@article{63264,
abstract = {{Abstract
In a smoothly bounded convex domain
Ω
⊂
R
n
${\Omega}\subset {\mathbb{R}}^{n}$
with n ≥ 1, a no-flux initial-boundary value problem for
u
t
=
Δ
u
ϕ
(
v
)
,
v
t
=
Δ
v
−
u
v
,
$$\begin{cases}_{t}={\Delta}\left(u\phi \left(v\right)\right),\quad \hfill \\ {v}_{t}={\Delta}v-uv,\quad \hfill \end{cases}$$
is considered under the assumption that near the origin, the function ϕ suitably generalizes the prototype given by
ϕ
(
ξ
)
=
ξ
α
,
ξ
∈
[
0
,
ξ
0
]
.
$$\phi \left(\xi \right)={\xi }^{\alpha },\qquad \xi \in \left[0,{\xi }_{0}\right].$$
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
C
(
T
)
≔
ess sup
t
∈
(
0
,
T
)
∫
Ω
u
(
⋅
,
t
)
ln
u
(
⋅
,
t
)
<
∞
for all
T
>
0
,
$$C\left(T\right){:=}\underset{t\in \left(0,T\right)}{\text{ess\,sup}}{\int }_{{\Omega}}u\left(\cdot ,t\right)\mathrm{ln}u\left(\cdot ,t\right){< }\infty \qquad \text{for\,all\,}T{ >}0,$$
with sup
T>0
C(T) < ∞ if α ∈ [1, 2].}},
author = {{Winkler, Michael}},
issn = {{2169-0375}},
journal = {{Advanced Nonlinear Studies}},
number = {{3}},
pages = {{592--615}},
publisher = {{Walter de Gruyter GmbH}},
title = {{{A degenerate migration-consumption model in domains of arbitrary dimension}}},
doi = {{10.1515/ans-2023-0131}},
volume = {{24}},
year = {{2024}},
}
@article{63310,
abstract = {{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 = {{Winkler, Michael}},
issn = {{2169-0375}},
journal = {{Advanced Nonlinear Studies}},
number = {{1}},
pages = {{88--117}},
publisher = {{Walter de Gruyter GmbH}},
title = {{{Chemotaxis-Stokes interaction with very weak diffusion enhancement: Blow-up exclusion via detection of absorption-induced entropy structures involving multiplicative couplings}}},
doi = {{10.1515/ans-2022-0004}},
volume = {{22}},
year = {{2022}},
}
@article{63340,
abstract = {{Abstract
The chemotaxis-growth system
($\star$)
{
u
t
=
D
Δ
u
-
χ
∇
⋅
(
u
∇
v
)
+
ρ
u
-
μ
u
α
,
v
t
=
d
Δ
v
-
κ
v
+
λ
u
{}\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.
is considered under homogeneous Neumann boundary conditions in smoothly bounded domains
Ω
⊂
ℝ
n
{\Omega\subset\mathbb{R}^{n}}
,
n
≥
1
{n\geq 1}
. For any choice of
α
>
1
{\alpha>1}
, 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
α
≥
2
-
2
n
{\alpha\geq 2-\frac{2}{n}}
, under an appropriate smallness assumption on χ, any such solution at least asymptotically exhibits relaxation by approaching the nontrivial spatially homogeneous steady state
(
(
ρ
μ
)
1
α
-
1
,
λ
κ
(
ρ
μ
)
1
α
-
1
)
{\bigl{(}\bigl{(}\frac{\rho}{\mu}\bigr{)}^{\frac{1}{\alpha-1}},\frac{\lambda}{% \kappa}\bigl{(}\frac{\rho}{\mu}\bigr{)}^{\frac{1}{\alpha-1}}\bigr{)}}
in the large time limit.}},
author = {{Winkler, Michael}},
issn = {{1536-1365}},
journal = {{Advanced Nonlinear Studies}},
number = {{4}},
pages = {{795--817}},
publisher = {{Walter de Gruyter GmbH}},
title = {{{Attractiveness of Constant States in Logistic-Type Keller–Segel Systems Involving Subquadratic Growth Restrictions}}},
doi = {{10.1515/ans-2020-2107}},
volume = {{20}},
year = {{2020}},
}