---
_id: '63383'
abstract:
- lang: eng
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\nThis
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\nThis 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).
"
author:
- first_name: Nicola
full_name: Bellomo, Nicola
last_name: Bellomo
- first_name: Michael
full_name: Winkler, Michael
id: '31496'
last_name: Winkler
citation:
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
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} }'
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.'
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.'
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.
short: N. Bellomo, M. Winkler, Transactions of the American Mathematical Society,
Series B 4 (2017) 31–67.
date_created: 2025-12-19T11:09:53Z
date_updated: 2025-12-19T11:10:17Z
doi: 10.1090/btran/17
intvolume: ' 4'
issue: '2'
language:
- iso: eng
page: 31-67
publication: Transactions of the American Mathematical Society, Series B
publication_identifier:
issn:
- 2330-0000
publication_status: published
publisher: American Mathematical Society (AMS)
status: public
title: Finite-time blow-up in a degenerate chemotaxis system with flux limitation
type: journal_article
user_id: '31496'
volume: 4
year: '2017'
...