Attractiveness of Constant States in Logistic-Type Keller–Segel Systems Involving Subquadratic Growth Restrictions - Winkler, Michael (Advanced nonlinear studies, 2020)
The chemotaxis-growth system is considered under homogeneous Neumann boundary conditions in smoothly bounded domains , . For any choice of , 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 , under an appropriate smallness assumption on χ, any such solution at least asymptotically exhibits relaxation by approaching the nontrivial spatially homogeneous steady state in the large time limit.