  </m:mrow>\r\n                                                   <m:mo>+</m:mo>\r\n
    \                                                  <m:mrow>\r\n                                                      <m:mi>ρ</m:mi>\r\n
    \                                                     <m:mo>⁢</m:mo>\r\n                                                      <m:mi>u</m:mi>\r\n
    \                                                  </m:mrow>\r\n                                                </m:mrow>\r\n
    \                                               <m:mo>-</m:mo>\r\n                                                <m:mrow>\r\n
    \                                                  <m:mi>μ</m:mi>\r\n                                                   <m:mo>⁢</m:mo>\r\n
    \                                                  <m:msup>\r\n                                                      <m:mi>u</m:mi>\r\n
    \                                                     <m:mi>α</m:mi>\r\n                                                   </m:msup>\r\n
    \                                               </m:mrow>\r\n                                             </m:mrow>\r\n
    \                                         </m:mrow>\r\n                                          <m:mo>,</m:mo>\r\n
    \                                      </m:mrow>\r\n                                    </m:mtd>\r\n
    \                                </m:mtr>\r\n                                 <m:mtr>\r\n
    \                                   <m:mtd columnalign=\"right\">\r\n                                       <m:msub>\r\n
    \                                         <m:mi>v</m:mi>\r\n                                          <m:mi>t</m:mi>\r\n
    \                                      </m:msub>\r\n                                    </m:mtd>\r\n
    \                                   <m:mtd columnalign=\"left\">\r\n                                       <m:mrow>\r\n
    \                                         <m:mi />\r\n                                          <m:mo>=</m:mo>\r\n
    \                                         <m:mrow>\r\n                                             <m:mrow>\r\n
    \                                               <m:mrow>\r\n                                                   <m:mi>d</m:mi>\r\n
    \                                                  <m:mo>⁢</m:mo>\r\n                                                   <m:mi
    mathvariant=\"normal\">Δ</m:mi>\r\n                                                   <m:mo>⁢</m:mo>\r\n
    \                                                  <m:mi>v</m:mi>\r\n                                                </m:mrow>\r\n
    \                                               <m:mo>-</m:mo>\r\n                                                <m:mrow>\r\n
    \                                                  <m:mi>κ</m:mi>\r\n                                                   <m:mo>⁢</m:mo>\r\n
    \                                                  <m:mi>v</m:mi>\r\n                                                </m:mrow>\r\n
    \                                            </m:mrow>\r\n                                             <m:mo>+</m:mo>\r\n
    \                                            <m:mrow>\r\n                                                <m:mi>λ</m:mi>\r\n
    \                                               <m:mo>⁢</m:mo>\r\n                                                <m:mi>u</m:mi>\r\n
    \                                            </m:mrow>\r\n                                          </m:mrow>\r\n
    \                                      </m:mrow>\r\n                                    </m:mtd>\r\n
    \                                </m:mtr>\r\n                              </m:mtable>\r\n
    \                          </m:mrow>\r\n                        </m:math>\r\n
    \                       <jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\"
    xlink:href=\"graphic/j_ans-2020-2107_fig_001.png\" />\r\n                        <jats:tex-math>{}\\left\\{\\begin{aligned}
    \\displaystyle{}u_{t}&amp;\\displaystyle=D\\Delta u-\\chi% \\nabla\\cdot(u\\nabla
    v)+\\rho u-\\mu u^{\\alpha},\\\\ \\displaystyle v_{t}&amp;\\displaystyle=d\\Delta
    v-\\kappa v+\\lambda u\\end{aligned}\\right.</jats:tex-math>\r\n                     </jats:alternatives>\r\n
    \                 </jats:disp-formula>\r\n               </jats:p>\r\n               <jats:p>is
    considered under homogeneous Neumann boundary conditions in smoothly bounded domains
    <jats:inline-formula id=\"j_ans-2020-2107_ineq_9999\">\r\n                     <jats:alternatives>\r\n
    \                       <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\r\n
    \                          <m:mrow>\r\n                              <m:mi mathvariant=\"normal\">Ω</m:mi>\r\n
    \                             <m:mo>⊂</m:mo>\r\n                              <m:msup>\r\n
    \                                <m:mi>ℝ</m:mi>\r\n                                 <m:mi>n</m:mi>\r\n
    \                             </m:msup>\r\n                           </m:mrow>\r\n
    \                       </m:math>\r\n                        <jats:inline-graphic
    xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ans-2020-2107_inl_001.png\"
    />\r\n                        <jats:tex-math>{\\Omega\\subset\\mathbb{R}^{n}}</jats:tex-math>\r\n
    \                    </jats:alternatives>\r\n                  </jats:inline-formula>,
    <jats:inline-formula id=\"j_ans-2020-2107_ineq_9998\">\r\n                     <jats:alternatives>\r\n
    \                       <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\r\n
    \                          <m:mrow>\r\n                              <m:mi>n</m:mi>\r\n
    \                             <m:mo>≥</m:mo>\r\n                              <m:mn>1</m:mn>\r\n
    \                          </m:mrow>\r\n                        </m:math>\r\n
    \                       <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\"
    xlink:href=\"graphic/j_ans-2020-2107_inl_002.png\" />\r\n                        <jats:tex-math>{n\\geq
    1}</jats:tex-math>\r\n                     </jats:alternatives>\r\n                  </jats:inline-formula>.
    For any choice of <jats:inline-formula id=\"j_ans-2020-2107_ineq_9997\">\r\n                     <jats:alternatives>\r\n
    \                       <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\r\n
    \                          <m:mrow>\r\n                              <m:mi>α</m:mi>\r\n
    \                             <m:mo>&gt;</m:mo>\r\n                              <m:mn>1</m:mn>\r\n
    \                          </m:mrow>\r\n                        </m:math>\r\n
    \                       <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\"
    xlink:href=\"graphic/j_ans-2020-2107_inl_003.png\" />\r\n                        <jats:tex-math>{\\alpha&gt;1}</jats:tex-math>\r\n
    \                    </jats:alternatives>\r\n                  </jats:inline-formula>,
    the literature provides a comprehensive result on global existence for widely
    arbitrary initial data within a suitably generalized solution concept, but the
    regularity properties of such solutions may be rather poor, as indicated by precedent
    results on the occurrence of finite-time blow-up in corresponding parabolic-elliptic
    simplifications. Based on the analysis of a certain eventual Lyapunov-type feature
    of ($\\star$), the present work shows that, whenever <jats:inline-formula id=\"j_ans-2020-2107_ineq_9996\">\r\n
    \                    <jats:alternatives>\r\n                        <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\r\n
    \                          <m:mrow>\r\n                              <m:mi>α</m:mi>\r\n
    \                             <m:mo>≥</m:mo>\r\n                              <m:mrow>\r\n
    \                                <m:mn>2</m:mn>\r\n                                 <m:mo>-</m:mo>\r\n
    \                                <m:mfrac>\r\n                                    <m:mn>2</m:mn>\r\n
    \                                   <m:mi>n</m:mi>\r\n                                 </m:mfrac>\r\n
    \                             </m:mrow>\r\n                           </m:mrow>\r\n
    \                       </m:math>\r\n                        <jats:inline-graphic
    xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ans-2020-2107_inl_004.png\"
    />\r\n                        <jats:tex-math>{\\alpha\\geq 2-\\frac{2}{n}}</jats:tex-math>\r\n
    \                    </jats:alternatives>\r\n                  </jats:inline-formula>,
    under an appropriate smallness assumption on χ, any such solution at least asymptotically
    exhibits relaxation by approaching the nontrivial spatially homogeneous steady
    state <jats:inline-formula id=\"j_ans-2020-2107_ineq_9995\">\r\n                     <jats:alternatives>\r\n
    \                       <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\r\n
    \                          <m:mrow>\r\n                              <m:mo maxsize=\"120%\"
    minsize=\"120%\">(</m:mo>\r\n                              <m:msup>\r\n                                 <m:mrow>\r\n
    \                                   <m:mo maxsize=\"120%\" minsize=\"120%\">(</m:mo>\r\n
    \                                   <m:mfrac>\r\n                                       <m:mi>ρ</m:mi>\r\n
    \                                      <m:mi>μ</m:mi>\r\n                                    </m:mfrac>\r\n
    \                                   <m:mo maxsize=\"120%\" minsize=\"120%\">)</m:mo>\r\n
    \                                </m:mrow>\r\n                                 <m:mfrac>\r\n
    \                                   <m:mn>1</m:mn>\r\n                                    <m:mrow>\r\n
    \                                      <m:mi>α</m:mi>\r\n                                       <m:mo>-</m:mo>\r\n
    \                                      <m:mn>1</m:mn>\r\n                                    </m:mrow>\r\n
    \                                </m:mfrac>\r\n                              </m:msup>\r\n
    \                             <m:mo>,</m:mo>\r\n                              <m:mrow>\r\n
    \                                <m:mfrac>\r\n                                    <m:mi>λ</m:mi>\r\n
    \                                   <m:mi>κ</m:mi>\r\n                                 </m:mfrac>\r\n
    \                                <m:mo>⁢</m:mo>\r\n                                 <m:msup>\r\n
    \                                   <m:mrow>\r\n                                       <m:mo
    maxsize=\"120%\" minsize=\"120%\">(</m:mo>\r\n                                       <m:mfrac>\r\n
    \                                         <m:mi>ρ</m:mi>\r\n                                          <m:mi>μ</m:mi>\r\n
    \                                      </m:mfrac>\r\n                                       <m:mo
    maxsize=\"120%\" minsize=\"120%\">)</m:mo>\r\n                                    </m:mrow>\r\n
    \                                   <m:mfrac>\r\n                                       <m:mn>1</m:mn>\r\n
    \                                      <m:mrow>\r\n                                          <m:mi>α</m:mi>\r\n
    \                                         <m:mo>-</m:mo>\r\n                                          <m:mn>1</m:mn>\r\n
    \                                      </m:mrow>\r\n                                    </m:mfrac>\r\n
    \                                </m:msup>\r\n                              </m:mrow>\r\n
    \                             <m:mo maxsize=\"120%\" minsize=\"120%\">)</m:mo>\r\n
    \                          </m:mrow>\r\n                        </m:math>\r\n
    \                       <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\"
    xlink:href=\"graphic/j_ans-2020-2107_inl_005.png\" />\r\n                        <jats:tex-math>{\\bigl{(}\\bigl{(}\\frac{\\rho}{\\mu}\\bigr{)}^{\\frac{1}{\\alpha-1}},\\frac{\\lambda}{%
    \\kappa}\\bigl{(}\\frac{\\rho}{\\mu}\\bigr{)}^{\\frac{1}{\\alpha-1}}\\bigr{)}}</jats:tex-math>\r\n
    \                    </jats:alternatives>\r\n                  </jats:inline-formula>
    in the large time limit.</jats:p>"
author:
- first_name: Michael
  full_name: Winkler, Michael
  id: '31496'
  last_name: Winkler
citation:
  ama: Winkler M. Attractiveness of Constant States in Logistic-Type Keller–Segel
    Systems Involving Subquadratic Growth Restrictions. <i>Advanced Nonlinear Studies</i>.
    2020;20(4):795-817. doi:<a href="https://doi.org/10.1515/ans-2020-2107">10.1515/ans-2020-2107</a>
  apa: Winkler, M. (2020). Attractiveness of Constant States in Logistic-Type Keller–Segel
    Systems Involving Subquadratic Growth Restrictions. <i>Advanced Nonlinear Studies</i>,
    <i>20</i>(4), 795–817. <a href="https://doi.org/10.1515/ans-2020-2107">https://doi.org/10.1515/ans-2020-2107</a>
  bibtex: '@article{Winkler_2020, title={Attractiveness of Constant States in Logistic-Type
    Keller–Segel Systems Involving Subquadratic Growth Restrictions}, volume={20},
    DOI={<a href="https://doi.org/10.1515/ans-2020-2107">10.1515/ans-2020-2107</a>},
    number={4}, journal={Advanced Nonlinear Studies}, publisher={Walter de Gruyter
    GmbH}, author={Winkler, Michael}, year={2020}, pages={795–817} }'
  chicago: 'Winkler, Michael. “Attractiveness of Constant States in Logistic-Type
    Keller–Segel Systems Involving Subquadratic Growth Restrictions.” <i>Advanced
    Nonlinear Studies</i> 20, no. 4 (2020): 795–817. <a href="https://doi.org/10.1515/ans-2020-2107">https://doi.org/10.1515/ans-2020-2107</a>.'
  ieee: 'M. Winkler, “Attractiveness of Constant States in Logistic-Type Keller–Segel
    Systems Involving Subquadratic Growth Restrictions,” <i>Advanced Nonlinear Studies</i>,
    vol. 20, no. 4, pp. 795–817, 2020, doi: <a href="https://doi.org/10.1515/ans-2020-2107">10.1515/ans-2020-2107</a>.'
  mla: Winkler, Michael. “Attractiveness of Constant States in Logistic-Type Keller–Segel
    Systems Involving Subquadratic Growth Restrictions.” <i>Advanced Nonlinear Studies</i>,
    vol. 20, no. 4, Walter de Gruyter GmbH, 2020, pp. 795–817, doi:<a href="https://doi.org/10.1515/ans-2020-2107">10.1515/ans-2020-2107</a>.
  short: M. Winkler, Advanced Nonlinear Studies 20 (2020) 795–817.
date_created: 2025-12-18T19:46:54Z
date_updated: 2025-12-18T19:58:22Z
doi: 10.1515/ans-2020-2107
intvolume: '        20'
issue: '4'
language:
- iso: eng
page: 795-817
publication: Advanced Nonlinear Studies
publication_identifier:
  issn:
  - 1536-1365
  - 2169-0375
publication_status: published
publisher: Walter de Gruyter GmbH
status: public
title: Attractiveness of Constant States in Logistic-Type Keller–Segel Systems Involving
  Subquadratic Growth Restrictions
type: journal_article
user_id: '31496'
volume: 20
year: '2020'
...
---
_id: '63342'
abstract:
- lang: eng
  text: "<jats:title>Abstract</jats:title><jats:p>In a bounded planar domain <jats:inline-formula><jats:alternatives><jats:tex-math>$\\varOmega
    $</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n
    \                 <mml:mi>Ω</mml:mi>\r\n                </mml:math></jats:alternatives></jats:inline-formula>
    with smooth boundary, the initial-boundary value problem of homogeneous Neumann
    type for the Keller-Segel-fluid system \r\n\t\t\t<jats:disp-formula><jats:alternatives><jats:tex-math>
    $$\\begin{aligned} \\left \\{ \\textstyle\\begin{array}{l@{\\quad }l} n_{t} +
    \\nabla \\cdot (nu) = \\Delta n - \\nabla \\cdot (n\\nabla c), &amp; x\\in \\varOmega
    , \\ t&gt;0, \\\\ 0 = \\Delta c -c+n, &amp; x\\in \\varOmega , \\ t&gt;0, \\end{array}\\displaystyle
    \\right . \\end{aligned}$$ </jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n
    \                 <mml:mrow>\r\n                    <mml:mo>{</mml:mo>\r\n                    <mml:mtable>\r\n
    \                     <mml:mtr>\r\n                        <mml:mtd>\r\n                          <mml:msub>\r\n
    \                           <mml:mi>n</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:mo>(</mml:mo>\r\n                          <mml:mi>n</mml:mi>\r\n
    \                         <mml:mi>u</mml:mi>\r\n                          <mml:mo>)</mml:mo>\r\n
    \                         <mml:mo>=</mml:mo>\r\n                          <mml:mi>Δ</mml:mi>\r\n
    \                         <mml:mi>n</mml:mi>\r\n                          <mml:mo>−</mml:mo>\r\n
    \                         <mml:mi>∇</mml:mi>\r\n                          <mml:mo>⋅</mml:mo>\r\n
    \                         <mml:mo>(</mml:mo>\r\n                          <mml:mi>n</mml:mi>\r\n
    \                         <mml:mi>∇</mml:mi>\r\n                          <mml:mi>c</mml:mi>\r\n
    \                         <mml:mo>)</mml:mo>\r\n                          <mml:mo>,</mml:mo>\r\n
    \                       </mml:mtd>\r\n                        <mml:mtd>\r\n                          <mml:mi>x</mml:mi>\r\n
    \                         <mml:mo>∈</mml:mo>\r\n                          <mml:mi>Ω</mml:mi>\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:mtd>\r\n                      </mml:mtr>\r\n                      <mml:mtr>\r\n
    \                       <mml:mtd>\r\n                          <mml:mn>0</mml:mn>\r\n
    \                         <mml:mo>=</mml:mo>\r\n                          <mml:mi>Δ</mml:mi>\r\n
    \                         <mml:mi>c</mml:mi>\r\n                          <mml:mo>−</mml:mo>\r\n
    \                         <mml:mi>c</mml:mi>\r\n                          <mml:mo>+</mml:mo>\r\n
    \                         <mml:mi>n</mml:mi>\r\n                          <mml:mo>,</mml:mo>\r\n
    \                       </mml:mtd>\r\n                        <mml:mtd>\r\n                          <mml:mi>x</mml:mi>\r\n
    \                         <mml:mo>∈</mml:mo>\r\n                          <mml:mi>Ω</mml:mi>\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:mtd>\r\n                      </mml:mtr>\r\n                    </mml:mtable>\r\n
    \                 </mml:mrow>\r\n                </mml:math></jats:alternatives></jats:disp-formula>
    is considered, where <jats:inline-formula><jats:alternatives><jats:tex-math>$u$</jats:tex-math><mml:math
    xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n                  <mml:mi>u</mml:mi>\r\n
    \               </mml:math></jats:alternatives></jats:inline-formula> is a given
    sufficiently smooth velocity field on <jats:inline-formula><jats:alternatives><jats:tex-math>$\\overline
    {\\varOmega }\\times [0,\\infty )$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n
    \                 <mml:mover>\r\n                    <mml:mi>Ω</mml:mi>\r\n                    <mml:mo>‾</mml:mo>\r\n
    \                 </mml:mover>\r\n                  <mml:mo>×</mml:mo>\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:math></jats:alternatives></jats:inline-formula> that is
    tangential on <jats:inline-formula><jats:alternatives><jats:tex-math>$\\partial
    \\varOmega $</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n
    \                 <mml:mi>∂</mml:mi>\r\n                  <mml:mi>Ω</mml:mi>\r\n
    \               </mml:math></jats:alternatives></jats:inline-formula> but not
    necessarily solenoidal.</jats:p><jats:p>It is firstly shown that for any choice
    of <jats:inline-formula><jats:alternatives><jats:tex-math>$n_{0}\\in C^{0}(\\overline
    {\\varOmega })$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n
    \                 <mml:msub>\r\n                    <mml:mi>n</mml:mi>\r\n                    <mml:mn>0</mml:mn>\r\n
    \                 </mml:msub>\r\n                  <mml:mo>∈</mml:mo>\r\n                  <mml:msup>\r\n
    \                   <mml:mi>C</mml:mi>\r\n                    <mml:mn>0</mml:mn>\r\n
    \                 </mml:msup>\r\n                  <mml:mo>(</mml:mo>\r\n                  <mml:mover>\r\n
    \                   <mml:mi>Ω</mml:mi>\r\n                    <mml:mo>‾</mml:mo>\r\n
    \                 </mml:mover>\r\n                  <mml:mo>)</mml:mo>\r\n                </mml:math></jats:alternatives></jats:inline-formula>
    with <jats:inline-formula><jats:alternatives><jats:tex-math>$\\int _{\\varOmega}n_{0}&lt;4\\pi
    $</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n
    \                 <mml:msub>\r\n                    <mml:mo>∫</mml:mo>\r\n                    <mml:mi>Ω</mml:mi>\r\n
    \                 </mml:msub>\r\n                  <mml:msub>\r\n                    <mml:mi>n</mml:mi>\r\n
    \                   <mml:mn>0</mml:mn>\r\n                  </mml:msub>\r\n                  <mml:mo>&lt;</mml:mo>\r\n
    \                 <mml:mn>4</mml:mn>\r\n                  <mml:mi>π</mml:mi>\r\n
    \               </mml:math></jats:alternatives></jats:inline-formula>, this problem
    admits a global classical solution with <jats:inline-formula><jats:alternatives><jats:tex-math>$n(\\cdot
    ,0)=n_{0}$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n
    \                 <mml:mi>n</mml:mi>\r\n                  <mml:mo>(</mml:mo>\r\n
    \                 <mml:mo>⋅</mml:mo>\r\n                  <mml:mo>,</mml:mo>\r\n
    \                 <mml:mn>0</mml:mn>\r\n                  <mml:mo>)</mml:mo>\r\n
    \                 <mml:mo>=</mml:mo>\r\n                  <mml:msub>\r\n                    <mml:mi>n</mml:mi>\r\n
    \                   <mml:mn>0</mml:mn>\r\n                  </mml:msub>\r\n                </mml:math></jats:alternatives></jats:inline-formula>,
    and that this solution is even bounded whenever <jats:inline-formula><jats:alternatives><jats:tex-math>$u$</jats:tex-math><mml:math
    xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n                  <mml:mi>u</mml:mi>\r\n
    \               </mml:math></jats:alternatives></jats:inline-formula> is bounded
    and <jats:inline-formula><jats:alternatives><jats:tex-math>$\\int _{\\varOmega}n_{0}&lt;2\\pi
    $</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n
    \                 <mml:msub>\r\n                    <mml:mo>∫</mml:mo>\r\n                    <mml:mi>Ω</mml:mi>\r\n
    \                 </mml:msub>\r\n                  <mml:msub>\r\n                    <mml:mi>n</mml:mi>\r\n
    \                   <mml:mn>0</mml:mn>\r\n                  </mml:msub>\r\n                  <mml:mo>&lt;</mml:mo>\r\n
    \                 <mml:mn>2</mml:mn>\r\n                  <mml:mi>π</mml:mi>\r\n
    \               </mml:math></jats:alternatives></jats:inline-formula>. Secondly,
    it is seen that for each <jats:inline-formula><jats:alternatives><jats:tex-math>$m&gt;4\\pi
    $</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n
    \                 <mml:mi>m</mml:mi>\r\n                  <mml:mo>&gt;</mml:mo>\r\n
    \                 <mml:mn>4</mml:mn>\r\n                  <mml:mi>π</mml:mi>\r\n
    \               </mml:math></jats:alternatives></jats:inline-formula> one can
    find a classical solution with <jats:inline-formula><jats:alternatives><jats:tex-math>$\\int
    _{\\varOmega}n(\\cdot ,0)=m$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n
    \                 <mml:msub>\r\n                    <mml:mo>∫</mml:mo>\r\n                    <mml:mi>Ω</mml:mi>\r\n
    \                 </mml:msub>\r\n                  <mml:mi>n</mml:mi>\r\n                  <mml:mo>(</mml:mo>\r\n
    \                 <mml:mo>⋅</mml:mo>\r\n                  <mml:mo>,</mml:mo>\r\n
    \                 <mml:mn>0</mml:mn>\r\n                  <mml:mo>)</mml:mo>\r\n
    \                 <mml:mo>=</mml:mo>\r\n                  <mml:mi>m</mml:mi>\r\n
    \               </mml:math></jats:alternatives></jats:inline-formula> which blows
    up in finite time, provided that <jats:inline-formula><jats:alternatives><jats:tex-math>$\\varOmega
    $</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n
    \                 <mml:mi>Ω</mml:mi>\r\n                </mml:math></jats:alternatives></jats:inline-formula>
    satisfies a technical assumption requiring <jats:inline-formula><jats:alternatives><jats:tex-math>$\\partial
    \\varOmega $</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n
    \                 <mml:mi>∂</mml:mi>\r\n                  <mml:mi>Ω</mml:mi>\r\n
    \               </mml:math></jats:alternatives></jats:inline-formula> to contain
    a line segment.</jats:p><jats:p>In particular, this indicates that the value <jats:inline-formula><jats:alternatives><jats:tex-math>$4\\pi
    $</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n
    \                 <mml:mn>4</mml:mn>\r\n                  <mml:mi>π</mml:mi>\r\n
    \               </mml:math></jats:alternatives></jats:inline-formula> of the critical
    mass for the corresponding fluid-free Keller-Segel system is left unchanged by
    any fluid interaction of the considered type, thus marking a considerable contrast
    to a recent result revealing some fluid-induced increase of critical blow-up masses
    in a related Cauchy problem in the entire plane.</jats:p>"
author:
- first_name: Michael
  full_name: Winkler, Michael
  id: '31496'
  last_name: Winkler
citation:
  ama: Winkler M. Can Fluid Interaction Influence the Critical Mass for Taxis-Driven
    Blow-up in Bounded Planar Domains? <i>Acta Applicandae Mathematicae</i>. 2020;169(1):577-591.
    doi:<a href="https://doi.org/10.1007/s10440-020-00312-2">10.1007/s10440-020-00312-2</a>
  apa: Winkler, M. (2020). Can Fluid Interaction Influence the Critical Mass for Taxis-Driven
    Blow-up in Bounded Planar Domains? <i>Acta Applicandae Mathematicae</i>, <i>169</i>(1),
    577–591. <a href="https://doi.org/10.1007/s10440-020-00312-2">https://doi.org/10.1007/s10440-020-00312-2</a>
  bibtex: '@article{Winkler_2020, title={Can Fluid Interaction Influence the Critical
    Mass for Taxis-Driven Blow-up in Bounded Planar Domains?}, volume={169}, DOI={<a
    href="https://doi.org/10.1007/s10440-020-00312-2">10.1007/s10440-020-00312-2</a>},
    number={1}, journal={Acta Applicandae Mathematicae}, publisher={Springer Science
    and Business Media LLC}, author={Winkler, Michael}, year={2020}, pages={577–591}
    }'
  chicago: 'Winkler, Michael. “Can Fluid Interaction Influence the Critical Mass for
    Taxis-Driven Blow-up in Bounded Planar Domains?” <i>Acta Applicandae Mathematicae</i>
    169, no. 1 (2020): 577–91. <a href="https://doi.org/10.1007/s10440-020-00312-2">https://doi.org/10.1007/s10440-020-00312-2</a>.'
  ieee: 'M. Winkler, “Can Fluid Interaction Influence the Critical Mass for Taxis-Driven
    Blow-up in Bounded Planar Domains?,” <i>Acta Applicandae Mathematicae</i>, vol.
    169, no. 1, pp. 577–591, 2020, doi: <a href="https://doi.org/10.1007/s10440-020-00312-2">10.1007/s10440-020-00312-2</a>.'
  mla: Winkler, Michael. “Can Fluid Interaction Influence the Critical Mass for Taxis-Driven
    Blow-up in Bounded Planar Domains?” <i>Acta Applicandae Mathematicae</i>, vol.
    169, no. 1, Springer Science and Business Media LLC, 2020, pp. 577–91, doi:<a
    href="https://doi.org/10.1007/s10440-020-00312-2">10.1007/s10440-020-00312-2</a>.
  short: M. Winkler, Acta Applicandae Mathematicae 169 (2020) 577–591.
date_created: 2025-12-18T19:47:51Z
date_updated: 2025-12-18T19:57:40Z
doi: 10.1007/s10440-020-00312-2
intvolume: '       169'
issue: '1'
language:
- iso: eng
page: 577-591
publication: Acta Applicandae Mathematicae
publication_identifier:
  issn:
  - 0167-8019
  - 1572-9036
publication_status: published
publisher: Springer Science and Business Media LLC
status: public
title: Can Fluid Interaction Influence the Critical Mass for Taxis-Driven Blow-up
  in Bounded Planar Domains?
type: journal_article
user_id: '31496'
volume: 169
year: '2020'
...
---
_id: '63336'
author:
- first_name: Michael
  full_name: Winkler, Michael
  id: '31496'
  last_name: Winkler
citation:
  ama: Winkler M. Blow-up profiles and life beyond blow-up in the fully parabolic
    Keller-Segel system. <i>Journal d’Analyse Mathématique</i>. 2020;141(2):585-624.
    doi:<a href="https://doi.org/10.1007/s11854-020-0109-4">10.1007/s11854-020-0109-4</a>
  apa: Winkler, M. (2020). Blow-up profiles and life beyond blow-up in the fully parabolic
    Keller-Segel system. <i>Journal d’Analyse Mathématique</i>, <i>141</i>(2), 585–624.
    <a href="https://doi.org/10.1007/s11854-020-0109-4">https://doi.org/10.1007/s11854-020-0109-4</a>
  bibtex: '@article{Winkler_2020, title={Blow-up profiles and life beyond blow-up
    in the fully parabolic Keller-Segel system}, volume={141}, DOI={<a href="https://doi.org/10.1007/s11854-020-0109-4">10.1007/s11854-020-0109-4</a>},
    number={2}, journal={Journal d’Analyse Mathématique}, publisher={Springer Science
    and Business Media LLC}, author={Winkler, Michael}, year={2020}, pages={585–624}
    }'
  chicago: 'Winkler, Michael. “Blow-up Profiles and Life beyond Blow-up in the Fully
    Parabolic Keller-Segel System.” <i>Journal d’Analyse Mathématique</i> 141, no.
    2 (2020): 585–624. <a href="https://doi.org/10.1007/s11854-020-0109-4">https://doi.org/10.1007/s11854-020-0109-4</a>.'
  ieee: 'M. Winkler, “Blow-up profiles and life beyond blow-up in the fully parabolic
    Keller-Segel system,” <i>Journal d’Analyse Mathématique</i>, vol. 141, no. 2,
    pp. 585–624, 2020, doi: <a href="https://doi.org/10.1007/s11854-020-0109-4">10.1007/s11854-020-0109-4</a>.'
  mla: Winkler, Michael. “Blow-up Profiles and Life beyond Blow-up in the Fully Parabolic
    Keller-Segel System.” <i>Journal d’Analyse Mathématique</i>, vol. 141, no. 2,
    Springer Science and Business Media LLC, 2020, pp. 585–624, doi:<a href="https://doi.org/10.1007/s11854-020-0109-4">10.1007/s11854-020-0109-4</a>.
  short: M. Winkler, Journal d’Analyse Mathématique 141 (2020) 585–624.
date_created: 2025-12-18T19:44:38Z
date_updated: 2025-12-18T19:56:40Z
doi: 10.1007/s11854-020-0109-4
intvolume: '       141'
issue: '2'
language:
- iso: eng
page: 585-624
publication: Journal d'Analyse Mathématique
publication_identifier:
  issn:
  - 0021-7670
  - 1565-8538
publication_status: published
publisher: Springer Science and Business Media LLC
status: public
title: Blow-up profiles and life beyond blow-up in the fully parabolic Keller-Segel
  system
type: journal_article
user_id: '31496'
volume: 141
year: '2020'
...
---
_id: '63329'
abstract:
- lang: eng
  text: "<jats:title>Abstract</jats:title><jats:p>Recent experimental work has revealed
    that interstitial fluid flow can mobilize two types of tumor cell migration mechanisms.
    One is a chemotactic-driven mechanism where chemokine (chemical component) bounded
    to the extracellular matrix (ECM) is released and skewed in the flow direction.
    This leads to higher chemical concentrations downstream which the tumor cells
    can sense and migrate toward. The other is a mechanism where the flowing fluid
    imposes a stress on the tumor cells which triggers them to go in the upstream
    direction. Researchers have suggested that these two migration modes possibly
    can play a role in metastatic behavior, i.e., the process where tumor cells are
    able to break loose from the primary tumor and move to nearby lymphatic vessels.
    In Waldeland and Evje (J Biomech 81:22–35, 2018), a mathematical cell–fluid model
    was put forward based on a mixture theory formulation. It was demonstrated that
    the model was able to capture the main characteristics of the two competing migration
    mechanisms. The objective of the current work is to seek deeper insight into certain
    qualitative aspects of these competing mechanisms by means of mathematical methods.
    For that purpose, we propose a simpler version of the cell–fluid model mentioned
    above but such that the two competing migration mechanisms are retained. An initial
    cell distribution in a one-dimensional slab is exposed to a constant fluid flow
    from one end to the other, consistent with the experimental setup. Then, we explore
    by means of analytical estimates the long-time behavior of the two competing migration
    mechanisms for two different scenarios: (i) when the initial cell volume fraction
    is low and (ii) when the initial cell volume fraction is high. In particular,
    it is demonstrated in a strict mathematical sense that for a sufficiently low
    initial cell volume fraction, the downstream migration dominates in the sense
    that the solution converges to a downstream-dominated steady state as time elapses.
    On the other hand, with a sufficiently high initial cell volume fraction, the
    upstream migration mechanism is the stronger in the sense that the solution converges
    to an upstream-dominated steady state.\r\n</jats:p>"
author:
- first_name: Steinar
  full_name: Evje, Steinar
  last_name: Evje
- first_name: Michael
  full_name: Winkler, Michael
  id: '31496'
  last_name: Winkler
citation:
  ama: Evje S, Winkler M. Mathematical Analysis of Two Competing Cancer Cell Migration
    Mechanisms Driven by Interstitial Fluid Flow. <i>Journal of Nonlinear Science</i>.
    2020;30(4):1809-1847. doi:<a href="https://doi.org/10.1007/s00332-020-09625-w">10.1007/s00332-020-09625-w</a>
  apa: Evje, S., &#38; Winkler, M. (2020). Mathematical Analysis of Two Competing
    Cancer Cell Migration Mechanisms Driven by Interstitial Fluid Flow. <i>Journal
    of Nonlinear Science</i>, <i>30</i>(4), 1809–1847. <a href="https://doi.org/10.1007/s00332-020-09625-w">https://doi.org/10.1007/s00332-020-09625-w</a>
  bibtex: '@article{Evje_Winkler_2020, title={Mathematical Analysis of Two Competing
    Cancer Cell Migration Mechanisms Driven by Interstitial Fluid Flow}, volume={30},
    DOI={<a href="https://doi.org/10.1007/s00332-020-09625-w">10.1007/s00332-020-09625-w</a>},
    number={4}, journal={Journal of Nonlinear Science}, publisher={Springer Science
    and Business Media LLC}, author={Evje, Steinar and Winkler, Michael}, year={2020},
    pages={1809–1847} }'
  chicago: 'Evje, Steinar, and Michael Winkler. “Mathematical Analysis of Two Competing
    Cancer Cell Migration Mechanisms Driven by Interstitial Fluid Flow.” <i>Journal
    of Nonlinear Science</i> 30, no. 4 (2020): 1809–47. <a href="https://doi.org/10.1007/s00332-020-09625-w">https://doi.org/10.1007/s00332-020-09625-w</a>.'
  ieee: 'S. Evje and M. Winkler, “Mathematical Analysis of Two Competing Cancer Cell
    Migration Mechanisms Driven by Interstitial Fluid Flow,” <i>Journal of Nonlinear
    Science</i>, vol. 30, no. 4, pp. 1809–1847, 2020, doi: <a href="https://doi.org/10.1007/s00332-020-09625-w">10.1007/s00332-020-09625-w</a>.'
  mla: Evje, Steinar, and Michael Winkler. “Mathematical Analysis of Two Competing
    Cancer Cell Migration Mechanisms Driven by Interstitial Fluid Flow.” <i>Journal
    of Nonlinear Science</i>, vol. 30, no. 4, Springer Science and Business Media
    LLC, 2020, pp. 1809–47, doi:<a href="https://doi.org/10.1007/s00332-020-09625-w">10.1007/s00332-020-09625-w</a>.
  short: S. Evje, M. Winkler, Journal of Nonlinear Science 30 (2020) 1809–1847.
date_created: 2025-12-18T19:38:02Z
date_updated: 2025-12-18T20:00:26Z
doi: 10.1007/s00332-020-09625-w
intvolume: '        30'
issue: '4'
language:
- iso: eng
page: 1809-1847
publication: Journal of Nonlinear Science
publication_identifier:
  issn:
  - 0938-8974
  - 1432-1467
publication_status: published
publisher: Springer Science and Business Media LLC
status: public
title: Mathematical Analysis of Two Competing Cancer Cell Migration Mechanisms Driven
  by Interstitial Fluid Flow
type: journal_article
user_id: '31496'
volume: 30
year: '2020'
...
---
_id: '63331'
abstract:
- lang: eng
  text: <jats:p> We consider a class of macroscopic models for the spatio-temporal
    evolution of urban crime, as originally going back to Ref. 29 [M. B. Short, M.
    R. D’Orsogna, V. B. Pasour, G. E. Tita, P. J. Brantingham, A. L. Bertozzi and
    L. B. Chayes, A statistical model of criminal behavior, Math. Models Methods Appl.
    Sci. 18 (2008) 1249–1267]. The focus here is on the question of how far a certain
    porous medium enhancement in the random diffusion of criminal agents may exert
    visible relaxation effects. It is shown that sufficient regularity of the non-negative
    source terms in the system and a sufficiently strong nonlinear enhancement ensure
    that a corresponding Neumann-type initial–boundary value problem, posed in a smoothly
    bounded planar convex domain, admits locally bounded solutions for a wide class
    of arbitrary initial data. Furthermore, this solution is globally bounded under
    mild additional conditions on the source terms. These results are supplemented
    by numerical evidence which illustrates smoothing effects in solutions with sharply
    structured initial data in the presence of such porous medium-type diffusion and
    support the existence of singular structures in the linear diffusion case, which
    is the type of diffusion proposed in Ref. 29. </jats:p>
author:
- first_name: Nancy
  full_name: Rodríguez, Nancy
  last_name: Rodríguez
- first_name: Michael
  full_name: Winkler, Michael
  id: '31496'
  last_name: Winkler
citation:
  ama: Rodríguez N, Winkler M. Relaxation by nonlinear diffusion enhancement in a
    two-dimensional cross-diffusion model for urban crime propagation. <i>Mathematical
    Models and Methods in Applied Sciences</i>. 2020;30(11):2105-2137. doi:<a href="https://doi.org/10.1142/s0218202520500396">10.1142/s0218202520500396</a>
  apa: Rodríguez, N., &#38; Winkler, M. (2020). Relaxation by nonlinear diffusion
    enhancement in a two-dimensional cross-diffusion model for urban crime propagation.
    <i>Mathematical Models and Methods in Applied Sciences</i>, <i>30</i>(11), 2105–2137.
    <a href="https://doi.org/10.1142/s0218202520500396">https://doi.org/10.1142/s0218202520500396</a>
  bibtex: '@article{Rodríguez_Winkler_2020, title={Relaxation by nonlinear diffusion
    enhancement in a two-dimensional cross-diffusion model for urban crime propagation},
    volume={30}, DOI={<a href="https://doi.org/10.1142/s0218202520500396">10.1142/s0218202520500396</a>},
    number={11}, journal={Mathematical Models and Methods in Applied Sciences}, publisher={World
    Scientific Pub Co Pte Ltd}, author={Rodríguez, Nancy and Winkler, Michael}, year={2020},
    pages={2105–2137} }'
  chicago: 'Rodríguez, Nancy, and Michael Winkler. “Relaxation by Nonlinear Diffusion
    Enhancement in a Two-Dimensional Cross-Diffusion Model for Urban Crime Propagation.”
    <i>Mathematical Models and Methods in Applied Sciences</i> 30, no. 11 (2020):
    2105–37. <a href="https://doi.org/10.1142/s0218202520500396">https://doi.org/10.1142/s0218202520500396</a>.'
  ieee: 'N. Rodríguez and M. Winkler, “Relaxation by nonlinear diffusion enhancement
    in a two-dimensional cross-diffusion model for urban crime propagation,” <i>Mathematical
    Models and Methods in Applied Sciences</i>, vol. 30, no. 11, pp. 2105–2137, 2020,
    doi: <a href="https://doi.org/10.1142/s0218202520500396">10.1142/s0218202520500396</a>.'
  mla: Rodríguez, Nancy, and Michael Winkler. “Relaxation by Nonlinear Diffusion Enhancement
    in a Two-Dimensional Cross-Diffusion Model for Urban Crime Propagation.” <i>Mathematical
    Models and Methods in Applied Sciences</i>, vol. 30, no. 11, World Scientific
    Pub Co Pte Ltd, 2020, pp. 2105–37, doi:<a href="https://doi.org/10.1142/s0218202520500396">10.1142/s0218202520500396</a>.
  short: N. Rodríguez, M. Winkler, Mathematical Models and Methods in Applied Sciences
    30 (2020) 2105–2137.
date_created: 2025-12-18T19:38:42Z
date_updated: 2025-12-18T20:00:53Z
doi: 10.1142/s0218202520500396
intvolume: '        30'
issue: '11'
language:
- iso: eng
page: 2105-2137
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: Relaxation by nonlinear diffusion enhancement in a two-dimensional cross-diffusion
  model for urban crime propagation
type: journal_article
user_id: '31496'
volume: 30
year: '2020'
...