[{"title":"Finite-time blow-up in a degenerate chemotaxis system with flux limitation","doi":"10.1090/btran/17","publisher":"American Mathematical Society (AMS)","date_updated":"2025-12-19T11:10:17Z","author":[{"first_name":"Nicola","last_name":"Bellomo","full_name":"Bellomo, Nicola"},{"first_name":"Michael","last_name":"Winkler","id":"31496","full_name":"Winkler, Michael"}],"date_created":"2025-12-19T11:09:53Z","volume":4,"year":"2017","citation":{"apa":"Bellomo, N., & Winkler, M. (2017). Finite-time blow-up in a degenerate chemotaxis system with flux limitation. Transactions of the American Mathematical Society, Series B, 4(2), 31–67. https://doi.org/10.1090/btran/17","bibtex":"@article{Bellomo_Winkler_2017, title={Finite-time blow-up in a degenerate chemotaxis system with flux limitation}, volume={4}, DOI={10.1090/btran/17}, number={2}, journal={Transactions of the American Mathematical Society, Series B}, publisher={American Mathematical Society (AMS)}, author={Bellomo, Nicola and Winkler, Michael}, year={2017}, pages={31–67} }","short":"N. Bellomo, M. Winkler, Transactions of the American Mathematical Society, Series B 4 (2017) 31–67.","mla":"Bellomo, Nicola, and Michael Winkler. “Finite-Time Blow-up in a Degenerate Chemotaxis System with Flux Limitation.” Transactions of the American Mathematical Society, Series B, vol. 4, no. 2, American Mathematical Society (AMS), 2017, pp. 31–67, doi:10.1090/btran/17.","ama":"Bellomo N, Winkler M. Finite-time blow-up in a degenerate chemotaxis system with flux limitation. Transactions of the American Mathematical Society, Series B. 2017;4(2):31-67. doi:10.1090/btran/17","ieee":"N. Bellomo and M. Winkler, “Finite-time blow-up in a degenerate chemotaxis system with flux limitation,” Transactions of the American Mathematical Society, Series B, vol. 4, no. 2, pp. 31–67, 2017, doi: 10.1090/btran/17.","chicago":"Bellomo, Nicola, and Michael Winkler. “Finite-Time Blow-up in a Degenerate Chemotaxis System with Flux Limitation.” Transactions of the American Mathematical Society, Series B 4, no. 2 (2017): 31–67. https://doi.org/10.1090/btran/17."},"page":"31-67","intvolume":" 4","publication_status":"published","publication_identifier":{"issn":["2330-0000"]},"issue":"2","language":[{"iso":"eng"}],"_id":"63383","user_id":"31496","abstract":[{"text":"

This paper is concerned with radially symmetric solutions of the parabolic-elliptic version of the Keller-Segel system with flux limitation, as given by \r\n\r\n \r\n \r\n \r\n \r\n (\\star)\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 \r\n \r\n \r\n \r\n u\r\n ∇\r\n u\r\n \r\n \r\n \r\n u\r\n 2\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 2\r\n \r\n \r\n \r\n \r\n \r\n )\r\n \r\n \r\n −\r\n χ\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 ∇\r\n v\r\n \r\n \r\n 1\r\n +\r\n \r\n |\r\n \r\n ∇\r\n v\r\n \r\n \r\n |\r\n \r\n 2\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 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 \r\n \r\n \r\n \r\n \r\n \r\n \r\n \\begin{equation}\\tag {\\star } \\begin {cases} u_t=\\nabla \\cdot \\Big (\\frac {u\\nabla u}{\\sqrt {u^2+|\\nabla u|^2}}\\Big ) - \\chi \\, \\nabla \\cdot \\Big (\\frac {u\\nabla v}{\\sqrt {1+|\\nabla v|^2}}\\Big ), \\\\[3pt] 0=\\Delta v - \\mu + u, \\end{cases} \\end{equation}\r\n \r\n\r\n\r\n under the initial condition \r\n\r\n \r\n \r\n u\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 u\r\n 0\r\n \r\n >\r\n 0\r\n \r\n u|_{t=0}=u_0>0\r\n \r\n\r\n and no-flux boundary conditions in a ball \r\n\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 \\Omega \\subset \\mathbb {R}^n\r\n \r\n\r\n, where \r\n\r\n \r\n \r\n χ\r\n >\r\n 0\r\n \r\n \\chi >0\r\n \r\n\r\n and \r\n\r\n \r\n \r\n μ\r\n :=\r\n \r\n 1\r\n \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 Ω\r\n \r\n \r\n u\r\n 0\r\n \r\n \r\n \\mu :=\\frac {1}{|\\Omega |} \\int _\\Omega u_0\r\n \r\n\r\n. A previous result of the authors [Comm. Partial Differential Equations 42 (2017), 436–473] has asserted global existence of bounded classical solutions for arbitrary positive radial initial data \r\n\r\n \r\n \r\n \r\n u\r\n 0\r\n \r\n ∈\r\n \r\n C\r\n 3\r\n \r\n (\r\n \r\n \r\n Ω\r\n ¯\r\n \r\n \r\n )\r\n \r\n u_0\\in C^3(\\bar \\Omega )\r\n \r\n\r\n when either \r\n\r\n \r\n \r\n n\r\n ≥\r\n 2\r\n \r\n n\\ge 2\r\n \r\n\r\n and \r\n\r\n \r\n \r\n χ\r\n >\r\n 1\r\n \r\n \\chi >1\r\n \r\n\r\n, or \r\n\r\n \r\n \r\n n\r\n =\r\n 1\r\n \r\n n=1\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 1\r\n \r\n (\r\n \r\n χ\r\n 2\r\n \r\n −\r\n 1\r\n \r\n )\r\n +\r\n \r\n \r\n \r\n \r\n \\int _\\Omega u_0>\\frac {1}{\\sqrt {(\\chi ^2-1)_+}}\r\n \r\n\r\n.

\r\n\r\n

This present paper shows that these conditions are essentially optimal: Indeed, it is shown that if the taxis coefficient satisfies \r\n\r\n \r\n \r\n χ\r\n >\r\n 1\r\n \r\n \\chi >1\r\n \r\n\r\n, then for any choice of \r\n\\[\r\n\r\n \r\n \r\n {\r\n \r\n \r\n \r\n m\r\n >\r\n \r\n 1\r\n \r\n \r\n χ\r\n 2\r\n \r\n −\r\n 1\r\n \r\n \r\n \r\n \r\n a\r\n m\r\n p\r\n ;\r\n \r\n if \r\n \r\n n\r\n =\r\n 1\r\n \r\n \r\n ,\r\n \r\n \r\n \r\n \r\n \r\n \r\n m\r\n >\r\n 0\r\n \r\n  is arbitrary\r\n \r\n \r\n \r\n a\r\n m\r\n p\r\n ;\r\n \r\n if \r\n \r\n n\r\n ≥\r\n 2\r\n \r\n \r\n ,\r\n \r\n \r\n \r\n \r\n \r\n \\begin {cases} m>\\frac {1}{\\sqrt {\\chi ^2-1}} & \\text {if $n=1$}, \\\\ \\text {$m>0$ is arbitrary} & \\text {if $n\\ge 2$}, \\end {cases}\r\n \r\n\r\n\\]\r\n there exist positive initial data \r\n\r\n \r\n \r\n \r\n u\r\n 0\r\n \r\n ∈\r\n \r\n C\r\n 3\r\n \r\n (\r\n \r\n \r\n Ω\r\n ¯\r\n \r\n \r\n )\r\n \r\n u_0\\in C^3(\\bar \\Omega )\r\n \r\n\r\n satisfying \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 \\int _\\Omega u_0=m\r\n \r\n\r\n which are such that for some \r\n\r\n \r\n \r\n T\r\n >\r\n 0\r\n \r\n T>0\r\n \r\n\r\n, (\r\n\r\n \r\n ⋆\r\n \\star\r\n \r\n\r\n) possesses a uniquely determined classical solution \r\n\r\n \r\n \r\n (\r\n u\r\n ,\r\n v\r\n )\r\n \r\n (u,v)\r\n \r\n\r\n in \r\n\r\n \r\n \r\n Ω\r\n ×\r\n (\r\n 0\r\n ,\r\n T\r\n )\r\n \r\n \\Omega \\times (0,T)\r\n \r\n\r\n blowing up at time \r\n\r\n \r\n T\r\n T\r\n \r\n\r\n in the sense that \r\n\r\n \r\n \r\n \r\n lim sup\r\n \r\n t\r\n ↗\r\n T\r\n \r\n \r\n ‖\r\n u\r\n (\r\n ⋅\r\n ,\r\n t\r\n )\r\n \r\n ‖\r\n \r\n \r\n L\r\n ∞\r\n \r\n (\r\n Ω\r\n )\r\n \r\n \r\n =\r\n ∞\r\n \r\n \\limsup _{t\\nearrow T} \\|u(\\cdot ,t)\\|_{L^\\infty (\\Omega )}=\\infty\r\n \r\n\r\n.

\r\n\r\n

This result is derived by means of a comparison argument applied to the doubly degenerate scalar parabolic equation satisfied by the mass accumulation function associated with (\r\n\r\n \r\n ⋆\r\n \\star\r\n \r\n\r\n).

","lang":"eng"}],"status":"public","type":"journal_article","publication":"Transactions of the American Mathematical Society, Series B"}]