# Attractiveness of Constant States in Logistic-Type Keller–Segel Systems Involving Subquadratic Growth Restrictions

Authors
• 1 Universität Paderborn, Warburger Str. 100, 33098 , (Germany)
Type
Published Article
Journal
Publisher
De Gruyter
Publication Date
Sep 03, 2020
Volume
20
Issue
4
Pages
795–817
Identifiers
DOI: 10.1515/ans-2020-2107
Source
De Gruyter
Keywords
The chemotaxis-growth system ($\star$){ut=D⁢Δ⁢u-χ⁢∇⋅(u⁢∇⁡v)+ρ⁢u-μ⁢uα,vt=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⊂ℝn{\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≥2-2n{\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 ((ρμ)1α-1,λκ⁢(ρμ)1α-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.