---
_id: '63330'
author:
- first_name: Genglin
  full_name: Li, Genglin
  last_name: Li
- first_name: Youshan
  full_name: Tao, Youshan
  last_name: Tao
- first_name: Michael
  full_name: Winkler, Michael
  id: '31496'
  last_name: Winkler
citation:
  ama: Li G, Tao Y, Winkler M. Large time behavior in a predator-prey system with
    indirect pursuit-evasion interaction. <i>Discrete and Continuous Dynamical Systems
    - B</i>. 2020;25(11):4383-4396. doi:<a href="https://doi.org/10.3934/dcdsb.2020102">10.3934/dcdsb.2020102</a>
  apa: Li, G., Tao, Y., &#38; Winkler, M. (2020). Large time behavior in a predator-prey
    system with indirect pursuit-evasion interaction. <i>Discrete and Continuous Dynamical
    Systems - B</i>, <i>25</i>(11), 4383–4396. <a href="https://doi.org/10.3934/dcdsb.2020102">https://doi.org/10.3934/dcdsb.2020102</a>
  bibtex: '@article{Li_Tao_Winkler_2020, title={Large time behavior in a predator-prey
    system with indirect pursuit-evasion interaction}, volume={25}, DOI={<a href="https://doi.org/10.3934/dcdsb.2020102">10.3934/dcdsb.2020102</a>},
    number={11}, journal={Discrete and Continuous Dynamical Systems - B}, publisher={American
    Institute of Mathematical Sciences (AIMS)}, author={Li, Genglin and Tao, Youshan
    and Winkler, Michael}, year={2020}, pages={4383–4396} }'
  chicago: 'Li, Genglin, Youshan Tao, and Michael Winkler. “Large Time Behavior in
    a Predator-Prey System with Indirect Pursuit-Evasion Interaction.” <i>Discrete
    and Continuous Dynamical Systems - B</i> 25, no. 11 (2020): 4383–96. <a href="https://doi.org/10.3934/dcdsb.2020102">https://doi.org/10.3934/dcdsb.2020102</a>.'
  ieee: 'G. Li, Y. Tao, and M. Winkler, “Large time behavior in a predator-prey system
    with indirect pursuit-evasion interaction,” <i>Discrete and Continuous Dynamical
    Systems - B</i>, vol. 25, no. 11, pp. 4383–4396, 2020, doi: <a href="https://doi.org/10.3934/dcdsb.2020102">10.3934/dcdsb.2020102</a>.'
  mla: Li, Genglin, et al. “Large Time Behavior in a Predator-Prey System with Indirect
    Pursuit-Evasion Interaction.” <i>Discrete and Continuous Dynamical Systems - B</i>,
    vol. 25, no. 11, American Institute of Mathematical Sciences (AIMS), 2020, pp.
    4383–96, doi:<a href="https://doi.org/10.3934/dcdsb.2020102">10.3934/dcdsb.2020102</a>.
  short: G. Li, Y. Tao, M. Winkler, Discrete and Continuous Dynamical Systems - B
    25 (2020) 4383–4396.
date_created: 2025-12-18T19:38:22Z
date_updated: 2025-12-18T20:00:40Z
doi: 10.3934/dcdsb.2020102
intvolume: '        25'
issue: '11'
language:
- iso: eng
page: 4383-4396
publication: Discrete and Continuous Dynamical Systems - B
publication_identifier:
  issn:
  - 1531-3492
  - 1553-524X
publication_status: published
publisher: American Institute of Mathematical Sciences (AIMS)
status: public
title: Large time behavior in a predator-prey system with indirect pursuit-evasion
  interaction
type: journal_article
user_id: '31496'
volume: 25
year: '2020'
...
---
_id: '63327'
article_number: '103257'
author:
- first_name: Michael
  full_name: Winkler, Michael
  id: '31496'
  last_name: Winkler
citation:
  ama: 'Winkler M. Global weak solutions in a three-dimensional Keller–Segel–Navier–Stokes
    system with gradient-dependent flux limitation. <i>Nonlinear Analysis: Real World
    Applications</i>. 2020;59. doi:<a href="https://doi.org/10.1016/j.nonrwa.2020.103257">10.1016/j.nonrwa.2020.103257</a>'
  apa: 'Winkler, M. (2020). Global weak solutions in a three-dimensional Keller–Segel–Navier–Stokes
    system with gradient-dependent flux limitation. <i>Nonlinear Analysis: Real World
    Applications</i>, <i>59</i>, Article 103257. <a href="https://doi.org/10.1016/j.nonrwa.2020.103257">https://doi.org/10.1016/j.nonrwa.2020.103257</a>'
  bibtex: '@article{Winkler_2020, title={Global weak solutions in a three-dimensional
    Keller–Segel–Navier–Stokes system with gradient-dependent flux limitation}, volume={59},
    DOI={<a href="https://doi.org/10.1016/j.nonrwa.2020.103257">10.1016/j.nonrwa.2020.103257</a>},
    number={103257}, journal={Nonlinear Analysis: Real World Applications}, publisher={Elsevier
    BV}, author={Winkler, Michael}, year={2020} }'
  chicago: 'Winkler, Michael. “Global Weak Solutions in a Three-Dimensional Keller–Segel–Navier–Stokes
    System with Gradient-Dependent Flux Limitation.” <i>Nonlinear Analysis: Real World
    Applications</i> 59 (2020). <a href="https://doi.org/10.1016/j.nonrwa.2020.103257">https://doi.org/10.1016/j.nonrwa.2020.103257</a>.'
  ieee: 'M. Winkler, “Global weak solutions in a three-dimensional Keller–Segel–Navier–Stokes
    system with gradient-dependent flux limitation,” <i>Nonlinear Analysis: Real World
    Applications</i>, vol. 59, Art. no. 103257, 2020, doi: <a href="https://doi.org/10.1016/j.nonrwa.2020.103257">10.1016/j.nonrwa.2020.103257</a>.'
  mla: 'Winkler, Michael. “Global Weak Solutions in a Three-Dimensional Keller–Segel–Navier–Stokes
    System with Gradient-Dependent Flux Limitation.” <i>Nonlinear Analysis: Real World
    Applications</i>, vol. 59, 103257, Elsevier BV, 2020, doi:<a href="https://doi.org/10.1016/j.nonrwa.2020.103257">10.1016/j.nonrwa.2020.103257</a>.'
  short: 'M. Winkler, Nonlinear Analysis: Real World Applications 59 (2020).'
date_created: 2025-12-18T19:36:51Z
date_updated: 2025-12-18T19:59:57Z
doi: 10.1016/j.nonrwa.2020.103257
intvolume: '        59'
language:
- iso: eng
publication: 'Nonlinear Analysis: Real World Applications'
publication_identifier:
  issn:
  - 1468-1218
publication_status: published
publisher: Elsevier BV
status: public
title: Global weak solutions in a three-dimensional Keller–Segel–Navier–Stokes system
  with gradient-dependent flux limitation
type: journal_article
user_id: '31496'
volume: 59
year: '2020'
...
---
_id: '63333'
article_number: '111870'
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. A critical virus production rate for blow-up suppression
    in a haptotaxis model for oncolytic virotherapy. <i>Nonlinear Analysis</i>. 2020;198.
    doi:<a href="https://doi.org/10.1016/j.na.2020.111870">10.1016/j.na.2020.111870</a>
  apa: Tao, Y., &#38; Winkler, M. (2020). A critical virus production rate for blow-up
    suppression in a haptotaxis model for oncolytic virotherapy. <i>Nonlinear Analysis</i>,
    <i>198</i>, Article 111870. <a href="https://doi.org/10.1016/j.na.2020.111870">https://doi.org/10.1016/j.na.2020.111870</a>
  bibtex: '@article{Tao_Winkler_2020, title={A critical virus production rate for
    blow-up suppression in a haptotaxis model for oncolytic virotherapy}, volume={198},
    DOI={<a href="https://doi.org/10.1016/j.na.2020.111870">10.1016/j.na.2020.111870</a>},
    number={111870}, journal={Nonlinear Analysis}, publisher={Elsevier BV}, author={Tao,
    Youshan and Winkler, Michael}, year={2020} }'
  chicago: Tao, Youshan, and Michael Winkler. “A Critical Virus Production Rate for
    Blow-up Suppression in a Haptotaxis Model for Oncolytic Virotherapy.” <i>Nonlinear
    Analysis</i> 198 (2020). <a href="https://doi.org/10.1016/j.na.2020.111870">https://doi.org/10.1016/j.na.2020.111870</a>.
  ieee: 'Y. Tao and M. Winkler, “A critical virus production rate for blow-up suppression
    in a haptotaxis model for oncolytic virotherapy,” <i>Nonlinear Analysis</i>, vol.
    198, Art. no. 111870, 2020, doi: <a href="https://doi.org/10.1016/j.na.2020.111870">10.1016/j.na.2020.111870</a>.'
  mla: Tao, Youshan, and Michael Winkler. “A Critical Virus Production Rate for Blow-up
    Suppression in a Haptotaxis Model for Oncolytic Virotherapy.” <i>Nonlinear Analysis</i>,
    vol. 198, 111870, Elsevier BV, 2020, doi:<a href="https://doi.org/10.1016/j.na.2020.111870">10.1016/j.na.2020.111870</a>.
  short: Y. Tao, M. Winkler, Nonlinear Analysis 198 (2020).
date_created: 2025-12-18T19:39:40Z
date_updated: 2025-12-18T20:01:18Z
doi: 10.1016/j.na.2020.111870
intvolume: '       198'
language:
- iso: eng
publication: Nonlinear Analysis
publication_identifier:
  issn:
  - 0362-546X
publication_status: published
publisher: Elsevier BV
status: public
title: A critical virus production rate for blow-up suppression in a haptotaxis model
  for oncolytic virotherapy
type: journal_article
user_id: '31496'
volume: 198
year: '2020'
...
---
_id: '63328'
article_number: '106785'
author:
- first_name: Michael
  full_name: Winkler, Michael
  id: '31496'
  last_name: Winkler
citation:
  ama: Winkler M. Boundedness in a three-dimensional Keller–Segel–Stokes system with
    subcritical sensitivity. <i>Applied Mathematics Letters</i>. 2020;112. doi:<a
    href="https://doi.org/10.1016/j.aml.2020.106785">10.1016/j.aml.2020.106785</a>
  apa: Winkler, M. (2020). Boundedness in a three-dimensional Keller–Segel–Stokes
    system with subcritical sensitivity. <i>Applied Mathematics Letters</i>, <i>112</i>,
    Article 106785. <a href="https://doi.org/10.1016/j.aml.2020.106785">https://doi.org/10.1016/j.aml.2020.106785</a>
  bibtex: '@article{Winkler_2020, title={Boundedness in a three-dimensional Keller–Segel–Stokes
    system with subcritical sensitivity}, volume={112}, DOI={<a href="https://doi.org/10.1016/j.aml.2020.106785">10.1016/j.aml.2020.106785</a>},
    number={106785}, journal={Applied Mathematics Letters}, publisher={Elsevier BV},
    author={Winkler, Michael}, year={2020} }'
  chicago: Winkler, Michael. “Boundedness in a Three-Dimensional Keller–Segel–Stokes
    System with Subcritical Sensitivity.” <i>Applied Mathematics Letters</i> 112 (2020).
    <a href="https://doi.org/10.1016/j.aml.2020.106785">https://doi.org/10.1016/j.aml.2020.106785</a>.
  ieee: 'M. Winkler, “Boundedness in a three-dimensional Keller–Segel–Stokes system
    with subcritical sensitivity,” <i>Applied Mathematics Letters</i>, vol. 112, Art.
    no. 106785, 2020, doi: <a href="https://doi.org/10.1016/j.aml.2020.106785">10.1016/j.aml.2020.106785</a>.'
  mla: Winkler, Michael. “Boundedness in a Three-Dimensional Keller–Segel–Stokes System
    with Subcritical Sensitivity.” <i>Applied Mathematics Letters</i>, vol. 112, 106785,
    Elsevier BV, 2020, doi:<a href="https://doi.org/10.1016/j.aml.2020.106785">10.1016/j.aml.2020.106785</a>.
  short: M. Winkler, Applied Mathematics Letters 112 (2020).
date_created: 2025-12-18T19:37:32Z
date_updated: 2025-12-18T20:00:10Z
doi: 10.1016/j.aml.2020.106785
intvolume: '       112'
language:
- iso: eng
publication: Applied Mathematics Letters
publication_identifier:
  issn:
  - 0893-9659
publication_status: published
publisher: Elsevier BV
status: public
title: Boundedness in a three-dimensional Keller–Segel–Stokes system with subcritical
  sensitivity
type: journal_article
user_id: '31496'
volume: 112
year: '2020'
...
---
_id: '63320'
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. Critical mass for infinite-time blow-up in a haptotaxis system
    with nonlinear zero-order interaction. <i>Discrete &#38;amp; Continuous Dynamical
    Systems - A</i>. 2020;41(1):439-454. doi:<a href="https://doi.org/10.3934/dcds.2020216">10.3934/dcds.2020216</a>
  apa: Tao, Y., &#38; Winkler, M. (2020). Critical mass for infinite-time blow-up
    in a haptotaxis system with nonlinear zero-order interaction. <i>Discrete &#38;amp;
    Continuous Dynamical Systems - A</i>, <i>41</i>(1), 439–454. <a href="https://doi.org/10.3934/dcds.2020216">https://doi.org/10.3934/dcds.2020216</a>
  bibtex: '@article{Tao_Winkler_2020, title={Critical mass for infinite-time blow-up
    in a haptotaxis system with nonlinear zero-order interaction}, volume={41}, DOI={<a
    href="https://doi.org/10.3934/dcds.2020216">10.3934/dcds.2020216</a>}, number={1},
    journal={Discrete &#38;amp; Continuous Dynamical Systems - A}, publisher={American
    Institute of Mathematical Sciences (AIMS)}, author={Tao, Youshan and Winkler,
    Michael}, year={2020}, pages={439–454} }'
  chicago: 'Tao, Youshan, and Michael Winkler. “Critical Mass for Infinite-Time Blow-up
    in a Haptotaxis System with Nonlinear Zero-Order Interaction.” <i>Discrete &#38;amp;
    Continuous Dynamical Systems - A</i> 41, no. 1 (2020): 439–54. <a href="https://doi.org/10.3934/dcds.2020216">https://doi.org/10.3934/dcds.2020216</a>.'
  ieee: 'Y. Tao and M. Winkler, “Critical mass for infinite-time blow-up in a haptotaxis
    system with nonlinear zero-order interaction,” <i>Discrete &#38;amp; Continuous
    Dynamical Systems - A</i>, vol. 41, no. 1, pp. 439–454, 2020, doi: <a href="https://doi.org/10.3934/dcds.2020216">10.3934/dcds.2020216</a>.'
  mla: Tao, Youshan, and Michael Winkler. “Critical Mass for Infinite-Time Blow-up
    in a Haptotaxis System with Nonlinear Zero-Order Interaction.” <i>Discrete &#38;amp;
    Continuous Dynamical Systems - A</i>, vol. 41, no. 1, American Institute of Mathematical
    Sciences (AIMS), 2020, pp. 439–54, doi:<a href="https://doi.org/10.3934/dcds.2020216">10.3934/dcds.2020216</a>.
  short: Y. Tao, M. Winkler, Discrete &#38;amp; Continuous Dynamical Systems - A 41
    (2020) 439–454.
date_created: 2025-12-18T19:33:59Z
date_updated: 2025-12-18T20:04:09Z
doi: 10.3934/dcds.2020216
intvolume: '        41'
issue: '1'
language:
- iso: eng
page: 439-454
publication: Discrete &amp; Continuous Dynamical Systems - A
publication_identifier:
  issn:
  - 1553-5231
publication_status: published
publisher: American Institute of Mathematical Sciences (AIMS)
status: public
title: Critical mass for infinite-time blow-up in a haptotaxis system with nonlinear
  zero-order interaction
type: journal_article
user_id: '31496'
volume: 41
year: '2020'
...
---
_id: '63335'
author:
- first_name: Michael
  full_name: Winkler, Michael
  id: '31496'
  last_name: Winkler
citation:
  ama: Winkler M. Small-Mass Solutions in the Two-Dimensional Keller--Segel System
    Coupled to the Navier--Stokes Equations. <i>SIAM Journal on Mathematical Analysis</i>.
    2020;52(2):2041-2080. doi:<a href="https://doi.org/10.1137/19m1264199">10.1137/19m1264199</a>
  apa: Winkler, M. (2020). Small-Mass Solutions in the Two-Dimensional Keller--Segel
    System Coupled to the Navier--Stokes Equations. <i>SIAM Journal on Mathematical
    Analysis</i>, <i>52</i>(2), 2041–2080. <a href="https://doi.org/10.1137/19m1264199">https://doi.org/10.1137/19m1264199</a>
  bibtex: '@article{Winkler_2020, title={Small-Mass Solutions in the Two-Dimensional
    Keller--Segel System Coupled to the Navier--Stokes Equations}, volume={52}, DOI={<a
    href="https://doi.org/10.1137/19m1264199">10.1137/19m1264199</a>}, number={2},
    journal={SIAM Journal on Mathematical Analysis}, publisher={Society for Industrial
    &#38; Applied Mathematics (SIAM)}, author={Winkler, Michael}, year={2020}, pages={2041–2080}
    }'
  chicago: 'Winkler, Michael. “Small-Mass Solutions in the Two-Dimensional Keller--Segel
    System Coupled to the Navier--Stokes Equations.” <i>SIAM Journal on Mathematical
    Analysis</i> 52, no. 2 (2020): 2041–80. <a href="https://doi.org/10.1137/19m1264199">https://doi.org/10.1137/19m1264199</a>.'
  ieee: 'M. Winkler, “Small-Mass Solutions in the Two-Dimensional Keller--Segel System
    Coupled to the Navier--Stokes Equations,” <i>SIAM Journal on Mathematical Analysis</i>,
    vol. 52, no. 2, pp. 2041–2080, 2020, doi: <a href="https://doi.org/10.1137/19m1264199">10.1137/19m1264199</a>.'
  mla: Winkler, Michael. “Small-Mass Solutions in the Two-Dimensional Keller--Segel
    System Coupled to the Navier--Stokes Equations.” <i>SIAM Journal on Mathematical
    Analysis</i>, vol. 52, no. 2, Society for Industrial &#38; Applied Mathematics
    (SIAM), 2020, pp. 2041–80, doi:<a href="https://doi.org/10.1137/19m1264199">10.1137/19m1264199</a>.
  short: M. Winkler, SIAM Journal on Mathematical Analysis 52 (2020) 2041–2080.
date_created: 2025-12-18T19:40:35Z
date_updated: 2025-12-18T20:01:42Z
doi: 10.1137/19m1264199
intvolume: '        52'
issue: '2'
language:
- iso: eng
page: 2041-2080
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: Small-Mass Solutions in the Two-Dimensional Keller--Segel System Coupled to
  the Navier--Stokes Equations
type: journal_article
user_id: '31496'
volume: 52
year: '2020'
...
---
_id: '63318'
abstract:
- lang: eng
  text: <jats:p>In a planar smoothly bounded domain<jats:inline-formula><jats:alternatives><jats:inline-graphic
    xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0956792520000133_inline1.png"
    /><jats:tex-math>$\Omega$</jats:tex-math></jats:alternatives></jats:inline-formula>,
    we consider the model for oncolytic virotherapy given by<jats:disp-formula id="S0956792520000133_udisp1"><jats:alternatives><jats:graphic
    xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image"
    xlink:href="S0956792520000133_eqnu1.png" /><jats:tex-math>$$\left\{ \begin{array}{l}
    u_t = \Delta u - \nabla \cdot (u\nabla v) - uz, \\[1mm] v_t = - (u+w)v, \\[1mm]
    w_t = d_w \Delta w - w + uz, \\[1mm] z_t = d_z \Delta z - z - uz + \beta w, \end{array}
    \right.$$</jats:tex-math></jats:alternatives></jats:disp-formula>with positive
    parameters<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink"
    mime-subtype="png" xlink:href="S0956792520000133_inline2.png" /><jats:tex-math>$
    D_w $</jats:tex-math></jats:alternatives></jats:inline-formula>,<jats:inline-formula><jats:alternatives><jats:inline-graphic
    xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0956792520000133_inline3.png"
    /><jats:tex-math>$ D_z $</jats:tex-math></jats:alternatives></jats:inline-formula>and<jats:inline-formula><jats:alternatives><jats:inline-graphic
    xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0956792520000133_inline4.png"
    /><jats:tex-math>$\beta$</jats:tex-math></jats:alternatives></jats:inline-formula>.
    It is firstly shown that whenever<jats:inline-formula><jats:alternatives><jats:inline-graphic
    xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0956792520000133_inline5.png"
    /><jats:tex-math>$\beta \lt 1$</jats:tex-math></jats:alternatives></jats:inline-formula>,
    for any choice of<jats:inline-formula><jats:alternatives><jats:inline-graphic
    xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0956792520000133_inline6.png"
    /><jats:tex-math>$M \gt 0$</jats:tex-math></jats:alternatives></jats:inline-formula>,
    one can find initial data such that the solution of an associated no-flux initial-boundary
    value problem, well known to exist globally actually for any choice of<jats:inline-formula><jats:alternatives><jats:inline-graphic
    xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0956792520000133_inline7.png"
    /><jats:tex-math>$\beta \gt 0$</jats:tex-math></jats:alternatives></jats:inline-formula>,
    satisfies<jats:disp-formula id="S0956792520000133_udisp2"><jats:alternatives><jats:graphic
    xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image"
    xlink:href="S0956792520000133_eqnu2.png" /><jats:tex-math>$$u\ge M \qquad \mbox{in
    } \Omega\times (0,\infty).$$</jats:tex-math></jats:alternatives></jats:disp-formula>If<jats:inline-formula><jats:alternatives><jats:inline-graphic
    xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0956792520000133_inline8.png"
    /><jats:tex-math>$\beta \gt 1$</jats:tex-math></jats:alternatives></jats:inline-formula>,
    however, then for arbitrary initial data the corresponding is seen to have the
    property that<jats:disp-formula id="S0956792520000133_udisp3"><jats:alternatives><jats:graphic
    xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image"
    xlink:href="S0956792520000133_eqnu3.png" /><jats:tex-math>$$\liminf_{t\to\infty}
    \inf_{x\in\Omega} u(x,t)\le \frac{1}{\beta-1}.$$</jats:tex-math></jats:alternatives></jats:disp-formula>This
    may be interpreted as indicating that<jats:inline-formula><jats:alternatives><jats:inline-graphic
    xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0956792520000133_inline9.png"
    /><jats:tex-math>$\beta$</jats:tex-math></jats:alternatives></jats:inline-formula>plays
    the role of a critical virus replication rate with regard to efficiency of the
    considered virotherapy, with corresponding threshold value given by<jats:inline-formula><jats:alternatives><jats:inline-graphic
    xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0956792520000133_inline10.png"
    /><jats:tex-math>$\beta = 1$</jats:tex-math></jats:alternatives></jats:inline-formula>.</jats:p>
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. A critical virus production rate for efficiency of oncolytic
    virotherapy. <i>European Journal of Applied Mathematics</i>. 2020;32(2):301-316.
    doi:<a href="https://doi.org/10.1017/s0956792520000133">10.1017/s0956792520000133</a>
  apa: TAO, Y., &#38; Winkler, M. (2020). A critical virus production rate for efficiency
    of oncolytic virotherapy. <i>European Journal of Applied Mathematics</i>, <i>32</i>(2),
    301–316. <a href="https://doi.org/10.1017/s0956792520000133">https://doi.org/10.1017/s0956792520000133</a>
  bibtex: '@article{TAO_Winkler_2020, title={A critical virus production rate for
    efficiency of oncolytic virotherapy}, volume={32}, DOI={<a href="https://doi.org/10.1017/s0956792520000133">10.1017/s0956792520000133</a>},
    number={2}, journal={European Journal of Applied Mathematics}, publisher={Cambridge
    University Press (CUP)}, author={TAO, YOUSHAN and Winkler, Michael}, year={2020},
    pages={301–316} }'
  chicago: 'TAO, YOUSHAN, and Michael Winkler. “A Critical Virus Production Rate for
    Efficiency of Oncolytic Virotherapy.” <i>European Journal of Applied Mathematics</i>
    32, no. 2 (2020): 301–16. <a href="https://doi.org/10.1017/s0956792520000133">https://doi.org/10.1017/s0956792520000133</a>.'
  ieee: 'Y. TAO and M. Winkler, “A critical virus production rate for efficiency of
    oncolytic virotherapy,” <i>European Journal of Applied Mathematics</i>, vol. 32,
    no. 2, pp. 301–316, 2020, doi: <a href="https://doi.org/10.1017/s0956792520000133">10.1017/s0956792520000133</a>.'
  mla: TAO, YOUSHAN, and Michael Winkler. “A Critical Virus Production Rate for Efficiency
    of Oncolytic Virotherapy.” <i>European Journal of Applied Mathematics</i>, vol.
    32, no. 2, Cambridge University Press (CUP), 2020, pp. 301–16, doi:<a href="https://doi.org/10.1017/s0956792520000133">10.1017/s0956792520000133</a>.
  short: Y. TAO, M. Winkler, European Journal of Applied Mathematics 32 (2020) 301–316.
date_created: 2025-12-18T19:33:01Z
date_updated: 2025-12-18T20:06:35Z
doi: 10.1017/s0956792520000133
intvolume: '        32'
issue: '2'
language:
- iso: eng
page: 301-316
publication: European Journal of Applied Mathematics
publication_identifier:
  issn:
  - 0956-7925
  - 1469-4425
publication_status: published
publisher: Cambridge University Press (CUP)
status: public
title: A critical virus production rate for efficiency of oncolytic virotherapy
type: journal_article
user_id: '31496'
volume: 32
year: '2020'
...
---
_id: '63314'
abstract:
- lang: eng
  text: <jats:p>We propose and study a class of parabolic-ordinary differential equation
    models involving chemotaxis and haptotaxis of a species following signals indirectly
    produced by another, non-motile one. The setting is motivated by cancer invasion
    mediated by interactions with the tumour microenvironment, but has much wider
    applicability, being able to comprise descriptions of biologically quite different
    problems. As a main mathematical feature constituting a core difference to both
    classical Keller–Segel chemotaxis systems and Chaplain–Lolas type chemotaxis–haptotaxis
    systems, the considered model accounts for certain types of indirect signal production
    mechanisms. The main results assert unique global classical solvability under
    suitably mild assumptions on the system parameter functions in associated spatially
    two-dimensional initial-boundary value problems. In particular, this rigorously
    confirms that at least in two-dimensional settings, the considered indirectness
    in signal production induces a significant blow-up suppressing tendency also in
    taxis systems substantially more general than some particular examples for which
    corresponding effects have recently been observed.</jats:p>
author:
- first_name: CHRISTINA
  full_name: SURULESCU, CHRISTINA
  last_name: SURULESCU
- first_name: Michael
  full_name: Winkler, Michael
  id: '31496'
  last_name: Winkler
citation:
  ama: SURULESCU C, Winkler M. Does indirectness of signal production reduce the explosion-supporting
    potential in chemotaxis–haptotaxis systems? Global classical solvability in a
    class of models for cancer invasion (and more). <i>European Journal of Applied
    Mathematics</i>. 2020;32(4):618-651. doi:<a href="https://doi.org/10.1017/s0956792520000236">10.1017/s0956792520000236</a>
  apa: SURULESCU, C., &#38; Winkler, M. (2020). Does indirectness of signal production
    reduce the explosion-supporting potential in chemotaxis–haptotaxis systems? Global
    classical solvability in a class of models for cancer invasion (and more). <i>European
    Journal of Applied Mathematics</i>, <i>32</i>(4), 618–651. <a href="https://doi.org/10.1017/s0956792520000236">https://doi.org/10.1017/s0956792520000236</a>
  bibtex: '@article{SURULESCU_Winkler_2020, title={Does indirectness of signal production
    reduce the explosion-supporting potential in chemotaxis–haptotaxis systems? Global
    classical solvability in a class of models for cancer invasion (and more)}, volume={32},
    DOI={<a href="https://doi.org/10.1017/s0956792520000236">10.1017/s0956792520000236</a>},
    number={4}, journal={European Journal of Applied Mathematics}, publisher={Cambridge
    University Press (CUP)}, author={SURULESCU, CHRISTINA and Winkler, Michael}, year={2020},
    pages={618–651} }'
  chicago: 'SURULESCU, CHRISTINA, and Michael Winkler. “Does Indirectness of Signal
    Production Reduce the Explosion-Supporting Potential in Chemotaxis–Haptotaxis
    Systems? Global Classical Solvability in a Class of Models for Cancer Invasion
    (and More).” <i>European Journal of Applied Mathematics</i> 32, no. 4 (2020):
    618–51. <a href="https://doi.org/10.1017/s0956792520000236">https://doi.org/10.1017/s0956792520000236</a>.'
  ieee: 'C. SURULESCU and M. Winkler, “Does indirectness of signal production reduce
    the explosion-supporting potential in chemotaxis–haptotaxis systems? Global classical
    solvability in a class of models for cancer invasion (and more),” <i>European
    Journal of Applied Mathematics</i>, vol. 32, no. 4, pp. 618–651, 2020, doi: <a
    href="https://doi.org/10.1017/s0956792520000236">10.1017/s0956792520000236</a>.'
  mla: SURULESCU, CHRISTINA, and Michael Winkler. “Does Indirectness of Signal Production
    Reduce the Explosion-Supporting Potential in Chemotaxis–Haptotaxis Systems? Global
    Classical Solvability in a Class of Models for Cancer Invasion (and More).” <i>European
    Journal of Applied Mathematics</i>, vol. 32, no. 4, Cambridge University Press
    (CUP), 2020, pp. 618–51, doi:<a href="https://doi.org/10.1017/s0956792520000236">10.1017/s0956792520000236</a>.
  short: C. SURULESCU, M. Winkler, European Journal of Applied Mathematics 32 (2020)
    618–651.
date_created: 2025-12-18T19:31:21Z
date_updated: 2025-12-18T20:06:05Z
doi: 10.1017/s0956792520000236
intvolume: '        32'
issue: '4'
language:
- iso: eng
page: 618-651
publication: European Journal of Applied Mathematics
publication_identifier:
  issn:
  - 0956-7925
  - 1469-4425
publication_status: published
publisher: Cambridge University Press (CUP)
status: public
title: Does indirectness of signal production reduce the explosion-supporting potential
  in chemotaxis–haptotaxis systems? Global classical solvability in a class of models
  for cancer invasion (and more)
type: journal_article
user_id: '31496'
volume: 32
year: '2020'
...
---
_id: '63265'
abstract:
- lang: eng
  text: "<jats:title>Abstract</jats:title><jats:p>The Cauchy problem in <jats:inline-formula><jats:alternatives><jats:tex-math>$${\\mathbb
    {R}}^n$$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n
    \                 <mml:msup>\r\n                    <mml:mrow>\r\n                      <mml:mi>R</mml:mi>\r\n
    \                   </mml:mrow>\r\n                    <mml:mi>n</mml:mi>\r\n
    \                 </mml:msup>\r\n                </mml:math></jats:alternatives></jats:inline-formula>,
    <jats:inline-formula><jats:alternatives><jats:tex-math>$$n\\ge 1$$</jats:tex-math><mml:math
    xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n                  <mml:mrow>\r\n
    \                   <mml:mi>n</mml:mi>\r\n                    <mml:mo>≥</mml:mo>\r\n
    \                   <mml:mn>1</mml:mn>\r\n                  </mml:mrow>\r\n                </mml:math></jats:alternatives></jats:inline-formula>,
    for the parabolic equation <jats:disp-formula><jats:alternatives><jats:tex-math>$$\\begin{aligned}
    u_t=u^p \\Delta u \\qquad \\qquad (\\star ) \\end{aligned}$$</jats:tex-math><mml:math
    xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n                  <mml:mrow>\r\n
    \                   <mml:mtable>\r\n                      <mml:mtr>\r\n                        <mml:mtd>\r\n
    \                         <mml:mrow>\r\n                            <mml:msub>\r\n
    \                             <mml:mi>u</mml:mi>\r\n                              <mml:mi>t</mml:mi>\r\n
    \                           </mml:msub>\r\n                            <mml:mo>=</mml:mo>\r\n
    \                           <mml:msup>\r\n                              <mml:mi>u</mml:mi>\r\n
    \                             <mml:mi>p</mml:mi>\r\n                            </mml:msup>\r\n
    \                           <mml:mi>Δ</mml:mi>\r\n                            <mml:mi>u</mml:mi>\r\n
    \                           <mml:mspace/>\r\n                            <mml:mspace/>\r\n
    \                           <mml:mrow>\r\n                              <mml:mo>(</mml:mo>\r\n
    \                             <mml:mo>⋆</mml:mo>\r\n                              <mml:mo>)</mml:mo>\r\n
    \                           </mml:mrow>\r\n                          </mml:mrow>\r\n
    \                       </mml:mtd>\r\n                      </mml:mtr>\r\n                    </mml:mtable>\r\n
    \                 </mml:mrow>\r\n                </mml:math></jats:alternatives></jats:disp-formula>is
    considered in the strongly degenerate regime <jats:inline-formula><jats:alternatives><jats:tex-math>$$p\\ge
    1$$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n
    \                 <mml:mrow>\r\n                    <mml:mi>p</mml:mi>\r\n                    <mml:mo>≥</mml:mo>\r\n
    \                   <mml:mn>1</mml:mn>\r\n                  </mml:mrow>\r\n                </mml:math></jats:alternatives></jats:inline-formula>.
    The focus is firstly on the case of positive continuous and bounded initial data,
    in which it is known that a minimal positive classical solution exists, and that
    this solution satisfies <jats:disp-formula><jats:alternatives><jats:tex-math>$$\\begin{aligned}
    t^\\frac{1}{p}\\Vert u(\\cdot ,t)\\Vert _{L^\\infty ({\\mathbb {R}}^n)} \\rightarrow
    \\infty \\quad \\hbox {as } t\\rightarrow \\infty . \\end{aligned}$$</jats:tex-math><mml:math
    xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n                  <mml:mrow>\r\n
    \                   <mml:mtable>\r\n                      <mml:mtr>\r\n                        <mml:mtd>\r\n
    \                         <mml:mrow>\r\n                            <mml:msup>\r\n
    \                             <mml:mi>t</mml:mi>\r\n                              <mml:mfrac>\r\n
    \                               <mml:mn>1</mml:mn>\r\n                                <mml:mi>p</mml:mi>\r\n
    \                             </mml:mfrac>\r\n                            </mml:msup>\r\n
    \                           <mml:msub>\r\n                              <mml:mrow>\r\n
    \                               <mml:mo>‖</mml:mo>\r\n                                <mml:mi>u</mml:mi>\r\n
    \                               <mml:mrow>\r\n                                  <mml:mo>(</mml:mo>\r\n
    \                                 <mml:mo>·</mml:mo>\r\n                                  <mml:mo>,</mml:mo>\r\n
    \                                 <mml:mi>t</mml:mi>\r\n                                  <mml:mo>)</mml:mo>\r\n
    \                               </mml:mrow>\r\n                                <mml:mo>‖</mml:mo>\r\n
    \                             </mml:mrow>\r\n                              <mml:mrow>\r\n
    \                               <mml:msup>\r\n                                  <mml:mi>L</mml:mi>\r\n
    \                                 <mml:mi>∞</mml:mi>\r\n                                </mml:msup>\r\n
    \                               <mml:mrow>\r\n                                  <mml:mo>(</mml:mo>\r\n
    \                                 <mml:msup>\r\n                                    <mml:mrow>\r\n
    \                                     <mml:mi>R</mml:mi>\r\n                                    </mml:mrow>\r\n
    \                                   <mml:mi>n</mml:mi>\r\n                                  </mml:msup>\r\n
    \                                 <mml:mo>)</mml:mo>\r\n                                </mml:mrow>\r\n
    \                             </mml:mrow>\r\n                            </mml:msub>\r\n
    \                           <mml:mo>→</mml:mo>\r\n                            <mml:mi>∞</mml:mi>\r\n
    \                           <mml:mspace/>\r\n                            <mml:mtext>as</mml:mtext>\r\n
    \                           <mml:mspace/>\r\n                            <mml:mi>t</mml:mi>\r\n
    \                           <mml:mo>→</mml:mo>\r\n                            <mml:mi>∞</mml:mi>\r\n
    \                           <mml:mo>.</mml:mo>\r\n                          </mml:mrow>\r\n
    \                       </mml:mtd>\r\n                      </mml:mtr>\r\n                    </mml:mtable>\r\n
    \                 </mml:mrow>\r\n                </mml:math></jats:alternatives></jats:disp-formula>The
    first result of this study complements this by asserting that given any positive
    <jats:inline-formula><jats:alternatives><jats:tex-math>$$f\\in C^0([0,\\infty
    ))$$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n
    \                 <mml:mrow>\r\n                    <mml:mi>f</mml:mi>\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:mrow>\r\n                      <mml:mo>(</mml:mo>\r\n
    \                     <mml:mrow>\r\n                        <mml:mo>[</mml:mo>\r\n
    \                       <mml:mn>0</mml:mn>\r\n                        <mml:mo>,</mml:mo>\r\n
    \                       <mml:mi>∞</mml:mi>\r\n                        <mml:mo>)</mml:mo>\r\n
    \                     </mml:mrow>\r\n                      <mml:mo>)</mml:mo>\r\n
    \                   </mml:mrow>\r\n                  </mml:mrow>\r\n                </mml:math></jats:alternatives></jats:inline-formula>
    fulfilling <jats:inline-formula><jats:alternatives><jats:tex-math>$$f(t)\\rightarrow
    +\\infty $$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n
    \                 <mml:mrow>\r\n                    <mml:mi>f</mml:mi>\r\n                    <mml:mo>(</mml:mo>\r\n
    \                   <mml:mi>t</mml:mi>\r\n                    <mml:mo>)</mml:mo>\r\n
    \                   <mml:mo>→</mml:mo>\r\n                    <mml:mo>+</mml:mo>\r\n
    \                   <mml:mi>∞</mml:mi>\r\n                  </mml:mrow>\r\n                </mml:math></jats:alternatives></jats:inline-formula>
    as <jats:inline-formula><jats:alternatives><jats:tex-math>$$t\\rightarrow \\infty
    $$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n
    \                 <mml:mrow>\r\n                    <mml:mi>t</mml:mi>\r\n                    <mml:mo>→</mml:mo>\r\n
    \                   <mml:mi>∞</mml:mi>\r\n                  </mml:mrow>\r\n                </mml:math></jats:alternatives></jats:inline-formula>
    one can find a positive nondecreasing function <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\phi
    \\in C^0([0,\\infty ))$$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n
    \                 <mml:mrow>\r\n                    <mml:mi>ϕ</mml:mi>\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:mrow>\r\n                      <mml:mo>(</mml:mo>\r\n
    \                     <mml:mrow>\r\n                        <mml:mo>[</mml:mo>\r\n
    \                       <mml:mn>0</mml:mn>\r\n                        <mml:mo>,</mml:mo>\r\n
    \                       <mml:mi>∞</mml:mi>\r\n                        <mml:mo>)</mml:mo>\r\n
    \                     </mml:mrow>\r\n                      <mml:mo>)</mml:mo>\r\n
    \                   </mml:mrow>\r\n                  </mml:mrow>\r\n                </mml:math></jats:alternatives></jats:inline-formula>
    such that whenever <jats:inline-formula><jats:alternatives><jats:tex-math>$$u_0\\in
    C^0({\\mathbb {R}}^n)$$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n
    \                 <mml:mrow>\r\n                    <mml:msub>\r\n                      <mml:mi>u</mml:mi>\r\n
    \                     <mml:mn>0</mml:mn>\r\n                    </mml:msub>\r\n
    \                   <mml: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:mrow>\r\n                      <mml:mo>(</mml:mo>\r\n
    \                     <mml:msup>\r\n                        <mml:mrow>\r\n                          <mml:mi>R</mml:mi>\r\n
    \                       </mml:mrow>\r\n                        <mml:mi>n</mml:mi>\r\n
    \                     </mml:msup>\r\n                      <mml:mo>)</mml:mo>\r\n
    \                   </mml:mrow>\r\n                  </mml:mrow>\r\n                </mml:math></jats:alternatives></jats:inline-formula>
    is radially symmetric with <jats:inline-formula><jats:alternatives><jats:tex-math>$$0&lt;
    u_0 &lt; \\phi (|\\cdot |)$$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n
    \                 <mml:mrow>\r\n                    <mml:mn>0</mml:mn>\r\n                    <mml:mo>&lt;</mml:mo>\r\n
    \                   <mml:msub>\r\n                      <mml:mi>u</mml:mi>\r\n
    \                     <mml:mn>0</mml:mn>\r\n                    </mml:msub>\r\n
    \                   <mml:mrow>\r\n                      <mml:mo>&lt;</mml:mo>\r\n
    \                     <mml:mi>ϕ</mml:mi>\r\n                      <mml:mo>(</mml:mo>\r\n
    \                     <mml:mo>|</mml:mo>\r\n                    </mml:mrow>\r\n
    \                   <mml:mo>·</mml:mo>\r\n                    <mml:mrow>\r\n                      <mml:mo>|</mml:mo>\r\n
    \                     <mml:mo>)</mml:mo>\r\n                    </mml:mrow>\r\n
    \                 </mml:mrow>\r\n                </mml:math></jats:alternatives></jats:inline-formula>,
    the corresponding minimal solution <jats:italic>u</jats:italic> satisfies <jats:disp-formula><jats:alternatives><jats:tex-math>$$\\begin{aligned}
    \\frac{t^\\frac{1}{p}\\Vert u(\\cdot ,t)\\Vert _{L^\\infty ({\\mathbb {R}}^n)}}{f(t)}
    \\rightarrow 0 \\quad \\hbox {as } t\\rightarrow \\infty . \\end{aligned}$$</jats:tex-math><mml:math
    xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n                  <mml:mrow>\r\n
    \                   <mml:mtable>\r\n                      <mml:mtr>\r\n                        <mml:mtd>\r\n
    \                         <mml:mrow>\r\n                            <mml:mfrac>\r\n
    \                             <mml:mrow>\r\n                                <mml:msup>\r\n
    \                                 <mml:mi>t</mml:mi>\r\n                                  <mml:mfrac>\r\n
    \                                   <mml:mn>1</mml:mn>\r\n                                    <mml:mi>p</mml:mi>\r\n
    \                                 </mml:mfrac>\r\n                                </mml:msup>\r\n
    \                               <mml:msub>\r\n                                  <mml:mrow>\r\n
    \                                   <mml:mo>‖</mml:mo>\r\n                                    <mml:mi>u</mml:mi>\r\n
    \                                   <mml:mrow>\r\n                                      <mml:mo>(</mml:mo>\r\n
    \                                     <mml:mo>·</mml:mo>\r\n                                      <mml:mo>,</mml:mo>\r\n
    \                                     <mml:mi>t</mml:mi>\r\n                                      <mml:mo>)</mml:mo>\r\n
    \                                   </mml:mrow>\r\n                                    <mml:mo>‖</mml:mo>\r\n
    \                                 </mml:mrow>\r\n                                  <mml:mrow>\r\n
    \                                   <mml:msup>\r\n                                      <mml:mi>L</mml:mi>\r\n
    \                                     <mml:mi>∞</mml:mi>\r\n                                    </mml:msup>\r\n
    \                                   <mml:mrow>\r\n                                      <mml:mo>(</mml:mo>\r\n
    \                                     <mml:msup>\r\n                                        <mml:mrow>\r\n
    \                                         <mml:mi>R</mml:mi>\r\n                                        </mml:mrow>\r\n
    \                                       <mml:mi>n</mml:mi>\r\n                                      </mml:msup>\r\n
    \                                     <mml:mo>)</mml:mo>\r\n                                    </mml:mrow>\r\n
    \                                 </mml:mrow>\r\n                                </mml:msub>\r\n
    \                             </mml:mrow>\r\n                              <mml:mrow>\r\n
    \                               <mml:mi>f</mml:mi>\r\n                                <mml:mo>(</mml:mo>\r\n
    \                               <mml:mi>t</mml:mi>\r\n                                <mml:mo>)</mml:mo>\r\n
    \                             </mml:mrow>\r\n                            </mml:mfrac>\r\n
    \                           <mml:mo>→</mml:mo>\r\n                            <mml:mn>0</mml:mn>\r\n
    \                           <mml:mspace/>\r\n                            <mml:mtext>as</mml:mtext>\r\n
    \                           <mml:mspace/>\r\n                            <mml:mi>t</mml:mi>\r\n
    \                           <mml:mo>→</mml:mo>\r\n                            <mml:mi>∞</mml:mi>\r\n
    \                           <mml:mo>.</mml:mo>\r\n                          </mml:mrow>\r\n
    \                       </mml:mtd>\r\n                      </mml:mtr>\r\n                    </mml:mtable>\r\n
    \                 </mml:mrow>\r\n                </mml:math></jats:alternatives></jats:disp-formula>Secondly,
    (<jats:inline-formula><jats:alternatives><jats:tex-math>$$\\star $$</jats:tex-math><mml:math
    xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n                  <mml:mo>⋆</mml:mo>\r\n
    \               </mml:math></jats:alternatives></jats:inline-formula>) is considered
    along with initial conditions involving nonnegative but not necessarily strictly
    positive bounded and continuous initial data <jats:inline-formula><jats:alternatives><jats:tex-math>$$u_0$$</jats:tex-math><mml:math
    xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n                  <mml:msub>\r\n
    \                   <mml:mi>u</mml:mi>\r\n                    <mml:mn>0</mml:mn>\r\n
    \                 </mml:msub>\r\n                </mml:math></jats:alternatives></jats:inline-formula>.
    It is shown that if the connected components of <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\{u_0&gt;0\\}$$</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:msub>\r\n                      <mml:mi>u</mml:mi>\r\n
    \                     <mml:mn>0</mml:mn>\r\n                    </mml:msub>\r\n
    \                   <mml:mo>&gt;</mml:mo>\r\n                    <mml:mn>0</mml:mn>\r\n
    \                   <mml:mo>}</mml:mo>\r\n                  </mml:mrow>\r\n                </mml:math></jats:alternatives></jats:inline-formula>
    comply with a condition reflecting some uniform boundedness property, then a corresponding
    uniquely determined continuous weak solution to (<jats:inline-formula><jats:alternatives><jats:tex-math>$$\\star
    $$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n
    \                 <mml:mo>⋆</mml:mo>\r\n                </mml:math></jats:alternatives></jats:inline-formula>)
    satisfies <jats:disp-formula><jats:alternatives><jats:tex-math>$$\\begin{aligned}
    0&lt; \\liminf _{t\\rightarrow \\infty } \\Big \\{ t^\\frac{1}{p} \\Vert u(\\cdot
    ,t)\\Vert _{L^\\infty ({\\mathbb {R}}^n)} \\Big \\} \\le \\limsup _{t\\rightarrow
    \\infty } \\Big \\{ t^\\frac{1}{p} \\Vert u(\\cdot ,t)\\Vert _{L^\\infty ({\\mathbb
    {R}}^n)} \\Big \\} &lt;\\infty . \\end{aligned}$$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n
    \                 <mml:mrow>\r\n                    <mml:mtable>\r\n                      <mml:mtr>\r\n
    \                       <mml:mtd>\r\n                          <mml:mrow>\r\n
    \                           <mml:mn>0</mml:mn>\r\n                            <mml:mo>&lt;</mml:mo>\r\n
    \                           <mml:munder>\r\n                              <mml:mo>lim
    inf</mml:mo>\r\n                              <mml:mrow>\r\n                                <mml:mi>t</mml:mi>\r\n
    \                               <mml:mo>→</mml:mo>\r\n                                <mml:mi>∞</mml:mi>\r\n
    \                             </mml:mrow>\r\n                            </mml:munder>\r\n
    \                           <mml:mrow>\r\n                              <mml:mo>{</mml:mo>\r\n
    \                           </mml:mrow>\r\n                            <mml:msup>\r\n
    \                             <mml:mi>t</mml:mi>\r\n                              <mml:mfrac>\r\n
    \                               <mml:mn>1</mml:mn>\r\n                                <mml:mi>p</mml:mi>\r\n
    \                             </mml:mfrac>\r\n                            </mml:msup>\r\n
    \                           <mml:msub>\r\n                              <mml:mrow>\r\n
    \                               <mml:mo>‖</mml:mo>\r\n                                <mml:mi>u</mml:mi>\r\n
    \                               <mml:mrow>\r\n                                  <mml:mo>(</mml:mo>\r\n
    \                                 <mml:mo>·</mml:mo>\r\n                                  <mml:mo>,</mml:mo>\r\n
    \                                 <mml:mi>t</mml:mi>\r\n                                  <mml:mo>)</mml:mo>\r\n
    \                               </mml:mrow>\r\n                                <mml:mo>‖</mml:mo>\r\n
    \                             </mml:mrow>\r\n                              <mml:mrow>\r\n
    \                               <mml:msup>\r\n                                  <mml:mi>L</mml:mi>\r\n
    \                                 <mml:mi>∞</mml:mi>\r\n                                </mml:msup>\r\n
    \                               <mml:mrow>\r\n                                  <mml:mo>(</mml:mo>\r\n
    \                                 <mml:msup>\r\n                                    <mml:mrow>\r\n
    \                                     <mml:mi>R</mml:mi>\r\n                                    </mml:mrow>\r\n
    \                                   <mml:mi>n</mml:mi>\r\n                                  </mml:msup>\r\n
    \                                 <mml:mo>)</mml:mo>\r\n                                </mml:mrow>\r\n
    \                             </mml:mrow>\r\n                            </mml:msub>\r\n
    \                           <mml:mrow>\r\n                              <mml:mo>}</mml:mo>\r\n
    \                           </mml:mrow>\r\n                            <mml:mo>≤</mml:mo>\r\n
    \                           <mml:munder>\r\n                              <mml:mo>lim
    sup</mml:mo>\r\n                              <mml:mrow>\r\n                                <mml:mi>t</mml:mi>\r\n
    \                               <mml:mo>→</mml:mo>\r\n                                <mml:mi>∞</mml:mi>\r\n
    \                             </mml:mrow>\r\n                            </mml:munder>\r\n
    \                           <mml:mrow>\r\n                              <mml:mo>{</mml:mo>\r\n
    \                           </mml:mrow>\r\n                            <mml:msup>\r\n
    \                             <mml:mi>t</mml:mi>\r\n                              <mml:mfrac>\r\n
    \                               <mml:mn>1</mml:mn>\r\n                                <mml:mi>p</mml:mi>\r\n
    \                             </mml:mfrac>\r\n                            </mml:msup>\r\n
    \                           <mml:msub>\r\n                              <mml:mrow>\r\n
    \                               <mml:mo>‖</mml:mo>\r\n                                <mml:mi>u</mml:mi>\r\n
    \                               <mml:mrow>\r\n                                  <mml:mo>(</mml:mo>\r\n
    \                                 <mml:mo>·</mml:mo>\r\n                                  <mml:mo>,</mml:mo>\r\n
    \                                 <mml:mi>t</mml:mi>\r\n                                  <mml:mo>)</mml:mo>\r\n
    \                               </mml:mrow>\r\n                                <mml:mo>‖</mml:mo>\r\n
    \                             </mml:mrow>\r\n                              <mml:mrow>\r\n
    \                               <mml:msup>\r\n                                  <mml:mi>L</mml:mi>\r\n
    \                                 <mml:mi>∞</mml:mi>\r\n                                </mml:msup>\r\n
    \                               <mml:mrow>\r\n                                  <mml:mo>(</mml:mo>\r\n
    \                                 <mml:msup>\r\n                                    <mml:mrow>\r\n
    \                                     <mml:mi>R</mml:mi>\r\n                                    </mml:mrow>\r\n
    \                                   <mml:mi>n</mml:mi>\r\n                                  </mml:msup>\r\n
    \                                 <mml:mo>)</mml:mo>\r\n                                </mml:mrow>\r\n
    \                             </mml:mrow>\r\n                            </mml:msub>\r\n
    \                           <mml:mrow>\r\n                              <mml:mo>}</mml:mo>\r\n
    \                           </mml:mrow>\r\n                            <mml:mo>&lt;</mml:mo>\r\n
    \                           <mml:mi>∞</mml:mi>\r\n                            <mml:mo>.</mml:mo>\r\n
    \                         </mml:mrow>\r\n                        </mml:mtd>\r\n
    \                     </mml:mtr>\r\n                    </mml:mtable>\r\n                  </mml:mrow>\r\n
    \               </mml:math></jats:alternatives></jats:disp-formula>Under a somewhat
    complementary hypothesis, particularly fulfilled if <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\{u_0&gt;0\\}$$</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:msub>\r\n                      <mml:mi>u</mml:mi>\r\n
    \                     <mml:mn>0</mml:mn>\r\n                    </mml:msub>\r\n
    \                   <mml:mo>&gt;</mml:mo>\r\n                    <mml:mn>0</mml:mn>\r\n
    \                   <mml:mo>}</mml:mo>\r\n                  </mml:mrow>\r\n                </mml:math></jats:alternatives></jats:inline-formula>
    contains components with arbitrarily small principal eigenvalues of the associated
    Dirichlet Laplacian, it is finally seen that (0.1) continues to hold also for
    such not everywhere positive weak solutions.</jats:p>"
author:
- first_name: Michael
  full_name: Winkler, Michael
  id: '31496'
  last_name: Winkler
citation:
  ama: Winkler M. Approaching Critical Decay in a Strongly Degenerate Parabolic Equation.
    <i>Journal of Dynamics and Differential Equations</i>. 2020;36(S1):3-23. doi:<a
    href="https://doi.org/10.1007/s10884-020-09892-x">10.1007/s10884-020-09892-x</a>
  apa: Winkler, M. (2020). Approaching Critical Decay in a Strongly Degenerate Parabolic
    Equation. <i>Journal of Dynamics and Differential Equations</i>, <i>36</i>(S1),
    3–23. <a href="https://doi.org/10.1007/s10884-020-09892-x">https://doi.org/10.1007/s10884-020-09892-x</a>
  bibtex: '@article{Winkler_2020, title={Approaching Critical Decay in a Strongly
    Degenerate Parabolic Equation}, volume={36}, DOI={<a href="https://doi.org/10.1007/s10884-020-09892-x">10.1007/s10884-020-09892-x</a>},
    number={S1}, journal={Journal of Dynamics and Differential Equations}, publisher={Springer
    Science and Business Media LLC}, author={Winkler, Michael}, year={2020}, pages={3–23}
    }'
  chicago: 'Winkler, Michael. “Approaching Critical Decay in a Strongly Degenerate
    Parabolic Equation.” <i>Journal of Dynamics and Differential Equations</i> 36,
    no. S1 (2020): 3–23. <a href="https://doi.org/10.1007/s10884-020-09892-x">https://doi.org/10.1007/s10884-020-09892-x</a>.'
  ieee: 'M. Winkler, “Approaching Critical Decay in a Strongly Degenerate Parabolic
    Equation,” <i>Journal of Dynamics and Differential Equations</i>, vol. 36, no.
    S1, pp. 3–23, 2020, doi: <a href="https://doi.org/10.1007/s10884-020-09892-x">10.1007/s10884-020-09892-x</a>.'
  mla: Winkler, Michael. “Approaching Critical Decay in a Strongly Degenerate Parabolic
    Equation.” <i>Journal of Dynamics and Differential Equations</i>, vol. 36, no.
    S1, Springer Science and Business Media LLC, 2020, pp. 3–23, doi:<a href="https://doi.org/10.1007/s10884-020-09892-x">10.1007/s10884-020-09892-x</a>.
  short: M. Winkler, Journal of Dynamics and Differential Equations 36 (2020) 3–23.
date_created: 2025-12-18T19:10:01Z
date_updated: 2025-12-18T20:10:07Z
doi: 10.1007/s10884-020-09892-x
intvolume: '        36'
issue: S1
language:
- iso: eng
page: 3-23
publication: Journal of Dynamics and Differential Equations
publication_identifier:
  issn:
  - 1040-7294
  - 1572-9222
publication_status: published
publisher: Springer Science and Business Media LLC
status: public
title: Approaching Critical Decay in a Strongly Degenerate Parabolic Equation
type: journal_article
user_id: '31496'
volume: 36
year: '2020'
...
---
_id: '63325'
abstract:
- lang: eng
  text: "<jats:title>Abstract</jats:title>\r\n               <jats:p>We consider the
    spatially 2D version of the model $$\\begin{equation*} \\qquad\\quad\\left\\{
    \\begin{array}{@{}rcll} n_t + u\\cdot\\nabla n &amp;=&amp; \\Delta n - \\nabla
    \\cdot \\big(nS(x,n,c) \\cdot \\nabla c \\big), \\qquad &amp;\\qquad x\\in \\Omega,
    \\ t&amp;gt;0, \\\\ c_t + u\\cdot \\nabla c &amp;=&amp; \\Delta c - n f(c), \\qquad
    &amp;\\qquad x\\in \\Omega, \\ t&amp;gt;0, \\\\ u_t &amp;=&amp; \\Delta u + \\nabla
    P + n\\nabla\\phi, \\qquad \\nabla\\cdot u=0, \\qquad &amp;\\qquad x\\in \\Omega,
    \\ t&amp;gt;0, \\end{array} \\right. \\qquad \\qquad (\\star) \\end{equation*}$$for
    nutrient taxis processes, possibly interacting with liquid environments. Here
    the particular focus is on the situation when the chemotactic sensitivity $S$
    is not a scalar function but rather attains general values in ${\\mathbb{R}}^{2\\times
    2}$, thus accounting for rotational flux components in accordance with experimental
    findings and recent modeling approaches. Reflecting significant new challenges
    that mainly stem from apparent loss of energy-like structures, especially for
    initial data with large size, the knowledge on ($\\star$) so far seems essentially
    restricted to results on global existence of certain generalized solutions with
    possibly quite poor boundedness and regularity properties; widely unaddressed
    seem aspects related to possible effects of such non-diagonal taxis mechanisms
    on the qualitative solution behavior, especially with regard to the fundamental
    question whether spatial structures may thereby be supported. The present work
    answers the latter in the negative in the following sense: under the assumptions
    that the initial data $(n_0,c_0,u_0)$ and the parameter functions $S$, $f$, and
    $\\phi$ are sufficiently smooth, and that $S$ is bounded and $f$ is positive on
    $(0,\\infty )$ with $f(0)=0$, it is shown that any nontrivial of these solutions
    eventually becomes smooth and satisfies $$\\begin{equation*} n(\\cdot,t)\\to -
    \\int_\\Omega n_0, \\quad c(\\cdot,t)\\to 0 \\quad \\text{and} \\quad u(\\cdot,t)\\to
    0 \\qquad \\text{as} \\ t\\to\\infty, \\end{equation*}$$uniformly with respect
    to $x\\in \\Omega$. By not requiring any smallness condition on the initial data,
    the latter seems new even in the corresponding fluid-free version obtained on
    letting $u\\equiv 0$ in ($\\star$).</jats:p>"
author:
- first_name: Michael
  full_name: Winkler, Michael
  id: '31496'
  last_name: Winkler
citation:
  ama: Winkler M. Can Rotational Fluxes Impede the Tendency Toward Spatial Homogeneity
    in Nutrient Taxis(-Stokes) Systems? <i>International Mathematics Research Notices</i>.
    2019;2021(11):8106-8152. doi:<a href="https://doi.org/10.1093/imrn/rnz056">10.1093/imrn/rnz056</a>
  apa: Winkler, M. (2019). Can Rotational Fluxes Impede the Tendency Toward Spatial
    Homogeneity in Nutrient Taxis(-Stokes) Systems? <i>International Mathematics Research
    Notices</i>, <i>2021</i>(11), 8106–8152. <a href="https://doi.org/10.1093/imrn/rnz056">https://doi.org/10.1093/imrn/rnz056</a>
  bibtex: '@article{Winkler_2019, title={Can Rotational Fluxes Impede the Tendency
    Toward Spatial Homogeneity in Nutrient Taxis(-Stokes) Systems?}, volume={2021},
    DOI={<a href="https://doi.org/10.1093/imrn/rnz056">10.1093/imrn/rnz056</a>}, number={11},
    journal={International Mathematics Research Notices}, publisher={Oxford University
    Press (OUP)}, author={Winkler, Michael}, year={2019}, pages={8106–8152} }'
  chicago: 'Winkler, Michael. “Can Rotational Fluxes Impede the Tendency Toward Spatial
    Homogeneity in Nutrient Taxis(-Stokes) Systems?” <i>International Mathematics
    Research Notices</i> 2021, no. 11 (2019): 8106–52. <a href="https://doi.org/10.1093/imrn/rnz056">https://doi.org/10.1093/imrn/rnz056</a>.'
  ieee: 'M. Winkler, “Can Rotational Fluxes Impede the Tendency Toward Spatial Homogeneity
    in Nutrient Taxis(-Stokes) Systems?,” <i>International Mathematics Research Notices</i>,
    vol. 2021, no. 11, pp. 8106–8152, 2019, doi: <a href="https://doi.org/10.1093/imrn/rnz056">10.1093/imrn/rnz056</a>.'
  mla: Winkler, Michael. “Can Rotational Fluxes Impede the Tendency Toward Spatial
    Homogeneity in Nutrient Taxis(-Stokes) Systems?” <i>International Mathematics
    Research Notices</i>, vol. 2021, no. 11, Oxford University Press (OUP), 2019,
    pp. 8106–52, doi:<a href="https://doi.org/10.1093/imrn/rnz056">10.1093/imrn/rnz056</a>.
  short: M. Winkler, International Mathematics Research Notices 2021 (2019) 8106–8152.
date_created: 2025-12-18T19:35:55Z
date_updated: 2025-12-18T19:59:29Z
doi: 10.1093/imrn/rnz056
intvolume: '      2021'
issue: '11'
language:
- iso: eng
page: 8106-8152
publication: International Mathematics Research Notices
publication_identifier:
  issn:
  - 1073-7928
  - 1687-0247
publication_status: published
publisher: Oxford University Press (OUP)
status: public
title: Can Rotational Fluxes Impede the Tendency Toward Spatial Homogeneity in Nutrient
  Taxis(-Stokes) Systems?
type: journal_article
user_id: '31496'
volume: 2021
year: '2019'
...
---
_id: '63337'
abstract:
- lang: eng
  text: <jats:title>Abstract</jats:title><jats:p>In bounded<jats:italic>n</jats:italic>-dimensional
    domains<jats:italic>Ω</jats:italic>, the Neumann problem for the parabolic equation</jats:p><jats:p><jats:disp-formula
    id="j_anona-2020-0013_eq_001"><jats:alternatives><jats:graphic xmlns:xlink="http://www.w3.org/1999/xlink"
    xlink:href="graphic/j_anona-2020-0013_eq_001.png" position="float" orientation="portrait"
    /><jats:tex-math>$$\begin{array}{} \displaystyle u_t = \nabla \cdot \Big( A(x,t)\cdot\nabla
    u\Big) + \nabla \cdot \Big(b(x,t)u\Big) - f(x,t,u)+g(x,t) \end{array}$$</jats:tex-math></jats:alternatives><jats:label>(*)</jats:label></jats:disp-formula></jats:p><jats:p>is
    considered for sufficiently regular matrix-valued<jats:italic>A</jats:italic>,
    vector-valued<jats:italic>b</jats:italic>and real valued<jats:italic>g</jats:italic>,
    and with<jats:italic>f</jats:italic>representing superlinear absorption in generalizing
    the prototypical choice given by<jats:italic>f</jats:italic>(⋅, ⋅,<jats:italic>s</jats:italic>)
    =<jats:italic>s<jats:sup>α</jats:sup></jats:italic>with<jats:italic>α</jats:italic>&gt;
    1. Problems of this form arise in a natural manner as sub-problems in several
    applications such as cross-diffusion systems either of Keller-Segel or of Shigesada-Kawasaki-Teramoto
    type in mathematical biology, and accordingly a natural space for initial data
    appears to be<jats:italic>L</jats:italic><jats:sup>1</jats:sup>(<jats:italic>Ω</jats:italic>).</jats:p><jats:p>The
    main objective thus consists in examining how far solutions can be constructed
    for initial data merely assumed to be integrable, with major challenges potentially
    resulting from the interplay between nonlinear degradation on the one hand, and
    the possibly destabilizing drift-type action on the other in such contexts. Especially,
    the applicability of well-established methods such as techniques relying on entropy-like
    structures available in some particular cases, for instance, seems quite limited
    in the present setting, as these typically rely on higher initial regularity properties.</jats:p><jats:p>The
    first of the main results shows that in the general framework of (*), nevertheless
    certain global very weak solutions can be constructed through a limit process
    involving smooth solutions to approximate variants thereof, provided that the
    ingredients of the latter satisfy appropriate assumptions with regard to their
    stabilization behavior.</jats:p><jats:p>The second and seemingly most substantial
    part of the paper develops a method by which it can be shown, under suitably stregthened
    hypotheses on the integrability of<jats:italic>b</jats:italic>and the degradation
    parameter<jats:italic>α</jats:italic>, that the solutions obtained above in fact
    form genuine weak solutions in a naturally defined sense. This is achieved by
    properly exploiting a weak integral inequality, as satisfied by the very weak
    solution at hand, through a testing procedure that appears to be novel and of
    potentially independent interest.</jats:p><jats:p>To underline the strength of
    this approach, both these general results are thereafter applied to two specific
    cross-diffusion systems. Inter alia, this leads to a statement on global solvability
    in a logistic Keller-Segel system under the assumption<jats:italic>α</jats:italic>&gt;<jats:inline-formula><jats:alternatives><jats:inline-graphic
    xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_anona-2020-0013_eq_002.png"
    /><jats:tex-math>$\begin{array}{} \frac{2n+4}{n+4} \end{array}$</jats:tex-math></jats:alternatives></jats:inline-formula>on
    the respective degradation rate which seems substantially milder than any previously
    found condition in the literature. Apart from that, for a Shigesada-Kawasaki-Teramoto
    system some apparently first results on global solvability for<jats:italic>L</jats:italic><jats:sup>1</jats:sup>initial
    data are derived.</jats:p>
author:
- first_name: Michael
  full_name: Winkler, Michael
  id: '31496'
  last_name: Winkler
citation:
  ama: Winkler M. The role of superlinear damping in the construction of solutions
    to drift-diffusion problems with initial data in L1. <i>Advances in Nonlinear
    Analysis</i>. 2019;9(1):526-566. doi:<a href="https://doi.org/10.1515/anona-2020-0013">10.1515/anona-2020-0013</a>
  apa: Winkler, M. (2019). The role of superlinear damping in the construction of
    solutions to drift-diffusion problems with initial data in L1. <i>Advances in
    Nonlinear Analysis</i>, <i>9</i>(1), 526–566. <a href="https://doi.org/10.1515/anona-2020-0013">https://doi.org/10.1515/anona-2020-0013</a>
  bibtex: '@article{Winkler_2019, title={The role of superlinear damping in the construction
    of solutions to drift-diffusion problems with initial data in L1}, volume={9},
    DOI={<a href="https://doi.org/10.1515/anona-2020-0013">10.1515/anona-2020-0013</a>},
    number={1}, journal={Advances in Nonlinear Analysis}, publisher={Walter de Gruyter
    GmbH}, author={Winkler, Michael}, year={2019}, pages={526–566} }'
  chicago: 'Winkler, Michael. “The Role of Superlinear Damping in the Construction
    of Solutions to Drift-Diffusion Problems with Initial Data in L1.” <i>Advances
    in Nonlinear Analysis</i> 9, no. 1 (2019): 526–66. <a href="https://doi.org/10.1515/anona-2020-0013">https://doi.org/10.1515/anona-2020-0013</a>.'
  ieee: 'M. Winkler, “The role of superlinear damping in the construction of solutions
    to drift-diffusion problems with initial data in L1,” <i>Advances in Nonlinear
    Analysis</i>, vol. 9, no. 1, pp. 526–566, 2019, doi: <a href="https://doi.org/10.1515/anona-2020-0013">10.1515/anona-2020-0013</a>.'
  mla: Winkler, Michael. “The Role of Superlinear Damping in the Construction of Solutions
    to Drift-Diffusion Problems with Initial Data in L1.” <i>Advances in Nonlinear
    Analysis</i>, vol. 9, no. 1, Walter de Gruyter GmbH, 2019, pp. 526–66, doi:<a
    href="https://doi.org/10.1515/anona-2020-0013">10.1515/anona-2020-0013</a>.
  short: M. Winkler, Advances in Nonlinear Analysis 9 (2019) 526–566.
date_created: 2025-12-18T19:45:09Z
date_updated: 2025-12-18T20:01:54Z
doi: 10.1515/anona-2020-0013
intvolume: '         9'
issue: '1'
language:
- iso: eng
page: 526-566
publication: Advances in Nonlinear Analysis
publication_identifier:
  issn:
  - 2191-950X
publication_status: published
publisher: Walter de Gruyter GmbH
status: public
title: The role of superlinear damping in the construction of solutions to drift-diffusion
  problems with initial data in L1
type: journal_article
user_id: '31496'
volume: 9
year: '2019'
...
---
_id: '63334'
author:
- first_name: Youshan
  full_name: Tao, Youshan
  last_name: Tao
- first_name: Michael
  full_name: Winkler, Michael
  id: '31496'
  last_name: Winkler
citation:
  ama: Tao Y, Winkler M. Global classical solutions to a doubly haptotactic cross-diffusion
    system modeling oncolytic virotherapy. <i>Journal of Differential Equations</i>.
    2019;268(9):4973-4997. doi:<a href="https://doi.org/10.1016/j.jde.2019.10.046">10.1016/j.jde.2019.10.046</a>
  apa: Tao, Y., &#38; Winkler, M. (2019). Global classical solutions to a doubly haptotactic
    cross-diffusion system modeling oncolytic virotherapy. <i>Journal of Differential
    Equations</i>, <i>268</i>(9), 4973–4997. <a href="https://doi.org/10.1016/j.jde.2019.10.046">https://doi.org/10.1016/j.jde.2019.10.046</a>
  bibtex: '@article{Tao_Winkler_2019, title={Global classical solutions to a doubly
    haptotactic cross-diffusion system modeling oncolytic virotherapy}, volume={268},
    DOI={<a href="https://doi.org/10.1016/j.jde.2019.10.046">10.1016/j.jde.2019.10.046</a>},
    number={9}, journal={Journal of Differential Equations}, publisher={Elsevier BV},
    author={Tao, Youshan and Winkler, Michael}, year={2019}, pages={4973–4997} }'
  chicago: 'Tao, Youshan, and Michael Winkler. “Global Classical Solutions to a Doubly
    Haptotactic Cross-Diffusion System Modeling Oncolytic Virotherapy.” <i>Journal
    of Differential Equations</i> 268, no. 9 (2019): 4973–97. <a href="https://doi.org/10.1016/j.jde.2019.10.046">https://doi.org/10.1016/j.jde.2019.10.046</a>.'
  ieee: 'Y. Tao and M. Winkler, “Global classical solutions to a doubly haptotactic
    cross-diffusion system modeling oncolytic virotherapy,” <i>Journal of Differential
    Equations</i>, vol. 268, no. 9, pp. 4973–4997, 2019, doi: <a href="https://doi.org/10.1016/j.jde.2019.10.046">10.1016/j.jde.2019.10.046</a>.'
  mla: Tao, Youshan, and Michael Winkler. “Global Classical Solutions to a Doubly
    Haptotactic Cross-Diffusion System Modeling Oncolytic Virotherapy.” <i>Journal
    of Differential Equations</i>, vol. 268, no. 9, Elsevier BV, 2019, pp. 4973–97,
    doi:<a href="https://doi.org/10.1016/j.jde.2019.10.046">10.1016/j.jde.2019.10.046</a>.
  short: Y. Tao, M. Winkler, Journal of Differential Equations 268 (2019) 4973–4997.
date_created: 2025-12-18T19:40:07Z
date_updated: 2025-12-18T20:01:29Z
doi: 10.1016/j.jde.2019.10.046
intvolume: '       268'
issue: '9'
language:
- iso: eng
page: 4973-4997
publication: Journal of Differential Equations
publication_identifier:
  issn:
  - 0022-0396
publication_status: published
publisher: Elsevier BV
status: public
title: Global classical solutions to a doubly haptotactic cross-diffusion system modeling
  oncolytic virotherapy
type: journal_article
user_id: '31496'
volume: 268
year: '2019'
...
---
_id: '63349'
author:
- first_name: Nicola
  full_name: Bellomo, Nicola
  last_name: Bellomo
- first_name: Kevin J.
  full_name: Painter, Kevin J.
  last_name: Painter
- first_name: Youshan
  full_name: Tao, Youshan
  last_name: Tao
- first_name: Michael
  full_name: Winkler, Michael
  id: '31496'
  last_name: Winkler
citation:
  ama: Bellomo N, Painter KJ, Tao Y, Winkler M. Occurrence vs. Absence of Taxis-Driven
    Instabilities in a May--Nowak Model for Virus Infection. <i>SIAM Journal on Applied
    Mathematics</i>. 2019;79(5):1990-2010. doi:<a href="https://doi.org/10.1137/19m1250261">10.1137/19m1250261</a>
  apa: Bellomo, N., Painter, K. J., Tao, Y., &#38; Winkler, M. (2019). Occurrence
    vs. Absence of Taxis-Driven Instabilities in a May--Nowak Model for Virus Infection.
    <i>SIAM Journal on Applied Mathematics</i>, <i>79</i>(5), 1990–2010. <a href="https://doi.org/10.1137/19m1250261">https://doi.org/10.1137/19m1250261</a>
  bibtex: '@article{Bellomo_Painter_Tao_Winkler_2019, title={Occurrence vs. Absence
    of Taxis-Driven Instabilities in a May--Nowak Model for Virus Infection}, volume={79},
    DOI={<a href="https://doi.org/10.1137/19m1250261">10.1137/19m1250261</a>}, number={5},
    journal={SIAM Journal on Applied Mathematics}, publisher={Society for Industrial
    &#38; Applied Mathematics (SIAM)}, author={Bellomo, Nicola and Painter, Kevin
    J. and Tao, Youshan and Winkler, Michael}, year={2019}, pages={1990–2010} }'
  chicago: 'Bellomo, Nicola, Kevin J. Painter, Youshan Tao, and Michael Winkler. “Occurrence
    vs. Absence of Taxis-Driven Instabilities in a May--Nowak Model for Virus Infection.”
    <i>SIAM Journal on Applied Mathematics</i> 79, no. 5 (2019): 1990–2010. <a href="https://doi.org/10.1137/19m1250261">https://doi.org/10.1137/19m1250261</a>.'
  ieee: 'N. Bellomo, K. J. Painter, Y. Tao, and M. Winkler, “Occurrence vs. Absence
    of Taxis-Driven Instabilities in a May--Nowak Model for Virus Infection,” <i>SIAM
    Journal on Applied Mathematics</i>, vol. 79, no. 5, pp. 1990–2010, 2019, doi:
    <a href="https://doi.org/10.1137/19m1250261">10.1137/19m1250261</a>.'
  mla: Bellomo, Nicola, et al. “Occurrence vs. Absence of Taxis-Driven Instabilities
    in a May--Nowak Model for Virus Infection.” <i>SIAM Journal on Applied Mathematics</i>,
    vol. 79, no. 5, Society for Industrial &#38; Applied Mathematics (SIAM), 2019,
    pp. 1990–2010, doi:<a href="https://doi.org/10.1137/19m1250261">10.1137/19m1250261</a>.
  short: N. Bellomo, K.J. Painter, Y. Tao, M. Winkler, SIAM Journal on Applied Mathematics
    79 (2019) 1990–2010.
date_created: 2025-12-19T10:47:32Z
date_updated: 2025-12-19T10:47:44Z
doi: 10.1137/19m1250261
intvolume: '        79'
issue: '5'
language:
- iso: eng
page: 1990-2010
publication: SIAM Journal on Applied Mathematics
publication_identifier:
  issn:
  - 0036-1399
  - 1095-712X
publication_status: published
publisher: Society for Industrial & Applied Mathematics (SIAM)
status: public
title: Occurrence vs. Absence of Taxis-Driven Instabilities in a May--Nowak Model
  for Virus Infection
type: journal_article
user_id: '31496'
volume: 79
year: '2019'
...
---
_id: '63355'
abstract:
- lang: eng
  text: "<jats:title>Abstract</jats:title><jats:p>This work studies the two‐species
    Shigesada–Kawasaki–Teramoto model with cross‐diffusion for one species, as given by\r\n<jats:disp-formula>\r\n</jats:disp-formula>with
    positive parameters  and , and nonnegative constants  and . Beyond some statements
    on global existence, the literature apparently provides only few results on qualitative
    behavior of solutions; in particular, questions related to boundedness as well
    as to large time asymptotics in <jats:ext-link xmlns:xlink=\"http://www.w3.org/1999/xlink\"
    xlink:href=\"#plms12276-disp-0001\" /> seem unsolved so far.</jats:p><jats:p>In
    the present paper it is <jats:italic>inter alia</jats:italic> shown that if  and
    \ is a bounded convex domain with smooth boundary, then whenever  and  are nonnegative,
    the associated Neumann initial‐boundary value problem for <jats:ext-link xmlns:xlink=\"http://www.w3.org/1999/xlink\"
    xlink:href=\"#plms12276-disp-0001\" /> possesses a global classical solution which
    in fact is bounded in the sense that\r\n<jats:disp-formula>\r\n</jats:disp-formula>Moreover,
    the asymptotic behavior of arbitrary nonnegative solutions enjoying the boundedness
    property is studied in the general situation when  is arbitrary and  no longer
    necessarily convex. If , then in both cases  and , an explicit smallness condition
    on  is identified as sufficient for stabilization of any nontrivial solutions
    toward a corresponding unique nontrivial spatially homogeneous steady state. If
    \ and , then without any further assumption all nonzero solutions are seen to
    approach the equilibrium (0,1). As a by‐product, this particularly improves previous
    knowledge on nonexistence of nonconstant equilibria of <jats:ext-link xmlns:xlink=\"http://www.w3.org/1999/xlink\"
    xlink:href=\"#plms12276-disp-0001\" />.</jats:p>"
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. Boundedness and stabilization in a population model with
    cross‐diffusion for one species. <i>Proceedings of the London Mathematical Society</i>.
    2019;119(6):1598-1632. doi:<a href="https://doi.org/10.1112/plms.12276">10.1112/plms.12276</a>
  apa: Tao, Y., &#38; Winkler, M. (2019). Boundedness and stabilization in a population
    model with cross‐diffusion for one species. <i>Proceedings of the London Mathematical
    Society</i>, <i>119</i>(6), 1598–1632. <a href="https://doi.org/10.1112/plms.12276">https://doi.org/10.1112/plms.12276</a>
  bibtex: '@article{Tao_Winkler_2019, title={Boundedness and stabilization in a population
    model with cross‐diffusion for one species}, volume={119}, DOI={<a href="https://doi.org/10.1112/plms.12276">10.1112/plms.12276</a>},
    number={6}, journal={Proceedings of the London Mathematical Society}, publisher={Wiley},
    author={Tao, Youshan and Winkler, Michael}, year={2019}, pages={1598–1632} }'
  chicago: 'Tao, Youshan, and Michael Winkler. “Boundedness and Stabilization in a
    Population Model with Cross‐diffusion for One Species.” <i>Proceedings of the
    London Mathematical Society</i> 119, no. 6 (2019): 1598–1632. <a href="https://doi.org/10.1112/plms.12276">https://doi.org/10.1112/plms.12276</a>.'
  ieee: 'Y. Tao and M. Winkler, “Boundedness and stabilization in a population model
    with cross‐diffusion for one species,” <i>Proceedings of the London Mathematical
    Society</i>, vol. 119, no. 6, pp. 1598–1632, 2019, doi: <a href="https://doi.org/10.1112/plms.12276">10.1112/plms.12276</a>.'
  mla: Tao, Youshan, and Michael Winkler. “Boundedness and Stabilization in a Population
    Model with Cross‐diffusion for One Species.” <i>Proceedings of the London Mathematical
    Society</i>, vol. 119, no. 6, Wiley, 2019, pp. 1598–632, doi:<a href="https://doi.org/10.1112/plms.12276">10.1112/plms.12276</a>.
  short: Y. Tao, M. Winkler, Proceedings of the London Mathematical Society 119 (2019)
    1598–1632.
date_created: 2025-12-19T10:54:01Z
date_updated: 2025-12-19T10:54:09Z
doi: 10.1112/plms.12276
intvolume: '       119'
issue: '6'
language:
- iso: eng
page: 1598-1632
publication: Proceedings of the London Mathematical Society
publication_identifier:
  issn:
  - 0024-6115
  - 1460-244X
publication_status: published
publisher: Wiley
status: public
title: Boundedness and stabilization in a population model with cross‐diffusion for
  one species
type: journal_article
user_id: '31496'
volume: 119
year: '2019'
...
---
_id: '63356'
abstract:
- lang: eng
  text: <jats:p> This work deals with a taxis cascade model for food consumption in
    two populations, namely foragers directly orienting their movement upward the
    gradients of food concentration and exploiters taking a parasitic strategy in
    search of food via tracking higher forager densities. As a consequence, the dynamics
    of both populations are adapted to the space distribution of food which is dynamically
    modified in time and space by the two populations. This model extends the classical
    one-species chemotaxis-consumption systems by additionally accounting for a second
    taxis mechanism coupled to the first in a consecutive manner. It is rigorously
    proved that for all suitably regular initial data, an associated Neumann-type
    initial-boundary value problem for the spatially one-dimensional version of this
    model possesses a globally defined bounded classical solution. Moreover, it is
    asserted that the considered two populations will approach spatially homogeneous
    distributions in the large time limit, provided that either the total population
    number of foragers or that of exploiters is appropriately small. </jats:p>
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. Large time behavior in a forager–exploiter model with different
    taxis strategies for two groups in search of food. <i>Mathematical Models and
    Methods in Applied Sciences</i>. 2019;29(11):2151-2182. doi:<a href="https://doi.org/10.1142/s021820251950043x">10.1142/s021820251950043x</a>
  apa: Tao, Y., &#38; Winkler, M. (2019). Large time behavior in a forager–exploiter
    model with different taxis strategies for two groups in search of food. <i>Mathematical
    Models and Methods in Applied Sciences</i>, <i>29</i>(11), 2151–2182. <a href="https://doi.org/10.1142/s021820251950043x">https://doi.org/10.1142/s021820251950043x</a>
  bibtex: '@article{Tao_Winkler_2019, title={Large time behavior in a forager–exploiter
    model with different taxis strategies for two groups in search of food}, volume={29},
    DOI={<a href="https://doi.org/10.1142/s021820251950043x">10.1142/s021820251950043x</a>},
    number={11}, journal={Mathematical Models and Methods in Applied Sciences}, publisher={World
    Scientific Pub Co Pte Ltd}, author={Tao, Youshan and Winkler, Michael}, year={2019},
    pages={2151–2182} }'
  chicago: 'Tao, Youshan, and Michael Winkler. “Large Time Behavior in a Forager–Exploiter
    Model with Different Taxis Strategies for Two Groups in Search of Food.” <i>Mathematical
    Models and Methods in Applied Sciences</i> 29, no. 11 (2019): 2151–82. <a href="https://doi.org/10.1142/s021820251950043x">https://doi.org/10.1142/s021820251950043x</a>.'
  ieee: 'Y. Tao and M. Winkler, “Large time behavior in a forager–exploiter model
    with different taxis strategies for two groups in search of food,” <i>Mathematical
    Models and Methods in Applied Sciences</i>, vol. 29, no. 11, pp. 2151–2182, 2019,
    doi: <a href="https://doi.org/10.1142/s021820251950043x">10.1142/s021820251950043x</a>.'
  mla: Tao, Youshan, and Michael Winkler. “Large Time Behavior in a Forager–Exploiter
    Model with Different Taxis Strategies for Two Groups in Search of Food.” <i>Mathematical
    Models and Methods in Applied Sciences</i>, vol. 29, no. 11, World Scientific
    Pub Co Pte Ltd, 2019, pp. 2151–82, doi:<a href="https://doi.org/10.1142/s021820251950043x">10.1142/s021820251950043x</a>.
  short: Y. Tao, M. Winkler, Mathematical Models and Methods in Applied Sciences 29
    (2019) 2151–2182.
date_created: 2025-12-19T10:54:36Z
date_updated: 2025-12-19T10:54:44Z
doi: 10.1142/s021820251950043x
intvolume: '        29'
issue: '11'
language:
- iso: eng
page: 2151-2182
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: Large time behavior in a forager–exploiter model with different taxis strategies
  for two groups in search of food
type: journal_article
user_id: '31496'
volume: 29
year: '2019'
...
---
_id: '63352'
author:
- first_name: Johannes
  full_name: Lankeit, Johannes
  last_name: Lankeit
- first_name: Michael
  full_name: Winkler, Michael
  id: '31496'
  last_name: Winkler
citation:
  ama: Lankeit J, Winkler M. Counterintuitive dependence of temporal asymptotics on
    initial decay in a nonlocal degenerate parabolic equation arising in game theory.
    <i>Israel Journal of Mathematics</i>. 2019;233(1):249-296. doi:<a href="https://doi.org/10.1007/s11856-019-1900-8">10.1007/s11856-019-1900-8</a>
  apa: Lankeit, J., &#38; Winkler, M. (2019). Counterintuitive dependence of temporal
    asymptotics on initial decay in a nonlocal degenerate parabolic equation arising
    in game theory. <i>Israel Journal of Mathematics</i>, <i>233</i>(1), 249–296.
    <a href="https://doi.org/10.1007/s11856-019-1900-8">https://doi.org/10.1007/s11856-019-1900-8</a>
  bibtex: '@article{Lankeit_Winkler_2019, title={Counterintuitive dependence of temporal
    asymptotics on initial decay in a nonlocal degenerate parabolic equation arising
    in game theory}, volume={233}, DOI={<a href="https://doi.org/10.1007/s11856-019-1900-8">10.1007/s11856-019-1900-8</a>},
    number={1}, journal={Israel Journal of Mathematics}, publisher={Springer Science
    and Business Media LLC}, author={Lankeit, Johannes and Winkler, Michael}, year={2019},
    pages={249–296} }'
  chicago: 'Lankeit, Johannes, and Michael Winkler. “Counterintuitive Dependence of
    Temporal Asymptotics on Initial Decay in a Nonlocal Degenerate Parabolic Equation
    Arising in Game Theory.” <i>Israel Journal of Mathematics</i> 233, no. 1 (2019):
    249–96. <a href="https://doi.org/10.1007/s11856-019-1900-8">https://doi.org/10.1007/s11856-019-1900-8</a>.'
  ieee: 'J. Lankeit and M. Winkler, “Counterintuitive dependence of temporal asymptotics
    on initial decay in a nonlocal degenerate parabolic equation arising in game theory,”
    <i>Israel Journal of Mathematics</i>, vol. 233, no. 1, pp. 249–296, 2019, doi:
    <a href="https://doi.org/10.1007/s11856-019-1900-8">10.1007/s11856-019-1900-8</a>.'
  mla: Lankeit, Johannes, and Michael Winkler. “Counterintuitive Dependence of Temporal
    Asymptotics on Initial Decay in a Nonlocal Degenerate Parabolic Equation Arising
    in Game Theory.” <i>Israel Journal of Mathematics</i>, vol. 233, no. 1, Springer
    Science and Business Media LLC, 2019, pp. 249–96, doi:<a href="https://doi.org/10.1007/s11856-019-1900-8">10.1007/s11856-019-1900-8</a>.
  short: J. Lankeit, M. Winkler, Israel Journal of Mathematics 233 (2019) 249–296.
date_created: 2025-12-19T10:51:24Z
date_updated: 2025-12-19T10:51:33Z
doi: 10.1007/s11856-019-1900-8
intvolume: '       233'
issue: '1'
language:
- iso: eng
page: 249-296
publication: Israel Journal of Mathematics
publication_identifier:
  issn:
  - 0021-2172
  - 1565-8511
publication_status: published
publisher: Springer Science and Business Media LLC
status: public
title: Counterintuitive dependence of temporal asymptotics on initial decay in a nonlocal
  degenerate parabolic equation arising in game theory
type: journal_article
user_id: '31496'
volume: 233
year: '2019'
...
---
_id: '63358'
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. A chemotaxis-haptotaxis system with haptoattractant remodeling:
    Boundedness enforced by mild saturation of signal production. <i>Communications
    on Pure &#38;amp; Applied Analysis</i>. 2019;18(4):2047-2067. doi:<a href="https://doi.org/10.3934/cpaa.2019092">10.3934/cpaa.2019092</a>'
  apa: 'Tao, Y., &#38; Winkler, M. (2019). A chemotaxis-haptotaxis system with haptoattractant
    remodeling: Boundedness enforced by mild saturation of signal production. <i>Communications
    on Pure &#38;amp; Applied Analysis</i>, <i>18</i>(4), 2047–2067. <a href="https://doi.org/10.3934/cpaa.2019092">https://doi.org/10.3934/cpaa.2019092</a>'
  bibtex: '@article{Tao_Winkler_2019, title={A chemotaxis-haptotaxis system with haptoattractant
    remodeling: Boundedness enforced by mild saturation of signal production}, volume={18},
    DOI={<a href="https://doi.org/10.3934/cpaa.2019092">10.3934/cpaa.2019092</a>},
    number={4}, journal={Communications on Pure &#38;amp; Applied Analysis}, publisher={American
    Institute of Mathematical Sciences (AIMS)}, author={Tao, Youshan and Winkler,
    Michael}, year={2019}, pages={2047–2067} }'
  chicago: 'Tao, Youshan, and Michael Winkler. “A Chemotaxis-Haptotaxis System with
    Haptoattractant Remodeling: Boundedness Enforced by Mild Saturation of Signal
    Production.” <i>Communications on Pure &#38;amp; Applied Analysis</i> 18, no.
    4 (2019): 2047–67. <a href="https://doi.org/10.3934/cpaa.2019092">https://doi.org/10.3934/cpaa.2019092</a>.'
  ieee: 'Y. Tao and M. Winkler, “A chemotaxis-haptotaxis system with haptoattractant
    remodeling: Boundedness enforced by mild saturation of signal production,” <i>Communications
    on Pure &#38;amp; Applied Analysis</i>, vol. 18, no. 4, pp. 2047–2067, 2019, doi:
    <a href="https://doi.org/10.3934/cpaa.2019092">10.3934/cpaa.2019092</a>.'
  mla: 'Tao, Youshan, and Michael Winkler. “A Chemotaxis-Haptotaxis System with Haptoattractant
    Remodeling: Boundedness Enforced by Mild Saturation of Signal Production.” <i>Communications
    on Pure &#38;amp; Applied Analysis</i>, vol. 18, no. 4, American Institute of
    Mathematical Sciences (AIMS), 2019, pp. 2047–67, doi:<a href="https://doi.org/10.3934/cpaa.2019092">10.3934/cpaa.2019092</a>.'
  short: Y. Tao, M. Winkler, Communications on Pure &#38;amp; Applied Analysis 18
    (2019) 2047–2067.
date_created: 2025-12-19T10:56:26Z
date_updated: 2025-12-19T10:56:33Z
doi: 10.3934/cpaa.2019092
intvolume: '        18'
issue: '4'
language:
- iso: eng
page: 2047-2067
publication: Communications on Pure &amp; Applied Analysis
publication_identifier:
  issn:
  - 1553-5258
publication_status: published
publisher: American Institute of Mathematical Sciences (AIMS)
status: public
title: 'A chemotaxis-haptotaxis system with haptoattractant remodeling: Boundedness
  enforced by mild saturation of signal production'
type: journal_article
user_id: '31496'
volume: 18
year: '2019'
...
---
_id: '63357'
author:
- first_name: Youshan
  full_name: Tao, Youshan
  last_name: Tao
- first_name: Michael
  full_name: Winkler, Michael
  id: '31496'
  last_name: Winkler
citation:
  ama: Tao Y, Winkler M. Global smooth solvability of a parabolic–elliptic nutrient
    taxis system in domains of arbitrary dimension. <i>Journal of Differential Equations</i>.
    2019;267(1):388-406. doi:<a href="https://doi.org/10.1016/j.jde.2019.01.014">10.1016/j.jde.2019.01.014</a>
  apa: Tao, Y., &#38; Winkler, M. (2019). Global smooth solvability of a parabolic–elliptic
    nutrient taxis system in domains of arbitrary dimension. <i>Journal of Differential
    Equations</i>, <i>267</i>(1), 388–406. <a href="https://doi.org/10.1016/j.jde.2019.01.014">https://doi.org/10.1016/j.jde.2019.01.014</a>
  bibtex: '@article{Tao_Winkler_2019, title={Global smooth solvability of a parabolic–elliptic
    nutrient taxis system in domains of arbitrary dimension}, volume={267}, DOI={<a
    href="https://doi.org/10.1016/j.jde.2019.01.014">10.1016/j.jde.2019.01.014</a>},
    number={1}, journal={Journal of Differential Equations}, publisher={Elsevier BV},
    author={Tao, Youshan and Winkler, Michael}, year={2019}, pages={388–406} }'
  chicago: 'Tao, Youshan, and Michael Winkler. “Global Smooth Solvability of a Parabolic–Elliptic
    Nutrient Taxis System in Domains of Arbitrary Dimension.” <i>Journal of Differential
    Equations</i> 267, no. 1 (2019): 388–406. <a href="https://doi.org/10.1016/j.jde.2019.01.014">https://doi.org/10.1016/j.jde.2019.01.014</a>.'
  ieee: 'Y. Tao and M. Winkler, “Global smooth solvability of a parabolic–elliptic
    nutrient taxis system in domains of arbitrary dimension,” <i>Journal of Differential
    Equations</i>, vol. 267, no. 1, pp. 388–406, 2019, doi: <a href="https://doi.org/10.1016/j.jde.2019.01.014">10.1016/j.jde.2019.01.014</a>.'
  mla: Tao, Youshan, and Michael Winkler. “Global Smooth Solvability of a Parabolic–Elliptic
    Nutrient Taxis System in Domains of Arbitrary Dimension.” <i>Journal of Differential
    Equations</i>, vol. 267, no. 1, Elsevier BV, 2019, pp. 388–406, doi:<a href="https://doi.org/10.1016/j.jde.2019.01.014">10.1016/j.jde.2019.01.014</a>.
  short: Y. Tao, M. Winkler, Journal of Differential Equations 267 (2019) 388–406.
date_created: 2025-12-19T10:55:53Z
date_updated: 2025-12-19T10:56:01Z
doi: 10.1016/j.jde.2019.01.014
intvolume: '       267'
issue: '1'
language:
- iso: eng
page: 388-406
publication: Journal of Differential Equations
publication_identifier:
  issn:
  - 0022-0396
publication_status: published
publisher: Elsevier BV
status: public
title: Global smooth solvability of a parabolic–elliptic nutrient taxis system in
  domains of arbitrary dimension
type: journal_article
user_id: '31496'
volume: 267
year: '2019'
...
---
_id: '63353'
author:
- first_name: Johannes
  full_name: Lankeit, Johannes
  last_name: Lankeit
- first_name: Michael
  full_name: Winkler, Michael
  id: '31496'
  last_name: Winkler
citation:
  ama: Lankeit J, Winkler M. Facing Low Regularity in Chemotaxis Systems. <i>Jahresbericht
    der Deutschen Mathematiker-Vereinigung</i>. 2019;122(1):35-64. doi:<a href="https://doi.org/10.1365/s13291-019-00210-z">10.1365/s13291-019-00210-z</a>
  apa: Lankeit, J., &#38; Winkler, M. (2019). Facing Low Regularity in Chemotaxis
    Systems. <i>Jahresbericht Der Deutschen Mathematiker-Vereinigung</i>, <i>122</i>(1),
    35–64. <a href="https://doi.org/10.1365/s13291-019-00210-z">https://doi.org/10.1365/s13291-019-00210-z</a>
  bibtex: '@article{Lankeit_Winkler_2019, title={Facing Low Regularity in Chemotaxis
    Systems}, volume={122}, DOI={<a href="https://doi.org/10.1365/s13291-019-00210-z">10.1365/s13291-019-00210-z</a>},
    number={1}, journal={Jahresbericht der Deutschen Mathematiker-Vereinigung}, publisher={Springer
    Fachmedien Wiesbaden GmbH}, author={Lankeit, Johannes and Winkler, Michael}, year={2019},
    pages={35–64} }'
  chicago: 'Lankeit, Johannes, and Michael Winkler. “Facing Low Regularity in Chemotaxis
    Systems.” <i>Jahresbericht Der Deutschen Mathematiker-Vereinigung</i> 122, no.
    1 (2019): 35–64. <a href="https://doi.org/10.1365/s13291-019-00210-z">https://doi.org/10.1365/s13291-019-00210-z</a>.'
  ieee: 'J. Lankeit and M. Winkler, “Facing Low Regularity in Chemotaxis Systems,”
    <i>Jahresbericht der Deutschen Mathematiker-Vereinigung</i>, vol. 122, no. 1,
    pp. 35–64, 2019, doi: <a href="https://doi.org/10.1365/s13291-019-00210-z">10.1365/s13291-019-00210-z</a>.'
  mla: Lankeit, Johannes, and Michael Winkler. “Facing Low Regularity in Chemotaxis
    Systems.” <i>Jahresbericht Der Deutschen Mathematiker-Vereinigung</i>, vol. 122,
    no. 1, Springer Fachmedien Wiesbaden GmbH, 2019, pp. 35–64, doi:<a href="https://doi.org/10.1365/s13291-019-00210-z">10.1365/s13291-019-00210-z</a>.
  short: J. Lankeit, M. Winkler, Jahresbericht Der Deutschen Mathematiker-Vereinigung
    122 (2019) 35–64.
date_created: 2025-12-19T10:52:04Z
date_updated: 2025-12-19T10:52:11Z
doi: 10.1365/s13291-019-00210-z
intvolume: '       122'
issue: '1'
language:
- iso: eng
page: 35-64
publication: Jahresbericht der Deutschen Mathematiker-Vereinigung
publication_identifier:
  issn:
  - 0012-0456
  - 1869-7135
publication_status: published
publisher: Springer Fachmedien Wiesbaden GmbH
status: public
title: Facing Low Regularity in Chemotaxis Systems
type: journal_article
user_id: '31496'
volume: 122
year: '2019'
...
---
_id: '63351'
author:
- first_name: Piotr
  full_name: Krzyżanowski, Piotr
  last_name: Krzyżanowski
- first_name: Michael
  full_name: Winkler, Michael
  id: '31496'
  last_name: Winkler
- first_name: Dariusz
  full_name: Wrzosek, Dariusz
  last_name: Wrzosek
citation:
  ama: 'Krzyżanowski P, Winkler M, Wrzosek D. Migration-driven benefit in a two-species
    nutrient taxis system. <i>Nonlinear Analysis: Real World Applications</i>. 2019;48:94-116.
    doi:<a href="https://doi.org/10.1016/j.nonrwa.2019.01.006">10.1016/j.nonrwa.2019.01.006</a>'
  apa: 'Krzyżanowski, P., Winkler, M., &#38; Wrzosek, D. (2019). Migration-driven
    benefit in a two-species nutrient taxis system. <i>Nonlinear Analysis: Real World
    Applications</i>, <i>48</i>, 94–116. <a href="https://doi.org/10.1016/j.nonrwa.2019.01.006">https://doi.org/10.1016/j.nonrwa.2019.01.006</a>'
  bibtex: '@article{Krzyżanowski_Winkler_Wrzosek_2019, title={Migration-driven benefit
    in a two-species nutrient taxis system}, volume={48}, DOI={<a href="https://doi.org/10.1016/j.nonrwa.2019.01.006">10.1016/j.nonrwa.2019.01.006</a>},
    journal={Nonlinear Analysis: Real World Applications}, publisher={Elsevier BV},
    author={Krzyżanowski, Piotr and Winkler, Michael and Wrzosek, Dariusz}, year={2019},
    pages={94–116} }'
  chicago: 'Krzyżanowski, Piotr, Michael Winkler, and Dariusz Wrzosek. “Migration-Driven
    Benefit in a Two-Species Nutrient Taxis System.” <i>Nonlinear Analysis: Real World
    Applications</i> 48 (2019): 94–116. <a href="https://doi.org/10.1016/j.nonrwa.2019.01.006">https://doi.org/10.1016/j.nonrwa.2019.01.006</a>.'
  ieee: 'P. Krzyżanowski, M. Winkler, and D. Wrzosek, “Migration-driven benefit in
    a two-species nutrient taxis system,” <i>Nonlinear Analysis: Real World Applications</i>,
    vol. 48, pp. 94–116, 2019, doi: <a href="https://doi.org/10.1016/j.nonrwa.2019.01.006">10.1016/j.nonrwa.2019.01.006</a>.'
  mla: 'Krzyżanowski, Piotr, et al. “Migration-Driven Benefit in a Two-Species Nutrient
    Taxis System.” <i>Nonlinear Analysis: Real World Applications</i>, vol. 48, Elsevier
    BV, 2019, pp. 94–116, doi:<a href="https://doi.org/10.1016/j.nonrwa.2019.01.006">10.1016/j.nonrwa.2019.01.006</a>.'
  short: 'P. Krzyżanowski, M. Winkler, D. Wrzosek, Nonlinear Analysis: Real World
    Applications 48 (2019) 94–116.'
date_created: 2025-12-19T10:50:49Z
date_updated: 2025-12-19T10:50:59Z
doi: 10.1016/j.nonrwa.2019.01.006
intvolume: '        48'
language:
- iso: eng
page: 94-116
publication: 'Nonlinear Analysis: Real World Applications'
publication_identifier:
  issn:
  - 1468-1218
publication_status: published
publisher: Elsevier BV
status: public
title: Migration-driven benefit in a two-species nutrient taxis system
type: journal_article
user_id: '31496'
volume: 48
year: '2019'
...