---
_id: '53331'
abstract:
- lang: eng
text: "In a ball $\\Omega
\\subset \\mathbb {R}^{n}$
with $n\\ge 2$, the chemotaxis system\r\n\\[
\\left\\{ \\begin{array}{@{}l} u_t = \\nabla \\cdot \\big( D(u)\\nabla u\\big)
+ \\nabla\\cdot \\big(\\dfrac{u}{v} \\nabla v\\big), \\\\ 0=\\Delta v - uv \\end{array}
\\right. \\]is considered along with no-flux boundary
conditions for $u$ and with prescribed constant positive
Dirichlet boundary data for $v$. It is shown that if $D\\in
C^{3}([0,\\infty ))$
is such that $0< D(\\xi
) \\le {K_D} (\\xi +1)^{-\\alpha }$
for all $\\xi >0$ with some ${K_D}>0$ and $\\alpha
>0$,
then for all initial data from a considerably large set of radial functions on
$\\Omega$, the corresponding initial-boundary
value problem admits a solution blowing up in finite time."
author:
- first_name: Yulan
full_name: Wang, Yulan
last_name: Wang
- first_name: Michael
full_name: Winkler, Michael
last_name: Winkler
citation:
ama: 'Wang Y, Winkler M. Finite-time blow-up in a repulsive chemotaxis-consumption
system. Proceedings of the Royal Society of Edinburgh: Section A Mathematics.
2022;153(4):1150-1166. doi:10.1017/prm.2022.39'
apa: 'Wang, Y., & Winkler, M. (2022). Finite-time blow-up in a repulsive chemotaxis-consumption
system. Proceedings of the Royal Society of Edinburgh: Section A Mathematics,
153(4), 1150–1166. https://doi.org/10.1017/prm.2022.39'
bibtex: '@article{Wang_Winkler_2022, title={Finite-time blow-up in a repulsive chemotaxis-consumption
system}, volume={153}, DOI={10.1017/prm.2022.39},
number={4}, journal={Proceedings of the Royal Society of Edinburgh: Section A
Mathematics}, publisher={Cambridge University Press (CUP)}, author={Wang, Yulan
and Winkler, Michael}, year={2022}, pages={1150–1166} }'
chicago: 'Wang, Yulan, and Michael Winkler. “Finite-Time Blow-up in a Repulsive
Chemotaxis-Consumption System.” Proceedings of the Royal Society of Edinburgh:
Section A Mathematics 153, no. 4 (2022): 1150–66. https://doi.org/10.1017/prm.2022.39.'
ieee: 'Y. Wang and M. Winkler, “Finite-time blow-up in a repulsive chemotaxis-consumption
system,” Proceedings of the Royal Society of Edinburgh: Section A Mathematics,
vol. 153, no. 4, pp. 1150–1166, 2022, doi: 10.1017/prm.2022.39.'
mla: 'Wang, Yulan, and Michael Winkler. “Finite-Time Blow-up in a Repulsive Chemotaxis-Consumption
System.” Proceedings of the Royal Society of Edinburgh: Section A Mathematics,
vol. 153, no. 4, Cambridge University Press (CUP), 2022, pp. 1150–66, doi:10.1017/prm.2022.39.'
short: 'Y. Wang, M. Winkler, Proceedings of the Royal Society of Edinburgh: Section
A Mathematics 153 (2022) 1150–1166.'
date_created: 2024-04-07T12:44:26Z
date_updated: 2024-04-07T12:44:30Z
doi: 10.1017/prm.2022.39
intvolume: ' 153'
issue: '4'
keyword:
- General Mathematics
language:
- iso: eng
page: 1150-1166
publication: 'Proceedings of the Royal Society of Edinburgh: Section A Mathematics'
publication_identifier:
issn:
- 0308-2105
- 1473-7124
publication_status: published
publisher: Cambridge University Press (CUP)
status: public
title: Finite-time blow-up in a repulsive chemotaxis-consumption system
type: journal_article
user_id: '31496'
volume: 153
year: '2022'
...
---
_id: '63274'
abstract:
- lang: eng
text: "In a ball $\\Omega
\\subset \\mathbb {R}^{n}$
with $n\\ge 2$, the chemotaxis system\r\n\\[
\\left\\{ \\begin{array}{@{}l} u_t = \\nabla \\cdot \\big( D(u)\\nabla u\\big)
+ \\nabla\\cdot \\big(\\dfrac{u}{v} \\nabla v\\big), \\\\ 0=\\Delta v - uv \\end{array}
\\right. \\]is considered along with no-flux boundary
conditions for $u$ and with prescribed constant positive
Dirichlet boundary data for $v$. It is shown that if $D\\in
C^{3}([0,\\infty ))$
is such that $0< D(\\xi
) \\le {K_D} (\\xi +1)^{-\\alpha }$
for all $\\xi >0$ with some ${K_D}>0$ and $\\alpha
>0$,
then for all initial data from a considerably large set of radial functions on
$\\Omega$, the corresponding initial-boundary
value problem admits a solution blowing up in finite time."
author:
- first_name: Yulan
full_name: Wang, Yulan
last_name: Wang
- first_name: Michael
full_name: Winkler, Michael
id: '31496'
last_name: Winkler
citation:
ama: 'Wang Y, Winkler M. Finite-time blow-up in a repulsive chemotaxis-consumption
system. Proceedings of the Royal Society of Edinburgh: Section A Mathematics.
2022;153(4):1150-1166. doi:10.1017/prm.2022.39'
apa: 'Wang, Y., & Winkler, M. (2022). Finite-time blow-up in a repulsive chemotaxis-consumption
system. Proceedings of the Royal Society of Edinburgh: Section A Mathematics,
153(4), 1150–1166. https://doi.org/10.1017/prm.2022.39'
bibtex: '@article{Wang_Winkler_2022, title={Finite-time blow-up in a repulsive chemotaxis-consumption
system}, volume={153}, DOI={10.1017/prm.2022.39},
number={4}, journal={Proceedings of the Royal Society of Edinburgh: Section A
Mathematics}, publisher={Cambridge University Press (CUP)}, author={Wang, Yulan
and Winkler, Michael}, year={2022}, pages={1150–1166} }'
chicago: 'Wang, Yulan, and Michael Winkler. “Finite-Time Blow-up in a Repulsive
Chemotaxis-Consumption System.” Proceedings of the Royal Society of Edinburgh:
Section A Mathematics 153, no. 4 (2022): 1150–66. https://doi.org/10.1017/prm.2022.39.'
ieee: 'Y. Wang and M. Winkler, “Finite-time blow-up in a repulsive chemotaxis-consumption
system,” Proceedings of the Royal Society of Edinburgh: Section A Mathematics,
vol. 153, no. 4, pp. 1150–1166, 2022, doi: 10.1017/prm.2022.39.'
mla: 'Wang, Yulan, and Michael Winkler. “Finite-Time Blow-up in a Repulsive Chemotaxis-Consumption
System.” Proceedings of the Royal Society of Edinburgh: Section A Mathematics,
vol. 153, no. 4, Cambridge University Press (CUP), 2022, pp. 1150–66, doi:10.1017/prm.2022.39.'
short: 'Y. Wang, M. Winkler, Proceedings of the Royal Society of Edinburgh: Section
A Mathematics 153 (2022) 1150–1166.'
date_created: 2025-12-18T19:14:20Z
date_updated: 2025-12-18T20:11:15Z
doi: 10.1017/prm.2022.39
intvolume: ' 153'
issue: '4'
language:
- iso: eng
page: 1150-1166
publication: 'Proceedings of the Royal Society of Edinburgh: Section A Mathematics'
publication_identifier:
issn:
- 0308-2105
- 1473-7124
publication_status: published
publisher: Cambridge University Press (CUP)
status: public
title: Finite-time blow-up in a repulsive chemotaxis-consumption system
type: journal_article
user_id: '31496'
volume: 153
year: '2022'
...
---
_id: '63369'
abstract:
- lang: eng
text: 'The paper studies large time behaviour of solutions to the Keller–Segel
system with quadratic degradation in a liquid environment, as given byunder Neumann boundary conditions in a
bounded domain Ω ⊂ ℝn,
where n ≥ 1 is arbitrary. It is shown that whenever
U : Ω × (0,∞)
→ ℝn
is a bounded and sufficiently regular solenoidal vector field any non-trivial
global bounded solution of (⋆) approaches the trivial
equilibrium at a rate that, with respect to the norm in either of the spaces L1(Ω)
and L∞(Ω),
can be controlled from above and below by appropriate multiples of 1/(t
+ 1). This underlines that, even up to this quantitative level of accuracy, the
large time behaviour in (⋆) is essentially independent
not only of the particular fluid flow, but also of any effect originating from
chemotactic cross-diffusion. The latter is in contrast to the corresponding Cauchy
problem, for which known results show that in the n
= 2 case the presence of chemotaxis can significantly enhance biomixing by reducing
the respective spatial L1 norms
of solutions.'
author:
- first_name: Xinru
full_name: Cao, Xinru
last_name: Cao
- first_name: Michael
full_name: Winkler, Michael
id: '31496'
last_name: Winkler
citation:
ama: 'Cao X, Winkler M. Sharp decay estimates in a bioconvection model with quadratic
degradation in bounded domains. Proceedings of the Royal Society of Edinburgh:
Section A Mathematics. 2018;148(5):939-955. doi:10.1017/s0308210518000057'
apa: 'Cao, X., & Winkler, M. (2018). Sharp decay estimates in a bioconvection
model with quadratic degradation in bounded domains. Proceedings of the Royal
Society of Edinburgh: Section A Mathematics, 148(5), 939–955. https://doi.org/10.1017/s0308210518000057'
bibtex: '@article{Cao_Winkler_2018, title={Sharp decay estimates in a bioconvection
model with quadratic degradation in bounded domains}, volume={148}, DOI={10.1017/s0308210518000057},
number={5}, journal={Proceedings of the Royal Society of Edinburgh: Section A
Mathematics}, publisher={Cambridge University Press (CUP)}, author={Cao, Xinru
and Winkler, Michael}, year={2018}, pages={939–955} }'
chicago: 'Cao, Xinru, and Michael Winkler. “Sharp Decay Estimates in a Bioconvection
Model with Quadratic Degradation in Bounded Domains.” Proceedings of the Royal
Society of Edinburgh: Section A Mathematics 148, no. 5 (2018): 939–55. https://doi.org/10.1017/s0308210518000057.'
ieee: 'X. Cao and M. Winkler, “Sharp decay estimates in a bioconvection model with
quadratic degradation in bounded domains,” Proceedings of the Royal Society
of Edinburgh: Section A Mathematics, vol. 148, no. 5, pp. 939–955, 2018, doi:
10.1017/s0308210518000057.'
mla: 'Cao, Xinru, and Michael Winkler. “Sharp Decay Estimates in a Bioconvection
Model with Quadratic Degradation in Bounded Domains.” Proceedings of the Royal
Society of Edinburgh: Section A Mathematics, vol. 148, no. 5, Cambridge University
Press (CUP), 2018, pp. 939–55, doi:10.1017/s0308210518000057.'
short: 'X. Cao, M. Winkler, Proceedings of the Royal Society of Edinburgh: Section
A Mathematics 148 (2018) 939–955.'
date_created: 2025-12-19T11:02:55Z
date_updated: 2025-12-19T11:03:03Z
doi: 10.1017/s0308210518000057
intvolume: ' 148'
issue: '5'
language:
- iso: eng
page: 939-955
publication: 'Proceedings of the Royal Society of Edinburgh: Section A Mathematics'
publication_identifier:
issn:
- 0308-2105
- 1473-7124
publication_status: published
publisher: Cambridge University Press (CUP)
status: public
title: Sharp decay estimates in a bioconvection model with quadratic degradation in
bounded domains
type: journal_article
user_id: '31496'
volume: 148
year: '2018'
...