[{"publication":"Studies in Applied Mathematics","abstract":[{"text":"AbstractWe give an overview of analytical results concerned with chemotaxis systems where the signal is absorbed. We recall results on existence and properties of solutions for the prototypical chemotaxis‐consumption model and various variants and review more recent findings on its ability to support the emergence of spatial structures.","lang":"eng"}],"keyword":["Applied Mathematics"],"language":[{"iso":"eng"}],"issue":"4","year":"2023","publisher":"Wiley","date_created":"2024-04-07T12:50:45Z","title":"Depleting the signal: Analysis of chemotaxis‐consumption models—A survey","type":"journal_article","status":"public","_id":"53338","user_id":"31496","publication_status":"published","publication_identifier":{"issn":["0022-2526","1467-9590"]},"citation":{"ieee":"J. Lankeit and M. Winkler, “Depleting the signal: Analysis of chemotaxis‐consumption models—A survey,” Studies in Applied Mathematics, vol. 151, no. 4, pp. 1197–1229, 2023, doi: 10.1111/sapm.12625.","chicago":"Lankeit, Johannes, and Michael Winkler. “Depleting the Signal: Analysis of Chemotaxis‐consumption Models—A Survey.” Studies in Applied Mathematics 151, no. 4 (2023): 1197–1229. https://doi.org/10.1111/sapm.12625.","ama":"Lankeit J, Winkler M. Depleting the signal: Analysis of chemotaxis‐consumption models—A survey. Studies in Applied Mathematics. 2023;151(4):1197-1229. doi:10.1111/sapm.12625","short":"J. Lankeit, M. Winkler, Studies in Applied Mathematics 151 (2023) 1197–1229.","bibtex":"@article{Lankeit_Winkler_2023, title={Depleting the signal: Analysis of chemotaxis‐consumption models—A survey}, volume={151}, DOI={10.1111/sapm.12625}, number={4}, journal={Studies in Applied Mathematics}, publisher={Wiley}, author={Lankeit, Johannes and Winkler, Michael}, year={2023}, pages={1197–1229} }","mla":"Lankeit, Johannes, and Michael Winkler. “Depleting the Signal: Analysis of Chemotaxis‐consumption Models—A Survey.” Studies in Applied Mathematics, vol. 151, no. 4, Wiley, 2023, pp. 1197–229, doi:10.1111/sapm.12625.","apa":"Lankeit, J., & Winkler, M. (2023). Depleting the signal: Analysis of chemotaxis‐consumption models—A survey. Studies in Applied Mathematics, 151(4), 1197–1229. https://doi.org/10.1111/sapm.12625"},"intvolume":" 151","page":"1197-1229","date_updated":"2025-12-18T20:16:04Z","author":[{"first_name":"Johannes","full_name":"Lankeit, Johannes","last_name":"Lankeit"},{"first_name":"Michael","full_name":"Winkler, Michael","id":"31496","last_name":"Winkler"}],"volume":151,"doi":"10.1111/sapm.12625"},{"status":"public","publication":"ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE","type":"journal_article","language":[{"iso":"eng"}],"article_number":"286","user_id":"31496","_id":"63243","citation":{"apa":"Colasuonno, F., & Winkler, M. (2023). Stability vs.~instability of singular steady states in the parabolic-elliptic Keller-Segel system on $\\R^n$. ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE, Article 286. https://doi.org/10.2422/2036-2145.202303_006","short":"F. Colasuonno, M. Winkler, ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE (2023).","bibtex":"@article{Colasuonno_Winkler_2023, title={Stability vs.~instability of singular steady states in the parabolic-elliptic Keller-Segel system on $\\R^n$}, DOI={10.2422/2036-2145.202303_006}, number={286}, journal={ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE}, publisher={Scuola Normale Superiore - Edizioni della Normale}, author={Colasuonno, Francesca and Winkler, Michael}, year={2023} }","mla":"Colasuonno, Francesca, and Michael Winkler. “Stability vs.~instability of Singular Steady States in the Parabolic-Elliptic Keller-Segel System on $\\R^n$.” ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE, 286, Scuola Normale Superiore - Edizioni della Normale, 2023, doi:10.2422/2036-2145.202303_006.","ama":"Colasuonno F, Winkler M. Stability vs.~instability of singular steady states in the parabolic-elliptic Keller-Segel system on $\\R^n$. ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE. Published online 2023. doi:10.2422/2036-2145.202303_006","chicago":"Colasuonno, Francesca, and Michael Winkler. “Stability vs.~instability of Singular Steady States in the Parabolic-Elliptic Keller-Segel System on $\\R^n$.” ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE, 2023. https://doi.org/10.2422/2036-2145.202303_006.","ieee":"F. Colasuonno and M. Winkler, “Stability vs.~instability of singular steady states in the parabolic-elliptic Keller-Segel system on $\\R^n$,” ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE, Art. no. 286, 2023, doi: 10.2422/2036-2145.202303_006."},"year":"2023","publication_identifier":{"issn":["2036-2145","0391-173X"]},"publication_status":"published","doi":"10.2422/2036-2145.202303_006","title":"Stability vs.~instability of singular steady states in the parabolic-elliptic Keller-Segel system on $\\R^n$","author":[{"last_name":"Colasuonno","full_name":"Colasuonno, Francesca","first_name":"Francesca"},{"last_name":"Winkler","full_name":"Winkler, Michael","id":"31496","first_name":"Michael"}],"date_created":"2025-12-18T18:58:40Z","date_updated":"2025-12-18T20:17:21Z","publisher":"Scuola Normale Superiore - Edizioni della Normale"},{"language":[{"iso":"eng"}],"_id":"35528","department":[{"_id":"34"},{"_id":"10"},{"_id":"90"}],"user_id":"15645","status":"public","publication":"Nonlinearity","type":"journal_article","title":"Radial solutions to a chemotaxis-consumption model involving prescribed signal concentrations on the boundary","date_updated":"2023-01-20T13:18:15Z","volume":35,"author":[{"last_name":"Lankeit","full_name":"Lankeit, Johannes","first_name":"Johannes"},{"full_name":"Winkler, Michael","id":"31496","last_name":"Winkler","first_name":"Michael"}],"date_created":"2023-01-09T15:33:38Z","year":"2022","page":"719-749","intvolume":" 35","citation":{"ieee":"J. Lankeit and M. Winkler, “Radial solutions to a chemotaxis-consumption model involving prescribed signal concentrations on the boundary,” Nonlinearity, vol. 35, pp. 719–749, 2022.","chicago":"Lankeit, Johannes, and Michael Winkler. “Radial Solutions to a Chemotaxis-Consumption Model Involving Prescribed Signal Concentrations on the Boundary.” Nonlinearity 35 (2022): 719–49.","ama":"Lankeit J, Winkler M. Radial solutions to a chemotaxis-consumption model involving prescribed signal concentrations on the boundary. Nonlinearity. 2022;35:719-749.","bibtex":"@article{Lankeit_Winkler_2022, title={Radial solutions to a chemotaxis-consumption model involving prescribed signal concentrations on the boundary}, volume={35}, journal={Nonlinearity}, author={Lankeit, Johannes and Winkler, Michael}, year={2022}, pages={719–749} }","mla":"Lankeit, Johannes, and Michael Winkler. “Radial Solutions to a Chemotaxis-Consumption Model Involving Prescribed Signal Concentrations on the Boundary.” Nonlinearity, vol. 35, 2022, pp. 719–49.","short":"J. Lankeit, M. Winkler, Nonlinearity 35 (2022) 719–749.","apa":"Lankeit, J., & Winkler, M. (2022). Radial solutions to a chemotaxis-consumption model involving prescribed signal concentrations on the boundary. Nonlinearity, 35, 719–749."}},{"year":"2022","intvolume":" 389","page":"439-489","citation":{"mla":"Winkler, Michael. “Reaction-Driven Relaxation in Threee-Dimensional Keller-Segel-Navier-Stokes Interaction.” Communications in Mathematical Physics, vol. 389, 2022, pp. 439–89.","bibtex":"@article{Winkler_2022, title={Reaction-driven relaxation in threee-dimensional Keller-Segel-Navier-Stokes interaction.}, volume={389}, journal={Communications in Mathematical Physics}, author={Winkler, Michael}, year={2022}, pages={439–489} }","short":"M. Winkler, Communications in Mathematical Physics 389 (2022) 439–489.","apa":"Winkler, M. (2022). Reaction-driven relaxation in threee-dimensional Keller-Segel-Navier-Stokes interaction. Communications in Mathematical Physics, 389, 439–489.","ama":"Winkler M. Reaction-driven relaxation in threee-dimensional Keller-Segel-Navier-Stokes interaction. Communications in Mathematical Physics. 2022;389:439-489.","chicago":"Winkler, Michael. “Reaction-Driven Relaxation in Threee-Dimensional Keller-Segel-Navier-Stokes Interaction.” Communications in Mathematical Physics 389 (2022): 439–89.","ieee":"M. Winkler, “Reaction-driven relaxation in threee-dimensional Keller-Segel-Navier-Stokes interaction.,” Communications in Mathematical Physics, vol. 389, pp. 439–489, 2022."},"title":"Reaction-driven relaxation in threee-dimensional Keller-Segel-Navier-Stokes interaction.","date_updated":"2023-01-20T13:17:37Z","volume":389,"author":[{"first_name":"Michael","last_name":"Winkler","full_name":"Winkler, Michael","id":"31496"}],"date_created":"2023-01-09T16:32:21Z","status":"public","publication":"Communications in Mathematical Physics","type":"journal_article","language":[{"iso":"eng"}],"_id":"35568","department":[{"_id":"34"},{"_id":"10"},{"_id":"90"}],"user_id":"15645"},{"title":"A family of mass-critical Keller-Segel systems.","date_updated":"2023-01-20T13:17:33Z","volume":124,"date_created":"2023-01-09T16:35:09Z","author":[{"first_name":"Michael","last_name":"Winkler","full_name":"Winkler, Michael","id":"31496"}],"year":"2022","intvolume":" 124","page":"133-181","citation":{"bibtex":"@article{Winkler_2022, title={A family of mass-critical Keller-Segel systems.}, volume={124}, journal={Proceedings of the London Mathematical Society}, author={Winkler, Michael}, year={2022}, pages={133–181} }","mla":"Winkler, Michael. “A Family of Mass-Critical Keller-Segel Systems.” Proceedings of the London Mathematical Society, vol. 124, 2022, pp. 133–81.","short":"M. Winkler, Proceedings of the London Mathematical Society 124 (2022) 133–181.","apa":"Winkler, M. (2022). A family of mass-critical Keller-Segel systems. Proceedings of the London Mathematical Society, 124, 133–181.","chicago":"Winkler, Michael. “A Family of Mass-Critical Keller-Segel Systems.” Proceedings of the London Mathematical Society 124 (2022): 133–81.","ieee":"M. Winkler, “A family of mass-critical Keller-Segel systems.,” Proceedings of the London Mathematical Society, vol. 124, pp. 133–181, 2022.","ama":"Winkler M. A family of mass-critical Keller-Segel systems. Proceedings of the London Mathematical Society. 2022;124:133-181."},"language":[{"iso":"eng"}],"_id":"35574","department":[{"_id":"34"},{"_id":"10"},{"_id":"90"}],"user_id":"15645","status":"public","publication":"Proceedings of the London Mathematical Society","type":"journal_article"},{"date_created":"2023-01-09T13:02:46Z","author":[{"last_name":" Kang","full_name":" Kang, Kyungkeun","first_name":"Kyungkeun"},{"first_name":"Jihoon","full_name":"Lee, Jihoon","last_name":"Lee"},{"id":"31496","full_name":"Winkler, Michael","last_name":"Winkler","first_name":"Michael"}],"volume":42,"date_updated":"2023-01-21T09:30:47Z","title":"Global weak solutions to a chemotaxis-Navier-Stokes system in $R^3$.","citation":{"bibtex":"@article{ Kang_Lee_Winkler_2022, title={Global weak solutions to a chemotaxis-Navier-Stokes system in $R^3$.}, volume={42}, journal={Discrete and Continuous Dynamical Systems}, author={ Kang, Kyungkeun and Lee, Jihoon and Winkler, Michael}, year={2022}, pages={5201–5222} }","short":"K. Kang, J. Lee, M. Winkler, Discrete and Continuous Dynamical Systems 42 (2022) 5201–5222.","mla":"Kang, Kyungkeun, et al. “Global Weak Solutions to a Chemotaxis-Navier-Stokes System in $R^3$.” Discrete and Continuous Dynamical Systems, vol. 42, 2022, pp. 5201–22.","apa":"Kang, K., Lee, J., & Winkler, M. (2022). Global weak solutions to a chemotaxis-Navier-Stokes system in $R^3$. Discrete and Continuous Dynamical Systems, 42, 5201–5222.","ama":"Kang K, Lee J, Winkler M. Global weak solutions to a chemotaxis-Navier-Stokes system in $R^3$. Discrete and Continuous Dynamical Systems. 2022;42:5201-5222.","ieee":"K. Kang, J. Lee, and M. Winkler, “Global weak solutions to a chemotaxis-Navier-Stokes system in $R^3$.,” Discrete and Continuous Dynamical Systems, vol. 42, pp. 5201–5222, 2022.","chicago":"Kang, Kyungkeun, Jihoon Lee, and Michael Winkler. “Global Weak Solutions to a Chemotaxis-Navier-Stokes System in $R^3$.” Discrete and Continuous Dynamical Systems 42 (2022): 5201–22."},"page":"5201-5222","intvolume":" 42","year":"2022","user_id":"15645","department":[{"_id":"34"},{"_id":"10"},{"_id":"90"}],"_id":"35483","language":[{"iso":"eng"}],"type":"journal_article","publication":"Discrete and Continuous Dynamical Systems","status":"public"},{"status":"public","type":"journal_article","publication":"Journal de Mathématiques Pures et Appliquées","language":[{"iso":"eng"}],"user_id":"15645","department":[{"_id":"34"},{"_id":"10"},{"_id":"90"}],"_id":"35481","citation":{"chicago":"Fuhrmann, Jan, Johannes Lankeit, and Michael Winkler. “A Double Critical Mass Phenomenon in a No-Flux-Dirichlet Keller-Segel System.” Journal de Mathématiques Pures et Appliquées 162 (2022): 124–51.","ieee":"J. Fuhrmann, J. Lankeit, and M. Winkler, “A double critical mass phenomenon in a no-flux-Dirichlet Keller-Segel system.,” Journal de Mathématiques Pures et Appliquées, vol. 162, pp. 124–151, 2022.","ama":"Fuhrmann J, Lankeit J, Winkler M. A double critical mass phenomenon in a no-flux-Dirichlet Keller-Segel system. Journal de Mathématiques Pures et Appliquées. 2022;162:124-151.","apa":"Fuhrmann, J., Lankeit, J., & Winkler, M. (2022). A double critical mass phenomenon in a no-flux-Dirichlet Keller-Segel system. 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Winkler, Journal de Mathématiques Pures et Appliquées 162 (2022) 124–151."},"page":"124-151","intvolume":" 162","year":"2022","title":"A double critical mass phenomenon in a no-flux-Dirichlet Keller-Segel system.","author":[{"full_name":"Fuhrmann, Jan","last_name":"Fuhrmann","first_name":"Jan"},{"first_name":"Johannes","last_name":"Lankeit","full_name":"Lankeit, Johannes"},{"first_name":"Michael","last_name":"Winkler","id":"31496","full_name":"Winkler, Michael"}],"date_created":"2023-01-09T12:45:23Z","volume":162,"date_updated":"2023-02-01T09:57:48Z"},{"title":"A smallness condition ensuring boundedness in a two-dimensional chemotaxis-Navier-Stokes system involving Dirichlet boundary conditions for the signal.","volume":38,"date_created":"2023-01-09T16:25:05Z","author":[{"first_name":"Yulan","full_name":"Wang, Yulan","last_name":"Wang"},{"first_name":"Michael","last_name":"Winkler","full_name":"Winkler, Michael","id":"31496"},{"first_name":"Zhaoyin","full_name":"Xiang, Zhaoyin","last_name":"Xiang"}],"date_updated":"2023-02-01T10:32:20Z","intvolume":" 38","page":"985-1001","citation":{"apa":"Wang, Y., Winkler, M., & Xiang, Z. (2022). A smallness condition ensuring boundedness in a two-dimensional chemotaxis-Navier-Stokes system involving Dirichlet boundary conditions for the signal. Acta Mathematica Sinica (English Series), 38, 985–1001.","short":"Y. Wang, M. Winkler, Z. Xiang, Acta Mathematica Sinica (English Series) 38 (2022) 985–1001.","mla":"Wang, Yulan, et al. “A Smallness Condition Ensuring Boundedness in a Two-Dimensional Chemotaxis-Navier-Stokes System Involving Dirichlet Boundary Conditions for the Signal.” Acta Mathematica Sinica (English Series), vol. 38, 2022, pp. 985–1001.","bibtex":"@article{Wang_Winkler_Xiang_2022, title={A smallness condition ensuring boundedness in a two-dimensional chemotaxis-Navier-Stokes system involving Dirichlet boundary conditions for the signal.}, volume={38}, journal={Acta Mathematica Sinica (English Series)}, author={Wang, Yulan and Winkler, Michael and Xiang, Zhaoyin}, year={2022}, pages={985–1001} }","ama":"Wang Y, Winkler M, Xiang Z. A smallness condition ensuring boundedness in a two-dimensional chemotaxis-Navier-Stokes system involving Dirichlet boundary conditions for the signal. Acta Mathematica Sinica (English Series). 2022;38:985-1001.","chicago":"Wang, Yulan, Michael Winkler, and Zhaoyin Xiang. “A Smallness Condition Ensuring Boundedness in a Two-Dimensional Chemotaxis-Navier-Stokes System Involving Dirichlet Boundary Conditions for the Signal.” Acta Mathematica Sinica (English Series) 38 (2022): 985–1001.","ieee":"Y. Wang, M. Winkler, and Z. Xiang, “A smallness condition ensuring boundedness in a two-dimensional chemotaxis-Navier-Stokes system involving Dirichlet boundary conditions for the signal.,” Acta Mathematica Sinica (English Series), vol. 38, pp. 985–1001, 2022."},"year":"2022","language":[{"iso":"eng"}],"department":[{"_id":"34"},{"_id":"10"},{"_id":"90"}],"user_id":"15645","_id":"35565","status":"public","publication":"Acta Mathematica Sinica (English Series)","type":"journal_article"},{"publication":"Analysis and Applications","type":"journal_article","status":"public","_id":"35560","department":[{"_id":"34"},{"_id":"10"},{"_id":"90"}],"user_id":"15645","language":[{"iso":"eng"}],"year":"2022","intvolume":" 20","page":"141-170","citation":{"ama":"Wang Y, Winkler M, Xiang Z. Global mass-preserving solutions to a chemotaxis-fluid model involving Dirichlet boundary conditions for the signal. Analysis and Applications. 2022;20:141-170.","chicago":"Wang, Yulan, Michael Winkler, and Zhaoyin Xiang. “Global Mass-Preserving Solutions to a Chemotaxis-Fluid Model Involving Dirichlet Boundary Conditions for the Signal.” Analysis and Applications 20 (2022): 141–70.","ieee":"Y. Wang, M. Winkler, and Z. Xiang, “Global mass-preserving solutions to a chemotaxis-fluid model involving Dirichlet boundary conditions for the signal.,” Analysis and Applications, vol. 20, pp. 141–170, 2022.","apa":"Wang, Y., Winkler, M., & Xiang, Z. (2022). Global mass-preserving solutions to a chemotaxis-fluid model involving Dirichlet boundary conditions for the signal. Analysis and Applications, 20, 141–170.","short":"Y. Wang, M. Winkler, Z. Xiang, Analysis and Applications 20 (2022) 141–170.","mla":"Wang, Yulan, et al. “Global Mass-Preserving Solutions to a Chemotaxis-Fluid Model Involving Dirichlet Boundary Conditions for the Signal.” Analysis and Applications, vol. 20, 2022, pp. 141–70.","bibtex":"@article{Wang_Winkler_Xiang_2022, title={Global mass-preserving solutions to a chemotaxis-fluid model involving Dirichlet boundary conditions for the signal.}, volume={20}, journal={Analysis and Applications}, author={Wang, Yulan and Winkler, Michael and Xiang, Zhaoyin}, year={2022}, pages={141–170} }"},"date_updated":"2023-02-01T10:29:44Z","volume":20,"date_created":"2023-01-09T16:21:59Z","author":[{"full_name":"Wang, Yulan","last_name":"Wang","first_name":"Yulan"},{"last_name":"Winkler","id":"31496","full_name":"Winkler, Michael","first_name":"Michael"},{"first_name":"Zhaoyin","last_name":"Xiang","full_name":"Xiang, Zhaoyin"}],"title":"Global mass-preserving solutions to a chemotaxis-fluid model involving Dirichlet boundary conditions for the signal."},{"type":"journal_article","publication":"Proceedings of the Royal Society of Edinburgh Section A: Mathematics","status":"public","_id":"35556","user_id":"15645","department":[{"_id":"34"},{"_id":"10"},{"_id":"90"}],"language":[{"iso":"eng"}],"year":"2022","citation":{"apa":"Tao, Y., & Winkler, M. (2022). Asymptotic stability of spatial homogeneity in a haptotaxis model for oncolytic virotherapy. Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 152, 81–101.","mla":"Tao, Youshan, and Michael Winkler. “Asymptotic Stability of Spatial Homogeneity in a Haptotaxis Model for Oncolytic Virotherapy.” Proceedings of the Royal Society of Edinburgh Section A: Mathematics, vol. 152, 2022, pp. 81–101.","bibtex":"@article{Tao_Winkler_2022, title={Asymptotic stability of spatial homogeneity in a haptotaxis model for oncolytic virotherapy.}, volume={152}, journal={Proceedings of the Royal Society of Edinburgh Section A: Mathematics}, author={Tao, Youshan and Winkler, Michael}, year={2022}, pages={81–101} }","short":"Y. Tao, M. Winkler, Proceedings of the Royal Society of Edinburgh Section A: Mathematics 152 (2022) 81–101.","chicago":"Tao, Youshan, and Michael Winkler. “Asymptotic Stability of Spatial Homogeneity in a Haptotaxis Model for Oncolytic Virotherapy.” Proceedings of the Royal Society of Edinburgh Section A: Mathematics 152 (2022): 81–101.","ieee":"Y. Tao and M. Winkler, “Asymptotic stability of spatial homogeneity in a haptotaxis model for oncolytic virotherapy.,” Proceedings of the Royal Society of Edinburgh Section A: Mathematics, vol. 152, pp. 81–101, 2022.","ama":"Tao Y, Winkler M. Asymptotic stability of spatial homogeneity in a haptotaxis model for oncolytic virotherapy. Proceedings of the Royal Society of Edinburgh Section A: Mathematics. 2022;152:81-101."},"intvolume":" 152","page":"81-101","date_updated":"2023-02-01T10:16:04Z","date_created":"2023-01-09T16:16:07Z","author":[{"first_name":"Youshan","full_name":"Tao, Youshan","last_name":"Tao"},{"id":"31496","full_name":"Winkler, Michael","last_name":"Winkler","first_name":"Michael"}],"volume":152,"title":"Asymptotic stability of spatial homogeneity in a haptotaxis model for oncolytic virotherapy."},{"title":"Nonnegative solutions to a doubly degenerate nutrient taxis system ","date_updated":"2023-02-01T10:09:37Z","author":[{"first_name":"Genglin","full_name":"Li, Genglin","last_name":"Li"},{"last_name":"Winkler","id":"31496","full_name":"Winkler, Michael","first_name":"Michael"}],"date_created":"2023-01-09T15:51:29Z","volume":21,"year":"2022","citation":{"ama":"Li G, Winkler M. Nonnegative solutions to a doubly degenerate nutrient taxis system . Communications on Pure and Applied Analysis. 2022;21:687-784.","ieee":"G. Li and M. Winkler, “Nonnegative solutions to a doubly degenerate nutrient taxis system ,” Communications on Pure and Applied Analysis, vol. 21, pp. 687–784, 2022.","chicago":"Li, Genglin, and Michael Winkler. “Nonnegative Solutions to a Doubly Degenerate Nutrient Taxis System .” Communications on Pure and Applied Analysis 21 (2022): 687–784.","bibtex":"@article{Li_Winkler_2022, title={Nonnegative solutions to a doubly degenerate nutrient taxis system }, volume={21}, journal={Communications on Pure and Applied Analysis}, author={Li, Genglin and Winkler, Michael}, year={2022}, pages={687–784} }","short":"G. Li, M. Winkler, Communications on Pure and Applied Analysis 21 (2022) 687–784.","mla":"Li, Genglin, and Michael Winkler. “Nonnegative Solutions to a Doubly Degenerate Nutrient Taxis System .” Communications on Pure and Applied Analysis, vol. 21, 2022, pp. 687–784.","apa":"Li, G., & Winkler, M. (2022). Nonnegative solutions to a doubly degenerate nutrient taxis system . Communications on Pure and Applied Analysis, 21, 687–784."},"page":"687-784","intvolume":" 21","language":[{"iso":"eng"}],"_id":"35532","user_id":"15645","department":[{"_id":"34"},{"_id":"10"},{"_id":"90"}],"status":"public","type":"journal_article","publication":"Communications on Pure and Applied Analysis"},{"abstract":[{"lang":"eng","text":"AbstractThe chemotaxis–Stokes systemnt+u⋅∇n=∇⋅(D(n)∇n)−∇⋅(nS(x,n,c)⋅∇c),ct+u⋅∇c=Δc−nc,ut=Δu+∇P+n∇Φ,∇⋅u=0,\\left\\{\\begin{array}{l}{n}_{t}+u\\cdot \\nabla n=\\nabla \\cdot (D\\left(n)\\nabla n)-\\nabla \\cdot (nS\\left(x,n,c)\\cdot \\nabla c),\\\\ {c}_{t}+u\\cdot \\nabla c=\\Delta c-nc,\\\\ {u}_{t}=\\Delta u+\\nabla P+n\\nabla \\Phi ,\\hspace{1.0em}\\nabla \\cdot u=0,\\end{array}\\right.is considered in a smoothly bounded convex domainΩ⊂R3\\Omega \\subset {{\\mathbb{R}}}^{3}, with given suitably regular functionsD:[0,∞)→[0,∞)D:{[}0,\\infty )\\to {[}0,\\infty ),S:Ω¯×[0,∞)×(0,∞)→R3×3S:\\overline{\\Omega }\\times {[}0,\\infty )\\times \\left(0,\\infty )\\to {{\\mathbb{R}}}^{3\\times 3}andΦ:Ω¯→R\\Phi :\\overline{\\Omega }\\to {\\mathbb{R}}such thatD>0D\\gt 0on(0,∞)\\left(0,\\infty ). It is shown that if with some nondecreasingS0:(0,∞)→(0,∞){S}_{0}:\\left(0,\\infty )\\to \\left(0,\\infty )we have∣S(x,n,c)∣≤S0(c)c12for all(x,n,c)∈Ω¯×[0,∞)×(0,∞),| S\\left(x,n,c)| \\le \\frac{{S}_{0}\\left(c)}{{c}^{\\tfrac{1}{2}}}\\hspace{1.0em}\\hspace{0.1em}\\text{for all}\\hspace{0.1em}\\hspace{0.33em}\\left(x,n,c)\\in \\overline{\\Omega }\\times {[}0,\\infty )\\times \\left(0,\\infty ),then for allM>0M\\gt 0there existsL(M)>0L\\left(M)\\gt 0such that wheneverliminfn→∞D(n)>L(M)andliminfn↘0D(n)n>0,\\mathop{\\mathrm{liminf}}\\limits_{n\\to \\infty }D\\left(n)\\gt L\\left(M)\\hspace{1.0em}\\hspace{0.1em}\\text{and}\\hspace{0.1em}\\hspace{1.0em}\\mathop{\\mathrm{liminf}}\\limits_{n\\searrow 0}\\frac{D\\left(n)}{n}\\gt 0,for all sufficiently regular initial data(n0,c0,u0)\\left({n}_{0},{c}_{0},{u}_{0})fulfilling‖c0‖L∞(Ω)≤M\\Vert {c}_{0}{\\Vert }_{{L}^{\\infty }\\left(\\Omega )}\\le Man associated no-flux/no-flux/Dirichlet initial-boundary value problem admits a global bounded weak solution, classical if additionallyD(0)>0D\\left(0)\\gt 0. When combined with previously known results, this particularly implies global existence of bounded solutions whenD(n)=nm−1D\\left(n)={n}^{m-1},n≥0n\\ge 0, with arbitrarym>1m\\gt 1, but beyond this asserts global boundedness also in the presence of diffusivities which exhibit arbitrarily slow divergence to+∞+\\inftyat large densities and of possibly singular chemotactic sensitivities."}],"status":"public","publication":"Advanced Nonlinear Studies","type":"journal_article","language":[{"iso":"eng"}],"_id":"63310","user_id":"31496","year":"2022","page":"88-117","intvolume":" 22","citation":{"apa":"Winkler, M. (2022). Chemotaxis-Stokes interaction with very weak diffusion enhancement: Blow-up exclusion via detection of absorption-induced entropy structures involving multiplicative couplings. Advanced Nonlinear Studies, 22(1), 88–117. https://doi.org/10.1515/ans-2022-0004","bibtex":"@article{Winkler_2022, title={Chemotaxis-Stokes interaction with very weak diffusion enhancement: Blow-up exclusion via detection of absorption-induced entropy structures involving multiplicative couplings}, volume={22}, DOI={10.1515/ans-2022-0004}, number={1}, journal={Advanced Nonlinear Studies}, publisher={Walter de Gruyter GmbH}, author={Winkler, Michael}, year={2022}, pages={88–117} }","short":"M. Winkler, Advanced Nonlinear Studies 22 (2022) 88–117.","mla":"Winkler, Michael. “Chemotaxis-Stokes Interaction with Very Weak Diffusion Enhancement: Blow-up Exclusion via Detection of Absorption-Induced Entropy Structures Involving Multiplicative Couplings.” Advanced Nonlinear Studies, vol. 22, no. 1, Walter de Gruyter GmbH, 2022, pp. 88–117, doi:10.1515/ans-2022-0004.","chicago":"Winkler, Michael. “Chemotaxis-Stokes Interaction with Very Weak Diffusion Enhancement: Blow-up Exclusion via Detection of Absorption-Induced Entropy Structures Involving Multiplicative Couplings.” Advanced Nonlinear Studies 22, no. 1 (2022): 88–117. https://doi.org/10.1515/ans-2022-0004.","ieee":"M. Winkler, “Chemotaxis-Stokes interaction with very weak diffusion enhancement: Blow-up exclusion via detection of absorption-induced entropy structures involving multiplicative couplings,” Advanced Nonlinear Studies, vol. 22, no. 1, pp. 88–117, 2022, doi: 10.1515/ans-2022-0004.","ama":"Winkler M. Chemotaxis-Stokes interaction with very weak diffusion enhancement: Blow-up exclusion via detection of absorption-induced entropy structures involving multiplicative couplings. Advanced Nonlinear Studies. 2022;22(1):88-117. doi:10.1515/ans-2022-0004"},"publication_identifier":{"issn":["2169-0375"]},"publication_status":"published","issue":"1","title":"Chemotaxis-Stokes interaction with very weak diffusion enhancement: Blow-up exclusion via detection of absorption-induced entropy structures involving multiplicative couplings","doi":"10.1515/ans-2022-0004","publisher":"Walter de Gruyter GmbH","date_updated":"2025-12-18T20:05:30Z","volume":22,"date_created":"2025-12-18T19:29:40Z","author":[{"first_name":"Michael","last_name":"Winkler","id":"31496","full_name":"Winkler, Michael"}]},{"user_id":"31496","_id":"63305","language":[{"iso":"eng"}],"article_number":"108","type":"journal_article","publication":"Calculus of Variations and Partial Differential Equations","status":"public","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"}],"author":[{"first_name":"Michael","full_name":"Winkler, Michael","id":"31496","last_name":"Winkler"}],"date_created":"2025-12-18T19:26:32Z","volume":61,"publisher":"Springer Science and Business Media LLC","date_updated":"2025-12-18T20:04:43Z","doi":"10.1007/s00526-021-02168-2","title":"Stabilization of arbitrary structures in a doubly degenerate reaction-diffusion system modeling bacterial motion on a nutrient-poor agar","issue":"3","publication_status":"published","publication_identifier":{"issn":["0944-2669","1432-0835"]},"citation":{"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.","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.","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","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","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} }","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)."},"intvolume":" 61","year":"2022"},{"citation":{"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","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.","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"},"intvolume":" 3","year":"2022","issue":"4","publication_status":"published","publication_identifier":{"issn":["2662-2963","2662-2971"]},"doi":"10.1007/s42985-022-00186-z","title":"Oscillatory decay in a degenerate parabolic equation","date_created":"2025-12-18T19:30:04Z","author":[{"last_name":"Winkler","full_name":"Winkler, Michael","id":"31496","first_name":"Michael"}],"volume":3,"publisher":"Springer Science and Business Media LLC","date_updated":"2025-12-18T20:05:38Z","status":"public","abstract":[{"lang":"eng","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 "}],"type":"journal_article","publication":"Partial Differential Equations and Applications","language":[{"iso":"eng"}],"article_number":"47","user_id":"31496","_id":"63311"},{"issue":"11","publication_status":"published","publication_identifier":{"issn":["1531-3492","1553-524X"]},"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","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.","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.","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.","short":"M. Winkler, Discrete and Continuous Dynamical Systems - B 27 (2022).","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","year":"2022","author":[{"id":"31496","full_name":"Winkler, Michael","last_name":"Winkler","first_name":"Michael"}],"date_created":"2025-12-18T19:30:32Z","volume":27,"publisher":"American Institute of Mathematical Sciences (AIMS)","date_updated":"2025-12-18T20:05:47Z","doi":"10.3934/dcdsb.2022009","title":"Approaching logarithmic singularities in quasilinear chemotaxis-consumption systems with signal-dependent sensitivities","type":"journal_article","publication":"Discrete and Continuous Dynamical Systems - B","status":"public","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 $\\end{document}</tex-math></inline-formula>.</p>"}],"user_id":"31496","_id":"63312","language":[{"iso":"eng"}],"article_number":"6565"},{"issue":"9","publication_identifier":{"issn":["0025-584X","1522-2616"]},"publication_status":"published","intvolume":" 295","page":"1840-1862","citation":{"apa":"Winkler, M. (2022). A unifying approach toward boundedness in Keller–Segel type cross‐diffusion systems via conditional L∞$L^\\infty$ estimates for taxis gradients. Mathematische Nachrichten, 295(9), 1840–1862. https://doi.org/10.1002/mana.202000403","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} }","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.","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"},"year":"2022","volume":295,"date_created":"2025-12-18T19:28:46Z","author":[{"last_name":"Winkler","full_name":"Winkler, Michael","id":"31496","first_name":"Michael"}],"publisher":"Wiley","date_updated":"2025-12-18T20:05:19Z","doi":"10.1002/mana.202000403","title":"A unifying approach toward boundedness in Keller–Segel type cross‐diffusion systems via conditional L∞$L^\\infty$ estimates for taxis gradients","publication":"Mathematische Nachrichten","type":"journal_article","status":"public","abstract":[{"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.","lang":"eng"}],"user_id":"31496","_id":"63309","language":[{"iso":"eng"}]},{"date_created":"2025-12-18T19:26:56Z","author":[{"first_name":"Michael","last_name":"Winkler","id":"31496","full_name":"Winkler, Michael"}],"volume":71,"date_updated":"2025-12-18T20:04:53Z","publisher":"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","issue":"4","publication_status":"published","publication_identifier":{"issn":["0022-2518"]},"citation":{"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","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.","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.","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"},"intvolume":" 71","page":"1437-1465","year":"2022","user_id":"31496","_id":"63306","language":[{"iso":"eng"}],"type":"journal_article","publication":"Indiana University Mathematics Journal","status":"public"},{"publication_identifier":{"issn":["1664-3607","1664-3615"]},"publication_status":"published","issue":"02","year":"2022","intvolume":" 13","citation":{"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.","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.","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","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.","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} }","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"},"date_updated":"2025-12-18T20:07:05Z","publisher":"World Scientific Pub Co Pte Ltd","volume":13,"author":[{"id":"31496","full_name":"Winkler, Michael","last_name":"Winkler","first_name":"Michael"}],"date_created":"2025-12-18T19:18:11Z","title":"Application of the Moser–Trudinger inequality in the construction of global solutions to a strongly degenerate migration model","doi":"10.1142/s1664360722500126","publication":"Bulletin of Mathematical Sciences","type":"journal_article","abstract":[{"lang":"eng","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. "}],"status":"public","_id":"63284","user_id":"31496","article_number":"2250012","language":[{"iso":"eng"}]},{"year":"2022","citation":{"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","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","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} }","short":"M. Winkler, Asymptotic Analysis 131 (2022) 33–57."},"intvolume":" 131","page":"33-57","publication_status":"published","publication_identifier":{"issn":["0921-7134","1875-8576"]},"issue":"1","title":"Exponential grow-up rates in a quasilinear Keller–Segel system","doi":"10.3233/asy-221765","date_updated":"2025-12-18T20:07:19Z","publisher":"SAGE Publications","author":[{"first_name":"Michael","last_name":"Winkler","full_name":"Winkler, Michael","id":"31496"}],"date_created":"2025-12-18T19:18:51Z","volume":131,"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 ( ⋆ ). "}],"status":"public","type":"journal_article","publication":"Asymptotic Analysis","language":[{"iso":"eng"}],"_id":"63286","user_id":"31496"},{"year":"2022","citation":{"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","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).","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","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.","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."},"intvolume":" 42","publication_status":"published","publication_identifier":{"issn":["1078-0947","1553-5231"]},"issue":"11","title":"Global weak solutions to a chemotaxis-Navier-Stokes system in $ \\mathbb{R}^3 $","doi":"10.3934/dcds.2022091","date_updated":"2025-12-18T20:08:21Z","publisher":"American Institute of Mathematical Sciences (AIMS)","date_created":"2025-12-18T19:22:04Z","author":[{"first_name":"Kyungkeun","last_name":"Kang","full_name":"Kang, Kyungkeun"},{"first_name":"Jihoon","last_name":"Lee","full_name":"Lee, Jihoon"},{"full_name":"Winkler, Michael","id":"31496","last_name":"Winkler","first_name":"Michael"}],"volume":42,"abstract":[{"text":"<p style='text-indent:20px;'>The Cauchy problem in <inline-formula><tex-math id=\"M2\">\\begin{document}$ \\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"}],"status":"public","type":"journal_article","publication":"Discrete and Continuous Dynamical Systems","article_number":"5201","language":[{"iso":"eng"}],"_id":"63293","user_id":"31496"}]