[{"_id":"63371","user_id":"31496","language":[{"iso":"eng"}],"type":"journal_article","publication":"European Journal of Applied Mathematics","abstract":[{"text":"<jats:p>Adhesion between cells and other cells (cell–cell adhesion) or other tissue components (cell–matrix adhesion) is an intrinsically non-local phenomenon. Consequently, a number of recently developed mathematical models for cell adhesion have taken the form of non-local partial differential equations, where the non-local term arises inside a spatial derivative. The mathematical properties of such a non-local gradient term are not yet well understood. Here we use sophisticated estimation techniques to show local and global existence of classical solutions for such examples of adhesion-type models, and we provide a uniform upper bound for the solutions. Further, we discuss the significance of these results to applications in cell sorting and in cancer invasion and support the theoretical results through numerical simulations.</jats:p>","lang":"eng"}],"status":"public","publisher":"Cambridge University Press (CUP)","date_updated":"2025-12-19T11:03:57Z","author":[{"first_name":"T.","full_name":"HILLEN, T.","last_name":"HILLEN"},{"first_name":"K. J.","full_name":"PAINTER, K. J.","last_name":"PAINTER"},{"full_name":"Winkler, Michael","id":"31496","last_name":"Winkler","first_name":"Michael"}],"date_created":"2025-12-19T11:03:50Z","volume":29,"title":"Global solvability and explicit bounds for non-local adhesion models","doi":"10.1017/s0956792517000328","publication_status":"published","publication_identifier":{"issn":["0956-7925","1469-4425"]},"issue":"4","year":"2017","citation":{"chicago":"HILLEN, T., K. J. PAINTER, and Michael Winkler. “Global Solvability and Explicit Bounds for Non-Local Adhesion Models.” <i>European Journal of Applied Mathematics</i> 29, no. 4 (2017): 645–84. <a href=\"https://doi.org/10.1017/s0956792517000328\">https://doi.org/10.1017/s0956792517000328</a>.","ieee":"T. HILLEN, K. J. PAINTER, and M. Winkler, “Global solvability and explicit bounds for non-local adhesion models,” <i>European Journal of Applied Mathematics</i>, vol. 29, no. 4, pp. 645–684, 2017, doi: <a href=\"https://doi.org/10.1017/s0956792517000328\">10.1017/s0956792517000328</a>.","ama":"HILLEN T, PAINTER KJ, Winkler M. Global solvability and explicit bounds for non-local adhesion models. <i>European Journal of Applied Mathematics</i>. 2017;29(4):645-684. doi:<a href=\"https://doi.org/10.1017/s0956792517000328\">10.1017/s0956792517000328</a>","short":"T. HILLEN, K.J. PAINTER, M. Winkler, European Journal of Applied Mathematics 29 (2017) 645–684.","bibtex":"@article{HILLEN_PAINTER_Winkler_2017, title={Global solvability and explicit bounds for non-local adhesion models}, volume={29}, DOI={<a href=\"https://doi.org/10.1017/s0956792517000328\">10.1017/s0956792517000328</a>}, number={4}, journal={European Journal of Applied Mathematics}, publisher={Cambridge University Press (CUP)}, author={HILLEN, T. and PAINTER, K. J. and Winkler, Michael}, year={2017}, pages={645–684} }","mla":"HILLEN, T., et al. “Global Solvability and Explicit Bounds for Non-Local Adhesion Models.” <i>European Journal of Applied Mathematics</i>, vol. 29, no. 4, Cambridge University Press (CUP), 2017, pp. 645–84, doi:<a href=\"https://doi.org/10.1017/s0956792517000328\">10.1017/s0956792517000328</a>.","apa":"HILLEN, T., PAINTER, K. J., &#38; Winkler, M. (2017). Global solvability and explicit bounds for non-local adhesion models. <i>European Journal of Applied Mathematics</i>, <i>29</i>(4), 645–684. <a href=\"https://doi.org/10.1017/s0956792517000328\">https://doi.org/10.1017/s0956792517000328</a>"},"page":"645-684","intvolume":"        29"},{"date_updated":"2025-12-19T11:04:46Z","publisher":"Springer Science and Business Media LLC","volume":289,"date_created":"2025-12-19T11:04:38Z","author":[{"first_name":"Yulan","last_name":"Wang","full_name":"Wang, Yulan"},{"last_name":"Winkler","id":"31496","full_name":"Winkler, Michael","first_name":"Michael"},{"first_name":"Zhaoyin","full_name":"Xiang, Zhaoyin","last_name":"Xiang"}],"title":"The small-convection limit in a two-dimensional chemotaxis-Navier–Stokes system","doi":"10.1007/s00209-017-1944-6","publication_identifier":{"issn":["0025-5874","1432-1823"]},"publication_status":"published","issue":"1-2","year":"2017","intvolume":"       289","page":"71-108","citation":{"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>","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>.","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} }","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>","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>.","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>."},"_id":"63372","user_id":"31496","language":[{"iso":"eng"}],"publication":"Mathematische Zeitschrift","type":"journal_article","status":"public"},{"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","author":[{"first_name":"Michael","id":"31496","full_name":"Winkler, Michael","last_name":"Winkler"}],"date_created":"2025-12-19T11:05:33Z","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>","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} }","short":"M. Winkler, Journal de Mathématiques Pures et Appliquées 112 (2017) 118–169.","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>.","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>"},"intvolume":"       112","page":"118-169","publication_status":"published","publication_identifier":{"issn":["0021-7824"]}},{"year":"2017","page":"331-358","intvolume":"        30","citation":{"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>.","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>.","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>.","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>"},"publication_identifier":{"issn":["1040-7294","1572-9222"]},"publication_status":"published","issue":"1","title":"One-Dimensional Super-Fast Diffusion: Persistence Versus Extinction Revisited—Extinction at Spatial Infinity","doi":"10.1007/s10884-017-9577-3","date_updated":"2025-12-19T11:07:30Z","publisher":"Springer Science and Business Media LLC","volume":30,"date_created":"2025-12-19T11:07:24Z","author":[{"last_name":"Winkler","id":"31496","full_name":"Winkler, Michael","first_name":"Michael"}],"status":"public","publication":"Journal of Dynamics and Differential Equations","type":"journal_article","language":[{"iso":"eng"}],"_id":"63378","user_id":"31496"},{"publication":"Journal of Differential Equations","type":"journal_article","status":"public","user_id":"31496","_id":"63379","language":[{"iso":"eng"}],"issue":"3","publication_identifier":{"issn":["0022-0396"]},"publication_status":"published","page":"2310-2350","intvolume":"       264","citation":{"short":"M. Winkler, Journal of Differential Equations 264 (2017) 2310–2350.","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} }","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>.","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>","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>.","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>.","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>"},"year":"2017","volume":264,"author":[{"first_name":"Michael","last_name":"Winkler","id":"31496","full_name":"Winkler, Michael"}],"date_created":"2025-12-19T11:07:52Z","publisher":"Elsevier BV","date_updated":"2025-12-19T11:07:59Z","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"},{"doi":"10.1090/btran/17","title":"Finite-time blow-up in a degenerate chemotaxis system with flux limitation","volume":4,"author":[{"first_name":"Nicola","last_name":"Bellomo","full_name":"Bellomo, Nicola"},{"full_name":"Winkler, Michael","id":"31496","last_name":"Winkler","first_name":"Michael"}],"date_created":"2025-12-19T11:09:53Z","publisher":"American Mathematical Society (AMS)","date_updated":"2025-12-19T11:10:17Z","intvolume":"         4","page":"31-67","citation":{"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>.","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>.","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>","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} }","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>.","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>"},"year":"2017","issue":"2","publication_identifier":{"issn":["2330-0000"]},"publication_status":"published","language":[{"iso":"eng"}],"user_id":"31496","_id":"63383","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"},{"publication_identifier":{"issn":["2469-9950","2469-9969"]},"publication_status":"published","issue":"5","year":"2017","intvolume":"        95","citation":{"bibtex":"@article{Thevenard_Boutigny_Güsken_Becerra_Ulysse_Shihab_Lemaître_Kim_Jeudy_Gourdon_2017, title={Spin transfer and spin-orbit torques in in-plane magnetized (Ga,Mn)As tracks}, volume={95}, DOI={<a href=\"https://doi.org/10.1103/physrevb.95.054422\">10.1103/physrevb.95.054422</a>}, number={5054422}, journal={Physical Review B}, publisher={American Physical Society (APS)}, author={Thevenard, L. and Boutigny, B. and Güsken, Nicholas Alexander and Becerra, L. and Ulysse, C. and Shihab, S. and Lemaître, A. and Kim, J.-V. and Jeudy, V. and Gourdon, C.}, year={2017} }","mla":"Thevenard, L., et al. “Spin Transfer and Spin-Orbit Torques in in-Plane Magnetized (Ga,Mn)As Tracks.” <i>Physical Review B</i>, vol. 95, no. 5, 054422, American Physical Society (APS), 2017, doi:<a href=\"https://doi.org/10.1103/physrevb.95.054422\">10.1103/physrevb.95.054422</a>.","short":"L. Thevenard, B. Boutigny, N.A. Güsken, L. Becerra, C. Ulysse, S. Shihab, A. Lemaître, J.-V. Kim, V. Jeudy, C. Gourdon, Physical Review B 95 (2017).","apa":"Thevenard, L., Boutigny, B., Güsken, N. A., Becerra, L., Ulysse, C., Shihab, S., Lemaître, A., Kim, J.-V., Jeudy, V., &#38; Gourdon, C. (2017). Spin transfer and spin-orbit torques in in-plane magnetized (Ga,Mn)As tracks. <i>Physical Review B</i>, <i>95</i>(5), Article 054422. <a href=\"https://doi.org/10.1103/physrevb.95.054422\">https://doi.org/10.1103/physrevb.95.054422</a>","chicago":"Thevenard, L., B. Boutigny, Nicholas Alexander Güsken, L. Becerra, C. Ulysse, S. Shihab, A. Lemaître, J.-V. Kim, V. Jeudy, and C. Gourdon. “Spin Transfer and Spin-Orbit Torques in in-Plane Magnetized (Ga,Mn)As Tracks.” <i>Physical Review B</i> 95, no. 5 (2017). <a href=\"https://doi.org/10.1103/physrevb.95.054422\">https://doi.org/10.1103/physrevb.95.054422</a>.","ieee":"L. Thevenard <i>et al.</i>, “Spin transfer and spin-orbit torques in in-plane magnetized (Ga,Mn)As tracks,” <i>Physical Review B</i>, vol. 95, no. 5, Art. no. 054422, 2017, doi: <a href=\"https://doi.org/10.1103/physrevb.95.054422\">10.1103/physrevb.95.054422</a>.","ama":"Thevenard L, Boutigny B, Güsken NA, et al. Spin transfer and spin-orbit torques in in-plane magnetized (Ga,Mn)As tracks. <i>Physical Review B</i>. 2017;95(5). doi:<a href=\"https://doi.org/10.1103/physrevb.95.054422\">10.1103/physrevb.95.054422</a>"},"publisher":"American Physical Society (APS)","date_updated":"2026-01-08T16:07:55Z","volume":95,"date_created":"2025-12-11T20:32:23Z","author":[{"full_name":"Thevenard, L.","last_name":"Thevenard","first_name":"L."},{"first_name":"B.","last_name":"Boutigny","full_name":"Boutigny, B."},{"full_name":"Güsken, Nicholas Alexander","id":"112030","last_name":"Güsken","orcid":"0000-0002-4816-0666","first_name":"Nicholas Alexander"},{"full_name":"Becerra, L.","last_name":"Becerra","first_name":"L."},{"first_name":"C.","last_name":"Ulysse","full_name":"Ulysse, C."},{"full_name":"Shihab, S.","last_name":"Shihab","first_name":"S."},{"first_name":"A.","full_name":"Lemaître, A.","last_name":"Lemaître"},{"first_name":"J.-V.","full_name":"Kim, J.-V.","last_name":"Kim"},{"last_name":"Jeudy","full_name":"Jeudy, V.","first_name":"V."},{"first_name":"C.","last_name":"Gourdon","full_name":"Gourdon, C."}],"title":"Spin transfer and spin-orbit torques in in-plane magnetized (Ga,Mn)As tracks","doi":"10.1103/physrevb.95.054422","publication":"Physical Review B","type":"journal_article","status":"public","_id":"63040","department":[{"_id":"623"},{"_id":"15"},{"_id":"230"}],"user_id":"112030","article_number":"054422","language":[{"iso":"eng"}]},{"_id":"59497","user_id":"112030","language":[{"iso":"eng"}],"publication":"Nanoscale","type":"journal_article","abstract":[{"lang":"eng","text":"<p>High-quality Al/InAs and Nb/InAs superconducting hybrid structure interfaces on catalyst free InAs nanowires.</p>"}],"status":"public","publisher":"Royal Society of Chemistry (RSC)","date_updated":"2026-01-08T16:08:19Z","volume":9,"author":[{"full_name":"Güsken, Nicholas Alexander","id":"112030","orcid":"0000-0002-4816-0666","last_name":"Güsken","first_name":"Nicholas Alexander"},{"full_name":"Rieger, Torsten","last_name":"Rieger","first_name":"Torsten"},{"last_name":"Zellekens","full_name":"Zellekens, Patrick","first_name":"Patrick"},{"first_name":"Benjamin","last_name":"Bennemann","full_name":"Bennemann, Benjamin"},{"first_name":"Elmar","full_name":"Neumann, Elmar","last_name":"Neumann"},{"full_name":"Lepsa, Mihail I.","last_name":"Lepsa","first_name":"Mihail I."},{"first_name":"Thomas","full_name":"Schäpers, Thomas","last_name":"Schäpers"},{"full_name":"Grützmacher, Detlev","last_name":"Grützmacher","first_name":"Detlev"}],"date_created":"2025-04-10T13:23:39Z","title":"MBE growth of Al/InAs and Nb/InAs superconducting hybrid nanowire structures","doi":"10.1039/c7nr03982d","publication_identifier":{"issn":["2040-3364","2040-3372"]},"publication_status":"published","issue":"43","year":"2017","intvolume":"         9","page":"16735-16741","citation":{"ama":"Güsken NA, Rieger T, Zellekens P, et al. MBE growth of Al/InAs and Nb/InAs superconducting hybrid nanowire structures. <i>Nanoscale</i>. 2017;9(43):16735-16741. doi:<a href=\"https://doi.org/10.1039/c7nr03982d\">10.1039/c7nr03982d</a>","ieee":"N. A. Güsken <i>et al.</i>, “MBE growth of Al/InAs and Nb/InAs superconducting hybrid nanowire structures,” <i>Nanoscale</i>, vol. 9, no. 43, pp. 16735–16741, 2017, doi: <a href=\"https://doi.org/10.1039/c7nr03982d\">10.1039/c7nr03982d</a>.","chicago":"Güsken, Nicholas Alexander, Torsten Rieger, Patrick Zellekens, Benjamin Bennemann, Elmar Neumann, Mihail I. Lepsa, Thomas Schäpers, and Detlev Grützmacher. “MBE Growth of Al/InAs and Nb/InAs Superconducting Hybrid Nanowire Structures.” <i>Nanoscale</i> 9, no. 43 (2017): 16735–41. <a href=\"https://doi.org/10.1039/c7nr03982d\">https://doi.org/10.1039/c7nr03982d</a>.","apa":"Güsken, N. A., Rieger, T., Zellekens, P., Bennemann, B., Neumann, E., Lepsa, M. I., Schäpers, T., &#38; Grützmacher, D. (2017). MBE growth of Al/InAs and Nb/InAs superconducting hybrid nanowire structures. <i>Nanoscale</i>, <i>9</i>(43), 16735–16741. <a href=\"https://doi.org/10.1039/c7nr03982d\">https://doi.org/10.1039/c7nr03982d</a>","short":"N.A. Güsken, T. Rieger, P. Zellekens, B. Bennemann, E. Neumann, M.I. Lepsa, T. Schäpers, D. Grützmacher, Nanoscale 9 (2017) 16735–16741.","bibtex":"@article{Güsken_Rieger_Zellekens_Bennemann_Neumann_Lepsa_Schäpers_Grützmacher_2017, title={MBE growth of Al/InAs and Nb/InAs superconducting hybrid nanowire structures}, volume={9}, DOI={<a href=\"https://doi.org/10.1039/c7nr03982d\">10.1039/c7nr03982d</a>}, number={43}, journal={Nanoscale}, publisher={Royal Society of Chemistry (RSC)}, author={Güsken, Nicholas Alexander and Rieger, Torsten and Zellekens, Patrick and Bennemann, Benjamin and Neumann, Elmar and Lepsa, Mihail I. and Schäpers, Thomas and Grützmacher, Detlev}, year={2017}, pages={16735–16741} }","mla":"Güsken, Nicholas Alexander, et al. “MBE Growth of Al/InAs and Nb/InAs Superconducting Hybrid Nanowire Structures.” <i>Nanoscale</i>, vol. 9, no. 43, Royal Society of Chemistry (RSC), 2017, pp. 16735–41, doi:<a href=\"https://doi.org/10.1039/c7nr03982d\">10.1039/c7nr03982d</a>."}},{"date_updated":"2026-03-10T16:05:32Z","date_created":"2025-02-23T16:17:16Z","author":[{"first_name":"Robert","last_name":"Wirth","full_name":"Wirth, Robert","id":"43040"}],"title":"Notions of India in A.J. Cronin’s Hatter’s Castle","year":"2017","place":"University of Reading, England","citation":{"ieee":"R. Wirth, “Notions of India in A.J. Cronin’s Hatter’s Castle,” 2017.","chicago":"Wirth, Robert. “Notions of India in A.J. Cronin’s Hatter’s Castle.” In <i>Imperial Middlebrow: Cross-Colonial Encounters and Expressions of Power in Middlebrow Literature and Culture, 1890-1940</i>. University of Reading, England, 2017.","apa":"Wirth, R. (2017). Notions of India in A.J. Cronin’s Hatter’s Castle. <i>Imperial Middlebrow: Cross-Colonial Encounters and Expressions of Power in Middlebrow Literature and Culture, 1890-1940</i>.","ama":"Wirth R. Notions of India in A.J. Cronin’s Hatter’s Castle. In: <i>Imperial Middlebrow: Cross-Colonial Encounters and Expressions of Power in Middlebrow Literature and Culture, 1890-1940</i>. ; 2017.","mla":"Wirth, Robert. “Notions of India in A.J. Cronin’s Hatter’s Castle.” <i>Imperial Middlebrow: Cross-Colonial Encounters and Expressions of Power in Middlebrow Literature and Culture, 1890-1940</i>, 2017.","short":"R. Wirth, in: Imperial Middlebrow: Cross-Colonial Encounters and Expressions of Power in Middlebrow Literature and Culture, 1890-1940, University of Reading, England, 2017.","bibtex":"@inproceedings{Wirth_2017, place={University of Reading, England}, title={Notions of India in A.J. Cronin’s Hatter’s Castle}, booktitle={Imperial Middlebrow: Cross-colonial encounters and expressions of power in middlebrow literature and culture, 1890-1940}, author={Wirth, Robert}, year={2017} }"},"_id":"58786","user_id":"43040","department":[{"_id":"381"}],"language":[{"iso":"eng"}],"type":"conference","publication":"Imperial Middlebrow: Cross-colonial encounters and expressions of power in middlebrow literature and culture, 1890-1940","status":"public"},{"year":"2017","place":"University of Portsmouth, England","citation":{"ieee":"R. Wirth, “Putting the ‘Great’ Back Into Great Britain,” 2017.","chicago":"Wirth, Robert. “Putting the ‘Great’ Back Into Great Britain.” In <i>Britain in Europe / Europe in Britain</i>. University of Portsmouth, England, 2017.","short":"R. Wirth, in: Britain in Europe / Europe in Britain, University of Portsmouth, England, 2017.","mla":"Wirth, Robert. “Putting the ‘Great’ Back Into Great Britain.” <i>Britain in Europe / Europe in Britain</i>, 2017.","bibtex":"@inproceedings{Wirth_2017, place={University of Portsmouth, England}, title={Putting the ‘Great’ Back Into Great Britain}, booktitle={Britain in Europe / Europe in Britain}, author={Wirth, Robert}, year={2017} }","apa":"Wirth, R. (2017). Putting the ‘Great’ Back Into Great Britain. <i>Britain in Europe / Europe in Britain</i>.","ama":"Wirth R. Putting the ‘Great’ Back Into Great Britain. In: <i>Britain in Europe / Europe in Britain</i>. ; 2017."},"title":"Putting the ‘Great’ Back Into Great Britain","date_updated":"2026-03-10T16:05:26Z","date_created":"2025-02-23T16:18:27Z","author":[{"first_name":"Robert","id":"43040","full_name":"Wirth, Robert","last_name":"Wirth"}],"status":"public","type":"conference","publication":"Britain in Europe / Europe in Britain","language":[{"iso":"eng"}],"_id":"58787","user_id":"43040","department":[{"_id":"381"}]},{"publication_identifier":{"issn":["0862-8424"]},"publication_status":"published","issue":"53","year":"2017","intvolume":"        27","page":"101-122","citation":{"chicago":"Procházka, Martin, and Petra Johana Poncarová, eds. <i>Cultural and Textual Appropriation in Alan Bissett’s Boyracers</i>. <i>Litteraria Pragensia: Studies in Literature and Culture</i>. Vol. 27, 2017.","ieee":"M. Procházka and P. J. Poncarová, Eds., <i>Cultural and Textual Appropriation in Alan Bissett’s Boyracers</i>, vol. 27, no. 53. 2017, pp. 101–122.","ama":"Procházka M, Poncarová PJ, eds. <i>Cultural and Textual Appropriation in Alan Bissett’s Boyracers</i>. Vol 27.; 2017:101-122.","apa":"Cultural and Textual Appropriation in Alan Bissett’s Boyracers. (2017). In M. Procházka &#38; P. J. Poncarová (Eds.), <i>Litteraria Pragensia: Studies in Literature and Culture</i> (Vol. 27, Issue 53, pp. 101–122).","bibtex":"@book{Procházka_Poncarová_2017, title={Cultural and Textual Appropriation in Alan Bissett’s Boyracers}, volume={27}, number={53}, journal={Litteraria Pragensia: Studies in Literature and Culture}, year={2017}, pages={101–122} }","mla":"Procházka, Martin, and Petra Johana Poncarová, editors. “Cultural and Textual Appropriation in Alan Bissett’s Boyracers.” <i>Litteraria Pragensia: Studies in Literature and Culture</i>, vol. 27, no. 53, 2017, pp. 101–22.","short":"M. Procházka, P.J. Poncarová, eds., Cultural and Textual Appropriation in Alan Bissett’s Boyracers, 2017."},"date_updated":"2026-03-10T16:09:07Z","volume":27,"date_created":"2025-02-23T15:52:24Z","title":"Cultural and Textual Appropriation in Alan Bissett's Boyracers","publication":"Litteraria Pragensia: Studies in Literature and Culture","type":"journal_editor","editor":[{"full_name":"Procházka, Martin","last_name":"Procházka","first_name":"Martin"},{"full_name":"Poncarová, Petra Johana","last_name":"Poncarová","first_name":"Petra Johana"}],"status":"public","_id":"58777","department":[{"_id":"381"}],"user_id":"43040","language":[{"iso":"eng"}]},{"status":"public","abstract":[{"lang":"eng","text":"<jats:title>Abstract</jats:title><jats:p>Manche boranbasierten frustrierten Lewis‐Paare spalten Wasserstoff – wie, war bislang unklar. Um Struktur und Reaktivität in Beziehung zu setzen, werden quantenchemische Untersuchungen und NMR‐Experimente kombiniert. Sind pK<jats:sub>a</jats:sub>‐Werte der Lewis‐Base bekannt, lässt sich damit Reaktivität vorhersagen.</jats:p>"}],"type":"journal_article","publication":"Nachrichten aus der Chemie","language":[{"iso":"eng"}],"user_id":"53339","department":[{"_id":"2"},{"_id":"389"}],"_id":"64888","citation":{"short":"J. Paradies, Nachrichten Aus Der Chemie 65 (2017) 118–122.","mla":"Paradies, Jan. “Reaktivität Verstehen, Ohne Die Katalysatorstruktur Zu Kennen.” <i>Nachrichten Aus Der Chemie</i>, vol. 65, no. 2, Wiley, 2017, pp. 118–22, doi:<a href=\"https://doi.org/10.1002/nadc.20174055283\">10.1002/nadc.20174055283</a>.","bibtex":"@article{Paradies_2017, title={Reaktivität verstehen, ohne die Katalysatorstruktur zu kennen}, volume={65}, DOI={<a href=\"https://doi.org/10.1002/nadc.20174055283\">10.1002/nadc.20174055283</a>}, number={2}, journal={Nachrichten aus der Chemie}, publisher={Wiley}, author={Paradies, Jan}, year={2017}, pages={118–122} }","apa":"Paradies, J. (2017). Reaktivität verstehen, ohne die Katalysatorstruktur zu kennen. <i>Nachrichten Aus Der Chemie</i>, <i>65</i>(2), 118–122. <a href=\"https://doi.org/10.1002/nadc.20174055283\">https://doi.org/10.1002/nadc.20174055283</a>","ama":"Paradies J. Reaktivität verstehen, ohne die Katalysatorstruktur zu kennen. <i>Nachrichten aus der Chemie</i>. 2017;65(2):118-122. doi:<a href=\"https://doi.org/10.1002/nadc.20174055283\">10.1002/nadc.20174055283</a>","chicago":"Paradies, Jan. “Reaktivität Verstehen, Ohne Die Katalysatorstruktur Zu Kennen.” <i>Nachrichten Aus Der Chemie</i> 65, no. 2 (2017): 118–22. <a href=\"https://doi.org/10.1002/nadc.20174055283\">https://doi.org/10.1002/nadc.20174055283</a>.","ieee":"J. Paradies, “Reaktivität verstehen, ohne die Katalysatorstruktur zu kennen,” <i>Nachrichten aus der Chemie</i>, vol. 65, no. 2, pp. 118–122, 2017, doi: <a href=\"https://doi.org/10.1002/nadc.20174055283\">10.1002/nadc.20174055283</a>."},"page":"118-122","intvolume":"        65","year":"2017","issue":"2","publication_status":"published","publication_identifier":{"issn":["1439-9598","1868-0054"]},"doi":"10.1002/nadc.20174055283","title":"Reaktivität verstehen, ohne die Katalysatorstruktur zu kennen","date_created":"2026-03-11T10:15:53Z","author":[{"orcid":"0000-0002-3698-668X","last_name":"Paradies","id":"53339","full_name":"Paradies, Jan","first_name":"Jan"}],"volume":65,"publisher":"Wiley","date_updated":"2026-03-11T10:15:59Z"},{"language":[{"iso":"eng"}],"_id":"64899","user_id":"53339","department":[{"_id":"2"},{"_id":"389"}],"status":"public","type":"journal_article","publication":"Nachrichten aus der Chemie","title":"Organische Chemie 2016","doi":"10.1002/nadc.20174059831","date_updated":"2026-03-11T10:26:39Z","publisher":"Wiley","author":[{"first_name":"Bernd F.","last_name":"Straub","full_name":"Straub, Bernd F."},{"first_name":"Jennifer","full_name":"Andexer, Jennifer","last_name":"Andexer"},{"first_name":"Christoph","last_name":"Arenz","full_name":"Arenz, Christoph"},{"first_name":"Uwe","full_name":"Beifuss, Uwe","last_name":"Beifuss"},{"full_name":"Beuerle, Florian","last_name":"Beuerle","first_name":"Florian"},{"first_name":"Malte","full_name":"Brasholz, Malte","last_name":"Brasholz"},{"full_name":"Breinbauer, Rolf","last_name":"Breinbauer","first_name":"Rolf"},{"last_name":"Ditrich","full_name":"Ditrich, Klaus","first_name":"Klaus"},{"first_name":"Tobias A. 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