[{"language":[{"iso":"eng"}],"_id":"63372","user_id":"31496","status":"public","publication":"Mathematische Zeitschrift","type":"journal_article","title":"The small-convection limit in a two-dimensional chemotaxis-Navier–Stokes system","doi":"10.1007/s00209-017-1944-6","publisher":"Springer Science and Business Media LLC","date_updated":"2025-12-19T11:04:46Z","volume":289,"author":[{"first_name":"Yulan","full_name":"Wang, Yulan","last_name":"Wang"},{"first_name":"Michael","full_name":"Winkler, Michael","id":"31496","last_name":"Winkler"},{"first_name":"Zhaoyin","full_name":"Xiang, Zhaoyin","last_name":"Xiang"}],"date_created":"2025-12-19T11:04:38Z","year":"2017","page":"71-108","intvolume":"       289","citation":{"ama":"Wang Y, Winkler M, Xiang Z. The small-convection limit in a two-dimensional chemotaxis-Navier–Stokes system. <i>Mathematische Zeitschrift</i>. 2017;289(1-2):71-108. doi:<a href=\"https://doi.org/10.1007/s00209-017-1944-6\">10.1007/s00209-017-1944-6</a>","ieee":"Y. Wang, M. Winkler, and Z. Xiang, “The small-convection limit in a two-dimensional chemotaxis-Navier–Stokes system,” <i>Mathematische Zeitschrift</i>, vol. 289, no. 1–2, pp. 71–108, 2017, doi: <a href=\"https://doi.org/10.1007/s00209-017-1944-6\">10.1007/s00209-017-1944-6</a>.","chicago":"Wang, Yulan, Michael Winkler, and Zhaoyin Xiang. “The Small-Convection Limit in a Two-Dimensional Chemotaxis-Navier–Stokes System.” <i>Mathematische Zeitschrift</i> 289, no. 1–2 (2017): 71–108. <a href=\"https://doi.org/10.1007/s00209-017-1944-6\">https://doi.org/10.1007/s00209-017-1944-6</a>.","bibtex":"@article{Wang_Winkler_Xiang_2017, title={The small-convection limit in a two-dimensional chemotaxis-Navier–Stokes system}, volume={289}, DOI={<a href=\"https://doi.org/10.1007/s00209-017-1944-6\">10.1007/s00209-017-1944-6</a>}, number={1–2}, journal={Mathematische Zeitschrift}, publisher={Springer Science and Business Media LLC}, author={Wang, Yulan and Winkler, Michael and Xiang, Zhaoyin}, year={2017}, pages={71–108} }","short":"Y. Wang, M. Winkler, Z. Xiang, Mathematische Zeitschrift 289 (2017) 71–108.","mla":"Wang, Yulan, et al. “The Small-Convection Limit in a Two-Dimensional Chemotaxis-Navier–Stokes System.” <i>Mathematische Zeitschrift</i>, vol. 289, no. 1–2, Springer Science and Business Media LLC, 2017, pp. 71–108, doi:<a href=\"https://doi.org/10.1007/s00209-017-1944-6\">10.1007/s00209-017-1944-6</a>.","apa":"Wang, Y., Winkler, M., &#38; Xiang, Z. (2017). The small-convection limit in a two-dimensional chemotaxis-Navier–Stokes system. <i>Mathematische Zeitschrift</i>, <i>289</i>(1–2), 71–108. <a href=\"https://doi.org/10.1007/s00209-017-1944-6\">https://doi.org/10.1007/s00209-017-1944-6</a>"},"publication_identifier":{"issn":["0025-5874","1432-1823"]},"publication_status":"published","issue":"1-2"},{"language":[{"iso":"eng"}],"_id":"63374","user_id":"31496","status":"public","type":"journal_article","publication":"Journal de Mathématiques Pures et Appliquées","title":"Singular structure formation in a degenerate haptotaxis model involving myopic diffusion","doi":"10.1016/j.matpur.2017.11.002","publisher":"Elsevier BV","date_updated":"2025-12-19T11:05:40Z","date_created":"2025-12-19T11:05:33Z","author":[{"first_name":"Michael","last_name":"Winkler","id":"31496","full_name":"Winkler, Michael"}],"volume":112,"year":"2017","citation":{"apa":"Winkler, M. (2017). Singular structure formation in a degenerate haptotaxis model involving myopic diffusion. <i>Journal de Mathématiques Pures et Appliquées</i>, <i>112</i>, 118–169. <a href=\"https://doi.org/10.1016/j.matpur.2017.11.002\">https://doi.org/10.1016/j.matpur.2017.11.002</a>","mla":"Winkler, Michael. “Singular Structure Formation in a Degenerate Haptotaxis Model Involving Myopic Diffusion.” <i>Journal de Mathématiques Pures et Appliquées</i>, vol. 112, Elsevier BV, 2017, pp. 118–69, doi:<a href=\"https://doi.org/10.1016/j.matpur.2017.11.002\">10.1016/j.matpur.2017.11.002</a>.","short":"M. Winkler, Journal de Mathématiques Pures et Appliquées 112 (2017) 118–169.","bibtex":"@article{Winkler_2017, title={Singular structure formation in a degenerate haptotaxis model involving myopic diffusion}, volume={112}, DOI={<a href=\"https://doi.org/10.1016/j.matpur.2017.11.002\">10.1016/j.matpur.2017.11.002</a>}, journal={Journal de Mathématiques Pures et Appliquées}, publisher={Elsevier BV}, author={Winkler, Michael}, year={2017}, pages={118–169} }","chicago":"Winkler, Michael. “Singular Structure Formation in a Degenerate Haptotaxis Model Involving Myopic Diffusion.” <i>Journal de Mathématiques Pures et Appliquées</i> 112 (2017): 118–69. <a href=\"https://doi.org/10.1016/j.matpur.2017.11.002\">https://doi.org/10.1016/j.matpur.2017.11.002</a>.","ieee":"M. Winkler, “Singular structure formation in a degenerate haptotaxis model involving myopic diffusion,” <i>Journal de Mathématiques Pures et Appliquées</i>, vol. 112, pp. 118–169, 2017, doi: <a href=\"https://doi.org/10.1016/j.matpur.2017.11.002\">10.1016/j.matpur.2017.11.002</a>.","ama":"Winkler M. Singular structure formation in a degenerate haptotaxis model involving myopic diffusion. <i>Journal de Mathématiques Pures et Appliquées</i>. 2017;112:118-169. doi:<a href=\"https://doi.org/10.1016/j.matpur.2017.11.002\">10.1016/j.matpur.2017.11.002</a>"},"page":"118-169","intvolume":"       112","publication_status":"published","publication_identifier":{"issn":["0021-7824"]}},{"author":[{"first_name":"Michael","id":"31496","full_name":"Winkler, Michael","last_name":"Winkler"}],"date_created":"2025-12-19T11:07:24Z","volume":30,"publisher":"Springer Science and Business Media LLC","date_updated":"2025-12-19T11:07:30Z","doi":"10.1007/s10884-017-9577-3","title":"One-Dimensional Super-Fast Diffusion: Persistence Versus Extinction Revisited—Extinction at Spatial Infinity","issue":"1","publication_status":"published","publication_identifier":{"issn":["1040-7294","1572-9222"]},"citation":{"ama":"Winkler M. One-Dimensional Super-Fast Diffusion: Persistence Versus Extinction Revisited—Extinction at Spatial Infinity. <i>Journal of Dynamics and Differential Equations</i>. 2017;30(1):331-358. doi:<a href=\"https://doi.org/10.1007/s10884-017-9577-3\">10.1007/s10884-017-9577-3</a>","ieee":"M. Winkler, “One-Dimensional Super-Fast Diffusion: Persistence Versus Extinction Revisited—Extinction at Spatial Infinity,” <i>Journal of Dynamics and Differential Equations</i>, vol. 30, no. 1, pp. 331–358, 2017, doi: <a href=\"https://doi.org/10.1007/s10884-017-9577-3\">10.1007/s10884-017-9577-3</a>.","chicago":"Winkler, Michael. “One-Dimensional Super-Fast Diffusion: Persistence Versus Extinction Revisited—Extinction at Spatial Infinity.” <i>Journal of Dynamics and Differential Equations</i> 30, no. 1 (2017): 331–58. <a href=\"https://doi.org/10.1007/s10884-017-9577-3\">https://doi.org/10.1007/s10884-017-9577-3</a>.","apa":"Winkler, M. (2017). One-Dimensional Super-Fast Diffusion: Persistence Versus Extinction Revisited—Extinction at Spatial Infinity. <i>Journal of Dynamics and Differential Equations</i>, <i>30</i>(1), 331–358. <a href=\"https://doi.org/10.1007/s10884-017-9577-3\">https://doi.org/10.1007/s10884-017-9577-3</a>","bibtex":"@article{Winkler_2017, title={One-Dimensional Super-Fast Diffusion: Persistence Versus Extinction Revisited—Extinction at Spatial Infinity}, volume={30}, DOI={<a href=\"https://doi.org/10.1007/s10884-017-9577-3\">10.1007/s10884-017-9577-3</a>}, number={1}, journal={Journal of Dynamics and Differential Equations}, publisher={Springer Science and Business Media LLC}, author={Winkler, Michael}, year={2017}, pages={331–358} }","short":"M. Winkler, Journal of Dynamics and Differential Equations 30 (2017) 331–358.","mla":"Winkler, Michael. “One-Dimensional Super-Fast Diffusion: Persistence Versus Extinction Revisited—Extinction at Spatial Infinity.” <i>Journal of Dynamics and Differential Equations</i>, vol. 30, no. 1, Springer Science and Business Media LLC, 2017, pp. 331–58, doi:<a href=\"https://doi.org/10.1007/s10884-017-9577-3\">10.1007/s10884-017-9577-3</a>."},"intvolume":"        30","page":"331-358","year":"2017","user_id":"31496","_id":"63378","language":[{"iso":"eng"}],"type":"journal_article","publication":"Journal of Dynamics and Differential Equations","status":"public"},{"type":"journal_article","publication":"Journal of Differential Equations","status":"public","user_id":"31496","_id":"63379","language":[{"iso":"eng"}],"issue":"3","publication_status":"published","publication_identifier":{"issn":["0022-0396"]},"citation":{"apa":"Winkler, M. (2017). Renormalized radial large-data solutions to the higher-dimensional Keller–Segel system with singular sensitivity and signal absorption. <i>Journal of Differential Equations</i>, <i>264</i>(3), 2310–2350. <a href=\"https://doi.org/10.1016/j.jde.2017.10.029\">https://doi.org/10.1016/j.jde.2017.10.029</a>","bibtex":"@article{Winkler_2017, title={Renormalized radial large-data solutions to the higher-dimensional Keller–Segel system with singular sensitivity and signal absorption}, volume={264}, DOI={<a href=\"https://doi.org/10.1016/j.jde.2017.10.029\">10.1016/j.jde.2017.10.029</a>}, number={3}, journal={Journal of Differential Equations}, publisher={Elsevier BV}, author={Winkler, Michael}, year={2017}, pages={2310–2350} }","short":"M. Winkler, Journal of Differential Equations 264 (2017) 2310–2350.","mla":"Winkler, Michael. “Renormalized Radial Large-Data Solutions to the Higher-Dimensional Keller–Segel System with Singular Sensitivity and Signal Absorption.” <i>Journal of Differential Equations</i>, vol. 264, no. 3, Elsevier BV, 2017, pp. 2310–50, doi:<a href=\"https://doi.org/10.1016/j.jde.2017.10.029\">10.1016/j.jde.2017.10.029</a>.","ama":"Winkler M. Renormalized radial large-data solutions to the higher-dimensional Keller–Segel system with singular sensitivity and signal absorption. <i>Journal of Differential Equations</i>. 2017;264(3):2310-2350. doi:<a href=\"https://doi.org/10.1016/j.jde.2017.10.029\">10.1016/j.jde.2017.10.029</a>","ieee":"M. Winkler, “Renormalized radial large-data solutions to the higher-dimensional Keller–Segel system with singular sensitivity and signal absorption,” <i>Journal of Differential Equations</i>, vol. 264, no. 3, pp. 2310–2350, 2017, doi: <a href=\"https://doi.org/10.1016/j.jde.2017.10.029\">10.1016/j.jde.2017.10.029</a>.","chicago":"Winkler, Michael. “Renormalized Radial Large-Data Solutions to the Higher-Dimensional Keller–Segel System with Singular Sensitivity and Signal Absorption.” <i>Journal of Differential Equations</i> 264, no. 3 (2017): 2310–50. <a href=\"https://doi.org/10.1016/j.jde.2017.10.029\">https://doi.org/10.1016/j.jde.2017.10.029</a>."},"intvolume":"       264","page":"2310-2350","year":"2017","date_created":"2025-12-19T11:07:52Z","author":[{"id":"31496","full_name":"Winkler, Michael","last_name":"Winkler","first_name":"Michael"}],"volume":264,"date_updated":"2025-12-19T11:07:59Z","publisher":"Elsevier BV","doi":"10.1016/j.jde.2017.10.029","title":"Renormalized radial large-data solutions to the higher-dimensional Keller–Segel system with singular sensitivity and signal absorption"},{"intvolume":"         4","page":"31-67","citation":{"mla":"Bellomo, Nicola, and Michael Winkler. “Finite-Time Blow-up in a Degenerate Chemotaxis System with Flux Limitation.” <i>Transactions of the American Mathematical Society, Series B</i>, vol. 4, no. 2, American Mathematical Society (AMS), 2017, pp. 31–67, doi:<a href=\"https://doi.org/10.1090/btran/17\">10.1090/btran/17</a>.","short":"N. Bellomo, M. Winkler, Transactions of the American Mathematical Society, Series B 4 (2017) 31–67.","bibtex":"@article{Bellomo_Winkler_2017, title={Finite-time blow-up in a degenerate chemotaxis system with flux limitation}, volume={4}, DOI={<a href=\"https://doi.org/10.1090/btran/17\">10.1090/btran/17</a>}, number={2}, journal={Transactions of the American Mathematical Society, Series B}, publisher={American Mathematical Society (AMS)}, author={Bellomo, Nicola and Winkler, Michael}, year={2017}, pages={31–67} }","apa":"Bellomo, N., &#38; Winkler, M. (2017). Finite-time blow-up in a degenerate chemotaxis system with flux limitation. <i>Transactions of the American Mathematical Society, Series B</i>, <i>4</i>(2), 31–67. <a href=\"https://doi.org/10.1090/btran/17\">https://doi.org/10.1090/btran/17</a>","ama":"Bellomo N, Winkler M. Finite-time blow-up in a degenerate chemotaxis system with flux limitation. <i>Transactions of the American Mathematical Society, Series B</i>. 2017;4(2):31-67. doi:<a href=\"https://doi.org/10.1090/btran/17\">10.1090/btran/17</a>","chicago":"Bellomo, Nicola, and Michael Winkler. “Finite-Time Blow-up in a Degenerate Chemotaxis System with Flux Limitation.” <i>Transactions of the American Mathematical Society, Series B</i> 4, no. 2 (2017): 31–67. <a href=\"https://doi.org/10.1090/btran/17\">https://doi.org/10.1090/btran/17</a>.","ieee":"N. Bellomo and M. Winkler, “Finite-time blow-up in a degenerate chemotaxis system with flux limitation,” <i>Transactions of the American Mathematical Society, Series B</i>, vol. 4, no. 2, pp. 31–67, 2017, doi: <a href=\"https://doi.org/10.1090/btran/17\">10.1090/btran/17</a>."},"year":"2017","issue":"2","publication_identifier":{"issn":["2330-0000"]},"publication_status":"published","doi":"10.1090/btran/17","title":"Finite-time blow-up in a degenerate chemotaxis system with flux limitation","volume":4,"author":[{"first_name":"Nicola","full_name":"Bellomo, Nicola","last_name":"Bellomo"},{"id":"31496","full_name":"Winkler, Michael","last_name":"Winkler","first_name":"Michael"}],"date_created":"2025-12-19T11:09:53Z","date_updated":"2025-12-19T11:10:17Z","publisher":"American Mathematical Society (AMS)","status":"public","abstract":[{"text":"<p>This paper is concerned with radially symmetric solutions of the parabolic-elliptic version of the Keller-Segel system with flux limitation, as given by <disp-formula content-type=\"math/mathml\">\r\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"StartLayout 1st Row  with Label left-parenthesis reverse-solidus star right-parenthesis EndLabel StartLayout Enlarged left-brace 1st Row  u Subscript t Baseline equals nabla dot left-parenthesis StartFraction u nabla u Over StartRoot u squared plus StartAbsoluteValue nabla u EndAbsoluteValue squared EndRoot EndFraction right-parenthesis minus chi nabla dot left-parenthesis StartFraction u nabla v Over StartRoot 1 plus StartAbsoluteValue nabla v EndAbsoluteValue squared EndRoot EndFraction right-parenthesis comma 2nd Row  0 equals normal upper Delta v minus mu plus u comma EndLayout EndLayout\">\r\n  <mml:semantics>\r\n    <mml:mtable side=\"left\" displaystyle=\"false\">\r\n      <mml:mlabeledtr>\r\n        <mml:mtd>\r\n          <mml:mtext>(\\star)</mml:mtext>\r\n        </mml:mtd>\r\n        <mml:mtd>\r\n          <mml:mrow>\r\n            <mml:mo>{</mml:mo>\r\n            <mml:mtable columnalign=\"left left\" rowspacing=\"0.5em 0.2em\" columnspacing=\"1em\" displaystyle=\"false\">\r\n              <mml:mtr>\r\n                <mml:mtd>\r\n                  <mml:msub>\r\n                    <mml:mi>u</mml:mi>\r\n                    <mml:mi>t</mml:mi>\r\n                  </mml:msub>\r\n                  <mml:mo>=</mml:mo>\r\n                  <mml:mi mathvariant=\"normal\">∇<!-- ∇ --></mml:mi>\r\n                  <mml:mo>⋅<!-- ⋅ --></mml:mo>\r\n                  <mml:mstyle scriptlevel=\"0\">\r\n                    <mml:mrow class=\"MJX-TeXAtom-ORD\">\r\n                      <mml:mo maxsize=\"1.623em\" minsize=\"1.623em\">(</mml:mo>\r\n                    </mml:mrow>\r\n                  </mml:mstyle>\r\n                  <mml:mfrac>\r\n                    <mml:mrow>\r\n                      <mml:mi>u</mml:mi>\r\n                      <mml:mi mathvariant=\"normal\">∇<!-- ∇ --></mml:mi>\r\n                      <mml:mi>u</mml:mi>\r\n                    </mml:mrow>\r\n                    <mml:msqrt>\r\n                      <mml:msup>\r\n                        <mml:mi>u</mml:mi>\r\n                        <mml:mn>2</mml:mn>\r\n                      </mml:msup>\r\n                      <mml:mo>+</mml:mo>\r\n                      <mml:mrow class=\"MJX-TeXAtom-ORD\">\r\n                        <mml:mo stretchy=\"false\">|</mml:mo>\r\n                      </mml:mrow>\r\n                      <mml:mi mathvariant=\"normal\">∇<!-- ∇ --></mml:mi>\r\n                      <mml:mi>u</mml:mi>\r\n                      <mml:msup>\r\n                        <mml:mrow class=\"MJX-TeXAtom-ORD\">\r\n                          <mml:mo stretchy=\"false\">|</mml:mo>\r\n                        </mml:mrow>\r\n                        <mml:mn>2</mml:mn>\r\n                      </mml:msup>\r\n                    </mml:msqrt>\r\n                  </mml:mfrac>\r\n                  <mml:mstyle scriptlevel=\"0\">\r\n                    <mml:mrow class=\"MJX-TeXAtom-ORD\">\r\n                      <mml:mo maxsize=\"1.623em\" minsize=\"1.623em\">)</mml:mo>\r\n                    </mml:mrow>\r\n                  </mml:mstyle>\r\n                  <mml:mo>−<!-- − --></mml:mo>\r\n                  <mml:mi>χ<!-- χ --></mml:mi>\r\n                  <mml:mspace width=\"thinmathspace\" />\r\n                  <mml:mi mathvariant=\"normal\">∇<!-- ∇ --></mml:mi>\r\n                  <mml:mo>⋅<!-- ⋅ --></mml:mo>\r\n                  <mml:mstyle scriptlevel=\"0\">\r\n                    <mml:mrow class=\"MJX-TeXAtom-ORD\">\r\n                      <mml:mo maxsize=\"1.623em\" minsize=\"1.623em\">(</mml:mo>\r\n                    </mml:mrow>\r\n                  </mml:mstyle>\r\n                  <mml:mfrac>\r\n                    <mml:mrow>\r\n                      <mml:mi>u</mml:mi>\r\n                      <mml:mi mathvariant=\"normal\">∇<!-- ∇ --></mml:mi>\r\n                      <mml:mi>v</mml:mi>\r\n                    </mml:mrow>\r\n                    <mml:msqrt>\r\n                      <mml:mn>1</mml:mn>\r\n                      <mml:mo>+</mml:mo>\r\n                      <mml:mrow class=\"MJX-TeXAtom-ORD\">\r\n                        <mml:mo stretchy=\"false\">|</mml:mo>\r\n                      </mml:mrow>\r\n                      <mml:mi mathvariant=\"normal\">∇<!-- ∇ --></mml:mi>\r\n                      <mml:mi>v</mml:mi>\r\n                      <mml:msup>\r\n                        <mml:mrow class=\"MJX-TeXAtom-ORD\">\r\n                          <mml:mo stretchy=\"false\">|</mml:mo>\r\n                        </mml:mrow>\r\n                        <mml:mn>2</mml:mn>\r\n                      </mml:msup>\r\n                    </mml:msqrt>\r\n                  </mml:mfrac>\r\n                  <mml:mstyle scriptlevel=\"0\">\r\n                    <mml:mrow class=\"MJX-TeXAtom-ORD\">\r\n                      <mml:mo maxsize=\"1.623em\" minsize=\"1.623em\">)</mml:mo>\r\n                    </mml:mrow>\r\n                  </mml:mstyle>\r\n                  <mml:mo>,</mml:mo>\r\n                </mml:mtd>\r\n              </mml:mtr>\r\n              <mml:mtr>\r\n                <mml:mtd>\r\n                  <mml:mn>0</mml:mn>\r\n                  <mml:mo>=</mml:mo>\r\n                  <mml:mi mathvariant=\"normal\">Δ<!-- Δ --></mml:mi>\r\n                  <mml:mi>v</mml:mi>\r\n                  <mml:mo>−<!-- − --></mml:mo>\r\n                  <mml:mi>μ<!-- μ --></mml:mi>\r\n                  <mml:mo>+</mml:mo>\r\n                  <mml:mi>u</mml:mi>\r\n                  <mml:mo>,</mml:mo>\r\n                </mml:mtd>\r\n              </mml:mtr>\r\n            </mml:mtable>\r\n            <mml:mo fence=\"true\" stretchy=\"true\" symmetric=\"true\" />\r\n          </mml:mrow>\r\n        </mml:mtd>\r\n      </mml:mlabeledtr>\r\n    </mml:mtable>\r\n    <mml:annotation encoding=\"application/x-tex\">\\begin{equation}\\tag {\\star } \\begin {cases} u_t=\\nabla \\cdot \\Big (\\frac {u\\nabla u}{\\sqrt {u^2+|\\nabla u|^2}}\\Big ) - \\chi \\, \\nabla \\cdot \\Big (\\frac {u\\nabla v}{\\sqrt {1+|\\nabla v|^2}}\\Big ), \\\\[3pt] 0=\\Delta v - \\mu + u, \\end{cases} \\end{equation}</mml:annotation>\r\n  </mml:semantics>\r\n</mml:math>\r\n</disp-formula>\r\n under the initial condition <inline-formula content-type=\"math/mathml\">\r\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"u vertical-bar Subscript t equals 0 Baseline equals u 0 greater-than 0\">\r\n  <mml:semantics>\r\n    <mml:mrow>\r\n      <mml:mi>u</mml:mi>\r\n      <mml:msub>\r\n        <mml:mrow class=\"MJX-TeXAtom-ORD\">\r\n          <mml:mo stretchy=\"false\">|</mml:mo>\r\n        </mml:mrow>\r\n        <mml:mrow class=\"MJX-TeXAtom-ORD\">\r\n          <mml:mi>t</mml:mi>\r\n          <mml:mo>=</mml:mo>\r\n          <mml:mn>0</mml:mn>\r\n        </mml:mrow>\r\n      </mml:msub>\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:mrow>\r\n    <mml:annotation encoding=\"application/x-tex\">u|_{t=0}=u_0&gt;0</mml:annotation>\r\n  </mml:semantics>\r\n</mml:math>\r\n</inline-formula> and no-flux boundary conditions in a ball <inline-formula content-type=\"math/mathml\">\r\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper Omega subset-of double-struck upper R Superscript n\">\r\n  <mml:semantics>\r\n    <mml:mrow>\r\n      <mml:mi mathvariant=\"normal\">Ω<!-- Ω --></mml:mi>\r\n      <mml:mo>⊂<!-- ⊂ --></mml:mo>\r\n      <mml:msup>\r\n        <mml:mrow class=\"MJX-TeXAtom-ORD\">\r\n          <mml:mi mathvariant=\"double-struck\">R</mml:mi>\r\n        </mml:mrow>\r\n        <mml:mi>n</mml:mi>\r\n      </mml:msup>\r\n    </mml:mrow>\r\n    <mml:annotation encoding=\"application/x-tex\">\\Omega \\subset \\mathbb {R}^n</mml:annotation>\r\n  </mml:semantics>\r\n</mml:math>\r\n</inline-formula>, where <inline-formula content-type=\"math/mathml\">\r\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"chi greater-than 0\">\r\n  <mml:semantics>\r\n    <mml:mrow>\r\n      <mml:mi>χ<!-- χ --></mml:mi>\r\n      <mml:mo>&gt;</mml:mo>\r\n      <mml:mn>0</mml:mn>\r\n    </mml:mrow>\r\n    <mml:annotation encoding=\"application/x-tex\">\\chi &gt;0</mml:annotation>\r\n  </mml:semantics>\r\n</mml:math>\r\n</inline-formula> and <inline-formula content-type=\"math/mathml\">\r\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"mu colon equals StartFraction 1 Over StartAbsoluteValue normal upper Omega EndAbsoluteValue EndFraction integral Underscript normal upper Omega Endscripts u 0\">\r\n  <mml:semantics>\r\n    <mml:mrow>\r\n      <mml:mi>μ<!-- μ --></mml:mi>\r\n      <mml:mo>:=</mml:mo>\r\n      <mml:mfrac>\r\n        <mml:mn>1</mml:mn>\r\n        <mml:mrow>\r\n          <mml:mrow class=\"MJX-TeXAtom-ORD\">\r\n            <mml:mo stretchy=\"false\">|</mml:mo>\r\n          </mml:mrow>\r\n          <mml:mi mathvariant=\"normal\">Ω<!-- Ω --></mml:mi>\r\n          <mml:mrow class=\"MJX-TeXAtom-ORD\">\r\n            <mml:mo stretchy=\"false\">|</mml:mo>\r\n          </mml:mrow>\r\n        </mml:mrow>\r\n      </mml:mfrac>\r\n      <mml:msub>\r\n        <mml:mo>∫<!-- ∫ --></mml:mo>\r\n        <mml:mi mathvariant=\"normal\">Ω<!-- Ω --></mml:mi>\r\n      </mml:msub>\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:annotation encoding=\"application/x-tex\">\\mu :=\\frac {1}{|\\Omega |} \\int _\\Omega u_0</mml:annotation>\r\n  </mml:semantics>\r\n</mml:math>\r\n</inline-formula>. A previous result of the authors [Comm. Partial Differential Equations 42 (2017), 436–473] has asserted global existence of bounded classical solutions for arbitrary positive radial initial data <inline-formula content-type=\"math/mathml\">\r\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"u 0 element-of upper C cubed left-parenthesis normal upper Omega overbar right-parenthesis\">\r\n  <mml:semantics>\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>3</mml:mn>\r\n      </mml:msup>\r\n      <mml:mo stretchy=\"false\">(</mml:mo>\r\n      <mml:mrow class=\"MJX-TeXAtom-ORD\">\r\n        <mml:mover>\r\n          <mml:mi mathvariant=\"normal\">Ω<!-- Ω --></mml:mi>\r\n          <mml:mo stretchy=\"false\">¯<!-- ¯ --></mml:mo>\r\n        </mml:mover>\r\n      </mml:mrow>\r\n      <mml:mo stretchy=\"false\">)</mml:mo>\r\n    </mml:mrow>\r\n    <mml:annotation encoding=\"application/x-tex\">u_0\\in C^3(\\bar \\Omega )</mml:annotation>\r\n  </mml:semantics>\r\n</mml:math>\r\n</inline-formula> when either <inline-formula content-type=\"math/mathml\">\r\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"n greater-than-or-equal-to 2\">\r\n  <mml:semantics>\r\n    <mml:mrow>\r\n      <mml:mi>n</mml:mi>\r\n      <mml:mo>≥<!-- ≥ --></mml:mo>\r\n      <mml:mn>2</mml:mn>\r\n    </mml:mrow>\r\n    <mml:annotation encoding=\"application/x-tex\">n\\ge 2</mml:annotation>\r\n  </mml:semantics>\r\n</mml:math>\r\n</inline-formula> and <inline-formula content-type=\"math/mathml\">\r\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"chi greater-than 1\">\r\n  <mml:semantics>\r\n    <mml:mrow>\r\n      <mml:mi>χ<!-- χ --></mml:mi>\r\n      <mml:mo>&gt;</mml:mo>\r\n      <mml:mn>1</mml:mn>\r\n    </mml:mrow>\r\n    <mml:annotation encoding=\"application/x-tex\">\\chi &gt;1</mml:annotation>\r\n  </mml:semantics>\r\n</mml:math>\r\n</inline-formula>, or <inline-formula content-type=\"math/mathml\">\r\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"n equals 1\">\r\n  <mml:semantics>\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:annotation encoding=\"application/x-tex\">n=1</mml:annotation>\r\n  </mml:semantics>\r\n</mml:math>\r\n</inline-formula> and <inline-formula content-type=\"math/mathml\">\r\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"integral Underscript normal upper Omega Endscripts u 0 greater-than StartFraction 1 Over StartRoot left-parenthesis chi squared minus 1 right-parenthesis Subscript plus Baseline EndRoot EndFraction\">\r\n  <mml:semantics>\r\n    <mml:mrow>\r\n      <mml:msub>\r\n        <mml:mo>∫<!-- ∫ --></mml:mo>\r\n        <mml:mi mathvariant=\"normal\">Ω<!-- Ω --></mml:mi>\r\n      </mml:msub>\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:mfrac>\r\n        <mml:mn>1</mml:mn>\r\n        <mml:msqrt>\r\n          <mml:mo stretchy=\"false\">(</mml:mo>\r\n          <mml:msup>\r\n            <mml:mi>χ<!-- χ --></mml:mi>\r\n            <mml:mn>2</mml:mn>\r\n          </mml:msup>\r\n          <mml:mo>−<!-- − --></mml:mo>\r\n          <mml:mn>1</mml:mn>\r\n          <mml:msub>\r\n            <mml:mo stretchy=\"false\">)</mml:mo>\r\n            <mml:mo>+</mml:mo>\r\n          </mml:msub>\r\n        </mml:msqrt>\r\n      </mml:mfrac>\r\n    </mml:mrow>\r\n    <mml:annotation encoding=\"application/x-tex\">\\int _\\Omega u_0&gt;\\frac {1}{\\sqrt {(\\chi ^2-1)_+}}</mml:annotation>\r\n  </mml:semantics>\r\n</mml:math>\r\n</inline-formula>.</p>\r\n\r\n<p>This present paper shows that these conditions are essentially optimal: Indeed, it is shown that if the taxis coefficient satisfies <inline-formula content-type=\"math/mathml\">\r\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"chi greater-than 1\">\r\n  <mml:semantics>\r\n    <mml:mrow>\r\n      <mml:mi>χ<!-- χ --></mml:mi>\r\n      <mml:mo>&gt;</mml:mo>\r\n      <mml:mn>1</mml:mn>\r\n    </mml:mrow>\r\n    <mml:annotation encoding=\"application/x-tex\">\\chi &gt;1</mml:annotation>\r\n  </mml:semantics>\r\n</mml:math>\r\n</inline-formula>, then for any choice of <disp-formula content-type=\"math/mathml\">\r\n\\[\r\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"StartLayout Enlarged left-brace 1st Row 1st Column m greater-than StartFraction 1 Over StartRoot chi squared minus 1 EndRoot EndFraction 2nd Column a m p semicolon if n equals 1 comma 2nd Row 1st Column m greater-than 0 is arbitrary 2nd Column a m p semicolon if n greater-than-or-equal-to 2 comma EndLayout\">\r\n  <mml:semantics>\r\n    <mml:mrow>\r\n      <mml:mo>{</mml:mo>\r\n      <mml:mtable columnalign=\"left left\" rowspacing=\".2em\" columnspacing=\"1em\" displaystyle=\"false\">\r\n        <mml:mtr>\r\n          <mml:mtd>\r\n            <mml:mi>m</mml:mi>\r\n            <mml:mo>&gt;</mml:mo>\r\n            <mml:mfrac>\r\n              <mml:mn>1</mml:mn>\r\n              <mml:msqrt>\r\n                <mml:msup>\r\n                  <mml:mi>χ<!-- χ --></mml:mi>\r\n                  <mml:mn>2</mml:mn>\r\n                </mml:msup>\r\n                <mml:mo>−<!-- − --></mml:mo>\r\n                <mml:mn>1</mml:mn>\r\n              </mml:msqrt>\r\n            </mml:mfrac>\r\n          </mml:mtd>\r\n          <mml:mtd>\r\n            <mml:mi>a</mml:mi>\r\n            <mml:mi>m</mml:mi>\r\n            <mml:mi>p</mml:mi>\r\n            <mml:mo>;</mml:mo>\r\n            <mml:mrow>\r\n              <mml:mtext>if </mml:mtext>\r\n              <mml:mrow class=\"MJX-TeXAtom-ORD\">\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:mrow>\r\n            <mml:mo>,</mml:mo>\r\n          </mml:mtd>\r\n        </mml:mtr>\r\n        <mml:mtr>\r\n          <mml:mtd>\r\n            <mml:mrow>\r\n              <mml:mrow class=\"MJX-TeXAtom-ORD\">\r\n                <mml:mi>m</mml:mi>\r\n                <mml:mo>&gt;</mml:mo>\r\n                <mml:mn>0</mml:mn>\r\n              </mml:mrow>\r\n              <mml:mtext> is arbitrary</mml:mtext>\r\n            </mml:mrow>\r\n          </mml:mtd>\r\n          <mml:mtd>\r\n            <mml:mi>a</mml:mi>\r\n            <mml:mi>m</mml:mi>\r\n            <mml:mi>p</mml:mi>\r\n            <mml:mo>;</mml:mo>\r\n            <mml:mrow>\r\n              <mml:mtext>if </mml:mtext>\r\n              <mml:mrow class=\"MJX-TeXAtom-ORD\">\r\n                <mml:mi>n</mml:mi>\r\n                <mml:mo>≥<!-- ≥ --></mml:mo>\r\n                <mml:mn>2</mml:mn>\r\n              </mml:mrow>\r\n            </mml:mrow>\r\n            <mml:mo>,</mml:mo>\r\n          </mml:mtd>\r\n        </mml:mtr>\r\n      </mml:mtable>\r\n      <mml:mo fence=\"true\" stretchy=\"true\" symmetric=\"true\" />\r\n    </mml:mrow>\r\n    <mml:annotation encoding=\"application/x-tex\">\\begin {cases} m&gt;\\frac {1}{\\sqrt {\\chi ^2-1}} &amp; \\text {if $n=1$}, \\\\ \\text {$m&gt;0$ is arbitrary} &amp; \\text {if $n\\ge 2$}, \\end {cases}</mml:annotation>\r\n  </mml:semantics>\r\n</mml:math>\r\n\\]\r\n</disp-formula> there exist positive initial data <inline-formula content-type=\"math/mathml\">\r\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"u 0 element-of upper C cubed left-parenthesis normal upper Omega overbar right-parenthesis\">\r\n  <mml:semantics>\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>3</mml:mn>\r\n      </mml:msup>\r\n      <mml:mo stretchy=\"false\">(</mml:mo>\r\n      <mml:mrow class=\"MJX-TeXAtom-ORD\">\r\n        <mml:mover>\r\n          <mml:mi mathvariant=\"normal\">Ω<!-- Ω --></mml:mi>\r\n          <mml:mo stretchy=\"false\">¯<!-- ¯ --></mml:mo>\r\n        </mml:mover>\r\n      </mml:mrow>\r\n      <mml:mo stretchy=\"false\">)</mml:mo>\r\n    </mml:mrow>\r\n    <mml:annotation encoding=\"application/x-tex\">u_0\\in C^3(\\bar \\Omega )</mml:annotation>\r\n  </mml:semantics>\r\n</mml:math>\r\n</inline-formula> satisfying <inline-formula content-type=\"math/mathml\">\r\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"integral Underscript normal upper Omega Endscripts u 0 equals m\">\r\n  <mml:semantics>\r\n    <mml:mrow>\r\n      <mml:msub>\r\n        <mml:mo>∫<!-- ∫ --></mml:mo>\r\n        <mml:mi mathvariant=\"normal\">Ω<!-- Ω --></mml:mi>\r\n      </mml:msub>\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:mi>m</mml:mi>\r\n    </mml:mrow>\r\n    <mml:annotation encoding=\"application/x-tex\">\\int _\\Omega u_0=m</mml:annotation>\r\n  </mml:semantics>\r\n</mml:math>\r\n</inline-formula> which are such that for some <inline-formula content-type=\"math/mathml\">\r\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper T greater-than 0\">\r\n  <mml:semantics>\r\n    <mml:mrow>\r\n      <mml:mi>T</mml:mi>\r\n      <mml:mo>&gt;</mml:mo>\r\n      <mml:mn>0</mml:mn>\r\n    </mml:mrow>\r\n    <mml:annotation encoding=\"application/x-tex\">T&gt;0</mml:annotation>\r\n  </mml:semantics>\r\n</mml:math>\r\n</inline-formula>, (<inline-formula content-type=\"math/mathml\">\r\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"star\">\r\n  <mml:semantics>\r\n    <mml:mo>⋆<!-- ⋆ --></mml:mo>\r\n    <mml:annotation encoding=\"application/x-tex\">\\star</mml:annotation>\r\n  </mml:semantics>\r\n</mml:math>\r\n</inline-formula>) possesses a uniquely determined classical solution <inline-formula content-type=\"math/mathml\">\r\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-parenthesis u comma v right-parenthesis\">\r\n  <mml:semantics>\r\n    <mml:mrow>\r\n      <mml:mo stretchy=\"false\">(</mml:mo>\r\n      <mml:mi>u</mml:mi>\r\n      <mml:mo>,</mml:mo>\r\n      <mml:mi>v</mml:mi>\r\n      <mml:mo stretchy=\"false\">)</mml:mo>\r\n    </mml:mrow>\r\n    <mml:annotation encoding=\"application/x-tex\">(u,v)</mml:annotation>\r\n  </mml:semantics>\r\n</mml:math>\r\n</inline-formula> in <inline-formula content-type=\"math/mathml\">\r\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper Omega times left-parenthesis 0 comma upper T right-parenthesis\">\r\n  <mml:semantics>\r\n    <mml:mrow>\r\n      <mml:mi mathvariant=\"normal\">Ω<!-- Ω --></mml:mi>\r\n      <mml:mo>×<!-- × --></mml:mo>\r\n      <mml:mo stretchy=\"false\">(</mml:mo>\r\n      <mml:mn>0</mml:mn>\r\n      <mml:mo>,</mml:mo>\r\n      <mml:mi>T</mml:mi>\r\n      <mml:mo stretchy=\"false\">)</mml:mo>\r\n    </mml:mrow>\r\n    <mml:annotation encoding=\"application/x-tex\">\\Omega \\times (0,T)</mml:annotation>\r\n  </mml:semantics>\r\n</mml:math>\r\n</inline-formula> blowing up at time <inline-formula content-type=\"math/mathml\">\r\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper T\">\r\n  <mml:semantics>\r\n    <mml:mi>T</mml:mi>\r\n    <mml:annotation encoding=\"application/x-tex\">T</mml:annotation>\r\n  </mml:semantics>\r\n</mml:math>\r\n</inline-formula> in the sense that <inline-formula content-type=\"math/mathml\">\r\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"limit sup double-vertical-bar u left-parenthesis dot comma t right-parenthesis double-vertical-bar Subscript upper L Sub Superscript normal infinity Subscript left-parenthesis normal upper Omega right-parenthesis Baseline equals normal infinity\">\r\n  <mml:semantics>\r\n    <mml:mrow>\r\n      <mml:munder>\r\n        <mml:mo movablelimits=\"true\" form=\"prefix\">lim sup</mml:mo>\r\n        <mml:mrow class=\"MJX-TeXAtom-ORD\">\r\n          <mml:mi>t</mml:mi>\r\n          <mml:mo stretchy=\"false\">↗<!-- ↗ --></mml:mo>\r\n          <mml:mi>T</mml:mi>\r\n        </mml:mrow>\r\n      </mml:munder>\r\n      <mml:mo fence=\"false\" stretchy=\"false\">‖<!-- ‖ --></mml:mo>\r\n      <mml:mi>u</mml:mi>\r\n      <mml:mo stretchy=\"false\">(</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 stretchy=\"false\">)</mml:mo>\r\n      <mml:msub>\r\n        <mml:mo fence=\"false\" stretchy=\"false\">‖<!-- ‖ --></mml:mo>\r\n        <mml:mrow class=\"MJX-TeXAtom-ORD\">\r\n          <mml:msup>\r\n            <mml:mi>L</mml:mi>\r\n            <mml:mi mathvariant=\"normal\">∞<!-- ∞ --></mml:mi>\r\n          </mml:msup>\r\n          <mml:mo stretchy=\"false\">(</mml:mo>\r\n          <mml:mi mathvariant=\"normal\">Ω<!-- Ω --></mml:mi>\r\n          <mml:mo stretchy=\"false\">)</mml:mo>\r\n        </mml:mrow>\r\n      </mml:msub>\r\n      <mml:mo>=</mml:mo>\r\n      <mml:mi mathvariant=\"normal\">∞<!-- ∞ --></mml:mi>\r\n    </mml:mrow>\r\n    <mml:annotation encoding=\"application/x-tex\">\\limsup _{t\\nearrow T} \\|u(\\cdot ,t)\\|_{L^\\infty (\\Omega )}=\\infty</mml:annotation>\r\n  </mml:semantics>\r\n</mml:math>\r\n</inline-formula>.</p>\r\n\r\n<p>This result is derived by means of a comparison argument applied to the doubly degenerate scalar parabolic equation satisfied by the mass accumulation function associated with (<inline-formula content-type=\"math/mathml\">\r\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"star\">\r\n  <mml:semantics>\r\n    <mml:mo>⋆<!-- ⋆ --></mml:mo>\r\n    <mml:annotation encoding=\"application/x-tex\">\\star</mml:annotation>\r\n  </mml:semantics>\r\n</mml:math>\r\n</inline-formula>).</p>","lang":"eng"}],"publication":"Transactions of the American Mathematical Society, Series B","type":"journal_article","language":[{"iso":"eng"}],"user_id":"31496","_id":"63383"}]