---
_id: '63264'
abstract:
- lang: eng
text: "Abstract\r\n In a smoothly
bounded convex domain \r\n
\ \r\n \r\n Ω\r\n
\ ⊂\r\n \r\n
\ \r\n R\r\n \r\n
\ \r\n n\r\n
\ \r\n \r\n
\ \r\n \r\n${\\Omega}\\subset
{\\mathbb{R}}^{n}$\r\n\r\n \r\n
\ \r\n
with n ≥ 1, a no-flux initial-boundary value problem
for\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 \r\n
\ t\r\n \r\n
\ \r\n =\r\n
\ Δ\r\n
\ \r\n
\ \r\n u\r\n
\ ϕ\r\n \r\n
\ (\r\n
\ \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 v\r\n
\ \r\n \r\n
\ t\r\n \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$$\\begin{cases}_{t}={\\Delta}\\left(u\\phi
\\left(v\\right)\\right),\\quad \\hfill \\\\ {v}_{t}={\\Delta}v-uv,\\quad \\hfill
\\end{cases}$$\r\n\r\n \r\n \r\n
\ is considered under the assumption that
near the origin, the function ϕ suitably generalizes
the prototype given by\r\n \r\n
\ \r\n ϕ\r\n
\ \r\n (\r\n
\ \r\n ξ\r\n
\ \r\n )\r\n \r\n =\r\n
\ \r\n \r\n
\ ξ\r\n \r\n
\ \r\n α\r\n
\ \r\n \r\n
\ ,\r\n \r\n ξ\r\n ∈\r\n
\ \r\n [\r\n
\ \r\n 0\r\n
\ ,\r\n \r\n
\ \r\n ξ\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$$\\phi \\left(\\xi \\right)={\\xi
}^{\\alpha },\\qquad \\xi \\in \\left[0,{\\xi }_{0}\\right].$$\r\n\r\n
\ \r\n \r\n
\ By means of separate approaches, it is shown
that in both cases α ∈ (0, 1) and α
∈ [1, 2] some global weak solutions exist which, inter alia, satisfy\r\n \r\n
\ \r\n C\r\n
\ \r\n (\r\n
\ \r\n T\r\n
\ \r\n )\r\n \r\n ≔\r\n
\ \r\n \r\n
\ ess sup\r\n \r\n
\ \r\n t\r\n
\ ∈\r\n \r\n
\ (\r\n \r\n
\ 0\r\n ,\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 Ω\r\n \r\n
\ \r\n u\r\n
\ \r\n (\r\n
\ \r\n ⋅\r\n
\ ,\r\n t\r\n
\ \r\n )\r\n \r\n ln\r\n
\ \r\n u\r\n
\ \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 for all \r\n
\ T\r\n >\r\n
\ 0\r\n ,\r\n
\ \r\n \r\n$$C\\left(T\\right){:=}\\underset{t\\in
\\left(0,T\\right)}{\\text{ess\\,sup}}{\\int }_{{\\Omega}}u\\left(\\cdot ,t\\right)\\mathrm{ln}u\\left(\\cdot
,t\\right){< }\\infty \\qquad \\text{for\\,all\\,}T{ >}0,$$\r\n\r\n
\ \r\n \r\n
\ with sup\r\n T>0\r\n
\ C(T) <
∞ if α ∈ [1, 2]."
author:
- first_name: Michael
full_name: Winkler, Michael
id: '31496'
last_name: Winkler
citation:
ama: Winkler M. A degenerate migration-consumption model in domains of arbitrary
dimension. Advanced Nonlinear Studies. 2024;24(3):592-615. doi:10.1515/ans-2023-0131
apa: Winkler, M. (2024). A degenerate migration-consumption model in domains of
arbitrary dimension. Advanced Nonlinear Studies, 24(3), 592–615.
https://doi.org/10.1515/ans-2023-0131
bibtex: '@article{Winkler_2024, title={A degenerate migration-consumption model
in domains of arbitrary dimension}, volume={24}, DOI={10.1515/ans-2023-0131},
number={3}, journal={Advanced Nonlinear Studies}, publisher={Walter de Gruyter
GmbH}, author={Winkler, Michael}, year={2024}, pages={592–615} }'
chicago: 'Winkler, Michael. “A Degenerate Migration-Consumption Model in Domains
of Arbitrary Dimension.” Advanced Nonlinear Studies 24, no. 3 (2024): 592–615.
https://doi.org/10.1515/ans-2023-0131.'
ieee: 'M. Winkler, “A degenerate migration-consumption model in domains of arbitrary
dimension,” Advanced Nonlinear Studies, vol. 24, no. 3, pp. 592–615, 2024,
doi: 10.1515/ans-2023-0131.'
mla: Winkler, Michael. “A Degenerate Migration-Consumption Model in Domains of Arbitrary
Dimension.” Advanced Nonlinear Studies, vol. 24, no. 3, Walter de Gruyter
GmbH, 2024, pp. 592–615, doi:10.1515/ans-2023-0131.
short: M. Winkler, Advanced Nonlinear Studies 24 (2024) 592–615.
date_created: 2025-12-18T19:09:41Z
date_updated: 2025-12-18T20:10:00Z
doi: 10.1515/ans-2023-0131
intvolume: ' 24'
issue: '3'
language:
- iso: eng
page: 592-615
publication: Advanced Nonlinear Studies
publication_identifier:
issn:
- 2169-0375
publication_status: published
publisher: Walter de Gruyter GmbH
status: public
title: A degenerate migration-consumption model in domains of arbitrary dimension
type: journal_article
user_id: '31496'
volume: 24
year: '2024'
...
---
_id: '63310'
abstract:
- lang: eng
text: AbstractThe chemotaxis–Stokes systemnt+u⋅∇n=∇⋅(D(n)∇n)−∇⋅(nS(x,n,c)⋅∇c),ct+u⋅∇c=Δc−nc,ut=Δu+∇P+n∇Φ,∇⋅u=0,\left\{\begin{array}{l}{n}_{t}+u\cdot
\nabla n=\nabla \cdot (D\left(n)\nabla n)-\nabla \cdot (nS\left(x,n,c)\cdot \nabla
c),\\ {c}_{t}+u\cdot \nabla c=\Delta c-nc,\\ {u}_{t}=\Delta u+\nabla P+n\nabla
\Phi ,\hspace{1.0em}\nabla \cdot u=0,\end{array}\right.is
considered in a smoothly bounded convex domainΩ⊂R3\Omega
\subset {{\mathbb{R}}}^{3},
with given suitably regular functionsD:[0,∞)→[0,∞)D:{[}0,\infty )\to {[}0,\infty
),S:Ω¯×[0,∞)×(0,∞)→R3×3S:\overline{\Omega
}\times {[}0,\infty )\times \left(0,\infty )\to {{\mathbb{R}}}^{3\times 3}andΦ:Ω¯→R\Phi
:\overline{\Omega }\to {\mathbb{R}}such
thatD>0D\gt
0on(0,∞)\left(0,\infty
). It is shown that
if with some nondecreasingS0:(0,∞)→(0,∞){S}_{0}:\left(0,\infty
)\to \left(0,\infty )we
have∣S(x,n,c)∣≤S0(c)c12for all(x,n,c)∈Ω¯×[0,∞)×(0,∞),|
S\left(x,n,c)| \le \frac{{S}_{0}\left(c)}{{c}^{\tfrac{1}{2}}}\hspace{1.0em}\hspace{0.1em}\text{for
all}\hspace{0.1em}\hspace{0.33em}\left(x,n,c)\in \overline{\Omega }\times {[}0,\infty
)\times \left(0,\infty ),then
for allM>0M\gt
0there existsL(M)>0L\left(M)\gt
0such that wheneverliminfn→∞D(n)>L(M)andliminfn↘0D(n)n>0,\mathop{\mathrm{liminf}}\limits_{n\to
\infty }D\left(n)\gt L\left(M)\hspace{1.0em}\hspace{0.1em}\text{and}\hspace{0.1em}\hspace{1.0em}\mathop{\mathrm{liminf}}\limits_{n\searrow
0}\frac{D\left(n)}{n}\gt 0,for
all sufficiently regular initial data(n0,c0,u0)\left({n}_{0},{c}_{0},{u}_{0})fulfilling‖c0‖L∞(Ω)≤M\Vert
{c}_{0}{\Vert }_{{L}^{\infty }\left(\Omega )}\le Man
associated no-flux/no-flux/Dirichlet initial-boundary value problem admits a global
bounded weak solution, classical if additionallyD(0)>0D\left(0)\gt
0. When combined with
previously known results, this particularly implies global existence of bounded
solutions whenD(n)=nm−1D\left(n)={n}^{m-1},n≥0n\ge
0, with arbitrarym>1m\gt
1, but beyond this asserts
global boundedness also in the presence of diffusivities which exhibit arbitrarily
slow divergence to+∞+\inftyat
large densities and of possibly singular chemotactic sensitivities.
author:
- first_name: Michael
full_name: Winkler, Michael
id: '31496'
last_name: Winkler
citation:
ama: 'Winkler M. Chemotaxis-Stokes interaction with very weak diffusion enhancement:
Blow-up exclusion via detection of absorption-induced entropy structures involving
multiplicative couplings. Advanced Nonlinear Studies. 2022;22(1):88-117.
doi:10.1515/ans-2022-0004'
apa: 'Winkler, M. (2022). Chemotaxis-Stokes interaction with very weak diffusion
enhancement: Blow-up exclusion via detection of absorption-induced entropy structures
involving multiplicative couplings. Advanced Nonlinear Studies, 22(1),
88–117. https://doi.org/10.1515/ans-2022-0004'
bibtex: '@article{Winkler_2022, title={Chemotaxis-Stokes interaction with very weak
diffusion enhancement: Blow-up exclusion via detection of absorption-induced entropy
structures involving multiplicative couplings}, volume={22}, DOI={10.1515/ans-2022-0004},
number={1}, journal={Advanced Nonlinear Studies}, publisher={Walter de Gruyter
GmbH}, author={Winkler, Michael}, year={2022}, pages={88–117} }'
chicago: 'Winkler, Michael. “Chemotaxis-Stokes Interaction with Very Weak Diffusion
Enhancement: Blow-up Exclusion via Detection of Absorption-Induced Entropy Structures
Involving Multiplicative Couplings.” Advanced Nonlinear Studies 22, no.
1 (2022): 88–117. https://doi.org/10.1515/ans-2022-0004.'
ieee: 'M. Winkler, “Chemotaxis-Stokes interaction with very weak diffusion enhancement:
Blow-up exclusion via detection of absorption-induced entropy structures involving
multiplicative couplings,” Advanced Nonlinear Studies, vol. 22, no. 1,
pp. 88–117, 2022, doi: 10.1515/ans-2022-0004.'
mla: 'Winkler, Michael. “Chemotaxis-Stokes Interaction with Very Weak Diffusion
Enhancement: Blow-up Exclusion via Detection of Absorption-Induced Entropy Structures
Involving Multiplicative Couplings.” Advanced Nonlinear Studies, vol. 22,
no. 1, Walter de Gruyter GmbH, 2022, pp. 88–117, doi:10.1515/ans-2022-0004.'
short: M. Winkler, Advanced Nonlinear Studies 22 (2022) 88–117.
date_created: 2025-12-18T19:29:40Z
date_updated: 2025-12-18T20:05:30Z
doi: 10.1515/ans-2022-0004
intvolume: ' 22'
issue: '1'
language:
- iso: eng
page: 88-117
publication: Advanced Nonlinear Studies
publication_identifier:
issn:
- 2169-0375
publication_status: published
publisher: Walter de Gruyter GmbH
status: public
title: 'Chemotaxis-Stokes interaction with very weak diffusion enhancement: Blow-up
exclusion via detection of absorption-induced entropy structures involving multiplicative
couplings'
type: journal_article
user_id: '31496'
volume: 22
year: '2022'
...
---
_id: '63340'
abstract:
- lang: eng
text: "Abstract\r\n The chemotaxis-growth
system\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 D\r\n
\ \r\n Δ\r\n \r\n
\ u\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
\ u\r\n
\ \r\n
\ \r\n ∇\r\n
\ \r\n
\ v\r\n
\ \r\n \r\n
\ )\r\n
\ \r\n \r\n
\ \r\n +\r\n
\ \r\n ρ\r\n
\ \r\n u\r\n
\ \r\n \r\n
\ -\r\n \r\n
\ μ\r\n \r\n
\ \r\n u\r\n
\ α\r\n \r\n
\ \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
\ \r\n \r\n
\ \r\n =\r\n
\ \r\n \r\n
\ \r\n d\r\n
\ \r\n Δ\r\n \r\n
\ v\r\n \r\n
\ -\r\n \r\n
\ κ\r\n \r\n
\ v\r\n \r\n
\ \r\n +\r\n
\ \r\n λ\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 {}\\left\\{\\begin{aligned}
\\displaystyle{}u_{t}&\\displaystyle=D\\Delta u-\\chi% \\nabla\\cdot(u\\nabla
v)+\\rho u-\\mu u^{\\alpha},\\\\ \\displaystyle v_{t}&\\displaystyle=d\\Delta
v-\\kappa v+\\lambda u\\end{aligned}\\right.\r\n \r\n
\ \r\n \r\n is
considered under homogeneous Neumann boundary conditions in smoothly bounded domains
\r\n \r\n
\ \r\n
\ \r\n Ω\r\n
\ ⊂\r\n \r\n
\ ℝ\r\n n\r\n
\ \r\n \r\n
\ \r\n \r\n {\\Omega\\subset\\mathbb{R}^{n}}\r\n
\ \r\n ,
\r\n \r\n
\ \r\n
\ \r\n n\r\n
\ ≥\r\n 1\r\n
\ \r\n \r\n
\ \r\n {n\\geq
1}\r\n \r\n .
For any choice of \r\n \r\n
\ \r\n
\ \r\n α\r\n
\ >\r\n 1\r\n
\ \r\n \r\n
\ \r\n {\\alpha>1}\r\n
\ \r\n ,
the literature provides a comprehensive result on global existence for widely
arbitrary initial data within a suitably generalized solution concept, but the
regularity properties of such solutions may be rather poor, as indicated by precedent
results on the occurrence of finite-time blow-up in corresponding parabolic-elliptic
simplifications. Based on the analysis of a certain eventual Lyapunov-type feature
of ($\\star$), the present work shows that, whenever \r\n
\ \r\n \r\n
\ \r\n α\r\n
\ ≥\r\n \r\n
\ 2\r\n -\r\n
\ \r\n 2\r\n
\ n\r\n \r\n
\ \r\n \r\n
\ \r\n \r\n {\\alpha\\geq 2-\\frac{2}{n}}\r\n
\ \r\n ,
under an appropriate smallness assumption on χ, any such solution at least asymptotically
exhibits relaxation by approaching the nontrivial spatially homogeneous steady
state \r\n \r\n
\ \r\n
\ \r\n (\r\n \r\n \r\n
\ (\r\n
\ \r\n ρ\r\n
\ μ\r\n \r\n
\ )\r\n
\ \r\n \r\n
\ 1\r\n \r\n
\ α\r\n -\r\n
\ 1\r\n \r\n
\ \r\n \r\n
\ ,\r\n \r\n
\ \r\n λ\r\n
\ κ\r\n \r\n
\ \r\n \r\n
\ \r\n (\r\n \r\n
\ ρ\r\n μ\r\n
\ \r\n )\r\n \r\n
\ \r\n 1\r\n
\ \r\n α\r\n
\ -\r\n 1\r\n
\ \r\n \r\n
\ \r\n \r\n
\ )\r\n
\ \r\n \r\n
\ \r\n {\\bigl{(}\\bigl{(}\\frac{\\rho}{\\mu}\\bigr{)}^{\\frac{1}{\\alpha-1}},\\frac{\\lambda}{%
\\kappa}\\bigl{(}\\frac{\\rho}{\\mu}\\bigr{)}^{\\frac{1}{\\alpha-1}}\\bigr{)}}\r\n
\ \r\n
in the large time limit."
author:
- first_name: Michael
full_name: Winkler, Michael
id: '31496'
last_name: Winkler
citation:
ama: Winkler M. Attractiveness of Constant States in Logistic-Type Keller–Segel
Systems Involving Subquadratic Growth Restrictions. Advanced Nonlinear Studies.
2020;20(4):795-817. doi:10.1515/ans-2020-2107
apa: Winkler, M. (2020). Attractiveness of Constant States in Logistic-Type Keller–Segel
Systems Involving Subquadratic Growth Restrictions. Advanced Nonlinear Studies,
20(4), 795–817. https://doi.org/10.1515/ans-2020-2107
bibtex: '@article{Winkler_2020, title={Attractiveness of Constant States in Logistic-Type
Keller–Segel Systems Involving Subquadratic Growth Restrictions}, volume={20},
DOI={10.1515/ans-2020-2107},
number={4}, journal={Advanced Nonlinear Studies}, publisher={Walter de Gruyter
GmbH}, author={Winkler, Michael}, year={2020}, pages={795–817} }'
chicago: 'Winkler, Michael. “Attractiveness of Constant States in Logistic-Type
Keller–Segel Systems Involving Subquadratic Growth Restrictions.” Advanced
Nonlinear Studies 20, no. 4 (2020): 795–817. https://doi.org/10.1515/ans-2020-2107.'
ieee: 'M. Winkler, “Attractiveness of Constant States in Logistic-Type Keller–Segel
Systems Involving Subquadratic Growth Restrictions,” Advanced Nonlinear Studies,
vol. 20, no. 4, pp. 795–817, 2020, doi: 10.1515/ans-2020-2107.'
mla: Winkler, Michael. “Attractiveness of Constant States in Logistic-Type Keller–Segel
Systems Involving Subquadratic Growth Restrictions.” Advanced Nonlinear Studies,
vol. 20, no. 4, Walter de Gruyter GmbH, 2020, pp. 795–817, doi:10.1515/ans-2020-2107.
short: M. Winkler, Advanced Nonlinear Studies 20 (2020) 795–817.
date_created: 2025-12-18T19:46:54Z
date_updated: 2025-12-18T19:58:22Z
doi: 10.1515/ans-2020-2107
intvolume: ' 20'
issue: '4'
language:
- iso: eng
page: 795-817
publication: Advanced Nonlinear Studies
publication_identifier:
issn:
- 1536-1365
- 2169-0375
publication_status: published
publisher: Walter de Gruyter GmbH
status: public
title: Attractiveness of Constant States in Logistic-Type Keller–Segel Systems Involving
Subquadratic Growth Restrictions
type: journal_article
user_id: '31496'
volume: 20
year: '2020'
...