{"status":"public","author":[{"last_name":"Ding","full_name":"Ding, Mengyao","first_name":"Mengyao"},{"full_name":"Winkler, Michael","last_name":"Winkler","id":"31496","first_name":"Michael"}],"publication_status":"published","doi":"10.1088/1361-6544/ad871a","abstract":[{"lang":"eng","text":"Abstract\r\n The Neumann problem for the Keller-Segel system \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 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 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 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 −\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n ∫\r\n Ω\r\n \r\n u\r\n d\r\n x\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 n-dimensional balls Ω with \r\n \r\n \r\n \r\n n\r\n ⩾\r\n 2\r\n \r\n \r\n , with suitably regular and radially symmetric, radially nonincreasing initial data u\r\n 0. The functions D and S are only assumed to belong to \r\n \r\n \r\n \r\n \r\n C\r\n 2\r\n \r\n (\r\n [\r\n 0\r\n ,\r\n ∞\r\n )\r\n )\r\n \r\n \r\n and to satisfy D > 0 and \r\n \r\n \r\n \r\n S\r\n ⩾\r\n 0\r\n \r\n \r\n on \r\n \r\n \r\n \r\n [\r\n 0\r\n ,\r\n ∞\r\n )\r\n \r\n \r\n as well as \r\n \r\n \r\n \r\n S\r\n (\r\n 0\r\n )\r\n =\r\n 0\r\n \r\n \r\n ; in particular, diffusivities with arbitrarily fast decay are included.\r\n In this general context, it is shown that it is merely the asymptotic behavior as \r\n \r\n \r\n \r\n ξ\r\n →\r\n ∞\r\n \r\n \r\n of the expression \r\n \r\n \r\n \r\n \r\n \r\n \r\n I\r\n \r\n (\r\n ξ\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 \r\n \r\n ξ\r\n \r\n 2\r\n n\r\n \r\n \r\n D\r\n \r\n (\r\n ξ\r\n )\r\n \r\n \r\n \r\n ,\r\n \r\n ξ\r\n >\r\n 0\r\n ,\r\n \r\n \r\n \r\n \r\n \r\n which decides about the occurrence of blow-up: Namely, it is seen that\r\n\r\n \r\n •\r\n if \r\n \r\n \r\n \r\n \r\n lim\r\n \r\n ξ\r\n →\r\n ∞\r\n \r\n \r\n I\r\n (\r\n ξ\r\n )\r\n =\r\n 0\r\n \r\n \r\n , then any such solution is global and bounded, that\r\n \r\n \r\n •\r\n if \r\n \r\n \r\n \r\n \r\n lim sup\r\n \r\n ξ\r\n →\r\n ∞\r\n \r\n \r\n I\r\n (\r\n ξ\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 Ω\r\n \r\n \r\n u\r\n 0\r\n \r\n \r\n \r\n is suitably small, then the corresponding solution is global and bounded, and that\r\n \r\n \r\n •\r\n if \r\n \r\n \r\n \r\n \r\n lim inf\r\n \r\n ξ\r\n →\r\n ∞\r\n \r\n \r\n I\r\n (\r\n ξ\r\n )\r\n >\r\n 0\r\n \r\n \r\n , then at each appropriately large mass level m, there exist radial initial data u\r\n 0 such that \r\n \r\n \r\n \r\n \r\n ∫\r\n Ω\r\n \r\n \r\n u\r\n 0\r\n \r\n =\r\n m\r\n \r\n \r\n , and that the associated solution blows up either in finite or in infinite time.\r\n \r\n \r\n \r\n This especially reveals the presence of critical mass phenomena whenever \r\n \r\n \r\n \r\n \r\n lim\r\n \r\n ξ\r\n →\r\n ∞\r\n \r\n \r\n I\r\n (\r\n ξ\r\n )\r\n ∈\r\n (\r\n 0\r\n ,\r\n ∞\r\n )\r\n \r\n \r\n exists."}],"language":[{"iso":"eng"}],"date_created":"2025-12-18T19:04:45Z","date_updated":"2025-12-18T20:13:49Z","intvolume":" 37","type":"journal_article","publication_identifier":{"issn":["0951-7715","1361-6544"]},"volume":37,"title":"Radial blow-up in quasilinear Keller-Segel systems: approaching the full picture","citation":{"chicago":"Ding, Mengyao, and Michael Winkler. “Radial Blow-up in Quasilinear Keller-Segel Systems: Approaching the Full Picture.” Nonlinearity 37, no. 12 (2024). https://doi.org/10.1088/1361-6544/ad871a.","apa":"Ding, M., & Winkler, M. (2024). Radial blow-up in quasilinear Keller-Segel systems: approaching the full picture. Nonlinearity, 37(12), Article 125006. https://doi.org/10.1088/1361-6544/ad871a","short":"M. Ding, M. Winkler, Nonlinearity 37 (2024).","ama":"Ding M, Winkler M. Radial blow-up in quasilinear Keller-Segel systems: approaching the full picture. Nonlinearity. 2024;37(12). doi:10.1088/1361-6544/ad871a","bibtex":"@article{Ding_Winkler_2024, title={Radial blow-up in quasilinear Keller-Segel systems: approaching the full picture}, volume={37}, DOI={10.1088/1361-6544/ad871a}, number={12125006}, journal={Nonlinearity}, publisher={IOP Publishing}, author={Ding, Mengyao and Winkler, Michael}, year={2024} }","mla":"Ding, Mengyao, and Michael Winkler. “Radial Blow-up in Quasilinear Keller-Segel Systems: Approaching the Full Picture.” Nonlinearity, vol. 37, no. 12, 125006, IOP Publishing, 2024, doi:10.1088/1361-6544/ad871a.","ieee":"M. Ding and M. Winkler, “Radial blow-up in quasilinear Keller-Segel systems: approaching the full picture,” Nonlinearity, vol. 37, no. 12, Art. no. 125006, 2024, doi: 10.1088/1361-6544/ad871a."},"_id":"63253","year":"2024","issue":"12","article_number":"125006","user_id":"31496","publisher":"IOP Publishing","publication":"Nonlinearity"}