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

The chemotaxis-growth system {u(t) = D Delta u - chi del . (u del v) + rho u - mu u(alpha), (*) v(t) = d Delta v - kappa v + lambda u is considered under homogeneous Neumann boundary conditions in smoothly bounded domains Omega subset of R-n, n >= 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 >= 2 -2/n under an appropriate smallness assumption on chi, any such solution at least asymptotically exhibits relaxation by approaching the nontrivial spatially homogeneous steady state ((rho/mu) 1/alpha-1, lambda/kappa (rho/mu) 1/alpha-1) in the large time limit.