@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}},
}