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

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.