---
_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: '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: '63311'
abstract:
- lang: eng
text: "AbstractThe Cauchy problem in $$\\mathbb
{R}^n$$\r\n
\ \r\n \r\n R\r\n
\ \r\n n\r\n
\ \r\n ,
$$n\\ge 1$$\r\n \r\n
\ n\r\n ≥\r\n
\ 1\r\n \r\n ,
for the degenerate parabolic equation $$\\begin{aligned}
u_t=u^p \\Delta u \\qquad \\qquad (\\star ) \\end{aligned}$$\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 u\r\n
\ p\r\n \r\n
\ Δ\r\n u\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 for $$p\\ge
1$$\r\n
\ \r\n p\r\n ≥\r\n
\ 1\r\n \r\n .
It is shown that given any positive $$f\\in
C^0([0,\\infty ))$$\r\n
\ \r\n f\r\n ∈\r\n
\ \r\n C\r\n
\ 0\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 $$g\\in C^0([0,\\infty
))$$\r\n
\ \r\n g\r\n ∈\r\n
\ \r\n C\r\n
\ 0\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
satisfying $$\\begin{aligned}
f(t)\\rightarrow + \\infty \\quad \\text{ and } \\quad g(t)\\rightarrow 0 \\qquad
\\text{ as } t\\rightarrow \\infty , \\end{aligned}$$\r\n \r\n
\ \r\n \r\n \r\n
\ \r\n f\r\n
\ (\r\n t\r\n
\ )\r\n →\r\n
\ +\r\n ∞\r\n
\ \r\n \r\n
\ and\r\n \r\n
\ \r\n g\r\n
\ (\r\n t\r\n
\ )\r\n →\r\n
\ 0\r\n \r\n
\ \r\n as\r\n
\ \r\n t\r\n
\ →\r\n ∞\r\n
\ ,\r\n \r\n
\ \r\n \r\n \r\n
\ \r\n one
can find positive and radially symmetric continuous initial data with the property
that the initial value problem for ($$\\star
$$\r\n
\ ⋆\r\n )
admits a positive classical solution such that $$\\begin{aligned}
t^\\frac{1}{p} \\Vert u(\\cdot ,t)\\Vert _{L^\\infty (\\mathbb {R}^n)} \\rightarrow
\\infty \\qquad \\text{ and } \\qquad \\Vert u(\\cdot ,t)\\Vert _{L^\\infty (\\mathbb
{R}^n)} \\rightarrow 0 \\qquad \\text{ as } t\\rightarrow \\infty , \\end{aligned}$$\r\n \r\n
\ \r\n \r\n \r\n
\ \r\n \r\n
\ t\r\n \r\n
\ 1\r\n p\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
\ ∞\r\n \r\n
\ \r\n (\r\n
\ \r\n \r\n
\ R\r\n \r\n
\ n\r\n \r\n
\ )\r\n \r\n
\ \r\n \r\n
\ →\r\n ∞\r\n
\ \r\n \r\n
\ and\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 ∞\r\n
\ \r\n \r\n
\ (\r\n \r\n
\ \r\n R\r\n
\ \r\n n\r\n
\ \r\n )\r\n
\ \r\n \r\n
\ \r\n →\r\n
\ 0\r\n \r\n
\ \r\n as\r\n
\ \r\n t\r\n
\ →\r\n ∞\r\n
\ ,\r\n \r\n
\ \r\n \r\n \r\n
\ \r\n but
that $$\\begin{aligned} \\liminf
_{t\\rightarrow \\infty } \\frac{t^\\frac{1}{p} \\Vert u(\\cdot ,t)\\Vert _{L^\\infty
(\\mathbb {R}^n)}}{f(t)} =0 \\end{aligned}$$\r\n
\ \r\n \r\n \r\n
\ \r\n \r\n
\ \r\n lim
inf\r\n \r\n t\r\n
\ →\r\n ∞\r\n
\ \r\n \r\n
\ \r\n \r\n
\ \r\n t\r\n
\ \r\n 1\r\n
\ p\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 ∞\r\n
\ \r\n \r\n
\ (\r\n \r\n
\ \r\n R\r\n
\ \r\n n\r\n
\ \r\n )\r\n
\ \r\n \r\n
\ \r\n \r\n
\ \r\n f\r\n
\ (\r\n t\r\n
\ )\r\n \r\n
\ \r\n =\r\n
\ 0\r\n \r\n
\ \r\n \r\n \r\n
\ \r\n and
$$\\begin{aligned} \\limsup
_{t\\rightarrow \\infty } \\frac{\\Vert u(\\cdot ,t)\\Vert _{L^\\infty (\\mathbb
{R}^n)}}{g(t)} =\\infty . \\end{aligned}$$\r\n
\ \r\n \r\n \r\n
\ \r\n \r\n
\ \r\n lim
sup\r\n \r\n t\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 ·\r\n
\ ,\r\n t\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 \r\n
\ \r\n R\r\n
\ \r\n n\r\n
\ \r\n )\r\n
\ \r\n \r\n
\ \r\n \r\n
\ g\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
\ \r\n "
article_number: '47'
author:
- first_name: Michael
full_name: Winkler, Michael
id: '31496'
last_name: Winkler
citation:
ama: Winkler M. Oscillatory decay in a degenerate parabolic equation. Partial
Differential Equations and Applications. 2022;3(4). doi:10.1007/s42985-022-00186-z
apa: Winkler, M. (2022). Oscillatory decay in a degenerate parabolic equation. Partial
Differential Equations and Applications, 3(4), Article 47. https://doi.org/10.1007/s42985-022-00186-z
bibtex: '@article{Winkler_2022, title={Oscillatory decay in a degenerate parabolic
equation}, volume={3}, DOI={10.1007/s42985-022-00186-z},
number={447}, journal={Partial Differential Equations and Applications}, publisher={Springer
Science and Business Media LLC}, author={Winkler, Michael}, year={2022} }'
chicago: Winkler, Michael. “Oscillatory Decay in a Degenerate Parabolic Equation.”
Partial Differential Equations and Applications 3, no. 4 (2022). https://doi.org/10.1007/s42985-022-00186-z.
ieee: 'M. Winkler, “Oscillatory decay in a degenerate parabolic equation,” Partial
Differential Equations and Applications, vol. 3, no. 4, Art. no. 47, 2022,
doi: 10.1007/s42985-022-00186-z.'
mla: Winkler, Michael. “Oscillatory Decay in a Degenerate Parabolic Equation.” Partial
Differential Equations and Applications, vol. 3, no. 4, 47, Springer Science
and Business Media LLC, 2022, doi:10.1007/s42985-022-00186-z.
short: M. Winkler, Partial Differential Equations and Applications 3 (2022).
date_created: 2025-12-18T19:30:04Z
date_updated: 2025-12-18T20:05:38Z
doi: 10.1007/s42985-022-00186-z
intvolume: ' 3'
issue: '4'
language:
- iso: eng
publication: Partial Differential Equations and Applications
publication_identifier:
issn:
- 2662-2963
- 2662-2971
publication_status: published
publisher: Springer Science and Business Media LLC
status: public
title: Oscillatory decay in a degenerate parabolic equation
type: journal_article
user_id: '31496'
volume: 3
year: '2022'
...
---
_id: '63312'
abstract:
- lang: eng
text: "<p style='text-indent:20px;'>The chemotaxis
system</p><p style='text-indent:20px;'><disp-formula> <label/>
<tex-math id=\"FE1\"> \\begin{document}$ \\begin{array}{l}\\left\\{ \\begin{array}{l}
\tu_t = \\nabla \\cdot \\big( D(u) \\nabla u \\big) - \\nabla \\cdot \\big( uS(x,
u, v)\\cdot \\nabla v\\big), \\\\ \tv_t = \\Delta v -uv, \\end{array} \\right.
\\end{array} $\\end{document} </tex-math></disp-formula></p><p
style='text-indent:20px;'>is considered in a bounded domain <inline-formula><tex-math
id=\"M1\">\\begin{document}$ \\Omega\\subset \\mathbb{R}^n $\\end{document}</tex-math></inline-formula>,
<inline-formula><tex-math id=\"M2\">\\begin{document}$ n\\ge 2 $\\end{document}</tex-math></inline-formula>,
with smooth boundary.</p><p style='text-indent:20px;'>It is shown
that if <inline-formula><tex-math id=\"M3\">\\begin{document}$ D:
[0, \\infty) \\to [0, \\infty) $\\end{document}</tex-math></inline-formula>
and <inline-formula><tex-math id=\"M4\">\\begin{document}$ S: \\overline{\\Omega}\\times
[0, \\infty)\\times (0, \\infty)\\to \\mathbb{R}^{n\\times n} $\\end{document}</tex-math></inline-formula>
are suitably smooth functions satisfying</p><p style='text-indent:20px;'><disp-formula>
<label/> <tex-math id=\"FE2\"> \\begin{document}$ \\begin{array}{l}D(u)
\\ge k_D u^{m-1} \t\\qquad {\\rm{for\\; all}}\\; u\\ge 0 \\end{array} $\\end{document}
</tex-math></disp-formula></p><p style='text-indent:20px;'>and</p><p
style='text-indent:20px;'><disp-formula> <label/> <tex-math
id=\"FE3\"> \\begin{document}$ \\begin{array}{l}|S(x, u, v)| \\le \\frac{S_0(v)}{v^\\alpha}
\\qquad {\\rm{for\\; all}}\\; (x, u, v)\\; \\in \\Omega\\times (0, \\infty)^2
\\end{array} $\\end{document} </tex-math></disp-formula></p><p
style='text-indent:20px;'>with some</p><p style='text-indent:20px;'><disp-formula>
<label/> <tex-math id=\"FE4\"> \\begin{document}$ \\begin{array}{l}m>\\frac{3n-2}{2n}
\t\\qquad {\\rm{and}}\\;\\alpha\\in [0, 1), \\end{array} $\\end{document} </tex-math></disp-formula></p><p
style='text-indent:20px;'>and with some <inline-formula><tex-math
id=\"M5\">\\begin{document}$ k_D>0 $\\end{document}</tex-math></inline-formula>
and nondecreasing <inline-formula><tex-math id=\"M6\">\\begin{document}$
S_0: (0, \\infty)\\to (0, \\infty) $\\end{document}</tex-math></inline-formula>,
then for all suitably regular initial data a corresponding no-flux type initial-boundary
value problem admits a global bounded weak solution which actually is smooth and
classical if <inline-formula><tex-math id=\"M7\">\\begin{document}$
D(0)>0 $\\end{document}</tex-math></inline-formula>.</p>"
article_number: '6565'
author:
- first_name: Michael
full_name: Winkler, Michael
id: '31496'
last_name: Winkler
citation:
ama: Winkler M. Approaching logarithmic singularities in quasilinear chemotaxis-consumption
systems with signal-dependent sensitivities. Discrete and Continuous Dynamical
Systems - B. 2022;27(11). doi:10.3934/dcdsb.2022009
apa: Winkler, M. (2022). Approaching logarithmic singularities in quasilinear chemotaxis-consumption
systems with signal-dependent sensitivities. Discrete and Continuous Dynamical
Systems - B, 27(11), Article 6565. https://doi.org/10.3934/dcdsb.2022009
bibtex: '@article{Winkler_2022, title={Approaching logarithmic singularities in
quasilinear chemotaxis-consumption systems with signal-dependent sensitivities},
volume={27}, DOI={10.3934/dcdsb.2022009},
number={116565}, journal={Discrete and Continuous Dynamical Systems - B}, publisher={American
Institute of Mathematical Sciences (AIMS)}, author={Winkler, Michael}, year={2022}
}'
chicago: Winkler, Michael. “Approaching Logarithmic Singularities in Quasilinear
Chemotaxis-Consumption Systems with Signal-Dependent Sensitivities.” Discrete
and Continuous Dynamical Systems - B 27, no. 11 (2022). https://doi.org/10.3934/dcdsb.2022009.
ieee: 'M. Winkler, “Approaching logarithmic singularities in quasilinear chemotaxis-consumption
systems with signal-dependent sensitivities,” Discrete and Continuous Dynamical
Systems - B, vol. 27, no. 11, Art. no. 6565, 2022, doi: 10.3934/dcdsb.2022009.'
mla: Winkler, Michael. “Approaching Logarithmic Singularities in Quasilinear Chemotaxis-Consumption
Systems with Signal-Dependent Sensitivities.” Discrete and Continuous Dynamical
Systems - B, vol. 27, no. 11, 6565, American Institute of Mathematical Sciences
(AIMS), 2022, doi:10.3934/dcdsb.2022009.
short: M. Winkler, Discrete and Continuous Dynamical Systems - B 27 (2022).
date_created: 2025-12-18T19:30:32Z
date_updated: 2025-12-18T20:05:47Z
doi: 10.3934/dcdsb.2022009
intvolume: ' 27'
issue: '11'
language:
- iso: eng
publication: Discrete and Continuous Dynamical Systems - B
publication_identifier:
issn:
- 1531-3492
- 1553-524X
publication_status: published
publisher: American Institute of Mathematical Sciences (AIMS)
status: public
title: Approaching logarithmic singularities in quasilinear chemotaxis-consumption
systems with signal-dependent sensitivities
type: journal_article
user_id: '31496'
volume: 27
year: '2022'
...
---
_id: '63309'
abstract:
- lang: eng
text: AbstractThis manuscript is concerned with
the problem of efficiently estimating chemotactic gradients, as forming a ubiquitous
issue of key importance in virtually any proof of boundedness features in Keller–Segel
type systems. A strategy is proposed which at its core relies on bounds for such
quantities, conditional in the sense of involving certain Lebesgue norms of solution
components that explicitly influence the signal evolution.Applications
of this procedure firstly provide apparently novel boundedness results for two
particular classes chemotaxis systems, and apart from that are shown to significantly
condense proofs for basically well‐known statements on boundedness in two further
Keller–Segel type problems.
author:
- first_name: Michael
full_name: Winkler, Michael
id: '31496'
last_name: Winkler
citation:
ama: Winkler M. A unifying approach toward boundedness in Keller–Segel type cross‐diffusion
systems via conditional L∞$L^\infty$ estimates for taxis gradients. Mathematische
Nachrichten. 2022;295(9):1840-1862. doi:10.1002/mana.202000403
apa: Winkler, M. (2022). A unifying approach toward boundedness in Keller–Segel
type cross‐diffusion systems via conditional L∞$L^\infty$ estimates for taxis
gradients. Mathematische Nachrichten, 295(9), 1840–1862. https://doi.org/10.1002/mana.202000403
bibtex: '@article{Winkler_2022, title={A unifying approach toward boundedness in
Keller–Segel type cross‐diffusion systems via conditional L∞$L^\infty$ estimates
for taxis gradients}, volume={295}, DOI={10.1002/mana.202000403},
number={9}, journal={Mathematische Nachrichten}, publisher={Wiley}, author={Winkler,
Michael}, year={2022}, pages={1840–1862} }'
chicago: 'Winkler, Michael. “A Unifying Approach toward Boundedness in Keller–Segel
Type Cross‐diffusion Systems via Conditional L∞$L^\infty$ Estimates for Taxis
Gradients.” Mathematische Nachrichten 295, no. 9 (2022): 1840–62. https://doi.org/10.1002/mana.202000403.'
ieee: 'M. Winkler, “A unifying approach toward boundedness in Keller–Segel type
cross‐diffusion systems via conditional L∞$L^\infty$ estimates for taxis gradients,”
Mathematische Nachrichten, vol. 295, no. 9, pp. 1840–1862, 2022, doi: 10.1002/mana.202000403.'
mla: Winkler, Michael. “A Unifying Approach toward Boundedness in Keller–Segel Type
Cross‐diffusion Systems via Conditional L∞$L^\infty$ Estimates for Taxis Gradients.”
Mathematische Nachrichten, vol. 295, no. 9, Wiley, 2022, pp. 1840–62, doi:10.1002/mana.202000403.
short: M. Winkler, Mathematische Nachrichten 295 (2022) 1840–1862.
date_created: 2025-12-18T19:28:46Z
date_updated: 2025-12-18T20:05:19Z
doi: 10.1002/mana.202000403
intvolume: ' 295'
issue: '9'
language:
- iso: eng
page: 1840-1862
publication: Mathematische Nachrichten
publication_identifier:
issn:
- 0025-584X
- 1522-2616
publication_status: published
publisher: Wiley
status: public
title: A unifying approach toward boundedness in Keller–Segel type cross‐diffusion
systems via conditional L∞$L^\infty$ estimates for taxis gradients
type: journal_article
user_id: '31496'
volume: 295
year: '2022'
...
---
_id: '63306'
author:
- first_name: Michael
full_name: Winkler, Michael
id: '31496'
last_name: Winkler
citation:
ama: Winkler M. A critical blow-up exponent for flux limiation in a Keller-Segel
system. Indiana University Mathematics Journal. 2022;71(4):1437-1465. doi:10.1512/iumj.2022.71.9042
apa: Winkler, M. (2022). A critical blow-up exponent for flux limiation in a Keller-Segel
system. Indiana University Mathematics Journal, 71(4), 1437–1465.
https://doi.org/10.1512/iumj.2022.71.9042
bibtex: '@article{Winkler_2022, title={A critical blow-up exponent for flux limiation
in a Keller-Segel system}, volume={71}, DOI={10.1512/iumj.2022.71.9042},
number={4}, journal={Indiana University Mathematics Journal}, publisher={Indiana
University Mathematics Journal}, author={Winkler, Michael}, year={2022}, pages={1437–1465}
}'
chicago: 'Winkler, Michael. “A Critical Blow-up Exponent for Flux Limiation in a
Keller-Segel System.” Indiana University Mathematics Journal 71, no. 4
(2022): 1437–65. https://doi.org/10.1512/iumj.2022.71.9042.'
ieee: 'M. Winkler, “A critical blow-up exponent for flux limiation in a Keller-Segel
system,” Indiana University Mathematics Journal, vol. 71, no. 4, pp. 1437–1465,
2022, doi: 10.1512/iumj.2022.71.9042.'
mla: Winkler, Michael. “A Critical Blow-up Exponent for Flux Limiation in a Keller-Segel
System.” Indiana University Mathematics Journal, vol. 71, no. 4, Indiana
University Mathematics Journal, 2022, pp. 1437–65, doi:10.1512/iumj.2022.71.9042.
short: M. Winkler, Indiana University Mathematics Journal 71 (2022) 1437–1465.
date_created: 2025-12-18T19:26:56Z
date_updated: 2025-12-18T20:04:53Z
doi: 10.1512/iumj.2022.71.9042
intvolume: ' 71'
issue: '4'
language:
- iso: eng
page: 1437-1465
publication: Indiana University Mathematics Journal
publication_identifier:
issn:
- 0022-2518
publication_status: published
publisher: Indiana University Mathematics Journal
status: public
title: A critical blow-up exponent for flux limiation in a Keller-Segel system
type: journal_article
user_id: '31496'
volume: 71
year: '2022'
...
---
_id: '63284'
abstract:
- lang: eng
text: ' A no-flux initial-boundary value problem for the cross-diffusion
system [Formula: see text] is considered in smoothly bounded domains [Formula:
see text] with [Formula: see text]. It is shown that whenever [Formula: see text]
is positive on [Formula: see text] and such that [Formula: see text] for some
[Formula: see text], for all suitably regular positive initial data a global very
weak solution, particularly preserving mass in its first component, can be constructed.
This extends previous results which either concentrate on non-degenerate analogs,
or are restricted to the special case [Formula: see text]. To
appropriately cope with the considerably stronger cross-degeneracies thus allowed
through [Formula: see text] when [Formula: see text] is large, in its core part
the analysis relies on the use of the Moser–Trudinger inequality in controlling
the respective diffusion rates [Formula: see text] from below. '
article_number: '2250012'
author:
- first_name: Michael
full_name: Winkler, Michael
id: '31496'
last_name: Winkler
citation:
ama: Winkler M. Application of the Moser–Trudinger inequality in the construction
of global solutions to a strongly degenerate migration model. Bulletin of Mathematical
Sciences. 2022;13(02). doi:10.1142/s1664360722500126
apa: Winkler, M. (2022). Application of the Moser–Trudinger inequality in the construction
of global solutions to a strongly degenerate migration model. Bulletin of Mathematical
Sciences, 13(02), Article 2250012. https://doi.org/10.1142/s1664360722500126
bibtex: '@article{Winkler_2022, title={Application of the Moser–Trudinger inequality
in the construction of global solutions to a strongly degenerate migration model},
volume={13}, DOI={10.1142/s1664360722500126},
number={022250012}, journal={Bulletin of Mathematical Sciences}, publisher={World
Scientific Pub Co Pte Ltd}, author={Winkler, Michael}, year={2022} }'
chicago: Winkler, Michael. “Application of the Moser–Trudinger Inequality in the
Construction of Global Solutions to a Strongly Degenerate Migration Model.” Bulletin
of Mathematical Sciences 13, no. 02 (2022). https://doi.org/10.1142/s1664360722500126.
ieee: 'M. Winkler, “Application of the Moser–Trudinger inequality in the construction
of global solutions to a strongly degenerate migration model,” Bulletin of
Mathematical Sciences, vol. 13, no. 02, Art. no. 2250012, 2022, doi: 10.1142/s1664360722500126.'
mla: Winkler, Michael. “Application of the Moser–Trudinger Inequality in the Construction
of Global Solutions to a Strongly Degenerate Migration Model.” Bulletin of
Mathematical Sciences, vol. 13, no. 02, 2250012, World Scientific Pub Co Pte
Ltd, 2022, doi:10.1142/s1664360722500126.
short: M. Winkler, Bulletin of Mathematical Sciences 13 (2022).
date_created: 2025-12-18T19:18:11Z
date_updated: 2025-12-18T20:07:05Z
doi: 10.1142/s1664360722500126
intvolume: ' 13'
issue: '02'
language:
- iso: eng
publication: Bulletin of Mathematical Sciences
publication_identifier:
issn:
- 1664-3607
- 1664-3615
publication_status: published
publisher: World Scientific Pub Co Pte Ltd
status: public
title: Application of the Moser–Trudinger inequality in the construction of global
solutions to a strongly degenerate migration model
type: journal_article
user_id: '31496'
volume: 13
year: '2022'
...
---
_id: '63286'
abstract:
- lang: eng
text: ' The chemotaxis system [Formula: see text] is considered in a ball
[Formula: see text]. It is shown that if [Formula: see text]
suitably generalizes the prototype given by [Formula: see text] with some [Formula:
see text], and if diffusion is suitably weak in the sense that [Formula: see text]
is such that there exist [Formula: see text] and [Formula: see text] fulfilling
[Formula: see text] then for appropriate choices of sufficiently concentrated
initial data, an associated no-flux initial-boundary value problem admits a global
classical solution [Formula: see text] which blows up in infinite time and satisfies
[Formula: see text] A major part of the proof is based on a comparison argument
involving explicitly constructed subsolutions to a scalar parabolic problem satisfied
by mass accumulation functions corresponding to solutions of ( ⋆ ). '
author:
- first_name: Michael
full_name: Winkler, Michael
id: '31496'
last_name: Winkler
citation:
ama: Winkler M. Exponential grow-up rates in a quasilinear Keller–Segel system.
Asymptotic Analysis. 2022;131(1):33-57. doi:10.3233/asy-221765
apa: Winkler, M. (2022). Exponential grow-up rates in a quasilinear Keller–Segel
system. Asymptotic Analysis, 131(1), 33–57. https://doi.org/10.3233/asy-221765
bibtex: '@article{Winkler_2022, title={Exponential grow-up rates in a quasilinear
Keller–Segel system}, volume={131}, DOI={10.3233/asy-221765},
number={1}, journal={Asymptotic Analysis}, publisher={SAGE Publications}, author={Winkler,
Michael}, year={2022}, pages={33–57} }'
chicago: 'Winkler, Michael. “Exponential Grow-up Rates in a Quasilinear Keller–Segel
System.” Asymptotic Analysis 131, no. 1 (2022): 33–57. https://doi.org/10.3233/asy-221765.'
ieee: 'M. Winkler, “Exponential grow-up rates in a quasilinear Keller–Segel system,”
Asymptotic Analysis, vol. 131, no. 1, pp. 33–57, 2022, doi: 10.3233/asy-221765.'
mla: Winkler, Michael. “Exponential Grow-up Rates in a Quasilinear Keller–Segel
System.” Asymptotic Analysis, vol. 131, no. 1, SAGE Publications, 2022,
pp. 33–57, doi:10.3233/asy-221765.
short: M. Winkler, Asymptotic Analysis 131 (2022) 33–57.
date_created: 2025-12-18T19:18:51Z
date_updated: 2025-12-18T20:07:19Z
doi: 10.3233/asy-221765
intvolume: ' 131'
issue: '1'
language:
- iso: eng
page: 33-57
publication: Asymptotic Analysis
publication_identifier:
issn:
- 0921-7134
- 1875-8576
publication_status: published
publisher: SAGE Publications
status: public
title: Exponential grow-up rates in a quasilinear Keller–Segel system
type: journal_article
user_id: '31496'
volume: 131
year: '2022'
...
---
_id: '63293'
abstract:
- lang: eng
text: "<p style='text-indent:20px;'>The Cauchy problem
in <inline-formula><tex-math id=\"M2\">\\begin{document}$ \\mathbb{R}^3
$\\end{document}</tex-math></inline-formula> for the chemotaxis-Navier–Stokes
system</p><p style='text-indent:20px;'><disp-formula> <label/>
<tex-math id=\"FE1\"> \\begin{document}$ \\begin{eqnarray*} \\left\\{ \\begin{array}{l}
\ n_t + u\\cdot\\nabla n = \\Delta n - \\nabla \\cdot (n\\nabla c), \\\\\tc_t
+ u\\cdot\\nabla c = \\Delta c - nc, \\\\ \tu_t + (u\\cdot\\nabla) u = \\Delta
u + \\nabla P + n\\nabla\\phi, \\qquad \\nabla \\cdot u = 0, \\ \t\\end{array}
\\right. \\end{eqnarray*} $\\end{document} </tex-math></disp-formula></p><p
style='text-indent:20px;'>is considered. Under suitable conditions on the initial
data <inline-formula><tex-math id=\"M3\">\\begin{document}$ (n_0,
c_0, u_0) $\\end{document}</tex-math></inline-formula>, with regard
to the crucial first component requiring that <inline-formula><tex-math
id=\"M4\">\\begin{document}$ n_0\\in L^1( \\mathbb{R}^3) $\\end{document}</tex-math></inline-formula>
be nonnegative and such that <inline-formula><tex-math id=\"M5\">\\begin{document}$
(n_0+1)\\ln (n_0+1) \\in L^1( \\mathbb{R}^3) $\\end{document}</tex-math></inline-formula>,
a globally defined weak solution with <inline-formula><tex-math id=\"M6\">\\begin{document}$
(n, c, u)|_{t = 0} = (n_0, c_0, u_0) $\\end{document}</tex-math></inline-formula>
is constructed. Apart from that, assuming that moreover <inline-formula><tex-math
id=\"M7\">\\begin{document}$ \\int_{ \\mathbb{R}^3} n_0(x) \\ln (1+|x|^2) dx
$\\end{document}</tex-math></inline-formula> is finite, it is shown
that a weak solution exists which enjoys further regularity features and preserves
mass in an appropriate sense.</p>"
article_number: '5201'
author:
- first_name: Kyungkeun
full_name: Kang, Kyungkeun
last_name: Kang
- first_name: Jihoon
full_name: Lee, Jihoon
last_name: Lee
- first_name: Michael
full_name: Winkler, Michael
id: '31496'
last_name: Winkler
citation:
ama: Kang K, Lee J, Winkler M. Global weak solutions to a chemotaxis-Navier-Stokes
system in $ \mathbb{R}^3 $. Discrete and Continuous Dynamical Systems.
2022;42(11). doi:10.3934/dcds.2022091
apa: Kang, K., Lee, J., & Winkler, M. (2022). Global weak solutions to a chemotaxis-Navier-Stokes
system in $ \mathbb{R}^3 $. Discrete and Continuous Dynamical Systems,
42(11), Article 5201. https://doi.org/10.3934/dcds.2022091
bibtex: '@article{Kang_Lee_Winkler_2022, title={Global weak solutions to a chemotaxis-Navier-Stokes
system in $ \mathbb{R}^3 $}, volume={42}, DOI={10.3934/dcds.2022091},
number={115201}, journal={Discrete and Continuous Dynamical Systems}, publisher={American
Institute of Mathematical Sciences (AIMS)}, author={Kang, Kyungkeun and Lee, Jihoon
and Winkler, Michael}, year={2022} }'
chicago: Kang, Kyungkeun, Jihoon Lee, and Michael Winkler. “Global Weak Solutions
to a Chemotaxis-Navier-Stokes System in $ \mathbb{R}^3 $.” Discrete and Continuous
Dynamical Systems 42, no. 11 (2022). https://doi.org/10.3934/dcds.2022091.
ieee: 'K. Kang, J. Lee, and M. Winkler, “Global weak solutions to a chemotaxis-Navier-Stokes
system in $ \mathbb{R}^3 $,” Discrete and Continuous Dynamical Systems,
vol. 42, no. 11, Art. no. 5201, 2022, doi: 10.3934/dcds.2022091.'
mla: Kang, Kyungkeun, et al. “Global Weak Solutions to a Chemotaxis-Navier-Stokes
System in $ \mathbb{R}^3 $.” Discrete and Continuous Dynamical Systems,
vol. 42, no. 11, 5201, American Institute of Mathematical Sciences (AIMS), 2022,
doi:10.3934/dcds.2022091.
short: K. Kang, J. Lee, M. Winkler, Discrete and Continuous Dynamical Systems 42
(2022).
date_created: 2025-12-18T19:22:04Z
date_updated: 2025-12-18T20:08:21Z
doi: 10.3934/dcds.2022091
intvolume: ' 42'
issue: '11'
language:
- iso: eng
publication: Discrete and Continuous Dynamical Systems
publication_identifier:
issn:
- 1078-0947
- 1553-5231
publication_status: published
publisher: American Institute of Mathematical Sciences (AIMS)
status: public
title: Global weak solutions to a chemotaxis-Navier-Stokes system in $ \mathbb{R}^3
$
type: journal_article
user_id: '31496'
volume: 42
year: '2022'
...
---
_id: '63290'
abstract:
- lang: eng
text: This paper proposes a review focused on exotic chemotaxis and cross-diffusion
models in complex environments. The term exotic is used to denote the dynamics
of models interacting with a time-evolving external system and, specifically,
models derived with the aim of describing the dynamics of living systems. The
presentation first, considers the derivation of phenomenological models of chemotaxis
and cross-diffusion models with particular attention on nonlinear characteristics.
Then, a variety of exotic models is presented with some hints toward the derivation
of new models, by accounting for a critical analysis looking ahead to perspectives.
The second part of the paper is devoted to a survey of analytical problems concerning
the application of models to the study of real world dynamics. Finally, the focus
shifts to research perspectives within the framework of a multiscale vision, where
different paths are examined to move from the dynamics at the microscopic scale
to collective behaviors at the macroscopic scale.
author:
- first_name: N.
full_name: Bellomo, N.
last_name: Bellomo
- first_name: N.
full_name: Outada, N.
last_name: Outada
- first_name: J.
full_name: Soler, J.
last_name: Soler
- first_name: Y.
full_name: Tao, Y.
last_name: Tao
- first_name: Michael
full_name: Winkler, Michael
id: '31496'
last_name: Winkler
citation:
ama: 'Bellomo N, Outada N, Soler J, Tao Y, Winkler M. Chemotaxis and cross-diffusion
models in complex environments: Models and analytic problems toward a multiscale
vision. Mathematical Models and Methods in Applied Sciences. 2022;32(04):713-792.
doi:10.1142/s0218202522500166'
apa: 'Bellomo, N., Outada, N., Soler, J., Tao, Y., & Winkler, M. (2022). Chemotaxis
and cross-diffusion models in complex environments: Models and analytic problems
toward a multiscale vision. Mathematical Models and Methods in Applied Sciences,
32(04), 713–792. https://doi.org/10.1142/s0218202522500166'
bibtex: '@article{Bellomo_Outada_Soler_Tao_Winkler_2022, title={Chemotaxis and cross-diffusion
models in complex environments: Models and analytic problems toward a multiscale
vision}, volume={32}, DOI={10.1142/s0218202522500166},
number={04}, journal={Mathematical Models and Methods in Applied Sciences}, publisher={World
Scientific Pub Co Pte Ltd}, author={Bellomo, N. and Outada, N. and Soler, J. and
Tao, Y. and Winkler, Michael}, year={2022}, pages={713–792} }'
chicago: 'Bellomo, N., N. Outada, J. Soler, Y. Tao, and Michael Winkler. “Chemotaxis
and Cross-Diffusion Models in Complex Environments: Models and Analytic Problems
toward a Multiscale Vision.” Mathematical Models and Methods in Applied Sciences
32, no. 04 (2022): 713–92. https://doi.org/10.1142/s0218202522500166.'
ieee: 'N. Bellomo, N. Outada, J. Soler, Y. Tao, and M. Winkler, “Chemotaxis and
cross-diffusion models in complex environments: Models and analytic problems toward
a multiscale vision,” Mathematical Models and Methods in Applied Sciences,
vol. 32, no. 04, pp. 713–792, 2022, doi: 10.1142/s0218202522500166.'
mla: 'Bellomo, N., et al. “Chemotaxis and Cross-Diffusion Models in Complex Environments:
Models and Analytic Problems toward a Multiscale Vision.” Mathematical Models
and Methods in Applied Sciences, vol. 32, no. 04, World Scientific Pub Co
Pte Ltd, 2022, pp. 713–92, doi:10.1142/s0218202522500166.'
short: N. Bellomo, N. Outada, J. Soler, Y. Tao, M. Winkler, Mathematical Models
and Methods in Applied Sciences 32 (2022) 713–792.
date_created: 2025-12-18T19:20:25Z
date_updated: 2025-12-18T20:07:51Z
doi: 10.1142/s0218202522500166
intvolume: ' 32'
issue: '04'
language:
- iso: eng
page: 713-792
publication: Mathematical Models and Methods in Applied Sciences
publication_identifier:
issn:
- 0218-2025
- 1793-6314
publication_status: published
publisher: World Scientific Pub Co Pte Ltd
status: public
title: 'Chemotaxis and cross-diffusion models in complex environments: Models and
analytic problems toward a multiscale vision'
type: journal_article
user_id: '31496'
volume: 32
year: '2022'
...
---
_id: '63295'
abstract:
- lang: eng
text: AbstractWe introduce a generalized concept
of solutions for reaction–diffusion systems and prove their global existence.
The only restriction on the reaction function beyond regularity, quasipositivity
and mass control is special in that it merely controls the growth of cross-absorptive
terms. The result covers nonlinear diffusion and does not rely on an entropy estimate.
article_number: '14'
author:
- first_name: Johannes
full_name: Lankeit, Johannes
last_name: Lankeit
- first_name: Michael
full_name: Winkler, Michael
id: '31496'
last_name: Winkler
citation:
ama: Lankeit J, Winkler M. Global existence in reaction–diffusion systems with mass
control under relaxed assumptions merely referring to cross-absorptive effects.
Journal of Evolution Equations. 2022;22(1). doi:10.1007/s00028-022-00768-9
apa: Lankeit, J., & Winkler, M. (2022). Global existence in reaction–diffusion
systems with mass control under relaxed assumptions merely referring to cross-absorptive
effects. Journal of Evolution Equations, 22(1), Article 14. https://doi.org/10.1007/s00028-022-00768-9
bibtex: '@article{Lankeit_Winkler_2022, title={Global existence in reaction–diffusion
systems with mass control under relaxed assumptions merely referring to cross-absorptive
effects}, volume={22}, DOI={10.1007/s00028-022-00768-9},
number={114}, journal={Journal of Evolution Equations}, publisher={Springer Science
and Business Media LLC}, author={Lankeit, Johannes and Winkler, Michael}, year={2022}
}'
chicago: Lankeit, Johannes, and Michael Winkler. “Global Existence in Reaction–Diffusion
Systems with Mass Control under Relaxed Assumptions Merely Referring to Cross-Absorptive
Effects.” Journal of Evolution Equations 22, no. 1 (2022). https://doi.org/10.1007/s00028-022-00768-9.
ieee: 'J. Lankeit and M. Winkler, “Global existence in reaction–diffusion systems
with mass control under relaxed assumptions merely referring to cross-absorptive
effects,” Journal of Evolution Equations, vol. 22, no. 1, Art. no. 14,
2022, doi: 10.1007/s00028-022-00768-9.'
mla: Lankeit, Johannes, and Michael Winkler. “Global Existence in Reaction–Diffusion
Systems with Mass Control under Relaxed Assumptions Merely Referring to Cross-Absorptive
Effects.” Journal of Evolution Equations, vol. 22, no. 1, 14, Springer
Science and Business Media LLC, 2022, doi:10.1007/s00028-022-00768-9.
short: J. Lankeit, M. Winkler, Journal of Evolution Equations 22 (2022).
date_created: 2025-12-18T19:22:46Z
date_updated: 2025-12-18T20:08:35Z
doi: 10.1007/s00028-022-00768-9
intvolume: ' 22'
issue: '1'
language:
- iso: eng
publication: Journal of Evolution Equations
publication_identifier:
issn:
- 1424-3199
- 1424-3202
publication_status: published
publisher: Springer Science and Business Media LLC
status: public
title: Global existence in reaction–diffusion systems with mass control under relaxed
assumptions merely referring to cross-absorptive effects
type: journal_article
user_id: '31496'
volume: 22
year: '2022'
...
---
_id: '63299'
author:
- first_name: Youshan
full_name: Tao, Youshan
last_name: Tao
- first_name: Michael
full_name: Winkler, Michael
id: '31496'
last_name: Winkler
citation:
ama: Tao Y, Winkler M. Existence Theory and Qualitative Analysis for a Fully Cross-Diffusive
Predator-Prey System. SIAM Journal on Mathematical Analysis. 2022;54(4):4806-4864.
doi:10.1137/21m1449841
apa: Tao, Y., & Winkler, M. (2022). Existence Theory and Qualitative Analysis
for a Fully Cross-Diffusive Predator-Prey System. SIAM Journal on Mathematical
Analysis, 54(4), 4806–4864. https://doi.org/10.1137/21m1449841
bibtex: '@article{Tao_Winkler_2022, title={Existence Theory and Qualitative Analysis
for a Fully Cross-Diffusive Predator-Prey System}, volume={54}, DOI={10.1137/21m1449841},
number={4}, journal={SIAM Journal on Mathematical Analysis}, publisher={Society
for Industrial & Applied Mathematics (SIAM)}, author={Tao, Youshan and Winkler,
Michael}, year={2022}, pages={4806–4864} }'
chicago: 'Tao, Youshan, and Michael Winkler. “Existence Theory and Qualitative Analysis
for a Fully Cross-Diffusive Predator-Prey System.” SIAM Journal on Mathematical
Analysis 54, no. 4 (2022): 4806–64. https://doi.org/10.1137/21m1449841.'
ieee: 'Y. Tao and M. Winkler, “Existence Theory and Qualitative Analysis for a Fully
Cross-Diffusive Predator-Prey System,” SIAM Journal on Mathematical Analysis,
vol. 54, no. 4, pp. 4806–4864, 2022, doi: 10.1137/21m1449841.'
mla: Tao, Youshan, and Michael Winkler. “Existence Theory and Qualitative Analysis
for a Fully Cross-Diffusive Predator-Prey System.” SIAM Journal on Mathematical
Analysis, vol. 54, no. 4, Society for Industrial & Applied Mathematics
(SIAM), 2022, pp. 4806–64, doi:10.1137/21m1449841.
short: Y. Tao, M. Winkler, SIAM Journal on Mathematical Analysis 54 (2022) 4806–4864.
date_created: 2025-12-18T19:24:16Z
date_updated: 2025-12-18T20:09:05Z
doi: 10.1137/21m1449841
intvolume: ' 54'
issue: '4'
language:
- iso: eng
page: 4806-4864
publication: SIAM Journal on Mathematical Analysis
publication_identifier:
issn:
- 0036-1410
- 1095-7154
publication_status: published
publisher: Society for Industrial & Applied Mathematics (SIAM)
status: public
title: Existence Theory and Qualitative Analysis for a Fully Cross-Diffusive Predator-Prey
System
type: journal_article
user_id: '31496'
volume: 54
year: '2022'
...
---
_id: '63298'
author:
- first_name: Angela
full_name: Stevens, Angela
last_name: Stevens
- first_name: Michael
full_name: Winkler, Michael
id: '31496'
last_name: Winkler
citation:
ama: Stevens A, Winkler M. Taxis-driven persistent localization in a degenerate
Keller-Segel system. Communications in Partial Differential Equations.
2022;47(12):2341-2362. doi:10.1080/03605302.2022.2122836
apa: Stevens, A., & Winkler, M. (2022). Taxis-driven persistent localization
in a degenerate Keller-Segel system. Communications in Partial Differential
Equations, 47(12), 2341–2362. https://doi.org/10.1080/03605302.2022.2122836
bibtex: '@article{Stevens_Winkler_2022, title={Taxis-driven persistent localization
in a degenerate Keller-Segel system}, volume={47}, DOI={10.1080/03605302.2022.2122836},
number={12}, journal={Communications in Partial Differential Equations}, publisher={Informa
UK Limited}, author={Stevens, Angela and Winkler, Michael}, year={2022}, pages={2341–2362}
}'
chicago: 'Stevens, Angela, and Michael Winkler. “Taxis-Driven Persistent Localization
in a Degenerate Keller-Segel System.” Communications in Partial Differential
Equations 47, no. 12 (2022): 2341–62. https://doi.org/10.1080/03605302.2022.2122836.'
ieee: 'A. Stevens and M. Winkler, “Taxis-driven persistent localization in a degenerate
Keller-Segel system,” Communications in Partial Differential Equations,
vol. 47, no. 12, pp. 2341–2362, 2022, doi: 10.1080/03605302.2022.2122836.'
mla: Stevens, Angela, and Michael Winkler. “Taxis-Driven Persistent Localization
in a Degenerate Keller-Segel System.” Communications in Partial Differential
Equations, vol. 47, no. 12, Informa UK Limited, 2022, pp. 2341–62, doi:10.1080/03605302.2022.2122836.
short: A. Stevens, M. Winkler, Communications in Partial Differential Equations
47 (2022) 2341–2362.
date_created: 2025-12-18T19:23:52Z
date_updated: 2025-12-18T20:08:58Z
doi: 10.1080/03605302.2022.2122836
intvolume: ' 47'
issue: '12'
language:
- iso: eng
page: 2341-2362
publication: Communications in Partial Differential Equations
publication_identifier:
issn:
- 0360-5302
- 1532-4133
publication_status: published
publisher: Informa UK Limited
status: public
title: Taxis-driven persistent localization in a degenerate Keller-Segel system
type: journal_article
user_id: '31496'
volume: 47
year: '2022'
...
---
_id: '63266'
abstract:
- lang: eng
text: "AbstractIn a ball $$\\Omega
=B_R(0)\\subset \\mathbb {R}^n$$\r\n
\ \r\n Ω\r\n =\r\n
\ \r\n B\r\n
\ R\r\n \r\n
\ \r\n (\r\n
\ 0\r\n )\r\n
\ \r\n ⊂\r\n
\ \r\n \r\n R\r\n
\ \r\n n\r\n
\ \r\n \r\n ,
$$n\\ge 2$$\r\n \r\n
\ n\r\n ≥\r\n
\ 2\r\n \r\n ,
the chemotaxis system $$\\begin{aligned}
\\left\\{ \\begin{array}{l}u_t = \\nabla \\cdot \\big ( D(u) \\nabla u \\big )
- \\nabla \\cdot \\big ( uS(u)\\nabla v\\big ), \\\\ 0 = \\Delta v - \\mu + u,
\\qquad \\mu =\\frac{1}{|\\Omega |} \\int _\\Omega u, \\end{array} \\right. \\qquad
\\qquad (\\star ) \\end{aligned}$$\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
\ D\r\n \r\n
\ (\r\n u\r\n
\ )\r\n \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 u\r\n
\ S\r\n \r\n
\ (\r\n u\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 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 μ\r\n
\ =\r\n \r\n
\ 1\r\n \r\n
\ |\r\n Ω\r\n
\ |\r\n \r\n
\ \r\n \r\n
\ ∫\r\n Ω\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 \r\n
\ (\r\n ⋆\r\n
\ )\r\n \r\n
\ \r\n \r\n
\ \r\n \r\n \r\n
\ is considered
under no-flux boundary conditions, with a focus on nonlinearities $$S\\in
C^2([0,\\infty ))$$\r\n
\ \r\n S\r\n ∈\r\n
\ \r\n C\r\n
\ 2\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
which exhibit super-algebraically fast decay in the sense that with some $$K_S>0,
\\beta \\in [0,1)$$\r\n
\ \r\n \r\n K\r\n
\ S\r\n \r\n
\ >\r\n 0\r\n
\ ,\r\n β\r\n
\ ∈\r\n \r\n [\r\n
\ 0\r\n ,\r\n
\ 1\r\n )\r\n
\ \r\n \r\n
and $$\\xi _0>0$$\r\n \r\n
\ \r\n ξ\r\n
\ 0\r\n \r\n
\ >\r\n 0\r\n
\ \r\n ,
$$\\begin{aligned} S(\\xi
)>0 \\quad \\text{ and } \\quad S'(\\xi ) \\le -K_S\\xi ^{-\\beta } S(\\xi
) \\qquad \\text{ for } \\text{ all } \\xi \\ge \\xi _0. \\end{aligned}$$\r\n \r\n
\ \r\n \r\n \r\n
\ \r\n S\r\n
\ \r\n (\r\n
\ ξ\r\n )\r\n
\ \r\n >\r\n
\ 0\r\n \r\n
\ \r\n and\r\n
\ \r\n \r\n
\ \r\n S\r\n
\ ′\r\n \r\n
\ \r\n (\r\n
\ ξ\r\n )\r\n
\ \r\n ≤\r\n
\ -\r\n \r\n
\ K\r\n S\r\n
\ \r\n \r\n
\ ξ\r\n \r\n
\ -\r\n β\r\n
\ \r\n \r\n
\ S\r\n \r\n
\ (\r\n ξ\r\n
\ )\r\n \r\n
\ \r\n \r\n
\ for\r\n \r\n
\ \r\n all\r\n
\ \r\n ξ\r\n
\ ≥\r\n \r\n
\ ξ\r\n 0\r\n
\ \r\n .\r\n
\ \r\n \r\n
\ \r\n \r\n \r\n
\ It is, inter
alia, shown that if furthermore $$D\\in
C^2((0,\\infty ))$$\r\n
\ \r\n D\r\n ∈\r\n
\ \r\n C\r\n
\ 2\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
is positive and suitably small in relation to S by
satisfying $$\\begin{aligned}
\\frac{\\xi S(\\xi )}{D(\\xi )} \\ge K_{SD}\\xi ^\\lambda \\qquad \\text{ for
} \\text{ all } \\xi \\ge \\xi _0 \\end{aligned}$$\r\n
\ \r\n \r\n \r\n
\ \r\n \r\n
\ \r\n \r\n
\ ξ\r\n S\r\n
\ (\r\n ξ\r\n
\ )\r\n \r\n
\ \r\n D\r\n
\ (\r\n ξ\r\n
\ )\r\n \r\n
\ \r\n ≥\r\n
\ \r\n K\r\n
\ \r\n SD\r\n
\ \r\n \r\n
\ \r\n ξ\r\n
\ λ\r\n \r\n
\ \r\n \r\n
\ for\r\n \r\n
\ \r\n all\r\n
\ \r\n ξ\r\n
\ ≥\r\n \r\n
\ ξ\r\n 0\r\n
\ \r\n \r\n
\ \r\n \r\n \r\n
\ \r\n with
some $$K_{SD}>0$$\r\n \r\n
\ \r\n K\r\n
\ \r\n SD\r\n
\ \r\n \r\n >\r\n
\ 0\r\n \r\n
and $$\\lambda >\\frac{2}{n}$$\r\n \r\n
\ λ\r\n >\r\n
\ \r\n 2\r\n
\ n\r\n \r\n
\ \r\n ,
then throughout a considerably large set of initial data, ($$\\star
$$\r\n
\ ⋆\r\n )
admits global classical solutions (u, v)
fulfilling $$\\begin{aligned}
\\frac{z(t)}{C} \\le \\Vert u(\\cdot ,t)\\Vert _{L^\\infty (\\Omega )} \\le Cz(t)
\\qquad \\text{ for } \\text{ all } t>0, \\end{aligned}$$\r\n \r\n
\ \r\n \r\n \r\n
\ \r\n \r\n
\ \r\n z\r\n
\ (\r\n t\r\n
\ )\r\n \r\n
\ C\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 ∞\r\n
\ \r\n \r\n
\ (\r\n Ω\r\n
\ )\r\n \r\n
\ \r\n \r\n
\ ≤\r\n C\r\n
\ z\r\n \r\n
\ (\r\n t\r\n
\ )\r\n \r\n
\ \r\n \r\n
\ for\r\n \r\n
\ \r\n all\r\n
\ \r\n t\r\n
\ >\r\n 0\r\n
\ ,\r\n \r\n
\ \r\n \r\n \r\n
\ \r\n with
some $$C=C^{(u,v)}\\ge
1$$\r\n
\ \r\n C\r\n =\r\n
\ \r\n C\r\n
\ \r\n (\r\n
\ u\r\n ,\r\n
\ v\r\n )\r\n
\ \r\n \r\n ≥\r\n
\ 1\r\n \r\n ,
where z denotes the solution of $$\\begin{aligned}
\\left\\{ \\begin{array}{l}z'(t) = z^2(t) \\cdot S\\big ( z(t)\\big ), \\qquad
t>0, \\\\ z(0)=\\xi _0, \\end{array} \\right. \\end{aligned}$$\r\n \r\n
\ \r\n \r\n \r\n
\ \r\n \r\n
\ \r\n \r\n
\ \r\n \r\n
\ \r\n z\r\n
\ ′\r\n \r\n
\ \r\n (\r\n
\ t\r\n )\r\n
\ \r\n =\r\n
\ \r\n z\r\n
\ 2\r\n \r\n
\ \r\n (\r\n
\ t\r\n )\r\n
\ \r\n ·\r\n
\ S\r\n \r\n
\ (\r\n \r\n
\ z\r\n \r\n
\ (\r\n t\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
\ \r\n \r\n
\ z\r\n \r\n
\ (\r\n 0\r\n
\ )\r\n \r\n
\ =\r\n \r\n
\ ξ\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 which
is seen to exist globally, and to satisfy $$z(t)\\rightarrow
+\\infty $$\r\n
\ \r\n z\r\n (\r\n
\ t\r\n )\r\n
\ →\r\n +\r\n
\ ∞\r\n \r\n
as $$t\\rightarrow \\infty
$$\r\n
\ \r\n t\r\n →\r\n
\ ∞\r\n \r\n .
As particular examples, exponentially and doubly exponentially decaying S
are found to imply corresponding infinite-time blow-up properties in ($$\\star
$$\r\n
\ ⋆\r\n )
at logarithmic and doubly logarithmic rates, respectively."
author:
- first_name: Michael
full_name: Winkler, Michael
id: '31496'
last_name: Winkler
citation:
ama: Winkler M. Slow Grow-up in a Quasilinear Keller–Segel System. Journal of
Dynamics and Differential Equations. 2022;36(2):1677-1702. doi:10.1007/s10884-022-10167-w
apa: Winkler, M. (2022). Slow Grow-up in a Quasilinear Keller–Segel System. Journal
of Dynamics and Differential Equations, 36(2), 1677–1702. https://doi.org/10.1007/s10884-022-10167-w
bibtex: '@article{Winkler_2022, title={Slow Grow-up in a Quasilinear Keller–Segel
System}, volume={36}, DOI={10.1007/s10884-022-10167-w},
number={2}, journal={Journal of Dynamics and Differential Equations}, publisher={Springer
Science and Business Media LLC}, author={Winkler, Michael}, year={2022}, pages={1677–1702}
}'
chicago: 'Winkler, Michael. “Slow Grow-up in a Quasilinear Keller–Segel System.”
Journal of Dynamics and Differential Equations 36, no. 2 (2022): 1677–1702.
https://doi.org/10.1007/s10884-022-10167-w.'
ieee: 'M. Winkler, “Slow Grow-up in a Quasilinear Keller–Segel System,” Journal
of Dynamics and Differential Equations, vol. 36, no. 2, pp. 1677–1702, 2022,
doi: 10.1007/s10884-022-10167-w.'
mla: Winkler, Michael. “Slow Grow-up in a Quasilinear Keller–Segel System.” Journal
of Dynamics and Differential Equations, vol. 36, no. 2, Springer Science and
Business Media LLC, 2022, pp. 1677–702, doi:10.1007/s10884-022-10167-w.
short: M. Winkler, Journal of Dynamics and Differential Equations 36 (2022) 1677–1702.
date_created: 2025-12-18T19:10:32Z
date_updated: 2025-12-18T20:10:14Z
doi: 10.1007/s10884-022-10167-w
intvolume: ' 36'
issue: '2'
language:
- iso: eng
page: 1677-1702
publication: Journal of Dynamics and Differential Equations
publication_identifier:
issn:
- 1040-7294
- 1572-9222
publication_status: published
publisher: Springer Science and Business Media LLC
status: public
title: Slow Grow-up in a Quasilinear Keller–Segel System
type: journal_article
user_id: '31496'
volume: 36
year: '2022'
...
---
_id: '63272'
author:
- first_name: Youshan
full_name: Tao, Youshan
last_name: Tao
- first_name: Michael
full_name: Winkler, Michael
id: '31496'
last_name: Winkler
citation:
ama: Tao Y, Winkler M. Global solutions to a Keller-Segel-consumption system involving
singularly signal-dependent motilities in domains of arbitrary dimension. Journal
of Differential Equations. 2022;343:390-418. doi:10.1016/j.jde.2022.10.022
apa: Tao, Y., & Winkler, M. (2022). Global solutions to a Keller-Segel-consumption
system involving singularly signal-dependent motilities in domains of arbitrary
dimension. Journal of Differential Equations, 343, 390–418. https://doi.org/10.1016/j.jde.2022.10.022
bibtex: '@article{Tao_Winkler_2022, title={Global solutions to a Keller-Segel-consumption
system involving singularly signal-dependent motilities in domains of arbitrary
dimension}, volume={343}, DOI={10.1016/j.jde.2022.10.022},
journal={Journal of Differential Equations}, publisher={Elsevier BV}, author={Tao,
Youshan and Winkler, Michael}, year={2022}, pages={390–418} }'
chicago: 'Tao, Youshan, and Michael Winkler. “Global Solutions to a Keller-Segel-Consumption
System Involving Singularly Signal-Dependent Motilities in Domains of Arbitrary
Dimension.” Journal of Differential Equations 343 (2022): 390–418. https://doi.org/10.1016/j.jde.2022.10.022.'
ieee: 'Y. Tao and M. Winkler, “Global solutions to a Keller-Segel-consumption system
involving singularly signal-dependent motilities in domains of arbitrary dimension,”
Journal of Differential Equations, vol. 343, pp. 390–418, 2022, doi: 10.1016/j.jde.2022.10.022.'
mla: Tao, Youshan, and Michael Winkler. “Global Solutions to a Keller-Segel-Consumption
System Involving Singularly Signal-Dependent Motilities in Domains of Arbitrary
Dimension.” Journal of Differential Equations, vol. 343, Elsevier BV, 2022,
pp. 390–418, doi:10.1016/j.jde.2022.10.022.
short: Y. Tao, M. Winkler, Journal of Differential Equations 343 (2022) 390–418.
date_created: 2025-12-18T19:13:04Z
date_updated: 2025-12-18T20:11:02Z
doi: 10.1016/j.jde.2022.10.022
intvolume: ' 343'
language:
- iso: eng
page: 390-418
publication: Journal of Differential Equations
publication_identifier:
issn:
- 0022-0396
publication_status: published
publisher: Elsevier BV
status: public
title: Global solutions to a Keller-Segel-consumption system involving singularly
signal-dependent motilities in domains of arbitrary dimension
type: journal_article
user_id: '31496'
volume: 343
year: '2022'
...
---
_id: '63268'
article_number: '113153'
author:
- first_name: Laurent
full_name: Desvillettes, Laurent
last_name: Desvillettes
- first_name: Philippe
full_name: Laurençot, Philippe
last_name: Laurençot
- first_name: Ariane
full_name: Trescases, Ariane
last_name: Trescases
- first_name: Michael
full_name: Winkler, Michael
id: '31496'
last_name: Winkler
citation:
ama: Desvillettes L, Laurençot P, Trescases A, Winkler M. Weak solutions to triangular
cross diffusion systems modeling chemotaxis with local sensing. Nonlinear Analysis.
2022;226. doi:10.1016/j.na.2022.113153
apa: Desvillettes, L., Laurençot, P., Trescases, A., & Winkler, M. (2022). Weak
solutions to triangular cross diffusion systems modeling chemotaxis with local
sensing. Nonlinear Analysis, 226, Article 113153. https://doi.org/10.1016/j.na.2022.113153
bibtex: '@article{Desvillettes_Laurençot_Trescases_Winkler_2022, title={Weak solutions
to triangular cross diffusion systems modeling chemotaxis with local sensing},
volume={226}, DOI={10.1016/j.na.2022.113153},
number={113153}, journal={Nonlinear Analysis}, publisher={Elsevier BV}, author={Desvillettes,
Laurent and Laurençot, Philippe and Trescases, Ariane and Winkler, Michael}, year={2022}
}'
chicago: Desvillettes, Laurent, Philippe Laurençot, Ariane Trescases, and Michael
Winkler. “Weak Solutions to Triangular Cross Diffusion Systems Modeling Chemotaxis
with Local Sensing.” Nonlinear Analysis 226 (2022). https://doi.org/10.1016/j.na.2022.113153.
ieee: 'L. Desvillettes, P. Laurençot, A. Trescases, and M. Winkler, “Weak solutions
to triangular cross diffusion systems modeling chemotaxis with local sensing,”
Nonlinear Analysis, vol. 226, Art. no. 113153, 2022, doi: 10.1016/j.na.2022.113153.'
mla: Desvillettes, Laurent, et al. “Weak Solutions to Triangular Cross Diffusion
Systems Modeling Chemotaxis with Local Sensing.” Nonlinear Analysis, vol.
226, 113153, Elsevier BV, 2022, doi:10.1016/j.na.2022.113153.
short: L. Desvillettes, P. Laurençot, A. Trescases, M. Winkler, Nonlinear Analysis
226 (2022).
date_created: 2025-12-18T19:11:16Z
date_updated: 2025-12-18T20:10:32Z
doi: 10.1016/j.na.2022.113153
intvolume: ' 226'
language:
- iso: eng
publication: Nonlinear Analysis
publication_identifier:
issn:
- 0362-546X
publication_status: published
publisher: Elsevier BV
status: public
title: Weak solutions to triangular cross diffusion systems modeling chemotaxis with
local sensing
type: journal_article
user_id: '31496'
volume: 226
year: '2022'
...
---
_id: '63278'
abstract:
- lang: eng
text: "Abstract\r\n The Neumann problem
for (0.1)$$ \\begin{align}& V_t = \\Delta V-aV+f(x,t) \\end{align}$$is considered
in bounded domains $\\Omega \\subset {\\mathbb {R}}^n$ with smooth boundary, where
$n\\ge 1$ and $a\\in {\\mathbb {R}}$. By means of a variational approach, a statement
on boundedness of the quantities $$ \\begin{eqnarray*} \\sup_{t\\in (0,T)} \\int_\\Omega
\\big|\\nabla V(\\cdot,t)\\big|^p L^{\\frac{n+p}{n+2}} \\Big( \\big|\\nabla V(\\cdot,t)\\big|
\\Big) \\end{eqnarray*}$$in dependence on the expressions (0.2)$$ \\begin{align}&
\\sup_{t\\in (0,T-\\tau)} \\int_t^{t+\\tau} \\int_\\Omega |f|^{\\frac{(n+2)p}{n+p}}
L\\big( |f|\\big) \\end{align}$$is derived for $p\\ge 2$, $\\tau>0$, and
$T\\ge 2\\tau $, provided that $L\\in C^0([0,\\infty ))$ is positive, strictly
increasing, unbounded, and slowly growing in the sense that $\\limsup _{s\\to
\\infty } \\frac {L(s^{\\lambda _0})}{L(s)} <\\infty $ for some $\\lambda
_0>1$. In the particular case when $p=n\\ge 2$, an additional condition
on growth of $L$, particularly satisfied by $L(\\xi ):=\\ln ^\\alpha (\\xi +b)$
whenever $b>0$ and $\\alpha>\\frac {(n+2)(n-1)}{2n}$, is identified
as sufficient to ensure that as a consequence of the above, bounds for theintegrals
in (0.2) even imply estimates for the spatio-temporal modulus of continuity of
solutions to (0.1). A subsequent application to the Keller–Segel system $$ \\begin{eqnarray*}
\\left\\{ \\begin{array}{l} u_t = \\nabla \\cdot \\big( D(v)\\nabla u\\big) -
\\nabla \\cdot \\big( uS(v)\\nabla v\\big) + ru - \\mu u^2, \\\\[1mm] v_t = \\Delta
v-v+u, \\end{array} \\right. \\end{eqnarray*}$$shows that when $n=2$, $r\\in {\\mathbb
{R}}$, $0<D\\in C^2([0,\\infty ))$, and $S\\in C^2([0,\\infty )) \\cap
W^{1,\\infty }((0,\\infty ))$ and thus especially in the presence of arbitrarily
strong diffusion degeneracies implied by rapid decay of $D$, any choice of $\\mu>0$
excludes blowup in the sense that for all suitably regular nonnegative initial
data, an associated initial-boundary value problem admits a global bounded classical
solution."
author:
- first_name: Michael
full_name: Winkler, Michael
id: '31496'
last_name: Winkler
citation:
ama: Winkler M. A Result on Parabolic Gradient Regularity in Orlicz Spaces and Application
to Absorption-Induced Blow-Up Prevention in a Keller–Segel-Type Cross-Diffusion
System. International Mathematics Research Notices. 2022;2023(19):16336-16393.
doi:10.1093/imrn/rnac286
apa: Winkler, M. (2022). A Result on Parabolic Gradient Regularity in Orlicz Spaces
and Application to Absorption-Induced Blow-Up Prevention in a Keller–Segel-Type
Cross-Diffusion System. International Mathematics Research Notices, 2023(19),
16336–16393. https://doi.org/10.1093/imrn/rnac286
bibtex: '@article{Winkler_2022, title={A Result on Parabolic Gradient Regularity
in Orlicz Spaces and Application to Absorption-Induced Blow-Up Prevention in a
Keller–Segel-Type Cross-Diffusion System}, volume={2023}, DOI={10.1093/imrn/rnac286},
number={19}, journal={International Mathematics Research Notices}, publisher={Oxford
University Press (OUP)}, author={Winkler, Michael}, year={2022}, pages={16336–16393}
}'
chicago: 'Winkler, Michael. “A Result on Parabolic Gradient Regularity in Orlicz
Spaces and Application to Absorption-Induced Blow-Up Prevention in a Keller–Segel-Type
Cross-Diffusion System.” International Mathematics Research Notices 2023,
no. 19 (2022): 16336–93. https://doi.org/10.1093/imrn/rnac286.'
ieee: 'M. Winkler, “A Result on Parabolic Gradient Regularity in Orlicz Spaces and
Application to Absorption-Induced Blow-Up Prevention in a Keller–Segel-Type Cross-Diffusion
System,” International Mathematics Research Notices, vol. 2023, no. 19,
pp. 16336–16393, 2022, doi: 10.1093/imrn/rnac286.'
mla: Winkler, Michael. “A Result on Parabolic Gradient Regularity in Orlicz Spaces
and Application to Absorption-Induced Blow-Up Prevention in a Keller–Segel-Type
Cross-Diffusion System.” International Mathematics Research Notices, vol.
2023, no. 19, Oxford University Press (OUP), 2022, pp. 16336–93, doi:10.1093/imrn/rnac286.
short: M. Winkler, International Mathematics Research Notices 2023 (2022) 16336–16393.
date_created: 2025-12-18T19:15:52Z
date_updated: 2025-12-18T20:11:43Z
doi: 10.1093/imrn/rnac286
intvolume: ' 2023'
issue: '19'
language:
- iso: eng
page: 16336-16393
publication: International Mathematics Research Notices
publication_identifier:
issn:
- 1073-7928
- 1687-0247
publication_status: published
publisher: Oxford University Press (OUP)
status: public
title: A Result on Parabolic Gradient Regularity in Orlicz Spaces and Application
to Absorption-Induced Blow-Up Prevention in a Keller–Segel-Type Cross-Diffusion
System
type: journal_article
user_id: '31496'
volume: 2023
year: '2022'
...
---
_id: '63279'
abstract:
- lang: eng
text: "\r\n In a smoothly bounded convex domain\r\n \r\n
\ \\Omega \\subset \\mathbb{R}^3\r\n
\ \r\n , we consider
the chemotaxis-Navier–Stokes model\r\n \r\n \r\n
\ \r\n \\begin{cases}
n_t + u\\cdot\\nabla n = \\Delta n - \\nabla \\cdot (n\\nabla c), & x\\in
\\Omega, \\, t>0, \\\\ c_t + u\\cdot\\nabla c = \\Delta c -nc, & x\\in
\\Omega, \\, t>0, \\\\ u_t + (u\\cdot\\nabla) u = \\Delta u + \\nabla P + n\\nabla\\Phi,
\\quad \\nabla\\cdot u=0, & x\\in \\Omega, \\, t>0, \\end{cases} \\quad
(\\star)\r\n \r\n \r\n
\ \r\n proposed by Goldstein et al.
to describe pattern formation in populations of aerobic bacteria interacting with
their liquid environment via transport and buoyancy. Known results have asserted
that under appropriate regularity assumptions on\r\n \r\n
\ \\Phi\r\n \r\n
\ and the initial data, a corresponding no-flux/no-flux/Dirichlet
initial-boundary value problem is globally solvable in a framework of so-called
weak energy solutions, and that any such solution eventually becomes smooth and
classical.\r\n \r\n \r\n Going
beyond this, the present work focuses on the possible extent of unboundedness
phenomena also on short timescales, and hence investigates in more detail the
set of times in\r\n \r\n (0,\\infty)\r\n
\ \r\n at which solutions
may develop singularities. The main results in this direction reveal the existence
of a global weak energy solution which coincides with a smooth function throughout\r\n
\ \r\n \\overline{\\Omega}\\times
E\r\n \r\n ,
where\r\n \r\n E\r\n
\ \r\n denotes a countable
union of open intervals which is such that\r\n \r\n
\ |(0,\\infty)\\setminus E|=0\r\n
\ \r\n . In particular,
this indicates that a similar feature of the unperturbed Navie–Stokes equations,
known as Leray’s structure theorem, persists even in the presence of the coupling
to the attractive and hence potentially destabilizing cross-diffusive mechanism
in the full system (\r\n \r\n \\star\r\n
\ \r\n ).\r\n "
author:
- first_name: Michael
full_name: Winkler, Michael
id: '31496'
last_name: Winkler
citation:
ama: Winkler M. Does Leray’s structure theorem withstand buoyancy-driven chemotaxis-fluid
interaction? Journal of the European Mathematical Society. 2022;25(4):1423-1456.
doi:10.4171/jems/1226
apa: Winkler, M. (2022). Does Leray’s structure theorem withstand buoyancy-driven
chemotaxis-fluid interaction? Journal of the European Mathematical Society,
25(4), 1423–1456. https://doi.org/10.4171/jems/1226
bibtex: '@article{Winkler_2022, title={Does Leray’s structure theorem withstand
buoyancy-driven chemotaxis-fluid interaction?}, volume={25}, DOI={10.4171/jems/1226},
number={4}, journal={Journal of the European Mathematical Society}, publisher={European
Mathematical Society - EMS - Publishing House GmbH}, author={Winkler, Michael},
year={2022}, pages={1423–1456} }'
chicago: 'Winkler, Michael. “Does Leray’s Structure Theorem Withstand Buoyancy-Driven
Chemotaxis-Fluid Interaction?” Journal of the European Mathematical Society
25, no. 4 (2022): 1423–56. https://doi.org/10.4171/jems/1226.'
ieee: 'M. Winkler, “Does Leray’s structure theorem withstand buoyancy-driven chemotaxis-fluid
interaction?,” Journal of the European Mathematical Society, vol. 25, no.
4, pp. 1423–1456, 2022, doi: 10.4171/jems/1226.'
mla: Winkler, Michael. “Does Leray’s Structure Theorem Withstand Buoyancy-Driven
Chemotaxis-Fluid Interaction?” Journal of the European Mathematical Society,
vol. 25, no. 4, European Mathematical Society - EMS - Publishing House GmbH, 2022,
pp. 1423–56, doi:10.4171/jems/1226.
short: M. Winkler, Journal of the European Mathematical Society 25 (2022) 1423–1456.
date_created: 2025-12-18T19:16:13Z
date_updated: 2025-12-18T20:11:51Z
doi: 10.4171/jems/1226
intvolume: ' 25'
issue: '4'
language:
- iso: eng
page: 1423-1456
publication: Journal of the European Mathematical Society
publication_identifier:
issn:
- 1435-9855
- 1435-9863
publication_status: published
publisher: European Mathematical Society - EMS - Publishing House GmbH
status: public
title: Does Leray’s structure theorem withstand buoyancy-driven chemotaxis-fluid interaction?
type: journal_article
user_id: '31496'
volume: 25
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: '63282'
abstract:
- lang: eng
text: ' The chemotaxis system [Formula: see text] is considered in a ball
[Formula: see text], [Formula: see text], where the positive function [Formula:
see text] reflects suitably weak diffusion by satisfying [Formula: see text] for
some [Formula: see text]. It is shown that whenever [Formula: see text] is positive
and satisfies [Formula: see text] as [Formula: see text], one can find a suitably
regular nonlinearity [Formula: see text] with the property that at each sufficiently
large mass level [Formula: see text] there exists a globally defined radially
symmetric classical solution to a Neumann-type boundary value problem for (⋆)
which satisfies [Formula: see text] '
article_number: '2250062'
author:
- first_name: Michael
full_name: Winkler, Michael
id: '31496'
last_name: Winkler
citation:
ama: Winkler M. Arbitrarily fast grow-up rates in quasilinear Keller–Segel systems.
Communications in Contemporary Mathematics. 2022;25(10). doi:10.1142/s0219199722500626
apa: Winkler, M. (2022). Arbitrarily fast grow-up rates in quasilinear Keller–Segel
systems. Communications in Contemporary Mathematics, 25(10), Article
2250062. https://doi.org/10.1142/s0219199722500626
bibtex: '@article{Winkler_2022, title={Arbitrarily fast grow-up rates in quasilinear
Keller–Segel systems}, volume={25}, DOI={10.1142/s0219199722500626},
number={102250062}, journal={Communications in Contemporary Mathematics}, publisher={World
Scientific Pub Co Pte Ltd}, author={Winkler, Michael}, year={2022} }'
chicago: Winkler, Michael. “Arbitrarily Fast Grow-up Rates in Quasilinear Keller–Segel
Systems.” Communications in Contemporary Mathematics 25, no. 10 (2022).
https://doi.org/10.1142/s0219199722500626.
ieee: 'M. Winkler, “Arbitrarily fast grow-up rates in quasilinear Keller–Segel systems,”
Communications in Contemporary Mathematics, vol. 25, no. 10, Art. no. 2250062,
2022, doi: 10.1142/s0219199722500626.'
mla: Winkler, Michael. “Arbitrarily Fast Grow-up Rates in Quasilinear Keller–Segel
Systems.” Communications in Contemporary Mathematics, vol. 25, no. 10,
2250062, World Scientific Pub Co Pte Ltd, 2022, doi:10.1142/s0219199722500626.
short: M. Winkler, Communications in Contemporary Mathematics 25 (2022).
date_created: 2025-12-18T19:17:23Z
date_updated: 2025-12-18T20:12:13Z
doi: 10.1142/s0219199722500626
intvolume: ' 25'
issue: '10'
language:
- iso: eng
publication: Communications in Contemporary Mathematics
publication_identifier:
issn:
- 0219-1997
- 1793-6683
publication_status: published
publisher: World Scientific Pub Co Pte Ltd
status: public
title: Arbitrarily fast grow-up rates in quasilinear Keller–Segel systems
type: journal_article
user_id: '31496'
volume: 25
year: '2022'
...