---
_id: '63246'
abstract:
- lang: eng
text: "Abstract\r\n \r\n The
hyperbolic-parabolic model\r\n \r\n \r\n
\ $$\\begin{aligned} \\left\\{ \\begin{array}{ll}
u_{tt} = u_{xx} - \\big (f(\\Theta )\\big )_x, \\qquad & x\\in \\Omega ,
\\ t>0, \\\\ \\Theta _t = \\Theta _{xx} - f(\\Theta ) u_{xt}, \\qquad &
\ x\\in \\Omega , \\ t>0, \\end{array} \\right. \\end{aligned}$$\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
\ \r\n tt\r\n
\ \r\n \r\n
\ =\r\n \r\n
\ u\r\n \r\n
\ xx\r\n \r\n
\ \r\n -\r\n
\ \r\n (\r\n
\ \r\n f\r\n
\ \r\n (\r\n
\ Θ\r\n )\r\n
\ \r\n \r\n
\ \r\n )\r\n
\ \r\n x\r\n
\ \r\n ,\r\n
\ \r\n \r\n
\ \r\n \r\n
\ \r\n x\r\n
\ ∈\r\n Ω\r\n
\ ,\r\n \r\n
\ t\r\n >\r\n
\ 0\r\n ,\r\n
\ \r\n \r\n
\ \r\n \r\n
\ \r\n \r\n
\ \r\n \r\n
\ Θ\r\n t\r\n
\ \r\n =\r\n
\ \r\n Θ\r\n
\ \r\n xx\r\n
\ \r\n \r\n
\ -\r\n f\r\n
\ \r\n (\r\n
\ Θ\r\n )\r\n
\ \r\n \r\n
\ u\r\n \r\n
\ xt\r\n \r\n
\ \r\n ,\r\n
\ \r\n \r\n
\ \r\n \r\n
\ \r\n x\r\n
\ ∈\r\n Ω\r\n
\ ,\r\n \r\n
\ t\r\n >\r\n
\ 0\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 for the evolution
of the displacement variable\r\n u\r\n
\ and the temperature\r\n \r\n
\ \r\n $$\\Theta
\\ge 0$$\r\n \r\n
\ \r\n Θ\r\n
\ ≥\r\n 0\r\n
\ \r\n \r\n
\ \r\n \r\n
\ during thermoelastic interaction in a one-dimensional bounded
interval\r\n \r\n \r\n
\ $$\\Omega $$\r\n \r\n Ω\r\n
\ \r\n \r\n
\ \r\n is considered.
Whereas the literature has provided comprehensive results on global solutions
for sufficiently regular initial data\r\n \r\n
\ \r\n $$(u_0,u_{0t},\\Theta
_0)=(u,u_t,\\Theta )|_{t=0}$$\r\n \r\n \r\n
\ \r\n (\r\n
\ \r\n u\r\n
\ 0\r\n \r\n
\ ,\r\n \r\n
\ u\r\n \r\n
\ 0\r\n t\r\n
\ \r\n \r\n
\ ,\r\n \r\n
\ Θ\r\n 0\r\n
\ \r\n )\r\n
\ \r\n =\r\n
\ \r\n (\r\n
\ u\r\n ,\r\n
\ \r\n u\r\n
\ t\r\n \r\n
\ ,\r\n Θ\r\n
\ )\r\n \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
\ \r\n \r\n
\ when\r\n \r\n \r\n
\ $$f\\equiv id$$\r\n \r\n \r\n
\ f\r\n ≡\r\n
\ i\r\n d\r\n
\ \r\n \r\n
\ \r\n \r\n
\ , it seems to have remained open so far how far a solution
theory can be built solely on the two fundamental physical principles of energy
conservation and entropy nondecrease. The present manuscript addresses this by
asserting global existence of weak solutions under assumptions which are energy-
and entropy-minimal in the sense of allowing for any initial data\r\n \r\n
\ \r\n $$u_0\\in
W_0^{1,2}(\\Omega )$$\r\n \r\n
\ \r\n \r\n
\ u\r\n 0\r\n
\ \r\n ∈\r\n
\ \r\n W\r\n
\ 0\r\n \r\n
\ 1\r\n ,\r\n
\ 2\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 \r\n \r\n
\ $$u_{0t} \\in L^2(\\Omega )$$\r\n
\ \r\n
\ \r\n \r\n
\ u\r\n \r\n
\ 0\r\n t\r\n
\ \r\n \r\n
\ ∈\r\n \r\n
\ L\r\n 2\r\n
\ \r\n \r\n
\ (\r\n Ω\r\n
\ )\r\n \r\n
\ \r\n \r\n
\ \r\n \r\n
\ and\r\n \r\n \r\n
\ $$0\\le \\Theta _0\\in L^1(\\Omega )$$\r\n
\ \r\n
\ \r\n 0\r\n
\ ≤\r\n \r\n
\ Θ\r\n 0\r\n
\ \r\n ∈\r\n
\ \r\n L\r\n
\ 1\r\n \r\n
\ \r\n (\r\n
\ Ω\r\n )\r\n
\ \r\n \r\n
\ \r\n \r\n
\ \r\n , and which
apply to arbitrary\r\n \r\n \r\n
\ $$f\\in C^1([0,\\infty ))$$\r\n
\ \r\n
\ \r\n f\r\n
\ ∈\r\n \r\n
\ C\r\n 1\r\n
\ \r\n \r\n
\ (\r\n \r\n
\ [\r\n 0\r\n
\ ,\r\n ∞\r\n
\ )\r\n \r\n
\ )\r\n \r\n
\ \r\n \r\n
\ \r\n \r\n
\ with\r\n \r\n \r\n
\ $$f(0)=0$$\r\n \r\n \r\n
\ f\r\n (\r\n
\ 0\r\n )\r\n
\ =\r\n 0\r\n
\ \r\n \r\n
\ \r\n \r\n
\ and\r\n \r\n \r\n
\ $$f'>0$$\r\n \r\n \r\n
\ \r\n f\r\n
\ ′\r\n \r\n
\ >\r\n 0\r\n
\ \r\n \r\n
\ \r\n \r\n
\ on\r\n \r\n \r\n
\ $$[0,\\infty )$$\r\n \r\n \r\n
\ [\r\n 0\r\n
\ ,\r\n ∞\r\n
\ )\r\n \r\n
\ \r\n \r\n
\ \r\n .\r\n "
article_number: '1'
author:
- first_name: Michael
full_name: Winkler, Michael
id: '31496'
last_name: Winkler
citation:
ama: Winkler M. Rough solutions in one-dimensional nonlinear thermoelasticity. Calculus
of Variations and Partial Differential Equations. 2025;65(1). doi:10.1007/s00526-025-03170-8
apa: Winkler, M. (2025). Rough solutions in one-dimensional nonlinear thermoelasticity.
Calculus of Variations and Partial Differential Equations, 65(1),
Article 1. https://doi.org/10.1007/s00526-025-03170-8
bibtex: '@article{Winkler_2025, title={Rough solutions in one-dimensional nonlinear
thermoelasticity}, volume={65}, DOI={10.1007/s00526-025-03170-8},
number={11}, journal={Calculus of Variations and Partial Differential Equations},
publisher={Springer Science and Business Media LLC}, author={Winkler, Michael},
year={2025} }'
chicago: Winkler, Michael. “Rough Solutions in One-Dimensional Nonlinear Thermoelasticity.”
Calculus of Variations and Partial Differential Equations 65, no. 1 (2025).
https://doi.org/10.1007/s00526-025-03170-8.
ieee: 'M. Winkler, “Rough solutions in one-dimensional nonlinear thermoelasticity,”
Calculus of Variations and Partial Differential Equations, vol. 65, no.
1, Art. no. 1, 2025, doi: 10.1007/s00526-025-03170-8.'
mla: Winkler, Michael. “Rough Solutions in One-Dimensional Nonlinear Thermoelasticity.”
Calculus of Variations and Partial Differential Equations, vol. 65, no.
1, 1, Springer Science and Business Media LLC, 2025, doi:10.1007/s00526-025-03170-8.
short: M. Winkler, Calculus of Variations and Partial Differential Equations 65
(2025).
date_created: 2025-12-18T19:01:02Z
date_updated: 2026-04-23T12:18:59Z
doi: 10.1007/s00526-025-03170-8
intvolume: ' 65'
issue: '1'
language:
- iso: eng
project:
- _id: '245'
name: 'FOR 5208: Modellbasierte Bestimmung nichtlinearer Eigenschaften von Piezokeramiken
für Leistungsschallanwendungen (NEPTUN)'
publication: Calculus of Variations and Partial Differential Equations
publication_identifier:
issn:
- 0944-2669
- 1432-0835
publication_status: published
publisher: Springer Science and Business Media LLC
status: public
title: Rough solutions in one-dimensional nonlinear thermoelasticity
type: journal_article
user_id: '31496'
volume: 65
year: '2025'
...
---
_id: '53324'
article_number: '180'
author:
- first_name: Jaewook
full_name: Ahn, Jaewook
last_name: Ahn
- first_name: Michael
full_name: Winkler, Michael
last_name: Winkler
citation:
ama: Ahn J, Winkler M. A critical exponent for blow-up in a two-dimensional chemotaxis-consumption
system. Calculus of Variations and Partial Differential Equations. 2023;62(6).
doi:10.1007/s00526-023-02523-5
apa: Ahn, J., & Winkler, M. (2023). A critical exponent for blow-up in a two-dimensional
chemotaxis-consumption system. Calculus of Variations and Partial Differential
Equations, 62(6), Article 180. https://doi.org/10.1007/s00526-023-02523-5
bibtex: '@article{Ahn_Winkler_2023, title={A critical exponent for blow-up in a
two-dimensional chemotaxis-consumption system}, volume={62}, DOI={10.1007/s00526-023-02523-5},
number={6180}, journal={Calculus of Variations and Partial Differential Equations},
publisher={Springer Science and Business Media LLC}, author={Ahn, Jaewook and
Winkler, Michael}, year={2023} }'
chicago: Ahn, Jaewook, and Michael Winkler. “A Critical Exponent for Blow-up in
a Two-Dimensional Chemotaxis-Consumption System.” Calculus of Variations and
Partial Differential Equations 62, no. 6 (2023). https://doi.org/10.1007/s00526-023-02523-5.
ieee: 'J. Ahn and M. Winkler, “A critical exponent for blow-up in a two-dimensional
chemotaxis-consumption system,” Calculus of Variations and Partial Differential
Equations, vol. 62, no. 6, Art. no. 180, 2023, doi: 10.1007/s00526-023-02523-5.'
mla: Ahn, Jaewook, and Michael Winkler. “A Critical Exponent for Blow-up in a Two-Dimensional
Chemotaxis-Consumption System.” Calculus of Variations and Partial Differential
Equations, vol. 62, no. 6, 180, Springer Science and Business Media LLC, 2023,
doi:10.1007/s00526-023-02523-5.
short: J. Ahn, M. Winkler, Calculus of Variations and Partial Differential Equations
62 (2023).
date_created: 2024-04-07T12:40:02Z
date_updated: 2024-04-07T12:40:06Z
doi: 10.1007/s00526-023-02523-5
intvolume: ' 62'
issue: '6'
keyword:
- Applied Mathematics
- Analysis
language:
- iso: eng
publication: Calculus of Variations and Partial Differential Equations
publication_identifier:
issn:
- 0944-2669
- 1432-0835
publication_status: published
publisher: Springer Science and Business Media LLC
status: public
title: A critical exponent for blow-up in a two-dimensional chemotaxis-consumption
system
type: journal_article
user_id: '31496'
volume: 62
year: '2023'
...
---
_id: '63267'
article_number: '180'
author:
- first_name: Jaewook
full_name: Ahn, Jaewook
last_name: Ahn
- first_name: Michael
full_name: Winkler, Michael
id: '31496'
last_name: Winkler
citation:
ama: Ahn J, Winkler M. A critical exponent for blow-up in a two-dimensional chemotaxis-consumption
system. Calculus of Variations and Partial Differential Equations. 2023;62(6).
doi:10.1007/s00526-023-02523-5
apa: Ahn, J., & Winkler, M. (2023). A critical exponent for blow-up in a two-dimensional
chemotaxis-consumption system. Calculus of Variations and Partial Differential
Equations, 62(6), Article 180. https://doi.org/10.1007/s00526-023-02523-5
bibtex: '@article{Ahn_Winkler_2023, title={A critical exponent for blow-up in a
two-dimensional chemotaxis-consumption system}, volume={62}, DOI={10.1007/s00526-023-02523-5},
number={6180}, journal={Calculus of Variations and Partial Differential Equations},
publisher={Springer Science and Business Media LLC}, author={Ahn, Jaewook and
Winkler, Michael}, year={2023} }'
chicago: Ahn, Jaewook, and Michael Winkler. “A Critical Exponent for Blow-up in
a Two-Dimensional Chemotaxis-Consumption System.” Calculus of Variations and
Partial Differential Equations 62, no. 6 (2023). https://doi.org/10.1007/s00526-023-02523-5.
ieee: 'J. Ahn and M. Winkler, “A critical exponent for blow-up in a two-dimensional
chemotaxis-consumption system,” Calculus of Variations and Partial Differential
Equations, vol. 62, no. 6, Art. no. 180, 2023, doi: 10.1007/s00526-023-02523-5.'
mla: Ahn, Jaewook, and Michael Winkler. “A Critical Exponent for Blow-up in a Two-Dimensional
Chemotaxis-Consumption System.” Calculus of Variations and Partial Differential
Equations, vol. 62, no. 6, 180, Springer Science and Business Media LLC, 2023,
doi:10.1007/s00526-023-02523-5.
short: J. Ahn, M. Winkler, Calculus of Variations and Partial Differential Equations
62 (2023).
date_created: 2025-12-18T19:10:55Z
date_updated: 2025-12-18T20:10:21Z
doi: 10.1007/s00526-023-02523-5
intvolume: ' 62'
issue: '6'
language:
- iso: eng
publication: Calculus of Variations and Partial Differential Equations
publication_identifier:
issn:
- 0944-2669
- 1432-0835
publication_status: published
publisher: Springer Science and Business Media LLC
status: public
title: A critical exponent for blow-up in a two-dimensional chemotaxis-consumption
system
type: journal_article
user_id: '31496'
volume: 62
year: '2023'
...
---
_id: '34677'
article_number: '96'
author:
- first_name: Tobias
full_name: Black, Tobias
id: '23686'
last_name: Black
orcid: 0000-0001-9963-0800
- first_name: Chunyan
full_name: Wu, Chunyan
last_name: Wu
citation:
ama: 'Black T, Wu C. Prescribed signal concentration on the boundary: eventual smoothness
in a chemotaxis-Navier–Stokes system with logistic proliferation. Calculus
of Variations and Partial Differential Equations. 2022;61(3). doi:10.1007/s00526-022-02201-y'
apa: 'Black, T., & Wu, C. (2022). Prescribed signal concentration on the boundary:
eventual smoothness in a chemotaxis-Navier–Stokes system with logistic proliferation.
Calculus of Variations and Partial Differential Equations, 61(3),
Article 96. https://doi.org/10.1007/s00526-022-02201-y'
bibtex: '@article{Black_Wu_2022, title={Prescribed signal concentration on the boundary:
eventual smoothness in a chemotaxis-Navier–Stokes system with logistic proliferation},
volume={61}, DOI={10.1007/s00526-022-02201-y},
number={396}, journal={Calculus of Variations and Partial Differential Equations},
publisher={Springer Science and Business Media LLC}, author={Black, Tobias and
Wu, Chunyan}, year={2022} }'
chicago: 'Black, Tobias, and Chunyan Wu. “Prescribed Signal Concentration on the
Boundary: Eventual Smoothness in a Chemotaxis-Navier–Stokes System with Logistic
Proliferation.” Calculus of Variations and Partial Differential Equations
61, no. 3 (2022). https://doi.org/10.1007/s00526-022-02201-y.'
ieee: 'T. Black and C. Wu, “Prescribed signal concentration on the boundary: eventual
smoothness in a chemotaxis-Navier–Stokes system with logistic proliferation,”
Calculus of Variations and Partial Differential Equations, vol. 61, no.
3, Art. no. 96, 2022, doi: 10.1007/s00526-022-02201-y.'
mla: 'Black, Tobias, and Chunyan Wu. “Prescribed Signal Concentration on the Boundary:
Eventual Smoothness in a Chemotaxis-Navier–Stokes System with Logistic Proliferation.”
Calculus of Variations and Partial Differential Equations, vol. 61, no.
3, 96, Springer Science and Business Media LLC, 2022, doi:10.1007/s00526-022-02201-y.'
short: T. Black, C. Wu, Calculus of Variations and Partial Differential Equations
61 (2022).
date_created: 2022-12-21T09:50:59Z
date_updated: 2023-07-10T11:37:27Z
department:
- _id: '34'
- _id: '10'
- _id: '90'
doi: 10.1007/s00526-022-02201-y
intvolume: ' 61'
issue: '3'
keyword:
- Applied Mathematics
- Analysis
language:
- iso: eng
publication: Calculus of Variations and Partial Differential Equations
publication_identifier:
issn:
- 0944-2669
- 1432-0835
publication_status: published
publisher: Springer Science and Business Media LLC
status: public
title: 'Prescribed signal concentration on the boundary: eventual smoothness in a
chemotaxis-Navier–Stokes system with logistic proliferation'
type: journal_article
user_id: '23686'
volume: 61
year: '2022'
...
---
_id: '63305'
abstract:
- lang: eng
text: "AbstractA no-flux initial-boundary value
problem for the doubly degenrate parabolic system $$\\begin{aligned}
\\left\\{ \\begin{array}{l} u_t = \\nabla \\cdot \\big ( uv\\nabla u\\big ) +
\\ell uv, \\\\ v_t = \\Delta v - uv, \\end{array} \\right. \\qquad \\qquad (\\star
) \\end{aligned}$$\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 \r\n
\ u\r\n v\r\n
\ ∇\r\n u\r\n
\ \r\n )\r\n
\ \r\n +\r\n
\ ℓ\r\n u\r\n
\ v\r\n ,\r\n
\ \r\n \r\n
\ \r\n \r\n
\ \r\n \r\n
\ \r\n \r\n
\ v\r\n t\r\n
\ \r\n =\r\n
\ Δ\r\n v\r\n
\ -\r\n u\r\n
\ v\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 \r\n
\ \r\n \r\n \r\n
\ \r\n is
considered in a smoothly bounded convex domain $$\\Omega
\\subset \\mathbb {R}^n$$\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 ,
with $$n\\ge 1$$\r\n \r\n
\ n\r\n ≥\r\n
\ 1\r\n \r\n
and $$\\ell \\ge 0$$\r\n \r\n
\ ℓ\r\n ≥\r\n
\ 0\r\n \r\n .
The first of the main results asserts that for nonnegative initial data $$(u_0,v_0)\\in
(L^\\infty (\\Omega ))^2$$\r\n
\ \r\n \r\n (\r\n
\ \r\n u\r\n
\ 0\r\n \r\n
\ ,\r\n \r\n
\ v\r\n 0\r\n
\ \r\n )\r\n
\ \r\n ∈\r\n
\ \r\n \r\n (\r\n
\ \r\n L\r\n
\ ∞\r\n \r\n
\ \r\n (\r\n
\ Ω\r\n )\r\n
\ \r\n )\r\n
\ \r\n 2\r\n
\ \r\n \r\n
with $$u_0\\not \\equiv
0$$\r\n
\ \r\n \r\n u\r\n
\ 0\r\n \r\n
\ ≢\r\n 0\r\n
\ \r\n ,
$$v_0\\not \\equiv 0$$\r\n \r\n
\ \r\n v\r\n
\ 0\r\n \r\n
\ ≢\r\n 0\r\n
\ \r\n
and $$\\sqrt{v_0}\\in W^{1,2}(\\Omega
)$$\r\n
\ \r\n \r\n \r\n
\ v\r\n 0\r\n
\ \r\n \r\n ∈\r\n
\ \r\n W\r\n
\ \r\n 1\r\n
\ ,\r\n 2\r\n
\ \r\n \r\n \r\n
\ (\r\n Ω\r\n
\ )\r\n \r\n
\ \r\n ,
there exists a global weak solution (u, v)
which, inter alia, belongs to $$C^0(\\overline{\\Omega
}\\times (0,\\infty )) \\times C^{2,1}(\\overline{\\Omega }\\times (0,\\infty
))$$\r\n
\ \r\n \r\n C\r\n
\ 0\r\n \r\n
\ \r\n (\r\n
\ \r\n Ω\r\n
\ ¯\r\n \r\n
\ ×\r\n \r\n
\ (\r\n 0\r\n
\ ,\r\n ∞\r\n
\ )\r\n \r\n
\ )\r\n \r\n
\ ×\r\n \r\n C\r\n
\ \r\n 2\r\n
\ ,\r\n 1\r\n
\ \r\n \r\n \r\n
\ (\r\n \r\n
\ Ω\r\n ¯\r\n
\ \r\n ×\r\n
\ \r\n (\r\n
\ 0\r\n ,\r\n
\ ∞\r\n )\r\n
\ \r\n )\r\n
\ \r\n \r\n
and satisfies $$\\sup _{t>0}
\\Vert u(\\cdot ,t)\\Vert _{L^p(\\Omega )}<\\infty $$\r\n \r\n
\ \r\n sup\r\n
\ \r\n t\r\n
\ >\r\n 0\r\n
\ \r\n \r\n \r\n
\ \r\n ‖\r\n
\ u\r\n \r\n
\ (\r\n ·\r\n
\ ,\r\n t\r\n
\ )\r\n \r\n
\ ‖\r\n \r\n
\ \r\n \r\n L\r\n
\ p\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
for all $$p\\in [1,p_0)$$\r\n \r\n
\ p\r\n ∈\r\n
\ [\r\n 1\r\n
\ ,\r\n \r\n p\r\n
\ 0\r\n \r\n
\ )\r\n \r\n
with $$p_0:=\\frac{n}{(n-2)_+}$$\r\n \r\n
\ \r\n p\r\n
\ 0\r\n \r\n
\ :\r\n =\r\n
\ \r\n n\r\n
\ \r\n \r\n (\r\n
\ n\r\n -\r\n
\ 2\r\n )\r\n
\ \r\n +\r\n
\ \r\n \r\n \r\n
\ . It is next
seen that for each of these solutions one can find $$u_\\infty
\\in \\bigcap _{p\\in [1,p_0)} L^p(\\Omega )$$\r\n
\ \r\n \r\n u\r\n
\ ∞\r\n \r\n
\ ∈\r\n \r\n ⋂\r\n
\ \r\n p\r\n
\ ∈\r\n [\r\n
\ 1\r\n ,\r\n
\ \r\n p\r\n
\ 0\r\n \r\n
\ )\r\n \r\n
\ \r\n \r\n L\r\n
\ p\r\n \r\n
\ \r\n (\r\n
\ Ω\r\n )\r\n
\ \r\n \r\n
such that, within an appropriate topological setting, $$(u(\\cdot
,t),v(\\cdot ,t))$$\r\n
\ \r\n (\r\n u\r\n
\ (\r\n ·\r\n
\ ,\r\n t\r\n
\ )\r\n ,\r\n
\ v\r\n (\r\n
\ ·\r\n ,\r\n
\ t\r\n )\r\n
\ )\r\n \r\n
approaches the equilibrium $$(u_\\infty
,0)$$\r\n
\ \r\n (\r\n \r\n
\ u\r\n ∞\r\n
\ \r\n ,\r\n
\ 0\r\n )\r\n
\ \r\n
in the large time limit. Finally, in the case $$n\\le
5$$\r\n
\ \r\n n\r\n ≤\r\n
\ 5\r\n \r\n
a result ensuring a certain stability property of any member in the uncountably
large family of steady states $$(u_0,0)$$\r\n \r\n
\ (\r\n \r\n u\r\n
\ 0\r\n \r\n
\ ,\r\n 0\r\n
\ )\r\n \r\n ,
with arbitrary and suitably regular $$u_0:\\Omega
\\rightarrow [0,\\infty )$$\r\n
\ \r\n \r\n u\r\n
\ 0\r\n \r\n
\ :\r\n Ω\r\n
\ →\r\n \r\n [\r\n
\ 0\r\n ,\r\n
\ ∞\r\n )\r\n
\ \r\n \r\n ,
is derived. This provides some rigorous evidence for the appropriateness of ($$\\star
$$\r\n
\ ⋆\r\n )
to model the emergence of a strikingly large variety of stable structures observed
in experiments on bacterial motion in nutrient-poor environments. Essential parts
of the analysis rely on the use of an apparently novel class of functional inequalities
to suitably cope with the doubly degenerate diffusion mechanism in ($$\\star
$$\r\n
\ ⋆\r\n )."
article_number: '108'
author:
- first_name: Michael
full_name: Winkler, Michael
id: '31496'
last_name: Winkler
citation:
ama: Winkler M. Stabilization of arbitrary structures in a doubly degenerate reaction-diffusion
system modeling bacterial motion on a nutrient-poor agar. Calculus of Variations
and Partial Differential Equations. 2022;61(3). doi:10.1007/s00526-021-02168-2
apa: Winkler, M. (2022). Stabilization of arbitrary structures in a doubly degenerate
reaction-diffusion system modeling bacterial motion on a nutrient-poor agar. Calculus
of Variations and Partial Differential Equations, 61(3), Article 108.
https://doi.org/10.1007/s00526-021-02168-2
bibtex: '@article{Winkler_2022, title={Stabilization of arbitrary structures in
a doubly degenerate reaction-diffusion system modeling bacterial motion on a nutrient-poor
agar}, volume={61}, DOI={10.1007/s00526-021-02168-2},
number={3108}, journal={Calculus of Variations and Partial Differential Equations},
publisher={Springer Science and Business Media LLC}, author={Winkler, Michael},
year={2022} }'
chicago: Winkler, Michael. “Stabilization of Arbitrary Structures in a Doubly Degenerate
Reaction-Diffusion System Modeling Bacterial Motion on a Nutrient-Poor Agar.”
Calculus of Variations and Partial Differential Equations 61, no. 3 (2022).
https://doi.org/10.1007/s00526-021-02168-2.
ieee: 'M. Winkler, “Stabilization of arbitrary structures in a doubly degenerate
reaction-diffusion system modeling bacterial motion on a nutrient-poor agar,”
Calculus of Variations and Partial Differential Equations, vol. 61, no.
3, Art. no. 108, 2022, doi: 10.1007/s00526-021-02168-2.'
mla: Winkler, Michael. “Stabilization of Arbitrary Structures in a Doubly Degenerate
Reaction-Diffusion System Modeling Bacterial Motion on a Nutrient-Poor Agar.”
Calculus of Variations and Partial Differential Equations, vol. 61, no.
3, 108, Springer Science and Business Media LLC, 2022, doi:10.1007/s00526-021-02168-2.
short: M. Winkler, Calculus of Variations and Partial Differential Equations 61
(2022).
date_created: 2025-12-18T19:26:32Z
date_updated: 2025-12-18T20:04:43Z
doi: 10.1007/s00526-021-02168-2
intvolume: ' 61'
issue: '3'
language:
- iso: eng
publication: Calculus of Variations and Partial Differential Equations
publication_identifier:
issn:
- 0944-2669
- 1432-0835
publication_status: published
publisher: Springer Science and Business Media LLC
status: public
title: Stabilization of arbitrary structures in a doubly degenerate reaction-diffusion
system modeling bacterial motion on a nutrient-poor agar
type: journal_article
user_id: '31496'
volume: 61
year: '2022'
...
---
_id: '63359'
article_number: '196'
author:
- first_name: Yulan
full_name: Wang, Yulan
last_name: Wang
- first_name: Michael
full_name: Winkler, Michael
id: '31496'
last_name: Winkler
- first_name: Zhaoyin
full_name: Xiang, Zhaoyin
last_name: Xiang
citation:
ama: Wang Y, Winkler M, Xiang Z. The fast signal diffusion limit in Keller–Segel(-fluid)
systems. Calculus of Variations and Partial Differential Equations. 2019;58(6).
doi:10.1007/s00526-019-1656-3
apa: Wang, Y., Winkler, M., & Xiang, Z. (2019). The fast signal diffusion limit
in Keller–Segel(-fluid) systems. Calculus of Variations and Partial Differential
Equations, 58(6), Article 196. https://doi.org/10.1007/s00526-019-1656-3
bibtex: '@article{Wang_Winkler_Xiang_2019, title={The fast signal diffusion limit
in Keller–Segel(-fluid) systems}, volume={58}, DOI={10.1007/s00526-019-1656-3},
number={6196}, journal={Calculus of Variations and Partial Differential Equations},
publisher={Springer Science and Business Media LLC}, author={Wang, Yulan and Winkler,
Michael and Xiang, Zhaoyin}, year={2019} }'
chicago: Wang, Yulan, Michael Winkler, and Zhaoyin Xiang. “The Fast Signal Diffusion
Limit in Keller–Segel(-Fluid) Systems.” Calculus of Variations and Partial
Differential Equations 58, no. 6 (2019). https://doi.org/10.1007/s00526-019-1656-3.
ieee: 'Y. Wang, M. Winkler, and Z. Xiang, “The fast signal diffusion limit in Keller–Segel(-fluid)
systems,” Calculus of Variations and Partial Differential Equations, vol.
58, no. 6, Art. no. 196, 2019, doi: 10.1007/s00526-019-1656-3.'
mla: Wang, Yulan, et al. “The Fast Signal Diffusion Limit in Keller–Segel(-Fluid)
Systems.” Calculus of Variations and Partial Differential Equations, vol.
58, no. 6, 196, Springer Science and Business Media LLC, 2019, doi:10.1007/s00526-019-1656-3.
short: Y. Wang, M. Winkler, Z. Xiang, Calculus of Variations and Partial Differential
Equations 58 (2019).
date_created: 2025-12-19T10:56:58Z
date_updated: 2025-12-19T10:57:05Z
doi: 10.1007/s00526-019-1656-3
intvolume: ' 58'
issue: '6'
language:
- iso: eng
publication: Calculus of Variations and Partial Differential Equations
publication_identifier:
issn:
- 0944-2669
- 1432-0835
publication_status: published
publisher: Springer Science and Business Media LLC
status: public
title: The fast signal diffusion limit in Keller–Segel(-fluid) systems
type: journal_article
user_id: '31496'
volume: 58
year: '2019'
...