[{"type":"journal_article","publication":"Calculus of Variations and Partial Differential Equations","abstract":[{"text":"AbstractA no-flux initial-boundary value problem for the doubly degenrate parabolic system $$\\begin{aligned} \\left\\{ \\begin{array}{l} u_t = \\nabla \\cdot \\big ( uv\\nabla u\\big ) + \\ell uv, \\\\ v_t = \\Delta v - uv, \\end{array} \\right. \\qquad \\qquad (\\star ) \\end{aligned}$$\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n u\r\n t\r\n \r\n =\r\n ∇\r\n ·\r\n \r\n (\r\n \r\n u\r\n v\r\n ∇\r\n u\r\n \r\n )\r\n \r\n +\r\n ℓ\r\n u\r\n v\r\n ,\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n v\r\n t\r\n \r\n =\r\n Δ\r\n v\r\n -\r\n u\r\n v\r\n ,\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n (\r\n ⋆\r\n )\r\n \r\n \r\n \r\n \r\n \r\n \r\n is considered in a smoothly bounded convex domain $$\\Omega \\subset \\mathbb {R}^n$$\r\n \r\n Ω\r\n ⊂\r\n \r\n \r\n R\r\n \r\n n\r\n \r\n \r\n , with $$n\\ge 1$$\r\n \r\n n\r\n ≥\r\n 1\r\n \r\n and $$\\ell \\ge 0$$\r\n \r\n ℓ\r\n ≥\r\n 0\r\n \r\n . The first of the main results asserts that for nonnegative initial data $$(u_0,v_0)\\in (L^\\infty (\\Omega ))^2$$\r\n \r\n \r\n (\r\n \r\n u\r\n 0\r\n \r\n ,\r\n \r\n v\r\n 0\r\n \r\n )\r\n \r\n ∈\r\n \r\n \r\n (\r\n \r\n L\r\n ∞\r\n \r\n \r\n (\r\n Ω\r\n )\r\n \r\n )\r\n \r\n 2\r\n \r\n \r\n with $$u_0\\not \\equiv 0$$\r\n \r\n \r\n u\r\n 0\r\n \r\n ≢\r\n 0\r\n \r\n , $$v_0\\not \\equiv 0$$\r\n \r\n \r\n v\r\n 0\r\n \r\n ≢\r\n 0\r\n \r\n and $$\\sqrt{v_0}\\in W^{1,2}(\\Omega )$$\r\n \r\n \r\n \r\n v\r\n 0\r\n \r\n \r\n ∈\r\n \r\n W\r\n \r\n 1\r\n ,\r\n 2\r\n \r\n \r\n \r\n (\r\n Ω\r\n )\r\n \r\n \r\n , there exists a global weak solution (u, v) which, inter alia, belongs to $$C^0(\\overline{\\Omega }\\times (0,\\infty )) \\times C^{2,1}(\\overline{\\Omega }\\times (0,\\infty ))$$\r\n \r\n \r\n C\r\n 0\r\n \r\n \r\n (\r\n \r\n Ω\r\n ¯\r\n \r\n ×\r\n \r\n (\r\n 0\r\n ,\r\n ∞\r\n )\r\n \r\n )\r\n \r\n ×\r\n \r\n C\r\n \r\n 2\r\n ,\r\n 1\r\n \r\n \r\n \r\n (\r\n \r\n Ω\r\n ¯\r\n \r\n ×\r\n \r\n (\r\n 0\r\n ,\r\n ∞\r\n )\r\n \r\n )\r\n \r\n \r\n and satisfies $$\\sup _{t>0} \\Vert u(\\cdot ,t)\\Vert _{L^p(\\Omega )}<\\infty $$\r\n \r\n \r\n sup\r\n \r\n t\r\n >\r\n 0\r\n \r\n \r\n \r\n \r\n ‖\r\n u\r\n \r\n (\r\n ·\r\n ,\r\n t\r\n )\r\n \r\n ‖\r\n \r\n \r\n \r\n L\r\n p\r\n \r\n \r\n (\r\n Ω\r\n )\r\n \r\n \r\n \r\n <\r\n ∞\r\n \r\n for all $$p\\in [1,p_0)$$\r\n \r\n p\r\n ∈\r\n [\r\n 1\r\n ,\r\n \r\n p\r\n 0\r\n \r\n )\r\n \r\n with $$p_0:=\\frac{n}{(n-2)_+}$$\r\n \r\n \r\n p\r\n 0\r\n \r\n :\r\n =\r\n \r\n n\r\n \r\n \r\n (\r\n n\r\n -\r\n 2\r\n )\r\n \r\n +\r\n \r\n \r\n \r\n . It is next seen that for each of these solutions one can find $$u_\\infty \\in \\bigcap _{p\\in [1,p_0)} L^p(\\Omega )$$\r\n \r\n \r\n u\r\n ∞\r\n \r\n ∈\r\n \r\n ⋂\r\n \r\n p\r\n ∈\r\n [\r\n 1\r\n ,\r\n \r\n p\r\n 0\r\n \r\n )\r\n \r\n \r\n \r\n L\r\n p\r\n \r\n \r\n (\r\n Ω\r\n )\r\n \r\n \r\n such that, within an appropriate topological setting, $$(u(\\cdot ,t),v(\\cdot ,t))$$\r\n \r\n (\r\n u\r\n (\r\n ·\r\n ,\r\n t\r\n )\r\n ,\r\n v\r\n (\r\n ·\r\n ,\r\n t\r\n )\r\n )\r\n \r\n approaches the equilibrium $$(u_\\infty ,0)$$\r\n \r\n (\r\n \r\n u\r\n ∞\r\n \r\n ,\r\n 0\r\n )\r\n \r\n in the large time limit. Finally, in the case $$n\\le 5$$\r\n \r\n n\r\n ≤\r\n 5\r\n \r\n a result ensuring a certain stability property of any member in the uncountably large family of steady states $$(u_0,0)$$\r\n \r\n (\r\n \r\n u\r\n 0\r\n \r\n ,\r\n 0\r\n )\r\n \r\n , with arbitrary and suitably regular $$u_0:\\Omega \\rightarrow [0,\\infty )$$\r\n \r\n \r\n u\r\n 0\r\n \r\n :\r\n Ω\r\n →\r\n \r\n [\r\n 0\r\n ,\r\n ∞\r\n )\r\n \r\n \r\n , is derived. This provides some rigorous evidence for the appropriateness of ($$\\star $$\r\n ⋆\r\n ) to model the emergence of a strikingly large variety of stable structures observed in experiments on bacterial motion in nutrient-poor environments. Essential parts of the analysis rely on the use of an apparently novel class of functional inequalities to suitably cope with the doubly degenerate diffusion mechanism in ($$\\star $$\r\n ⋆\r\n ).","lang":"eng"}],"status":"public","_id":"63305","user_id":"31496","article_number":"108","language":[{"iso":"eng"}],"publication_status":"published","publication_identifier":{"issn":["0944-2669","1432-0835"]},"issue":"3","year":"2022","citation":{"ieee":"M. Winkler, “Stabilization of arbitrary structures in a doubly degenerate reaction-diffusion system modeling bacterial motion on a nutrient-poor agar,” Calculus of Variations and Partial Differential Equations, vol. 61, no. 3, Art. no. 108, 2022, doi: 10.1007/s00526-021-02168-2.","chicago":"Winkler, Michael. “Stabilization of Arbitrary Structures in a Doubly Degenerate Reaction-Diffusion System Modeling Bacterial Motion on a Nutrient-Poor Agar.” Calculus of Variations and Partial Differential Equations 61, no. 3 (2022). https://doi.org/10.1007/s00526-021-02168-2.","ama":"Winkler M. Stabilization of arbitrary structures in a doubly degenerate reaction-diffusion system modeling bacterial motion on a nutrient-poor agar. Calculus of Variations and Partial Differential Equations. 2022;61(3). doi:10.1007/s00526-021-02168-2","mla":"Winkler, Michael. “Stabilization of Arbitrary Structures in a Doubly Degenerate Reaction-Diffusion System Modeling Bacterial Motion on a Nutrient-Poor Agar.” Calculus of Variations and Partial Differential Equations, vol. 61, no. 3, 108, Springer Science and Business Media LLC, 2022, doi:10.1007/s00526-021-02168-2.","short":"M. Winkler, Calculus of Variations and Partial Differential Equations 61 (2022).","bibtex":"@article{Winkler_2022, title={Stabilization of arbitrary structures in a doubly degenerate reaction-diffusion system modeling bacterial motion on a nutrient-poor agar}, volume={61}, DOI={10.1007/s00526-021-02168-2}, number={3108}, journal={Calculus of Variations and Partial Differential Equations}, publisher={Springer Science and Business Media LLC}, author={Winkler, Michael}, year={2022} }","apa":"Winkler, M. (2022). Stabilization of arbitrary structures in a doubly degenerate reaction-diffusion system modeling bacterial motion on a nutrient-poor agar. Calculus of Variations and Partial Differential Equations, 61(3), Article 108. https://doi.org/10.1007/s00526-021-02168-2"},"intvolume":" 61","publisher":"Springer Science and Business Media LLC","date_updated":"2025-12-18T20:04:43Z","date_created":"2025-12-18T19:26:32Z","author":[{"first_name":"Michael","id":"31496","full_name":"Winkler, Michael","last_name":"Winkler"}],"volume":61,"title":"Stabilization of arbitrary structures in a doubly degenerate reaction-diffusion system modeling bacterial motion on a nutrient-poor agar","doi":"10.1007/s00526-021-02168-2"},{"language":[{"iso":"eng"}],"article_number":"47","user_id":"31496","_id":"63311","status":"public","abstract":[{"text":"AbstractThe Cauchy problem in $$\\mathbb {R}^n$$\r\n \r\n \r\n R\r\n \r\n n\r\n \r\n , $$n\\ge 1$$\r\n \r\n n\r\n ≥\r\n 1\r\n \r\n , for the degenerate parabolic equation $$\\begin{aligned} u_t=u^p \\Delta u \\qquad \\qquad (\\star ) \\end{aligned}$$\r\n \r\n \r\n \r\n \r\n \r\n \r\n u\r\n t\r\n \r\n =\r\n \r\n u\r\n p\r\n \r\n Δ\r\n u\r\n \r\n \r\n \r\n (\r\n ⋆\r\n )\r\n \r\n \r\n \r\n \r\n \r\n \r\n is considered for $$p\\ge 1$$\r\n \r\n p\r\n ≥\r\n 1\r\n \r\n . It is shown that given any positive $$f\\in C^0([0,\\infty ))$$\r\n \r\n f\r\n ∈\r\n \r\n C\r\n 0\r\n \r\n \r\n (\r\n \r\n [\r\n 0\r\n ,\r\n ∞\r\n )\r\n \r\n )\r\n \r\n \r\n and $$g\\in C^0([0,\\infty ))$$\r\n \r\n g\r\n ∈\r\n \r\n C\r\n 0\r\n \r\n \r\n (\r\n \r\n [\r\n 0\r\n ,\r\n ∞\r\n )\r\n \r\n )\r\n \r\n \r\n satisfying $$\\begin{aligned} f(t)\\rightarrow + \\infty \\quad \\text{ and } \\quad g(t)\\rightarrow 0 \\qquad \\text{ as } t\\rightarrow \\infty , \\end{aligned}$$\r\n \r\n \r\n \r\n \r\n \r\n f\r\n (\r\n t\r\n )\r\n →\r\n +\r\n ∞\r\n \r\n \r\n and\r\n \r\n \r\n g\r\n (\r\n t\r\n )\r\n →\r\n 0\r\n \r\n \r\n as\r\n \r\n t\r\n →\r\n ∞\r\n ,\r\n \r\n \r\n \r\n \r\n \r\n one can find positive and radially symmetric continuous initial data with the property that the initial value problem for ($$\\star $$\r\n ⋆\r\n ) admits a positive classical solution such that $$\\begin{aligned} t^\\frac{1}{p} \\Vert u(\\cdot ,t)\\Vert _{L^\\infty (\\mathbb {R}^n)} \\rightarrow \\infty \\qquad \\text{ and } \\qquad \\Vert u(\\cdot ,t)\\Vert _{L^\\infty (\\mathbb {R}^n)} \\rightarrow 0 \\qquad \\text{ as } t\\rightarrow \\infty , \\end{aligned}$$\r\n \r\n \r\n \r\n \r\n \r\n \r\n t\r\n \r\n 1\r\n p\r\n \r\n \r\n \r\n \r\n ‖\r\n u\r\n \r\n (\r\n ·\r\n ,\r\n t\r\n )\r\n \r\n ‖\r\n \r\n \r\n \r\n L\r\n ∞\r\n \r\n \r\n (\r\n \r\n \r\n R\r\n \r\n n\r\n \r\n )\r\n \r\n \r\n \r\n →\r\n ∞\r\n \r\n \r\n and\r\n \r\n \r\n \r\n \r\n ‖\r\n u\r\n \r\n (\r\n ·\r\n ,\r\n t\r\n )\r\n \r\n ‖\r\n \r\n \r\n \r\n L\r\n ∞\r\n \r\n \r\n (\r\n \r\n \r\n R\r\n \r\n n\r\n \r\n )\r\n \r\n \r\n \r\n →\r\n 0\r\n \r\n \r\n as\r\n \r\n t\r\n →\r\n ∞\r\n ,\r\n \r\n \r\n \r\n \r\n \r\n but that $$\\begin{aligned} \\liminf _{t\\rightarrow \\infty } \\frac{t^\\frac{1}{p} \\Vert u(\\cdot ,t)\\Vert _{L^\\infty (\\mathbb {R}^n)}}{f(t)} =0 \\end{aligned}$$\r\n \r\n \r\n \r\n \r\n \r\n \r\n lim inf\r\n \r\n t\r\n →\r\n ∞\r\n \r\n \r\n \r\n \r\n \r\n t\r\n \r\n 1\r\n p\r\n \r\n \r\n \r\n \r\n ‖\r\n u\r\n \r\n (\r\n ·\r\n ,\r\n t\r\n )\r\n \r\n ‖\r\n \r\n \r\n \r\n L\r\n ∞\r\n \r\n \r\n (\r\n \r\n \r\n R\r\n \r\n n\r\n \r\n )\r\n \r\n \r\n \r\n \r\n \r\n f\r\n (\r\n t\r\n )\r\n \r\n \r\n =\r\n 0\r\n \r\n \r\n \r\n \r\n \r\n and $$\\begin{aligned} \\limsup _{t\\rightarrow \\infty } \\frac{\\Vert u(\\cdot ,t)\\Vert _{L^\\infty (\\mathbb {R}^n)}}{g(t)} =\\infty . \\end{aligned}$$\r\n \r\n \r\n \r\n \r\n \r\n \r\n lim sup\r\n \r\n t\r\n →\r\n ∞\r\n \r\n \r\n \r\n \r\n \r\n ‖\r\n u\r\n \r\n (\r\n ·\r\n ,\r\n t\r\n )\r\n \r\n ‖\r\n \r\n \r\n \r\n L\r\n ∞\r\n \r\n \r\n (\r\n \r\n \r\n R\r\n \r\n n\r\n \r\n )\r\n \r\n \r\n \r\n \r\n g\r\n (\r\n t\r\n )\r\n \r\n \r\n =\r\n ∞\r\n .\r\n \r\n \r\n \r\n \r\n \r\n ","lang":"eng"}],"publication":"Partial Differential Equations and Applications","type":"journal_article","doi":"10.1007/s42985-022-00186-z","title":"Oscillatory decay in a degenerate parabolic equation","volume":3,"author":[{"first_name":"Michael","id":"31496","full_name":"Winkler, Michael","last_name":"Winkler"}],"date_created":"2025-12-18T19:30:04Z","date_updated":"2025-12-18T20:05:38Z","publisher":"Springer Science and Business Media LLC","intvolume":" 3","citation":{"ieee":"M. Winkler, “Oscillatory decay in a degenerate parabolic equation,” Partial Differential Equations and Applications, vol. 3, no. 4, Art. no. 47, 2022, doi: 10.1007/s42985-022-00186-z.","chicago":"Winkler, Michael. “Oscillatory Decay in a Degenerate Parabolic Equation.” Partial Differential Equations and Applications 3, no. 4 (2022). https://doi.org/10.1007/s42985-022-00186-z.","ama":"Winkler M. Oscillatory decay in a degenerate parabolic equation. Partial Differential Equations and Applications. 2022;3(4). doi:10.1007/s42985-022-00186-z","short":"M. Winkler, Partial Differential Equations and Applications 3 (2022).","bibtex":"@article{Winkler_2022, title={Oscillatory decay in a degenerate parabolic equation}, volume={3}, DOI={10.1007/s42985-022-00186-z}, number={447}, journal={Partial Differential Equations and Applications}, publisher={Springer Science and Business Media LLC}, author={Winkler, Michael}, year={2022} }","mla":"Winkler, Michael. “Oscillatory Decay in a Degenerate Parabolic Equation.” Partial Differential Equations and Applications, vol. 3, no. 4, 47, Springer Science and Business Media LLC, 2022, doi:10.1007/s42985-022-00186-z.","apa":"Winkler, M. (2022). Oscillatory decay in a degenerate parabolic equation. Partial Differential Equations and Applications, 3(4), Article 47. https://doi.org/10.1007/s42985-022-00186-z"},"year":"2022","issue":"4","publication_identifier":{"issn":["2662-2963","2662-2971"]},"publication_status":"published"},{"publication_status":"published","publication_identifier":{"issn":["1531-3492","1553-524X"]},"issue":"11","year":"2022","citation":{"ama":"Winkler M. Approaching logarithmic singularities in quasilinear chemotaxis-consumption systems with signal-dependent sensitivities. Discrete and Continuous Dynamical Systems - B. 2022;27(11). doi:10.3934/dcdsb.2022009","chicago":"Winkler, Michael. “Approaching Logarithmic Singularities in Quasilinear Chemotaxis-Consumption Systems with Signal-Dependent Sensitivities.” Discrete and Continuous Dynamical Systems - B 27, no. 11 (2022). https://doi.org/10.3934/dcdsb.2022009.","ieee":"M. Winkler, “Approaching logarithmic singularities in quasilinear chemotaxis-consumption systems with signal-dependent sensitivities,” Discrete and Continuous Dynamical Systems - B, vol. 27, no. 11, Art. no. 6565, 2022, doi: 10.3934/dcdsb.2022009.","short":"M. Winkler, Discrete and Continuous Dynamical Systems - B 27 (2022).","mla":"Winkler, Michael. “Approaching Logarithmic Singularities in Quasilinear Chemotaxis-Consumption Systems with Signal-Dependent Sensitivities.” Discrete and Continuous Dynamical Systems - B, vol. 27, no. 11, 6565, American Institute of Mathematical Sciences (AIMS), 2022, doi:10.3934/dcdsb.2022009.","bibtex":"@article{Winkler_2022, title={Approaching logarithmic singularities in quasilinear chemotaxis-consumption systems with signal-dependent sensitivities}, volume={27}, DOI={10.3934/dcdsb.2022009}, number={116565}, journal={Discrete and Continuous Dynamical Systems - B}, publisher={American Institute of Mathematical Sciences (AIMS)}, author={Winkler, Michael}, year={2022} }","apa":"Winkler, M. (2022). Approaching logarithmic singularities in quasilinear chemotaxis-consumption systems with signal-dependent sensitivities. Discrete and Continuous Dynamical Systems - B, 27(11), Article 6565. https://doi.org/10.3934/dcdsb.2022009"},"intvolume":" 27","publisher":"American Institute of Mathematical Sciences (AIMS)","date_updated":"2025-12-18T20:05:47Z","date_created":"2025-12-18T19:30:32Z","author":[{"first_name":"Michael","id":"31496","full_name":"Winkler, Michael","last_name":"Winkler"}],"volume":27,"title":"Approaching logarithmic singularities in quasilinear chemotaxis-consumption systems with signal-dependent sensitivities","doi":"10.3934/dcdsb.2022009","type":"journal_article","publication":"Discrete and Continuous Dynamical Systems - B","abstract":[{"lang":"eng","text":"<p style='text-indent:20px;'>The chemotaxis system</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id=\"FE1\"> \\begin{document}$ \\begin{array}{l}\\left\\{ \\begin{array}{l} \tu_t = \\nabla \\cdot \\big( D(u) \\nabla u \\big) - \\nabla \\cdot \\big( uS(x, u, v)\\cdot \\nabla v\\big), \\\\ \tv_t = \\Delta v -uv, \\end{array} \\right. \\end{array} $\\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>is considered in a bounded domain <inline-formula><tex-math id=\"M1\">\\begin{document}$ \\Omega\\subset \\mathbb{R}^n $\\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id=\"M2\">\\begin{document}$ n\\ge 2 $\\end{document}</tex-math></inline-formula>, with smooth boundary.</p><p style='text-indent:20px;'>It is shown that if <inline-formula><tex-math id=\"M3\">\\begin{document}$ D: [0, \\infty) \\to [0, \\infty) $\\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id=\"M4\">\\begin{document}$ S: \\overline{\\Omega}\\times [0, \\infty)\\times (0, \\infty)\\to \\mathbb{R}^{n\\times n} $\\end{document}</tex-math></inline-formula> are suitably smooth functions satisfying</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id=\"FE2\"> \\begin{document}$ \\begin{array}{l}D(u) \\ge k_D u^{m-1} \t\\qquad {\\rm{for\\; all}}\\; u\\ge 0 \\end{array} $\\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>and</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id=\"FE3\"> \\begin{document}$ \\begin{array}{l}|S(x, u, v)| \\le \\frac{S_0(v)}{v^\\alpha} \\qquad {\\rm{for\\; all}}\\; (x, u, v)\\; \\in \\Omega\\times (0, \\infty)^2 \\end{array} $\\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>with some</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id=\"FE4\"> \\begin{document}$ \\begin{array}{l}m&gt;\\frac{3n-2}{2n} \t\\qquad {\\rm{and}}\\;\\alpha\\in [0, 1), \\end{array} $\\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>and with some <inline-formula><tex-math id=\"M5\">\\begin{document}$ k_D&gt;0 $\\end{document}</tex-math></inline-formula> and nondecreasing <inline-formula><tex-math id=\"M6\">\\begin{document}$ S_0: (0, \\infty)\\to (0, \\infty) $\\end{document}</tex-math></inline-formula>, then for all suitably regular initial data a corresponding no-flux type initial-boundary value problem admits a global bounded weak solution which actually is smooth and classical if <inline-formula><tex-math id=\"M7\">\\begin{document}$ D(0)&gt;0 $\\end{document}</tex-math></inline-formula>.</p>"}],"status":"public","_id":"63312","user_id":"31496","article_number":"6565","language":[{"iso":"eng"}]},{"_id":"63309","user_id":"31496","language":[{"iso":"eng"}],"type":"journal_article","publication":"Mathematische Nachrichten","abstract":[{"lang":"eng","text":"AbstractThis manuscript is concerned with the problem of efficiently estimating chemotactic gradients, as forming a ubiquitous issue of key importance in virtually any proof of boundedness features in Keller–Segel type systems. A strategy is proposed which at its core relies on bounds for such quantities, conditional in the sense of involving certain Lebesgue norms of solution components that explicitly influence the signal evolution.Applications of this procedure firstly provide apparently novel boundedness results for two particular classes chemotaxis systems, and apart from that are shown to significantly condense proofs for basically well‐known statements on boundedness in two further Keller–Segel type problems."}],"status":"public","publisher":"Wiley","date_updated":"2025-12-18T20:05:19Z","date_created":"2025-12-18T19:28:46Z","author":[{"first_name":"Michael","full_name":"Winkler, Michael","id":"31496","last_name":"Winkler"}],"volume":295,"title":"A unifying approach toward boundedness in Keller–Segel type cross‐diffusion systems via conditional L∞$L^\\infty$ estimates for taxis gradients","doi":"10.1002/mana.202000403","publication_status":"published","publication_identifier":{"issn":["0025-584X","1522-2616"]},"issue":"9","year":"2022","citation":{"chicago":"Winkler, Michael. “A Unifying Approach toward Boundedness in Keller–Segel Type Cross‐diffusion Systems via Conditional L∞$L^\\infty$ Estimates for Taxis Gradients.” Mathematische Nachrichten 295, no. 9 (2022): 1840–62. https://doi.org/10.1002/mana.202000403.","ieee":"M. Winkler, “A unifying approach toward boundedness in Keller–Segel type cross‐diffusion systems via conditional L∞$L^\\infty$ estimates for taxis gradients,” Mathematische Nachrichten, vol. 295, no. 9, pp. 1840–1862, 2022, doi: 10.1002/mana.202000403.","ama":"Winkler M. A unifying approach toward boundedness in Keller–Segel type cross‐diffusion systems via conditional L∞$L^\\infty$ estimates for taxis gradients. Mathematische Nachrichten. 2022;295(9):1840-1862. doi:10.1002/mana.202000403","short":"M. Winkler, Mathematische Nachrichten 295 (2022) 1840–1862.","mla":"Winkler, Michael. “A Unifying Approach toward Boundedness in Keller–Segel Type Cross‐diffusion Systems via Conditional L∞$L^\\infty$ Estimates for Taxis Gradients.” Mathematische Nachrichten, vol. 295, no. 9, Wiley, 2022, pp. 1840–62, doi:10.1002/mana.202000403.","bibtex":"@article{Winkler_2022, title={A unifying approach toward boundedness in Keller–Segel type cross‐diffusion systems via conditional L∞$L^\\infty$ estimates for taxis gradients}, volume={295}, DOI={10.1002/mana.202000403}, number={9}, journal={Mathematische Nachrichten}, publisher={Wiley}, author={Winkler, Michael}, year={2022}, pages={1840–1862} }","apa":"Winkler, M. (2022). A unifying approach toward boundedness in Keller–Segel type cross‐diffusion systems via conditional L∞$L^\\infty$ estimates for taxis gradients. Mathematische Nachrichten, 295(9), 1840–1862. https://doi.org/10.1002/mana.202000403"},"page":"1840-1862","intvolume":" 295"},{"language":[{"iso":"eng"}],"user_id":"31496","_id":"63306","status":"public","type":"journal_article","publication":"Indiana University Mathematics Journal","doi":"10.1512/iumj.2022.71.9042","title":"A critical blow-up exponent for flux limiation in a Keller-Segel system","author":[{"first_name":"Michael","id":"31496","full_name":"Winkler, Michael","last_name":"Winkler"}],"date_created":"2025-12-18T19:26:56Z","volume":71,"date_updated":"2025-12-18T20:04:53Z","publisher":"Indiana University Mathematics Journal","citation":{"ama":"Winkler M. A critical blow-up exponent for flux limiation in a Keller-Segel system. Indiana University Mathematics Journal. 2022;71(4):1437-1465. doi:10.1512/iumj.2022.71.9042","ieee":"M. Winkler, “A critical blow-up exponent for flux limiation in a Keller-Segel system,” Indiana University Mathematics Journal, vol. 71, no. 4, pp. 1437–1465, 2022, doi: 10.1512/iumj.2022.71.9042.","chicago":"Winkler, Michael. “A Critical Blow-up Exponent for Flux Limiation in a Keller-Segel System.” Indiana University Mathematics Journal 71, no. 4 (2022): 1437–65. https://doi.org/10.1512/iumj.2022.71.9042.","bibtex":"@article{Winkler_2022, title={A critical blow-up exponent for flux limiation in a Keller-Segel system}, volume={71}, DOI={10.1512/iumj.2022.71.9042}, number={4}, journal={Indiana University Mathematics Journal}, publisher={Indiana University Mathematics Journal}, author={Winkler, Michael}, year={2022}, pages={1437–1465} }","mla":"Winkler, Michael. “A Critical Blow-up Exponent for Flux Limiation in a Keller-Segel System.” Indiana University Mathematics Journal, vol. 71, no. 4, Indiana University Mathematics Journal, 2022, pp. 1437–65, doi:10.1512/iumj.2022.71.9042.","short":"M. Winkler, Indiana University Mathematics Journal 71 (2022) 1437–1465.","apa":"Winkler, M. (2022). A critical blow-up exponent for flux limiation in a Keller-Segel system. Indiana University Mathematics Journal, 71(4), 1437–1465. https://doi.org/10.1512/iumj.2022.71.9042"},"intvolume":" 71","page":"1437-1465","year":"2022","issue":"4","publication_status":"published","publication_identifier":{"issn":["0022-2518"]}},{"user_id":"31496","_id":"63284","language":[{"iso":"eng"}],"article_number":"2250012","publication":"Bulletin of Mathematical Sciences","type":"journal_article","status":"public","abstract":[{"text":" A no-flux initial-boundary value problem for the cross-diffusion system [Formula: see text] is considered in smoothly bounded domains [Formula: see text] with [Formula: see text]. It is shown that whenever [Formula: see text] is positive on [Formula: see text] and such that [Formula: see text] for some [Formula: see text], for all suitably regular positive initial data a global very weak solution, particularly preserving mass in its first component, can be constructed. This extends previous results which either concentrate on non-degenerate analogs, or are restricted to the special case [Formula: see text]. To appropriately cope with the considerably stronger cross-degeneracies thus allowed through [Formula: see text] when [Formula: see text] is large, in its core part the analysis relies on the use of the Moser–Trudinger inequality in controlling the respective diffusion rates [Formula: see text] from below. ","lang":"eng"}],"volume":13,"date_created":"2025-12-18T19:18:11Z","author":[{"first_name":"Michael","last_name":"Winkler","full_name":"Winkler, Michael","id":"31496"}],"date_updated":"2025-12-18T20:07:05Z","publisher":"World Scientific Pub Co Pte Ltd","doi":"10.1142/s1664360722500126","title":"Application of the Moser–Trudinger inequality in the construction of global solutions to a strongly degenerate migration model","issue":"02","publication_identifier":{"issn":["1664-3607","1664-3615"]},"publication_status":"published","intvolume":" 13","citation":{"ieee":"M. Winkler, “Application of the Moser–Trudinger inequality in the construction of global solutions to a strongly degenerate migration model,” Bulletin of Mathematical Sciences, vol. 13, no. 02, Art. no. 2250012, 2022, doi: 10.1142/s1664360722500126.","chicago":"Winkler, Michael. “Application of the Moser–Trudinger Inequality in the Construction of Global Solutions to a Strongly Degenerate Migration Model.” Bulletin of Mathematical Sciences 13, no. 02 (2022). https://doi.org/10.1142/s1664360722500126.","ama":"Winkler M. Application of the Moser–Trudinger inequality in the construction of global solutions to a strongly degenerate migration model. Bulletin of Mathematical Sciences. 2022;13(02). doi:10.1142/s1664360722500126","bibtex":"@article{Winkler_2022, title={Application of the Moser–Trudinger inequality in the construction of global solutions to a strongly degenerate migration model}, volume={13}, DOI={10.1142/s1664360722500126}, number={022250012}, journal={Bulletin of Mathematical Sciences}, publisher={World Scientific Pub Co Pte Ltd}, author={Winkler, Michael}, year={2022} }","short":"M. Winkler, Bulletin of Mathematical Sciences 13 (2022).","mla":"Winkler, Michael. “Application of the Moser–Trudinger Inequality in the Construction of Global Solutions to a Strongly Degenerate Migration Model.” Bulletin of Mathematical Sciences, vol. 13, no. 02, 2250012, World Scientific Pub Co Pte Ltd, 2022, doi:10.1142/s1664360722500126.","apa":"Winkler, M. (2022). Application of the Moser–Trudinger inequality in the construction of global solutions to a strongly degenerate migration model. Bulletin of Mathematical Sciences, 13(02), Article 2250012. https://doi.org/10.1142/s1664360722500126"},"year":"2022"},{"publication":"Asymptotic Analysis","type":"journal_article","status":"public","abstract":[{"lang":"eng","text":" The chemotaxis system [Formula: see text] is considered in a ball [Formula: see text]. It is shown that if [Formula: see text] suitably generalizes the prototype given by [Formula: see text] with some [Formula: see text], and if diffusion is suitably weak in the sense that [Formula: see text] is such that there exist [Formula: see text] and [Formula: see text] fulfilling [Formula: see text] then for appropriate choices of sufficiently concentrated initial data, an associated no-flux initial-boundary value problem admits a global classical solution [Formula: see text] which blows up in infinite time and satisfies [Formula: see text] A major part of the proof is based on a comparison argument involving explicitly constructed subsolutions to a scalar parabolic problem satisfied by mass accumulation functions corresponding to solutions of ( ⋆ ). "}],"user_id":"31496","_id":"63286","language":[{"iso":"eng"}],"issue":"1","publication_identifier":{"issn":["0921-7134","1875-8576"]},"publication_status":"published","intvolume":" 131","page":"33-57","citation":{"short":"M. Winkler, Asymptotic Analysis 131 (2022) 33–57.","mla":"Winkler, Michael. “Exponential Grow-up Rates in a Quasilinear Keller–Segel System.” Asymptotic Analysis, vol. 131, no. 1, SAGE Publications, 2022, pp. 33–57, doi:10.3233/asy-221765.","bibtex":"@article{Winkler_2022, title={Exponential grow-up rates in a quasilinear Keller–Segel system}, volume={131}, DOI={10.3233/asy-221765}, number={1}, journal={Asymptotic Analysis}, publisher={SAGE Publications}, author={Winkler, Michael}, year={2022}, pages={33–57} }","apa":"Winkler, M. (2022). Exponential grow-up rates in a quasilinear Keller–Segel system. Asymptotic Analysis, 131(1), 33–57. https://doi.org/10.3233/asy-221765","chicago":"Winkler, Michael. “Exponential Grow-up Rates in a Quasilinear Keller–Segel System.” Asymptotic Analysis 131, no. 1 (2022): 33–57. https://doi.org/10.3233/asy-221765.","ieee":"M. Winkler, “Exponential grow-up rates in a quasilinear Keller–Segel system,” Asymptotic Analysis, vol. 131, no. 1, pp. 33–57, 2022, doi: 10.3233/asy-221765.","ama":"Winkler M. Exponential grow-up rates in a quasilinear Keller–Segel system. Asymptotic Analysis. 2022;131(1):33-57. doi:10.3233/asy-221765"},"year":"2022","volume":131,"author":[{"first_name":"Michael","last_name":"Winkler","id":"31496","full_name":"Winkler, Michael"}],"date_created":"2025-12-18T19:18:51Z","publisher":"SAGE Publications","date_updated":"2025-12-18T20:07:19Z","doi":"10.3233/asy-221765","title":"Exponential grow-up rates in a quasilinear Keller–Segel system"},{"status":"public","abstract":[{"text":"<p style='text-indent:20px;'>The Cauchy problem in <inline-formula><tex-math id=\"M2\">\\begin{document}$ \\mathbb{R}^3 $\\end{document}</tex-math></inline-formula> for the chemotaxis-Navier–Stokes system</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id=\"FE1\"> \\begin{document}$ \\begin{eqnarray*} \\left\\{ \\begin{array}{l} n_t + u\\cdot\\nabla n = \\Delta n - \\nabla \\cdot (n\\nabla c), \\\\\tc_t + u\\cdot\\nabla c = \\Delta c - nc, \\\\ \tu_t + (u\\cdot\\nabla) u = \\Delta u + \\nabla P + n\\nabla\\phi, \\qquad \\nabla \\cdot u = 0, \\ \t\\end{array} \\right. \\end{eqnarray*} $\\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>is considered. Under suitable conditions on the initial data <inline-formula><tex-math id=\"M3\">\\begin{document}$ (n_0, c_0, u_0) $\\end{document}</tex-math></inline-formula>, with regard to the crucial first component requiring that <inline-formula><tex-math id=\"M4\">\\begin{document}$ n_0\\in L^1( \\mathbb{R}^3) $\\end{document}</tex-math></inline-formula> be nonnegative and such that <inline-formula><tex-math id=\"M5\">\\begin{document}$ (n_0+1)\\ln (n_0+1) \\in L^1( \\mathbb{R}^3) $\\end{document}</tex-math></inline-formula>, a globally defined weak solution with <inline-formula><tex-math id=\"M6\">\\begin{document}$ (n, c, u)|_{t = 0} = (n_0, c_0, u_0) $\\end{document}</tex-math></inline-formula> is constructed. Apart from that, assuming that moreover <inline-formula><tex-math id=\"M7\">\\begin{document}$ \\int_{ \\mathbb{R}^3} n_0(x) \\ln (1+|x|^2) dx $\\end{document}</tex-math></inline-formula> is finite, it is shown that a weak solution exists which enjoys further regularity features and preserves mass in an appropriate sense.</p>","lang":"eng"}],"type":"journal_article","publication":"Discrete and Continuous Dynamical Systems","language":[{"iso":"eng"}],"article_number":"5201","user_id":"31496","_id":"63293","citation":{"ama":"Kang K, Lee J, Winkler M. Global weak solutions to a chemotaxis-Navier-Stokes system in $  \\mathbb{R}^3 $. Discrete and Continuous Dynamical Systems. 2022;42(11). doi:10.3934/dcds.2022091","chicago":"Kang, Kyungkeun, Jihoon Lee, and Michael Winkler. “Global Weak Solutions to a Chemotaxis-Navier-Stokes System in $  \\mathbb{R}^3 $.” Discrete and Continuous Dynamical Systems 42, no. 11 (2022). https://doi.org/10.3934/dcds.2022091.","ieee":"K. Kang, J. Lee, and M. Winkler, “Global weak solutions to a chemotaxis-Navier-Stokes system in $  \\mathbb{R}^3 $,” Discrete and Continuous Dynamical Systems, vol. 42, no. 11, Art. no. 5201, 2022, doi: 10.3934/dcds.2022091.","mla":"Kang, Kyungkeun, et al. “Global Weak Solutions to a Chemotaxis-Navier-Stokes System in $  \\mathbb{R}^3 $.” Discrete and Continuous Dynamical Systems, vol. 42, no. 11, 5201, American Institute of Mathematical Sciences (AIMS), 2022, doi:10.3934/dcds.2022091.","bibtex":"@article{Kang_Lee_Winkler_2022, title={Global weak solutions to a chemotaxis-Navier-Stokes system in $  \\mathbb{R}^3 $}, volume={42}, DOI={10.3934/dcds.2022091}, number={115201}, journal={Discrete and Continuous Dynamical Systems}, publisher={American Institute of Mathematical Sciences (AIMS)}, author={Kang, Kyungkeun and Lee, Jihoon and Winkler, Michael}, year={2022} }","short":"K. Kang, J. Lee, M. Winkler, Discrete and Continuous Dynamical Systems 42 (2022).","apa":"Kang, K., Lee, J., & Winkler, M. (2022). Global weak solutions to a chemotaxis-Navier-Stokes system in $  \\mathbb{R}^3 $. Discrete and Continuous Dynamical Systems, 42(11), Article 5201. https://doi.org/10.3934/dcds.2022091"},"intvolume":" 42","year":"2022","issue":"11","publication_status":"published","publication_identifier":{"issn":["1078-0947","1553-5231"]},"doi":"10.3934/dcds.2022091","title":"Global weak solutions to a chemotaxis-Navier-Stokes system in $ \\mathbb{R}^3 $","author":[{"first_name":"Kyungkeun","last_name":"Kang","full_name":"Kang, Kyungkeun"},{"last_name":"Lee","full_name":"Lee, Jihoon","first_name":"Jihoon"},{"last_name":"Winkler","full_name":"Winkler, Michael","id":"31496","first_name":"Michael"}],"date_created":"2025-12-18T19:22:04Z","volume":42,"publisher":"American Institute of Mathematical Sciences (AIMS)","date_updated":"2025-12-18T20:08:21Z"},{"title":"Chemotaxis and cross-diffusion models in complex environments: Models and analytic problems toward a multiscale vision","doi":"10.1142/s0218202522500166","date_updated":"2025-12-18T20:07:51Z","publisher":"World Scientific Pub Co Pte Ltd","volume":32,"author":[{"first_name":"N.","last_name":"Bellomo","full_name":"Bellomo, N."},{"last_name":"Outada","full_name":"Outada, N.","first_name":"N."},{"full_name":"Soler, J.","last_name":"Soler","first_name":"J."},{"last_name":"Tao","full_name":"Tao, Y.","first_name":"Y."},{"first_name":"Michael","last_name":"Winkler","full_name":"Winkler, Michael","id":"31496"}],"date_created":"2025-12-18T19:20:25Z","year":"2022","intvolume":" 32","page":"713-792","citation":{"short":"N. Bellomo, N. Outada, J. Soler, Y. Tao, M. Winkler, Mathematical Models and Methods in Applied Sciences 32 (2022) 713–792.","bibtex":"@article{Bellomo_Outada_Soler_Tao_Winkler_2022, title={Chemotaxis and cross-diffusion models in complex environments: Models and analytic problems toward a multiscale vision}, volume={32}, DOI={10.1142/s0218202522500166}, number={04}, journal={Mathematical Models and Methods in Applied Sciences}, publisher={World Scientific Pub Co Pte Ltd}, author={Bellomo, N. and Outada, N. and Soler, J. and Tao, Y. and Winkler, Michael}, year={2022}, pages={713–792} }","mla":"Bellomo, N., et al. “Chemotaxis and Cross-Diffusion Models in Complex Environments: Models and Analytic Problems toward a Multiscale Vision.” Mathematical Models and Methods in Applied Sciences, vol. 32, no. 04, World Scientific Pub Co Pte Ltd, 2022, pp. 713–92, doi:10.1142/s0218202522500166.","apa":"Bellomo, N., Outada, N., Soler, J., Tao, Y., & Winkler, M. (2022). Chemotaxis and cross-diffusion models in complex environments: Models and analytic problems toward a multiscale vision. Mathematical Models and Methods in Applied Sciences, 32(04), 713–792. https://doi.org/10.1142/s0218202522500166","ama":"Bellomo N, Outada N, Soler J, Tao Y, Winkler M. Chemotaxis and cross-diffusion models in complex environments: Models and analytic problems toward a multiscale vision. Mathematical Models and Methods in Applied Sciences. 2022;32(04):713-792. doi:10.1142/s0218202522500166","chicago":"Bellomo, N., N. Outada, J. Soler, Y. Tao, and Michael Winkler. “Chemotaxis and Cross-Diffusion Models in Complex Environments: Models and Analytic Problems toward a Multiscale Vision.” Mathematical Models and Methods in Applied Sciences 32, no. 04 (2022): 713–92. https://doi.org/10.1142/s0218202522500166.","ieee":"N. Bellomo, N. Outada, J. Soler, Y. Tao, and M. Winkler, “Chemotaxis and cross-diffusion models in complex environments: Models and analytic problems toward a multiscale vision,” Mathematical Models and Methods in Applied Sciences, vol. 32, no. 04, pp. 713–792, 2022, doi: 10.1142/s0218202522500166."},"publication_identifier":{"issn":["0218-2025","1793-6314"]},"publication_status":"published","issue":"04","language":[{"iso":"eng"}],"_id":"63290","user_id":"31496","abstract":[{"lang":"eng","text":" This paper proposes a review focused on exotic chemotaxis and cross-diffusion models in complex environments. The term exotic is used to denote the dynamics of models interacting with a time-evolving external system and, specifically, models derived with the aim of describing the dynamics of living systems. The presentation first, considers the derivation of phenomenological models of chemotaxis and cross-diffusion models with particular attention on nonlinear characteristics. Then, a variety of exotic models is presented with some hints toward the derivation of new models, by accounting for a critical analysis looking ahead to perspectives. The second part of the paper is devoted to a survey of analytical problems concerning the application of models to the study of real world dynamics. Finally, the focus shifts to research perspectives within the framework of a multiscale vision, where different paths are examined to move from the dynamics at the microscopic scale to collective behaviors at the macroscopic scale. "}],"status":"public","publication":"Mathematical Models and Methods in Applied Sciences","type":"journal_article"},{"issue":"1","publication_status":"published","publication_identifier":{"issn":["1424-3199","1424-3202"]},"citation":{"apa":"Lankeit, J., & Winkler, M. (2022). Global existence in reaction–diffusion systems with mass control under relaxed assumptions merely referring to cross-absorptive effects. Journal of Evolution Equations, 22(1), Article 14. https://doi.org/10.1007/s00028-022-00768-9","short":"J. Lankeit, M. Winkler, Journal of Evolution Equations 22 (2022).","bibtex":"@article{Lankeit_Winkler_2022, title={Global existence in reaction–diffusion systems with mass control under relaxed assumptions merely referring to cross-absorptive effects}, volume={22}, DOI={10.1007/s00028-022-00768-9}, number={114}, journal={Journal of Evolution Equations}, publisher={Springer Science and Business Media LLC}, author={Lankeit, Johannes and Winkler, Michael}, year={2022} }","mla":"Lankeit, Johannes, and Michael Winkler. “Global Existence in Reaction–Diffusion Systems with Mass Control under Relaxed Assumptions Merely Referring to Cross-Absorptive Effects.” Journal of Evolution Equations, vol. 22, no. 1, 14, Springer Science and Business Media LLC, 2022, doi:10.1007/s00028-022-00768-9.","ama":"Lankeit J, Winkler M. Global existence in reaction–diffusion systems with mass control under relaxed assumptions merely referring to cross-absorptive effects. Journal of Evolution Equations. 2022;22(1). doi:10.1007/s00028-022-00768-9","chicago":"Lankeit, Johannes, and Michael Winkler. “Global Existence in Reaction–Diffusion Systems with Mass Control under Relaxed Assumptions Merely Referring to Cross-Absorptive Effects.” Journal of Evolution Equations 22, no. 1 (2022). https://doi.org/10.1007/s00028-022-00768-9.","ieee":"J. Lankeit and M. Winkler, “Global existence in reaction–diffusion systems with mass control under relaxed assumptions merely referring to cross-absorptive effects,” Journal of Evolution Equations, vol. 22, no. 1, Art. no. 14, 2022, doi: 10.1007/s00028-022-00768-9."},"intvolume":" 22","year":"2022","date_created":"2025-12-18T19:22:46Z","author":[{"first_name":"Johannes","full_name":"Lankeit, Johannes","last_name":"Lankeit"},{"first_name":"Michael","last_name":"Winkler","full_name":"Winkler, Michael","id":"31496"}],"volume":22,"publisher":"Springer Science and Business Media LLC","date_updated":"2025-12-18T20:08:35Z","doi":"10.1007/s00028-022-00768-9","title":"Global existence in reaction–diffusion systems with mass control under relaxed assumptions merely referring to cross-absorptive effects","type":"journal_article","publication":"Journal of Evolution Equations","status":"public","abstract":[{"text":"AbstractWe introduce a generalized concept of solutions for reaction–diffusion systems and prove their global existence. The only restriction on the reaction function beyond regularity, quasipositivity and mass control is special in that it merely controls the growth of cross-absorptive terms. The result covers nonlinear diffusion and does not rely on an entropy estimate.","lang":"eng"}],"user_id":"31496","_id":"63295","language":[{"iso":"eng"}],"article_number":"14"},{"user_id":"31496","_id":"63299","language":[{"iso":"eng"}],"type":"journal_article","publication":"SIAM Journal on Mathematical Analysis","status":"public","author":[{"last_name":"Tao","full_name":"Tao, Youshan","first_name":"Youshan"},{"first_name":"Michael","last_name":"Winkler","full_name":"Winkler, Michael","id":"31496"}],"date_created":"2025-12-18T19:24:16Z","volume":54,"date_updated":"2025-12-18T20:09:05Z","publisher":"Society for Industrial & Applied Mathematics (SIAM)","doi":"10.1137/21m1449841","title":"Existence Theory and Qualitative Analysis for a Fully Cross-Diffusive Predator-Prey System","issue":"4","publication_status":"published","publication_identifier":{"issn":["0036-1410","1095-7154"]},"citation":{"apa":"Tao, Y., & Winkler, M. (2022). Existence Theory and Qualitative Analysis for a Fully Cross-Diffusive Predator-Prey System. SIAM Journal on Mathematical Analysis, 54(4), 4806–4864. https://doi.org/10.1137/21m1449841","bibtex":"@article{Tao_Winkler_2022, title={Existence Theory and Qualitative Analysis for a Fully Cross-Diffusive Predator-Prey System}, volume={54}, DOI={10.1137/21m1449841}, number={4}, journal={SIAM Journal on Mathematical Analysis}, publisher={Society for Industrial & Applied Mathematics (SIAM)}, author={Tao, Youshan and Winkler, Michael}, year={2022}, pages={4806–4864} }","short":"Y. Tao, M. Winkler, SIAM Journal on Mathematical Analysis 54 (2022) 4806–4864.","mla":"Tao, Youshan, and Michael Winkler. “Existence Theory and Qualitative Analysis for a Fully Cross-Diffusive Predator-Prey System.” SIAM Journal on Mathematical Analysis, vol. 54, no. 4, Society for Industrial & Applied Mathematics (SIAM), 2022, pp. 4806–64, doi:10.1137/21m1449841.","chicago":"Tao, Youshan, and Michael Winkler. “Existence Theory and Qualitative Analysis for a Fully Cross-Diffusive Predator-Prey System.” SIAM Journal on Mathematical Analysis 54, no. 4 (2022): 4806–64. https://doi.org/10.1137/21m1449841.","ieee":"Y. Tao and M. Winkler, “Existence Theory and Qualitative Analysis for a Fully Cross-Diffusive Predator-Prey System,” SIAM Journal on Mathematical Analysis, vol. 54, no. 4, pp. 4806–4864, 2022, doi: 10.1137/21m1449841.","ama":"Tao Y, Winkler M. Existence Theory and Qualitative Analysis for a Fully Cross-Diffusive Predator-Prey System. SIAM Journal on Mathematical Analysis. 2022;54(4):4806-4864. doi:10.1137/21m1449841"},"page":"4806-4864","intvolume":" 54","year":"2022"},{"publication_status":"published","publication_identifier":{"issn":["0360-5302","1532-4133"]},"issue":"12","year":"2022","citation":{"short":"A. Stevens, M. Winkler, Communications in Partial Differential Equations 47 (2022) 2341–2362.","bibtex":"@article{Stevens_Winkler_2022, title={Taxis-driven persistent localization in a degenerate Keller-Segel system}, volume={47}, DOI={10.1080/03605302.2022.2122836}, number={12}, journal={Communications in Partial Differential Equations}, publisher={Informa UK Limited}, author={Stevens, Angela and Winkler, Michael}, year={2022}, pages={2341–2362} }","mla":"Stevens, Angela, and Michael Winkler. “Taxis-Driven Persistent Localization in a Degenerate Keller-Segel System.” Communications in Partial Differential Equations, vol. 47, no. 12, Informa UK Limited, 2022, pp. 2341–62, doi:10.1080/03605302.2022.2122836.","apa":"Stevens, A., & Winkler, M. (2022). Taxis-driven persistent localization in a degenerate Keller-Segel system. Communications in Partial Differential Equations, 47(12), 2341–2362. https://doi.org/10.1080/03605302.2022.2122836","ama":"Stevens A, Winkler M. Taxis-driven persistent localization in a degenerate Keller-Segel system. Communications in Partial Differential Equations. 2022;47(12):2341-2362. doi:10.1080/03605302.2022.2122836","ieee":"A. Stevens and M. Winkler, “Taxis-driven persistent localization in a degenerate Keller-Segel system,” Communications in Partial Differential Equations, vol. 47, no. 12, pp. 2341–2362, 2022, doi: 10.1080/03605302.2022.2122836.","chicago":"Stevens, Angela, and Michael Winkler. “Taxis-Driven Persistent Localization in a Degenerate Keller-Segel System.” Communications in Partial Differential Equations 47, no. 12 (2022): 2341–62. https://doi.org/10.1080/03605302.2022.2122836."},"page":"2341-2362","intvolume":" 47","publisher":"Informa UK Limited","date_updated":"2025-12-18T20:08:58Z","date_created":"2025-12-18T19:23:52Z","author":[{"full_name":"Stevens, Angela","last_name":"Stevens","first_name":"Angela"},{"first_name":"Michael","last_name":"Winkler","full_name":"Winkler, Michael","id":"31496"}],"volume":47,"title":"Taxis-driven persistent localization in a degenerate Keller-Segel system","doi":"10.1080/03605302.2022.2122836","type":"journal_article","publication":"Communications in Partial Differential Equations","status":"public","_id":"63298","user_id":"31496","language":[{"iso":"eng"}]},{"publisher":"Springer Science and Business Media LLC","date_updated":"2025-12-18T20:10:14Z","volume":36,"author":[{"last_name":"Winkler","id":"31496","full_name":"Winkler, Michael","first_name":"Michael"}],"date_created":"2025-12-18T19:10:32Z","title":"Slow Grow-up in a Quasilinear Keller–Segel System","doi":"10.1007/s10884-022-10167-w","publication_identifier":{"issn":["1040-7294","1572-9222"]},"publication_status":"published","issue":"2","year":"2022","intvolume":" 36","page":"1677-1702","citation":{"apa":"Winkler, M. (2022). Slow Grow-up in a Quasilinear Keller–Segel System. Journal of Dynamics and Differential Equations, 36(2), 1677–1702. https://doi.org/10.1007/s10884-022-10167-w","short":"M. Winkler, Journal of Dynamics and Differential Equations 36 (2022) 1677–1702.","bibtex":"@article{Winkler_2022, title={Slow Grow-up in a Quasilinear Keller–Segel System}, volume={36}, DOI={10.1007/s10884-022-10167-w}, number={2}, journal={Journal of Dynamics and Differential Equations}, publisher={Springer Science and Business Media LLC}, author={Winkler, Michael}, year={2022}, pages={1677–1702} }","mla":"Winkler, Michael. “Slow Grow-up in a Quasilinear Keller–Segel System.” Journal of Dynamics and Differential Equations, vol. 36, no. 2, Springer Science and Business Media LLC, 2022, pp. 1677–702, doi:10.1007/s10884-022-10167-w.","ieee":"M. Winkler, “Slow Grow-up in a Quasilinear Keller–Segel System,” Journal of Dynamics and Differential Equations, vol. 36, no. 2, pp. 1677–1702, 2022, doi: 10.1007/s10884-022-10167-w.","chicago":"Winkler, Michael. “Slow Grow-up in a Quasilinear Keller–Segel System.” Journal of Dynamics and Differential Equations 36, no. 2 (2022): 1677–1702. https://doi.org/10.1007/s10884-022-10167-w.","ama":"Winkler M. Slow Grow-up in a Quasilinear Keller–Segel System. Journal of Dynamics and Differential Equations. 2022;36(2):1677-1702. doi:10.1007/s10884-022-10167-w"},"_id":"63266","user_id":"31496","language":[{"iso":"eng"}],"publication":"Journal of Dynamics and Differential Equations","type":"journal_article","abstract":[{"lang":"eng","text":"AbstractIn a ball $$\\Omega =B_R(0)\\subset \\mathbb {R}^n$$\r\n \r\n Ω\r\n =\r\n \r\n B\r\n R\r\n \r\n \r\n (\r\n 0\r\n )\r\n \r\n ⊂\r\n \r\n \r\n R\r\n \r\n n\r\n \r\n \r\n , $$n\\ge 2$$\r\n \r\n n\r\n ≥\r\n 2\r\n \r\n , the chemotaxis system $$\\begin{aligned} \\left\\{ \\begin{array}{l}u_t = \\nabla \\cdot \\big ( D(u) \\nabla u \\big ) - \\nabla \\cdot \\big ( uS(u)\\nabla v\\big ), \\\\ 0 = \\Delta v - \\mu + u, \\qquad \\mu =\\frac{1}{|\\Omega |} \\int _\\Omega u, \\end{array} \\right. \\qquad \\qquad (\\star ) \\end{aligned}$$\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n u\r\n t\r\n \r\n =\r\n ∇\r\n ·\r\n \r\n (\r\n \r\n D\r\n \r\n (\r\n u\r\n )\r\n \r\n ∇\r\n u\r\n \r\n )\r\n \r\n -\r\n ∇\r\n ·\r\n \r\n (\r\n \r\n u\r\n S\r\n \r\n (\r\n u\r\n )\r\n \r\n ∇\r\n v\r\n \r\n )\r\n \r\n ,\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n 0\r\n =\r\n Δ\r\n v\r\n -\r\n μ\r\n +\r\n u\r\n ,\r\n \r\n μ\r\n =\r\n \r\n 1\r\n \r\n |\r\n Ω\r\n |\r\n \r\n \r\n \r\n ∫\r\n Ω\r\n \r\n u\r\n ,\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n (\r\n ⋆\r\n )\r\n \r\n \r\n \r\n \r\n \r\n \r\n is considered under no-flux boundary conditions, with a focus on nonlinearities $$S\\in C^2([0,\\infty ))$$\r\n \r\n S\r\n ∈\r\n \r\n C\r\n 2\r\n \r\n \r\n (\r\n \r\n [\r\n 0\r\n ,\r\n ∞\r\n )\r\n \r\n )\r\n \r\n \r\n which exhibit super-algebraically fast decay in the sense that with some $$K_S>0, \\beta \\in [0,1)$$\r\n \r\n \r\n K\r\n S\r\n \r\n >\r\n 0\r\n ,\r\n β\r\n ∈\r\n \r\n [\r\n 0\r\n ,\r\n 1\r\n )\r\n \r\n \r\n and $$\\xi _0>0$$\r\n \r\n \r\n ξ\r\n 0\r\n \r\n >\r\n 0\r\n \r\n , $$\\begin{aligned} S(\\xi )>0 \\quad \\text{ and } \\quad S'(\\xi ) \\le -K_S\\xi ^{-\\beta } S(\\xi ) \\qquad \\text{ for } \\text{ all } \\xi \\ge \\xi _0. \\end{aligned}$$\r\n \r\n \r\n \r\n \r\n \r\n S\r\n \r\n (\r\n ξ\r\n )\r\n \r\n >\r\n 0\r\n \r\n \r\n and\r\n \r\n \r\n \r\n S\r\n ′\r\n \r\n \r\n (\r\n ξ\r\n )\r\n \r\n ≤\r\n -\r\n \r\n K\r\n S\r\n \r\n \r\n ξ\r\n \r\n -\r\n β\r\n \r\n \r\n S\r\n \r\n (\r\n ξ\r\n )\r\n \r\n \r\n \r\n for\r\n \r\n \r\n all\r\n \r\n ξ\r\n ≥\r\n \r\n ξ\r\n 0\r\n \r\n .\r\n \r\n \r\n \r\n \r\n \r\n It is, inter alia, shown that if furthermore $$D\\in C^2((0,\\infty ))$$\r\n \r\n D\r\n ∈\r\n \r\n C\r\n 2\r\n \r\n \r\n (\r\n \r\n (\r\n 0\r\n ,\r\n ∞\r\n )\r\n \r\n )\r\n \r\n \r\n is positive and suitably small in relation to S by satisfying $$\\begin{aligned} \\frac{\\xi S(\\xi )}{D(\\xi )} \\ge K_{SD}\\xi ^\\lambda \\qquad \\text{ for } \\text{ all } \\xi \\ge \\xi _0 \\end{aligned}$$\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n ξ\r\n S\r\n (\r\n ξ\r\n )\r\n \r\n \r\n D\r\n (\r\n ξ\r\n )\r\n \r\n \r\n ≥\r\n \r\n K\r\n \r\n SD\r\n \r\n \r\n \r\n ξ\r\n λ\r\n \r\n \r\n \r\n for\r\n \r\n \r\n all\r\n \r\n ξ\r\n ≥\r\n \r\n ξ\r\n 0\r\n \r\n \r\n \r\n \r\n \r\n \r\n with some $$K_{SD}>0$$\r\n \r\n \r\n K\r\n \r\n SD\r\n \r\n \r\n >\r\n 0\r\n \r\n and $$\\lambda >\\frac{2}{n}$$\r\n \r\n λ\r\n >\r\n \r\n 2\r\n n\r\n \r\n \r\n , then throughout a considerably large set of initial data, ($$\\star $$\r\n ⋆\r\n ) admits global classical solutions (u, v) fulfilling $$\\begin{aligned} \\frac{z(t)}{C} \\le \\Vert u(\\cdot ,t)\\Vert _{L^\\infty (\\Omega )} \\le Cz(t) \\qquad \\text{ for } \\text{ all } t>0, \\end{aligned}$$\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n z\r\n (\r\n t\r\n )\r\n \r\n C\r\n \r\n ≤\r\n \r\n \r\n ‖\r\n u\r\n \r\n (\r\n ·\r\n ,\r\n t\r\n )\r\n \r\n ‖\r\n \r\n \r\n \r\n L\r\n ∞\r\n \r\n \r\n (\r\n Ω\r\n )\r\n \r\n \r\n \r\n ≤\r\n C\r\n z\r\n \r\n (\r\n t\r\n )\r\n \r\n \r\n \r\n for\r\n \r\n \r\n all\r\n \r\n t\r\n >\r\n 0\r\n ,\r\n \r\n \r\n \r\n \r\n \r\n with some $$C=C^{(u,v)}\\ge 1$$\r\n \r\n C\r\n =\r\n \r\n C\r\n \r\n (\r\n u\r\n ,\r\n v\r\n )\r\n \r\n \r\n ≥\r\n 1\r\n \r\n , where z denotes the solution of $$\\begin{aligned} \\left\\{ \\begin{array}{l}z'(t) = z^2(t) \\cdot S\\big ( z(t)\\big ), \\qquad t>0, \\\\ z(0)=\\xi _0, \\end{array} \\right. \\end{aligned}$$\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n z\r\n ′\r\n \r\n \r\n (\r\n t\r\n )\r\n \r\n =\r\n \r\n z\r\n 2\r\n \r\n \r\n (\r\n t\r\n )\r\n \r\n ·\r\n S\r\n \r\n (\r\n \r\n z\r\n \r\n (\r\n t\r\n )\r\n \r\n \r\n )\r\n \r\n ,\r\n \r\n t\r\n >\r\n 0\r\n ,\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n z\r\n \r\n (\r\n 0\r\n )\r\n \r\n =\r\n \r\n ξ\r\n 0\r\n \r\n ,\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n which is seen to exist globally, and to satisfy $$z(t)\\rightarrow +\\infty $$\r\n \r\n z\r\n (\r\n t\r\n )\r\n →\r\n +\r\n ∞\r\n \r\n as $$t\\rightarrow \\infty $$\r\n \r\n t\r\n →\r\n ∞\r\n \r\n . As particular examples, exponentially and doubly exponentially decaying S are found to imply corresponding infinite-time blow-up properties in ($$\\star $$\r\n ⋆\r\n ) at logarithmic and doubly logarithmic rates, respectively."}],"status":"public"},{"publication_status":"published","publication_identifier":{"issn":["0022-0396"]},"year":"2022","citation":{"mla":"Tao, Youshan, and Michael Winkler. “Global Solutions to a Keller-Segel-Consumption System Involving Singularly Signal-Dependent Motilities in Domains of Arbitrary Dimension.” Journal of Differential Equations, vol. 343, Elsevier BV, 2022, pp. 390–418, doi:10.1016/j.jde.2022.10.022.","bibtex":"@article{Tao_Winkler_2022, title={Global solutions to a Keller-Segel-consumption system involving singularly signal-dependent motilities in domains of arbitrary dimension}, volume={343}, DOI={10.1016/j.jde.2022.10.022}, journal={Journal of Differential Equations}, publisher={Elsevier BV}, author={Tao, Youshan and Winkler, Michael}, year={2022}, pages={390–418} }","short":"Y. Tao, M. Winkler, Journal of Differential Equations 343 (2022) 390–418.","apa":"Tao, Y., & Winkler, M. (2022). Global solutions to a Keller-Segel-consumption system involving singularly signal-dependent motilities in domains of arbitrary dimension. Journal of Differential Equations, 343, 390–418. https://doi.org/10.1016/j.jde.2022.10.022","ieee":"Y. Tao and M. Winkler, “Global solutions to a Keller-Segel-consumption system involving singularly signal-dependent motilities in domains of arbitrary dimension,” Journal of Differential Equations, vol. 343, pp. 390–418, 2022, doi: 10.1016/j.jde.2022.10.022.","chicago":"Tao, Youshan, and Michael Winkler. “Global Solutions to a Keller-Segel-Consumption System Involving Singularly Signal-Dependent Motilities in Domains of Arbitrary Dimension.” Journal of Differential Equations 343 (2022): 390–418. https://doi.org/10.1016/j.jde.2022.10.022.","ama":"Tao Y, Winkler M. Global solutions to a Keller-Segel-consumption system involving singularly signal-dependent motilities in domains of arbitrary dimension. Journal of Differential Equations. 2022;343:390-418. doi:10.1016/j.jde.2022.10.022"},"page":"390-418","intvolume":" 343","publisher":"Elsevier BV","date_updated":"2025-12-18T20:11:02Z","author":[{"full_name":"Tao, Youshan","last_name":"Tao","first_name":"Youshan"},{"first_name":"Michael","last_name":"Winkler","full_name":"Winkler, Michael","id":"31496"}],"date_created":"2025-12-18T19:13:04Z","volume":343,"title":"Global solutions to a Keller-Segel-consumption system involving singularly signal-dependent motilities in domains of arbitrary dimension","doi":"10.1016/j.jde.2022.10.022","type":"journal_article","publication":"Journal of Differential Equations","status":"public","_id":"63272","user_id":"31496","language":[{"iso":"eng"}]},{"type":"journal_article","publication":"Nonlinear Analysis","status":"public","user_id":"31496","_id":"63268","language":[{"iso":"eng"}],"article_number":"113153","publication_status":"published","publication_identifier":{"issn":["0362-546X"]},"citation":{"bibtex":"@article{Desvillettes_Laurençot_Trescases_Winkler_2022, title={Weak solutions to triangular cross diffusion systems modeling chemotaxis with local sensing}, volume={226}, DOI={10.1016/j.na.2022.113153}, number={113153}, journal={Nonlinear Analysis}, publisher={Elsevier BV}, author={Desvillettes, Laurent and Laurençot, Philippe and Trescases, Ariane and Winkler, Michael}, year={2022} }","short":"L. Desvillettes, P. Laurençot, A. Trescases, M. Winkler, Nonlinear Analysis 226 (2022).","mla":"Desvillettes, Laurent, et al. “Weak Solutions to Triangular Cross Diffusion Systems Modeling Chemotaxis with Local Sensing.” Nonlinear Analysis, vol. 226, 113153, Elsevier BV, 2022, doi:10.1016/j.na.2022.113153.","apa":"Desvillettes, L., Laurençot, P., Trescases, A., & Winkler, M. (2022). Weak solutions to triangular cross diffusion systems modeling chemotaxis with local sensing. Nonlinear Analysis, 226, Article 113153. https://doi.org/10.1016/j.na.2022.113153","ieee":"L. Desvillettes, P. Laurençot, A. Trescases, and M. Winkler, “Weak solutions to triangular cross diffusion systems modeling chemotaxis with local sensing,” Nonlinear Analysis, vol. 226, Art. no. 113153, 2022, doi: 10.1016/j.na.2022.113153.","chicago":"Desvillettes, Laurent, Philippe Laurençot, Ariane Trescases, and Michael Winkler. “Weak Solutions to Triangular Cross Diffusion Systems Modeling Chemotaxis with Local Sensing.” Nonlinear Analysis 226 (2022). https://doi.org/10.1016/j.na.2022.113153.","ama":"Desvillettes L, Laurençot P, Trescases A, Winkler M. Weak solutions to triangular cross diffusion systems modeling chemotaxis with local sensing. Nonlinear Analysis. 2022;226. doi:10.1016/j.na.2022.113153"},"intvolume":" 226","year":"2022","date_created":"2025-12-18T19:11:16Z","author":[{"last_name":"Desvillettes","full_name":"Desvillettes, Laurent","first_name":"Laurent"},{"last_name":"Laurençot","full_name":"Laurençot, Philippe","first_name":"Philippe"},{"last_name":"Trescases","full_name":"Trescases, Ariane","first_name":"Ariane"},{"id":"31496","full_name":"Winkler, Michael","last_name":"Winkler","first_name":"Michael"}],"volume":226,"date_updated":"2025-12-18T20:10:32Z","publisher":"Elsevier BV","doi":"10.1016/j.na.2022.113153","title":"Weak solutions to triangular cross diffusion systems modeling chemotaxis with local sensing"},{"_id":"63278","user_id":"31496","language":[{"iso":"eng"}],"publication":"International Mathematics Research Notices","type":"journal_article","abstract":[{"text":"Abstract\r\n The Neumann problem for (0.1)$$ \\begin{align}& V_t = \\Delta V-aV+f(x,t) \\end{align}$$is considered in bounded domains $\\Omega \\subset {\\mathbb {R}}^n$ with smooth boundary, where $n\\ge 1$ and $a\\in {\\mathbb {R}}$. By means of a variational approach, a statement on boundedness of the quantities $$ \\begin{eqnarray*} \\sup_{t\\in (0,T)} \\int_\\Omega \\big|\\nabla V(\\cdot,t)\\big|^p L^{\\frac{n+p}{n+2}} \\Big( \\big|\\nabla V(\\cdot,t)\\big| \\Big) \\end{eqnarray*}$$in dependence on the expressions (0.2)$$ \\begin{align}& \\sup_{t\\in (0,T-\\tau)} \\int_t^{t+\\tau} \\int_\\Omega |f|^{\\frac{(n+2)p}{n+p}} L\\big( |f|\\big) \\end{align}$$is derived for $p\\ge 2$, $\\tau&gt;0$, and $T\\ge 2\\tau $, provided that $L\\in C^0([0,\\infty ))$ is positive, strictly increasing, unbounded, and slowly growing in the sense that $\\limsup _{s\\to \\infty } \\frac {L(s^{\\lambda _0})}{L(s)} &lt;\\infty $ for some $\\lambda _0&gt;1$. In the particular case when $p=n\\ge 2$, an additional condition on growth of $L$, particularly satisfied by $L(\\xi ):=\\ln ^\\alpha (\\xi +b)$ whenever $b&gt;0$ and $\\alpha&gt;\\frac {(n+2)(n-1)}{2n}$, is identified as sufficient to ensure that as a consequence of the above, bounds for theintegrals in (0.2) even imply estimates for the spatio-temporal modulus of continuity of solutions to (0.1). A subsequent application to the Keller–Segel system $$ \\begin{eqnarray*} \\left\\{ \\begin{array}{l} u_t = \\nabla \\cdot \\big( D(v)\\nabla u\\big) - \\nabla \\cdot \\big( uS(v)\\nabla v\\big) + ru - \\mu u^2, \\\\[1mm] v_t = \\Delta v-v+u, \\end{array} \\right. \\end{eqnarray*}$$shows that when $n=2$, $r\\in {\\mathbb {R}}$, $0&lt;D\\in C^2([0,\\infty ))$, and $S\\in C^2([0,\\infty )) \\cap W^{1,\\infty }((0,\\infty ))$ and thus especially in the presence of arbitrarily strong diffusion degeneracies implied by rapid decay of $D$, any choice of $\\mu&gt;0$ excludes blowup in the sense that for all suitably regular nonnegative initial data, an associated initial-boundary value problem admits a global bounded classical solution.","lang":"eng"}],"status":"public","publisher":"Oxford University Press (OUP)","date_updated":"2025-12-18T20:11:43Z","volume":2023,"author":[{"first_name":"Michael","last_name":"Winkler","full_name":"Winkler, Michael","id":"31496"}],"date_created":"2025-12-18T19:15:52Z","title":"A Result on Parabolic Gradient Regularity in Orlicz Spaces and Application to Absorption-Induced Blow-Up Prevention in a Keller–Segel-Type Cross-Diffusion System","doi":"10.1093/imrn/rnac286","publication_identifier":{"issn":["1073-7928","1687-0247"]},"publication_status":"published","issue":"19","year":"2022","page":"16336-16393","intvolume":" 2023","citation":{"chicago":"Winkler, Michael. “A Result on Parabolic Gradient Regularity in Orlicz Spaces and Application to Absorption-Induced Blow-Up Prevention in a Keller–Segel-Type Cross-Diffusion System.” International Mathematics Research Notices 2023, no. 19 (2022): 16336–93. https://doi.org/10.1093/imrn/rnac286.","ieee":"M. Winkler, “A Result on Parabolic Gradient Regularity in Orlicz Spaces and Application to Absorption-Induced Blow-Up Prevention in a Keller–Segel-Type Cross-Diffusion System,” International Mathematics Research Notices, vol. 2023, no. 19, pp. 16336–16393, 2022, doi: 10.1093/imrn/rnac286.","ama":"Winkler M. A Result on Parabolic Gradient Regularity in Orlicz Spaces and Application to Absorption-Induced Blow-Up Prevention in a Keller–Segel-Type Cross-Diffusion System. International Mathematics Research Notices. 2022;2023(19):16336-16393. doi:10.1093/imrn/rnac286","apa":"Winkler, M. (2022). A Result on Parabolic Gradient Regularity in Orlicz Spaces and Application to Absorption-Induced Blow-Up Prevention in a Keller–Segel-Type Cross-Diffusion System. International Mathematics Research Notices, 2023(19), 16336–16393. https://doi.org/10.1093/imrn/rnac286","mla":"Winkler, Michael. “A Result on Parabolic Gradient Regularity in Orlicz Spaces and Application to Absorption-Induced Blow-Up Prevention in a Keller–Segel-Type Cross-Diffusion System.” International Mathematics Research Notices, vol. 2023, no. 19, Oxford University Press (OUP), 2022, pp. 16336–93, doi:10.1093/imrn/rnac286.","bibtex":"@article{Winkler_2022, title={A Result on Parabolic Gradient Regularity in Orlicz Spaces and Application to Absorption-Induced Blow-Up Prevention in a Keller–Segel-Type Cross-Diffusion System}, volume={2023}, DOI={10.1093/imrn/rnac286}, number={19}, journal={International Mathematics Research Notices}, publisher={Oxford University Press (OUP)}, author={Winkler, Michael}, year={2022}, pages={16336–16393} }","short":"M. Winkler, International Mathematics Research Notices 2023 (2022) 16336–16393."}},{"type":"journal_article","publication":"Journal of the European Mathematical Society","abstract":[{"lang":"eng","text":"\r\n In a smoothly bounded convex domain\r\n \r\n \\Omega \\subset \\mathbb{R}^3\r\n \r\n , we consider the chemotaxis-Navier–Stokes model\r\n \r\n \r\n \r\n \\begin{cases} n_t + u\\cdot\\nabla n = \\Delta n - \\nabla \\cdot (n\\nabla c), & x\\in \\Omega, \\, t>0, \\\\ c_t + u\\cdot\\nabla c = \\Delta c -nc, & x\\in \\Omega, \\, t>0, \\\\ u_t + (u\\cdot\\nabla) u = \\Delta u + \\nabla P + n\\nabla\\Phi, \\quad \\nabla\\cdot u=0, & x\\in \\Omega, \\, t>0, \\end{cases} \\quad (\\star)\r\n \r\n \r\n \r\n proposed by Goldstein et al. to describe pattern formation in populations of aerobic bacteria interacting with their liquid environment via transport and buoyancy. Known results have asserted that under appropriate regularity assumptions on\r\n \r\n \\Phi\r\n \r\n and the initial data, a corresponding no-flux/no-flux/Dirichlet initial-boundary value problem is globally solvable in a framework of so-called weak energy solutions, and that any such solution eventually becomes smooth and classical.\r\n \r\n \r\n Going beyond this, the present work focuses on the possible extent of unboundedness phenomena also on short timescales, and hence investigates in more detail the set of times in\r\n \r\n (0,\\infty)\r\n \r\n at which solutions may develop singularities. The main results in this direction reveal the existence of a global weak energy solution which coincides with a smooth function throughout\r\n \r\n \\overline{\\Omega}\\times E\r\n \r\n , where\r\n \r\n E\r\n \r\n denotes a countable union of open intervals which is such that\r\n \r\n |(0,\\infty)\\setminus E|=0\r\n \r\n . In particular, this indicates that a similar feature of the unperturbed Navie–Stokes equations, known as Leray’s structure theorem, persists even in the presence of the coupling to the attractive and hence potentially destabilizing cross-diffusive mechanism in the full system (\r\n \r\n \\star\r\n \r\n ).\r\n "}],"status":"public","_id":"63279","user_id":"31496","language":[{"iso":"eng"}],"publication_status":"published","publication_identifier":{"issn":["1435-9855","1435-9863"]},"issue":"4","year":"2022","citation":{"bibtex":"@article{Winkler_2022, title={Does Leray’s structure theorem withstand buoyancy-driven chemotaxis-fluid interaction?}, volume={25}, DOI={10.4171/jems/1226}, number={4}, journal={Journal of the European Mathematical Society}, publisher={European Mathematical Society - EMS - Publishing House GmbH}, author={Winkler, Michael}, year={2022}, pages={1423–1456} }","short":"M. Winkler, Journal of the European Mathematical Society 25 (2022) 1423–1456.","mla":"Winkler, Michael. “Does Leray’s Structure Theorem Withstand Buoyancy-Driven Chemotaxis-Fluid Interaction?” Journal of the European Mathematical Society, vol. 25, no. 4, European Mathematical Society - EMS - Publishing House GmbH, 2022, pp. 1423–56, doi:10.4171/jems/1226.","apa":"Winkler, M. (2022). Does Leray’s structure theorem withstand buoyancy-driven chemotaxis-fluid interaction? Journal of the European Mathematical Society, 25(4), 1423–1456. https://doi.org/10.4171/jems/1226","ama":"Winkler M. Does Leray’s structure theorem withstand buoyancy-driven chemotaxis-fluid interaction? Journal of the European Mathematical Society. 2022;25(4):1423-1456. doi:10.4171/jems/1226","ieee":"M. Winkler, “Does Leray’s structure theorem withstand buoyancy-driven chemotaxis-fluid interaction?,” Journal of the European Mathematical Society, vol. 25, no. 4, pp. 1423–1456, 2022, doi: 10.4171/jems/1226.","chicago":"Winkler, Michael. “Does Leray’s Structure Theorem Withstand Buoyancy-Driven Chemotaxis-Fluid Interaction?” Journal of the European Mathematical Society 25, no. 4 (2022): 1423–56. https://doi.org/10.4171/jems/1226."},"page":"1423-1456","intvolume":" 25","date_updated":"2025-12-18T20:11:51Z","publisher":"European Mathematical Society - EMS - Publishing House GmbH","author":[{"first_name":"Michael","last_name":"Winkler","full_name":"Winkler, Michael","id":"31496"}],"date_created":"2025-12-18T19:16:13Z","volume":25,"title":"Does Leray’s structure theorem withstand buoyancy-driven chemotaxis-fluid interaction?","doi":"10.4171/jems/1226"},{"doi":"10.1017/prm.2022.39","title":"Finite-time blow-up in a repulsive chemotaxis-consumption system","volume":153,"date_created":"2025-12-18T19:14:20Z","author":[{"full_name":"Wang, Yulan","last_name":"Wang","first_name":"Yulan"},{"first_name":"Michael","last_name":"Winkler","id":"31496","full_name":"Winkler, Michael"}],"date_updated":"2025-12-18T20:11:15Z","publisher":"Cambridge University Press (CUP)","page":"1150-1166","intvolume":" 153","citation":{"ama":"Wang Y, Winkler M. Finite-time blow-up in a repulsive chemotaxis-consumption system. Proceedings of the Royal Society of Edinburgh: Section A Mathematics. 2022;153(4):1150-1166. doi:10.1017/prm.2022.39","chicago":"Wang, Yulan, and Michael Winkler. “Finite-Time Blow-up in a Repulsive Chemotaxis-Consumption System.” Proceedings of the Royal Society of Edinburgh: Section A Mathematics 153, no. 4 (2022): 1150–66. https://doi.org/10.1017/prm.2022.39.","ieee":"Y. Wang and M. Winkler, “Finite-time blow-up in a repulsive chemotaxis-consumption system,” Proceedings of the Royal Society of Edinburgh: Section A Mathematics, vol. 153, no. 4, pp. 1150–1166, 2022, doi: 10.1017/prm.2022.39.","apa":"Wang, Y., & Winkler, M. (2022). Finite-time blow-up in a repulsive chemotaxis-consumption system. Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 153(4), 1150–1166. https://doi.org/10.1017/prm.2022.39","bibtex":"@article{Wang_Winkler_2022, title={Finite-time blow-up in a repulsive chemotaxis-consumption system}, volume={153}, DOI={10.1017/prm.2022.39}, number={4}, journal={Proceedings of the Royal Society of Edinburgh: Section A Mathematics}, publisher={Cambridge University Press (CUP)}, author={Wang, Yulan and Winkler, Michael}, year={2022}, pages={1150–1166} }","short":"Y. Wang, M. Winkler, Proceedings of the Royal Society of Edinburgh: Section A Mathematics 153 (2022) 1150–1166.","mla":"Wang, Yulan, and Michael Winkler. “Finite-Time Blow-up in a Repulsive Chemotaxis-Consumption System.” Proceedings of the Royal Society of Edinburgh: Section A Mathematics, vol. 153, no. 4, Cambridge University Press (CUP), 2022, pp. 1150–66, doi:10.1017/prm.2022.39."},"year":"2022","issue":"4","publication_identifier":{"issn":["0308-2105","1473-7124"]},"publication_status":"published","language":[{"iso":"eng"}],"user_id":"31496","_id":"63274","status":"public","abstract":[{"lang":"eng","text":"In a ball $\\Omega \\subset \\mathbb {R}^{n}$ with $n\\ge 2$, the chemotaxis system\r\n\\[ \\left\\{ \\begin{array}{@{}l} u_t = \\nabla \\cdot \\big( D(u)\\nabla u\\big) + \\nabla\\cdot \\big(\\dfrac{u}{v} \\nabla v\\big), \\\\ 0=\\Delta v - uv \\end{array} \\right. \\]is considered along with no-flux boundary conditions for $u$ and with prescribed constant positive Dirichlet boundary data for $v$. It is shown that if $D\\in C^{3}([0,\\infty ))$ is such that $0< D(\\xi ) \\le {K_D} (\\xi +1)^{-\\alpha }$ for all $\\xi >0$ with some ${K_D}>0$ and $\\alpha >0$, then for all initial data from a considerably large set of radial functions on $\\Omega$, the corresponding initial-boundary value problem admits a solution blowing up in finite time."}],"publication":"Proceedings of the Royal Society of Edinburgh: Section A Mathematics","type":"journal_article"},{"intvolume":" 25","citation":{"ama":"Winkler M. Arbitrarily fast grow-up rates in quasilinear Keller–Segel systems. Communications in Contemporary Mathematics. 2022;25(10). doi:10.1142/s0219199722500626","ieee":"M. Winkler, “Arbitrarily fast grow-up rates in quasilinear Keller–Segel systems,” Communications in Contemporary Mathematics, vol. 25, no. 10, Art. no. 2250062, 2022, doi: 10.1142/s0219199722500626.","chicago":"Winkler, Michael. “Arbitrarily Fast Grow-up Rates in Quasilinear Keller–Segel Systems.” Communications in Contemporary Mathematics 25, no. 10 (2022). https://doi.org/10.1142/s0219199722500626.","apa":"Winkler, M. (2022). Arbitrarily fast grow-up rates in quasilinear Keller–Segel systems. Communications in Contemporary Mathematics, 25(10), Article 2250062. https://doi.org/10.1142/s0219199722500626","bibtex":"@article{Winkler_2022, title={Arbitrarily fast grow-up rates in quasilinear Keller–Segel systems}, volume={25}, DOI={10.1142/s0219199722500626}, number={102250062}, journal={Communications in Contemporary Mathematics}, publisher={World Scientific Pub Co Pte Ltd}, author={Winkler, Michael}, year={2022} }","mla":"Winkler, Michael. “Arbitrarily Fast Grow-up Rates in Quasilinear Keller–Segel Systems.” Communications in Contemporary Mathematics, vol. 25, no. 10, 2250062, World Scientific Pub Co Pte Ltd, 2022, doi:10.1142/s0219199722500626.","short":"M. Winkler, Communications in Contemporary Mathematics 25 (2022)."},"year":"2022","issue":"10","publication_identifier":{"issn":["0219-1997","1793-6683"]},"publication_status":"published","doi":"10.1142/s0219199722500626","title":"Arbitrarily fast grow-up rates in quasilinear Keller–Segel systems","volume":25,"author":[{"first_name":"Michael","last_name":"Winkler","id":"31496","full_name":"Winkler, Michael"}],"date_created":"2025-12-18T19:17:23Z","publisher":"World Scientific Pub Co Pte Ltd","date_updated":"2025-12-18T20:12:13Z","status":"public","abstract":[{"lang":"eng","text":" The chemotaxis system [Formula: see text] is considered in a ball [Formula: see text], [Formula: see text], where the positive function [Formula: see text] reflects suitably weak diffusion by satisfying [Formula: see text] for some [Formula: see text]. It is shown that whenever [Formula: see text] is positive and satisfies [Formula: see text] as [Formula: see text], one can find a suitably regular nonlinearity [Formula: see text] with the property that at each sufficiently large mass level [Formula: see text] there exists a globally defined radially symmetric classical solution to a Neumann-type boundary value problem for (⋆) which satisfies [Formula: see text] "}],"publication":"Communications in Contemporary Mathematics","type":"journal_article","language":[{"iso":"eng"}],"article_number":"2250062","user_id":"31496","_id":"63282"},{"_id":"40212","department":[{"_id":"52"}],"user_id":"93461","language":[{"iso":"eng"}],"publication":"IKMT 2022; 13. GMM/ETG-Symposium","type":"conference","status":"public","date_updated":"2025-12-19T12:44:31Z","date_created":"2023-01-26T08:51:40Z","author":[{"first_name":"Barnabas","id":"93461","full_name":"Haucke-Korber, Barnabas","last_name":"Haucke-Korber","orcid":"0000-0003-0862-2069"},{"first_name":"Maximilian","orcid":"0000-0001-5427-9527","last_name":"Schenke","full_name":"Schenke, Maximilian","id":"52638"},{"first_name":"Oliver","full_name":"Wallscheid, Oliver","id":"11291","orcid":"https://orcid.org/0000-0001-9362-8777","last_name":"Wallscheid"}],"title":"Reinforcement Learning-Based Deep Q Direct Torque Control with Adaptable Switching Frequency Towards Six-Step Operation of Permanent Magnet Synchronous Motors","year":"2022","page":"1-6","citation":{"apa":"Haucke-Korber, B., Schenke, M., & Wallscheid, O. (2022). Reinforcement Learning-Based Deep Q Direct Torque Control with Adaptable Switching Frequency Towards Six-Step Operation of Permanent Magnet Synchronous Motors. IKMT 2022; 13. GMM/ETG-Symposium, 1–6.","bibtex":"@inproceedings{Haucke-Korber_Schenke_Wallscheid_2022, title={Reinforcement Learning-Based Deep Q Direct Torque Control with Adaptable Switching Frequency Towards Six-Step Operation of Permanent Magnet Synchronous Motors}, booktitle={IKMT 2022; 13. GMM/ETG-Symposium}, author={Haucke-Korber, Barnabas and Schenke, Maximilian and Wallscheid, Oliver}, year={2022}, pages={1–6} }","short":"B. Haucke-Korber, M. Schenke, O. Wallscheid, in: IKMT 2022; 13. GMM/ETG-Symposium, 2022, pp. 1–6.","mla":"Haucke-Korber, Barnabas, et al. “Reinforcement Learning-Based Deep Q Direct Torque Control with Adaptable Switching Frequency Towards Six-Step Operation of Permanent Magnet Synchronous Motors.” IKMT 2022; 13. GMM/ETG-Symposium, 2022, pp. 1–6.","ieee":"B. Haucke-Korber, M. Schenke, and O. Wallscheid, “Reinforcement Learning-Based Deep Q Direct Torque Control with Adaptable Switching Frequency Towards Six-Step Operation of Permanent Magnet Synchronous Motors,” in IKMT 2022; 13. GMM/ETG-Symposium, 2022, pp. 1–6.","chicago":"Haucke-Korber, Barnabas, Maximilian Schenke, and Oliver Wallscheid. “Reinforcement Learning-Based Deep Q Direct Torque Control with Adaptable Switching Frequency Towards Six-Step Operation of Permanent Magnet Synchronous Motors.” In IKMT 2022; 13. GMM/ETG-Symposium, 1–6, 2022.","ama":"Haucke-Korber B, Schenke M, Wallscheid O. Reinforcement Learning-Based Deep Q Direct Torque Control with Adaptable Switching Frequency Towards Six-Step Operation of Permanent Magnet Synchronous Motors. In: IKMT 2022; 13. GMM/ETG-Symposium. ; 2022:1-6."}}]