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

The chemotaxis-growth system(⋆)\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 \Omega\subset\mathbb{R}^{n}, n\geq 1. For any choice of \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 (⋆), the present work shows that, whenever \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 \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.