[{"doi":"10.4171/aihpc/141","language":[{"iso":"eng"}],"intvolume":"        42","date_updated":"2025-12-18T20:12:43Z","publication_status":"published","publication_identifier":{"issn":["0294-1449","1873-1430"]},"author":[{"first_name":"Michael","last_name":"Winkler","full_name":"Winkler, Michael","id":"31496"}],"title":"Logarithmically refined Gagliardo–Nirenberg interpolation and application to blow-up exclusion in a singular chemotaxis–consumption system","year":"2024","type":"journal_article","date_created":"2025-12-18T19:00:24Z","abstract":[{"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>","lang":"eng"}],"publication":"Annales de l'Institut Henri Poincaré C, Analyse non linéaire","issue":"6","volume":42,"user_id":"31496","_id":"63245","publisher":"European Mathematical Society - EMS - Publishing House GmbH","page":"1601-1630","status":"public","citation":{"short":"M. Winkler, Annales de l’Institut Henri Poincaré C, Analyse Non Linéaire 42 (2024) 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>.","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>","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>.","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>","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} }","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>."}},{"keyword":["Mathematical Physics","Analysis","Applied Mathematics"],"type":"journal_article","date_created":"2024-04-07T12:34:35Z","publication":"Annales de l'Institut Henri Poincaré C, Analyse non linéaire","citation":{"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>.","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>","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} }","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>","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>.","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>.","short":"M. Winkler, Annales de l’Institut Henri Poincaré C, Analyse Non Linéaire (2023)."},"user_id":"31496","doi":"10.4171/aihpc/73","_id":"53320","language":[{"iso":"eng"}],"publisher":"European Mathematical Society - EMS - Publishing House GmbH","publication_status":"published","date_updated":"2024-04-07T12:36:00Z","status":"public","year":"2023","title":"A quantitative strong parabolic maximum principle and application to a taxis-type migration–consumption model involving signal-dependent degenerate diffusion","author":[{"full_name":"Winkler, Michael","last_name":"Winkler","first_name":"Michael"}],"publication_identifier":{"issn":["0294-1449","1873-1430"]}},{"type":"journal_article","date_created":"2025-12-18T19:08:10Z","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>"}],"issue":"1","publication":"Annales de l'Institut Henri Poincaré C, Analyse non linéaire","doi":"10.4171/aihpc/73","language":[{"iso":"eng"}],"publication_status":"published","date_updated":"2025-12-18T20:14:52Z","intvolume":"        41","year":"2023","title":"A quantitative strong parabolic maximum principle and application to a taxis-type migration–consumption model involving signal-dependent degenerate diffusion","author":[{"id":"31496","first_name":"Michael","last_name":"Winkler","full_name":"Winkler, Michael"}],"publication_identifier":{"issn":["0294-1449","1873-1430"]},"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>","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} }","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>.","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>.","short":"M. Winkler, Annales de l’Institut Henri Poincaré C, Analyse Non Linéaire 41 (2023) 95–127.","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>","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>."},"user_id":"31496","volume":41,"page":"95-127","_id":"63261","publisher":"European Mathematical Society - EMS - Publishing House GmbH","status":"public"},{"author":[{"first_name":"Michael","last_name":"Winkler","full_name":"Winkler, Michael","id":"31496"}],"publication_identifier":{"issn":["0294-1449","1873-1430"]},"year":"2019","title":"Global solvability and stabilization in a two-dimensional cross-diffusion system modeling urban crime propagation","intvolume":"        36","date_updated":"2025-12-19T10:58:37Z","publication_status":"published","language":[{"iso":"eng"}],"doi":"10.1016/j.anihpc.2019.02.004","issue":"6","publication":"Annales de l'Institut Henri Poincaré C, Analyse non linéaire","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>"}],"date_created":"2025-12-19T10:58:29Z","type":"journal_article","status":"public","_id":"63362","publisher":"European Mathematical Society - EMS - Publishing House GmbH","page":"1747-1790","volume":36,"user_id":"31496","citation":{"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>.","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>","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>.","short":"M. Winkler, Annales de l’Institut Henri Poincaré C, Analyse Non Linéaire 36 (2019) 1747–1790.","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>.","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} }","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>"}}]