---
_id: '63245'
abstract:
- lang: eng
  text: "<jats:p>\r\n            A family of interpolation inequalities is derived,
    which differ from estimates of classical Gagliardo–Nirenberg type through the
    appearance of certain logarithmic deviations from standard Lebesgue norms in zero-order
    expressions. Optimality of the obtained inequalities is shown. A subsequent application
    reveals that when posed under homogeneous Neumann boundary conditions in smoothly
    bounded planar domains and with suitably regular initial data, for any choice
    of \r\n            <jats:inline-formula>\r\n              <jats:tex-math>\\alpha&gt;0</jats:tex-math>\r\n
    \           </jats:inline-formula>\r\n             the Keller–Segel-type migration–consumption
    system \r\n            <jats:inline-formula>\r\n              <jats:tex-math>u_{t}
    = \\Delta (uv^{-\\alpha})</jats:tex-math>\r\n            </jats:inline-formula>\r\n
    \           , \r\n            <jats:inline-formula>\r\n              <jats:tex-math>v_{t}
    = \\Delta v-uv</jats:tex-math>\r\n            </jats:inline-formula>\r\n            ,
    admits a global classical solution.\r\n          </jats:p>"
author:
- first_name: Michael
  full_name: Winkler, Michael
  id: '31496'
  last_name: Winkler
citation:
  ama: Winkler M. Logarithmically refined Gagliardo–Nirenberg interpolation and application
    to blow-up exclusion in a singular chemotaxis–consumption system. <i>Annales de
    l’Institut Henri Poincaré C, Analyse non linéaire</i>. 2024;42(6):1601-1630. doi:<a
    href="https://doi.org/10.4171/aihpc/141">10.4171/aihpc/141</a>
  apa: Winkler, M. (2024). Logarithmically refined Gagliardo–Nirenberg interpolation
    and application to blow-up exclusion in a singular chemotaxis–consumption system.
    <i>Annales de l’Institut Henri Poincaré C, Analyse Non Linéaire</i>, <i>42</i>(6),
    1601–1630. <a href="https://doi.org/10.4171/aihpc/141">https://doi.org/10.4171/aihpc/141</a>
  bibtex: '@article{Winkler_2024, title={Logarithmically refined Gagliardo–Nirenberg
    interpolation and application to blow-up exclusion in a singular chemotaxis–consumption
    system}, volume={42}, DOI={<a href="https://doi.org/10.4171/aihpc/141">10.4171/aihpc/141</a>},
    number={6}, journal={Annales de l’Institut Henri Poincaré C, Analyse non linéaire},
    publisher={European Mathematical Society - EMS - Publishing House GmbH}, author={Winkler,
    Michael}, year={2024}, pages={1601–1630} }'
  chicago: 'Winkler, Michael. “Logarithmically Refined Gagliardo–Nirenberg Interpolation
    and Application to Blow-up Exclusion in a Singular Chemotaxis–Consumption System.”
    <i>Annales de l’Institut Henri Poincaré C, Analyse Non Linéaire</i> 42, no. 6
    (2024): 1601–30. <a href="https://doi.org/10.4171/aihpc/141">https://doi.org/10.4171/aihpc/141</a>.'
  ieee: 'M. Winkler, “Logarithmically refined Gagliardo–Nirenberg interpolation and
    application to blow-up exclusion in a singular chemotaxis–consumption system,”
    <i>Annales de l’Institut Henri Poincaré C, Analyse non linéaire</i>, vol. 42,
    no. 6, pp. 1601–1630, 2024, doi: <a href="https://doi.org/10.4171/aihpc/141">10.4171/aihpc/141</a>.'
  mla: Winkler, Michael. “Logarithmically Refined Gagliardo–Nirenberg Interpolation
    and Application to Blow-up Exclusion in a Singular Chemotaxis–Consumption System.”
    <i>Annales de l’Institut Henri Poincaré C, Analyse Non Linéaire</i>, vol. 42,
    no. 6, European Mathematical Society - EMS - Publishing House GmbH, 2024, pp.
    1601–30, doi:<a href="https://doi.org/10.4171/aihpc/141">10.4171/aihpc/141</a>.
  short: M. Winkler, Annales de l’Institut Henri Poincaré C, Analyse Non Linéaire
    42 (2024) 1601–1630.
date_created: 2025-12-18T19:00:24Z
date_updated: 2025-12-18T20:12:43Z
doi: 10.4171/aihpc/141
intvolume: '        42'
issue: '6'
language:
- iso: eng
page: 1601-1630
publication: Annales de l'Institut Henri Poincaré C, Analyse non linéaire
publication_identifier:
  issn:
  - 0294-1449
  - 1873-1430
publication_status: published
publisher: European Mathematical Society - EMS - Publishing House GmbH
status: public
title: Logarithmically refined Gagliardo–Nirenberg interpolation and application to
  blow-up exclusion in a singular chemotaxis–consumption system
type: journal_article
user_id: '31496'
volume: 42
year: '2024'
...
---
_id: '53320'
author:
- first_name: Michael
  full_name: Winkler, Michael
  last_name: Winkler
citation:
  ama: Winkler M. A quantitative strong parabolic maximum principle and application
    to a taxis-type migration–consumption model involving signal-dependent degenerate
    diffusion. <i>Annales de l’Institut Henri Poincaré C, Analyse non linéaire</i>.
    Published online 2023. doi:<a href="https://doi.org/10.4171/aihpc/73">10.4171/aihpc/73</a>
  apa: Winkler, M. (2023). A quantitative strong parabolic maximum principle and application
    to a taxis-type migration–consumption model involving signal-dependent degenerate
    diffusion. <i>Annales de l’Institut Henri Poincaré C, Analyse Non Linéaire</i>.
    <a href="https://doi.org/10.4171/aihpc/73">https://doi.org/10.4171/aihpc/73</a>
  bibtex: '@article{Winkler_2023, title={A quantitative strong parabolic maximum principle
    and application to a taxis-type migration–consumption model involving signal-dependent
    degenerate diffusion}, DOI={<a href="https://doi.org/10.4171/aihpc/73">10.4171/aihpc/73</a>},
    journal={Annales de l’Institut Henri Poincaré C, Analyse non linéaire}, publisher={European
    Mathematical Society - EMS - Publishing House GmbH}, author={Winkler, Michael},
    year={2023} }'
  chicago: Winkler, Michael. “A Quantitative Strong Parabolic Maximum Principle and
    Application to a Taxis-Type Migration–Consumption Model Involving Signal-Dependent
    Degenerate Diffusion.” <i>Annales de l’Institut Henri Poincaré C, Analyse Non
    Linéaire</i>, 2023. <a href="https://doi.org/10.4171/aihpc/73">https://doi.org/10.4171/aihpc/73</a>.
  ieee: 'M. Winkler, “A quantitative strong parabolic maximum principle and application
    to a taxis-type migration–consumption model involving signal-dependent degenerate
    diffusion,” <i>Annales de l’Institut Henri Poincaré C, Analyse non linéaire</i>,
    2023, doi: <a href="https://doi.org/10.4171/aihpc/73">10.4171/aihpc/73</a>.'
  mla: Winkler, Michael. “A Quantitative Strong Parabolic Maximum Principle and Application
    to a Taxis-Type Migration–Consumption Model Involving Signal-Dependent Degenerate
    Diffusion.” <i>Annales de l’Institut Henri Poincaré C, Analyse Non Linéaire</i>,
    European Mathematical Society - EMS - Publishing House GmbH, 2023, doi:<a href="https://doi.org/10.4171/aihpc/73">10.4171/aihpc/73</a>.
  short: M. Winkler, Annales de l’Institut Henri Poincaré C, Analyse Non Linéaire
    (2023).
date_created: 2024-04-07T12:34:35Z
date_updated: 2024-04-07T12:36:00Z
doi: 10.4171/aihpc/73
keyword:
- Mathematical Physics
- Analysis
- Applied Mathematics
language:
- iso: eng
publication: Annales de l'Institut Henri Poincaré C, Analyse non linéaire
publication_identifier:
  issn:
  - 0294-1449
  - 1873-1430
publication_status: published
publisher: European Mathematical Society - EMS - Publishing House GmbH
status: public
title: A quantitative strong parabolic maximum principle and application to a taxis-type
  migration–consumption model involving signal-dependent degenerate diffusion
type: journal_article
user_id: '31496'
year: '2023'
...
---
_id: '63261'
abstract:
- lang: eng
  text: "<jats:p>\r\n            The taxis-type migration–consumption model accounting
    for signal-dependent motilities, as given by \r\n            <jats:inline-formula>\r\n
    \             <jats:tex-math>u_{t} = \\Delta (u\\phi(v))</jats:tex-math>\r\n            </jats:inline-formula>\r\n
    \           , \r\n            <jats:inline-formula>\r\n              <jats:tex-math>v_{t}
    = \\Delta v-uv</jats:tex-math>\r\n            </jats:inline-formula>\r\n            ,
    is considered for suitably smooth functions \r\n            <jats:inline-formula>\r\n
    \             <jats:tex-math>\\phi\\colon[0,\\infty)\\to\\R</jats:tex-math>\r\n
    \           </jats:inline-formula>\r\n             which are such that \r\n            <jats:inline-formula>\r\n
    \             <jats:tex-math>\\phi&gt;0</jats:tex-math>\r\n            </jats:inline-formula>\r\n
    \            on \r\n            <jats:inline-formula>\r\n              <jats:tex-math>(0,\\infty)</jats:tex-math>\r\n
    \           </jats:inline-formula>\r\n            , but that in addition \r\n
    \           <jats:inline-formula>\r\n              <jats:tex-math>\\phi(0)=0</jats:tex-math>\r\n
    \           </jats:inline-formula>\r\n             with \r\n            <jats:inline-formula>\r\n
    \             <jats:tex-math>\\phi'(0)&gt;0</jats:tex-math>\r\n            </jats:inline-formula>\r\n
    \           . In order to appropriately cope with the diffusion degeneracies thereby
    included, this study separately examines the Neumann problem for the linear equation
    \r\n            <jats:inline-formula>\r\n              <jats:tex-math>V_{t} =
    \\Delta V + \\nabla\\cdot ( a(x,t)V) + b(x,t)V</jats:tex-math>\r\n            </jats:inline-formula>\r\n
    \            and establishes a statement on how pointwise positive lower bounds
    for nonnegative solutions depend on the supremum and the mass of the initial data,
    and on integrability features of \r\n            <jats:inline-formula>\r\n              <jats:tex-math>a</jats:tex-math>\r\n
    \           </jats:inline-formula>\r\n             and \r\n            <jats:inline-formula>\r\n
    \             <jats:tex-math>b</jats:tex-math>\r\n            </jats:inline-formula>\r\n
    \           . This is thereafter used as a key tool in the derivation of a result
    on global existence of solutions to the equation above, smooth and classical for
    positive times, under the mere assumption that the suitably regular initial data
    be nonnegative in both components. Apart from that, these solutions are seen to
    stabilize toward some equilibrium, and as a qualitative effect genuinely due to
    degeneracy in diffusion, a criterion on initial smallness of the second component
    is identified as sufficient for this limit state to be spatially nonconstant.\r\n
    \         </jats:p>"
author:
- first_name: Michael
  full_name: Winkler, Michael
  id: '31496'
  last_name: Winkler
citation:
  ama: Winkler M. A quantitative strong parabolic maximum principle and application
    to a taxis-type migration–consumption model involving signal-dependent degenerate
    diffusion. <i>Annales de l’Institut Henri Poincaré C, Analyse non linéaire</i>.
    2023;41(1):95-127. doi:<a href="https://doi.org/10.4171/aihpc/73">10.4171/aihpc/73</a>
  apa: Winkler, M. (2023). A quantitative strong parabolic maximum principle and application
    to a taxis-type migration–consumption model involving signal-dependent degenerate
    diffusion. <i>Annales de l’Institut Henri Poincaré C, Analyse Non Linéaire</i>,
    <i>41</i>(1), 95–127. <a href="https://doi.org/10.4171/aihpc/73">https://doi.org/10.4171/aihpc/73</a>
  bibtex: '@article{Winkler_2023, title={A quantitative strong parabolic maximum principle
    and application to a taxis-type migration–consumption model involving signal-dependent
    degenerate diffusion}, volume={41}, DOI={<a href="https://doi.org/10.4171/aihpc/73">10.4171/aihpc/73</a>},
    number={1}, journal={Annales de l’Institut Henri Poincaré C, Analyse non linéaire},
    publisher={European Mathematical Society - EMS - Publishing House GmbH}, author={Winkler,
    Michael}, year={2023}, pages={95–127} }'
  chicago: 'Winkler, Michael. “A Quantitative Strong Parabolic Maximum Principle and
    Application to a Taxis-Type Migration–Consumption Model Involving Signal-Dependent
    Degenerate Diffusion.” <i>Annales de l’Institut Henri Poincaré C, Analyse Non
    Linéaire</i> 41, no. 1 (2023): 95–127. <a href="https://doi.org/10.4171/aihpc/73">https://doi.org/10.4171/aihpc/73</a>.'
  ieee: 'M. Winkler, “A quantitative strong parabolic maximum principle and application
    to a taxis-type migration–consumption model involving signal-dependent degenerate
    diffusion,” <i>Annales de l’Institut Henri Poincaré C, Analyse non linéaire</i>,
    vol. 41, no. 1, pp. 95–127, 2023, doi: <a href="https://doi.org/10.4171/aihpc/73">10.4171/aihpc/73</a>.'
  mla: Winkler, Michael. “A Quantitative Strong Parabolic Maximum Principle and Application
    to a Taxis-Type Migration–Consumption Model Involving Signal-Dependent Degenerate
    Diffusion.” <i>Annales de l’Institut Henri Poincaré C, Analyse Non Linéaire</i>,
    vol. 41, no. 1, European Mathematical Society - EMS - Publishing House GmbH, 2023,
    pp. 95–127, doi:<a href="https://doi.org/10.4171/aihpc/73">10.4171/aihpc/73</a>.
  short: M. Winkler, Annales de l’Institut Henri Poincaré C, Analyse Non Linéaire
    41 (2023) 95–127.
date_created: 2025-12-18T19:08:10Z
date_updated: 2025-12-18T20:14:52Z
doi: 10.4171/aihpc/73
intvolume: '        41'
issue: '1'
language:
- iso: eng
page: 95-127
publication: Annales de l'Institut Henri Poincaré C, Analyse non linéaire
publication_identifier:
  issn:
  - 0294-1449
  - 1873-1430
publication_status: published
publisher: European Mathematical Society - EMS - Publishing House GmbH
status: public
title: A quantitative strong parabolic maximum principle and application to a taxis-type
  migration–consumption model involving signal-dependent degenerate diffusion
type: journal_article
user_id: '31496'
volume: 41
year: '2023'
...
---
_id: '63362'
abstract:
- lang: eng
  text: "<jats:p>The system</jats:p>\r\n          <jats:p>\r\n            <jats:disp-formula>\r\n
    \             <jats:tex-math>\\left\\{\\begin{matrix} u_{t} = \\mathrm{\\Delta
    }u−\\chi \\mathrm{∇} \\cdot \\left(\\frac{u}{v}\\mathrm{∇}v\\right)−uv + B_{1}(x,t),
    \\\\ v_{t} = \\mathrm{\\Delta }v + uv−v + B_{2}(x,t), \\\\  \\end{matrix}\\right.\\:\\:(
    \\star )</jats:tex-math>\r\n            </jats:disp-formula>\r\n          </jats:p>\r\n
    \         <jats:p>\r\n            is considered in a disk \r\n            <jats:inline-formula>\r\n
    \             <jats:tex-math>\\mathrm{\\Omega } \\subset \\mathbb{R}^{2}</jats:tex-math>\r\n
    \           </jats:inline-formula>\r\n            , with a positive parameter
    \r\n            <jats:inline-formula>\r\n              <jats:tex-math>χ</jats:tex-math>\r\n
    \           </jats:inline-formula>\r\n             and given nonnegative and suitably
    regular functions \r\n            <jats:inline-formula>\r\n              <jats:tex-math>B_{1}</jats:tex-math>\r\n
    \           </jats:inline-formula>\r\n             and \r\n            <jats:inline-formula>\r\n
    \             <jats:tex-math>B_{2}</jats:tex-math>\r\n            </jats:inline-formula>\r\n
    \            defined on \r\n            <jats:inline-formula>\r\n              <jats:tex-math>\\mathrm{\\Omega
    } \\times (0,\\infty )</jats:tex-math>\r\n            </jats:inline-formula>\r\n
    \           . In the particular version obtained when \r\n            <jats:inline-formula>\r\n
    \             <jats:tex-math>\\chi  = 2</jats:tex-math>\r\n            </jats:inline-formula>\r\n
    \           ,  (\r\n            <jats:inline-formula>\r\n              <jats:tex-math>\\star</jats:tex-math>\r\n
    \           </jats:inline-formula>\r\n            ) was proposed in [31] as a
    model for crime propagation in urban regions.\r\n          </jats:p>\r\n          <jats:p>\r\n
    \           Within a suitable generalized framework, it is shown that under mild
    assumptions on the parameter functions and the initial data the no-flux initial-boundary
    value problem for (\r\n            <jats:inline-formula>\r\n              <jats:tex-math>\\star</jats:tex-math>\r\n
    \           </jats:inline-formula>\r\n            ) possesses at least one global
    solution in the case when all model ingredients are radially symmetric with respect
    to the center of \r\n            <jats:inline-formula>\r\n              <jats:tex-math>Ω</jats:tex-math>\r\n
    \           </jats:inline-formula>\r\n            . Moreover, under an additional
    hypothesis on stabilization of the given external source terms in both equations,
    these solutions are shown to approach the solution of an elliptic boundary value
    problem in an appropriate sense.\r\n          </jats:p>\r\n          <jats:p>The
    analysis is based on deriving a priori estimates for a family of approximate problems,
    in a first step achieving some spatially global but weak initial regularity information
    which in a series of spatially localized arguments is thereafter successively
    improved.</jats:p>\r\n          <jats:p>\r\n            To the best of our knowledge,
    this is the first result on global existence of solutions to the two-dimensional
    version of the full original system  (\r\n            <jats:inline-formula>\r\n
    \             <jats:tex-math>\\star</jats:tex-math>\r\n            </jats:inline-formula>\r\n
    \           ) for arbitrarily large values of \r\n            <jats:inline-formula>\r\n
    \             <jats:tex-math>χ</jats:tex-math>\r\n            </jats:inline-formula>\r\n
    \           .\r\n          </jats:p>"
author:
- first_name: Michael
  full_name: Winkler, Michael
  id: '31496'
  last_name: Winkler
citation:
  ama: Winkler M. Global solvability and stabilization in a two-dimensional cross-diffusion
    system modeling urban crime propagation. <i>Annales de l’Institut Henri Poincaré
    C, Analyse non linéaire</i>. 2019;36(6):1747-1790. doi:<a href="https://doi.org/10.1016/j.anihpc.2019.02.004">10.1016/j.anihpc.2019.02.004</a>
  apa: Winkler, M. (2019). Global solvability and stabilization in a two-dimensional
    cross-diffusion system modeling urban crime propagation. <i>Annales de l’Institut
    Henri Poincaré C, Analyse Non Linéaire</i>, <i>36</i>(6), 1747–1790. <a href="https://doi.org/10.1016/j.anihpc.2019.02.004">https://doi.org/10.1016/j.anihpc.2019.02.004</a>
  bibtex: '@article{Winkler_2019, title={Global solvability and stabilization in a
    two-dimensional cross-diffusion system modeling urban crime propagation}, volume={36},
    DOI={<a href="https://doi.org/10.1016/j.anihpc.2019.02.004">10.1016/j.anihpc.2019.02.004</a>},
    number={6}, journal={Annales de l’Institut Henri Poincaré C, Analyse non linéaire},
    publisher={European Mathematical Society - EMS - Publishing House GmbH}, author={Winkler,
    Michael}, year={2019}, pages={1747–1790} }'
  chicago: 'Winkler, Michael. “Global Solvability and Stabilization in a Two-Dimensional
    Cross-Diffusion System Modeling Urban Crime Propagation.” <i>Annales de l’Institut
    Henri Poincaré C, Analyse Non Linéaire</i> 36, no. 6 (2019): 1747–90. <a href="https://doi.org/10.1016/j.anihpc.2019.02.004">https://doi.org/10.1016/j.anihpc.2019.02.004</a>.'
  ieee: 'M. Winkler, “Global solvability and stabilization in a two-dimensional cross-diffusion
    system modeling urban crime propagation,” <i>Annales de l’Institut Henri Poincaré
    C, Analyse non linéaire</i>, vol. 36, no. 6, pp. 1747–1790, 2019, doi: <a href="https://doi.org/10.1016/j.anihpc.2019.02.004">10.1016/j.anihpc.2019.02.004</a>.'
  mla: Winkler, Michael. “Global Solvability and Stabilization in a Two-Dimensional
    Cross-Diffusion System Modeling Urban Crime Propagation.” <i>Annales de l’Institut
    Henri Poincaré C, Analyse Non Linéaire</i>, vol. 36, no. 6, European Mathematical
    Society - EMS - Publishing House GmbH, 2019, pp. 1747–90, doi:<a href="https://doi.org/10.1016/j.anihpc.2019.02.004">10.1016/j.anihpc.2019.02.004</a>.
  short: M. Winkler, Annales de l’Institut Henri Poincaré C, Analyse Non Linéaire
    36 (2019) 1747–1790.
date_created: 2025-12-19T10:58:29Z
date_updated: 2025-12-19T10:58:37Z
doi: 10.1016/j.anihpc.2019.02.004
intvolume: '        36'
issue: '6'
language:
- iso: eng
page: 1747-1790
publication: Annales de l'Institut Henri Poincaré C, Analyse non linéaire
publication_identifier:
  issn:
  - 0294-1449
  - 1873-1430
publication_status: published
publisher: European Mathematical Society - EMS - Publishing House GmbH
status: public
title: Global solvability and stabilization in a two-dimensional cross-diffusion system
  modeling urban crime propagation
type: journal_article
user_id: '31496'
volume: 36
year: '2019'
...