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Winkler, “Singular structure formation in a degenerate haptotaxis model involving myopic diffusion,” Journal de Mathématiques Pures et Appliquées, vol. 112, pp. 118–169, 2017, doi: 10.1016/j.matpur.2017.11.002."},"publication":"Journal de Mathématiques Pures et Appliquées","type":"journal_article","date_created":"2025-12-19T11:05:33Z"},{"doi":"10.1007/s10884-017-9577-3","user_id":"31496","volume":30,"page":"331-358","_id":"63378","language":[{"iso":"eng"}],"publisher":"Springer Science and Business Media LLC","date_updated":"2025-12-19T11:07:30Z","publication_status":"published","intvolume":" 30","title":"One-Dimensional Super-Fast Diffusion: Persistence Versus Extinction Revisited—Extinction at Spatial Infinity","year":"2017","status":"public","author":[{"id":"31496","first_name":"Michael","last_name":"Winkler","full_name":"Winkler, Michael"}],"publication_identifier":{"issn":["1040-7294","1572-9222"]},"type":"journal_article","date_created":"2025-12-19T11:07:24Z","issue":"1","publication":"Journal of Dynamics and Differential Equations","citation":{"ieee":"M. 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(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"},"status":"public","volume":4,"user_id":"31496","_id":"63383","publisher":"American Mathematical Society (AMS)","page":"31-67","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).

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High-quality Al/InAs and Nb/InAs superconducting hybrid structure interfaces on catalyst free InAs nanowires.

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