Eventual smoothness and stabilization of global weak solutions in parabolic–elliptic chemotaxis systems with logarithmic sensitivity

A parabolic–elliptic chemotaxis system with non-negative chemotactic sensitivity χ∕v and positive chemical diffusion coefficient η is considered in a smooth bounded domain Ω⊂R ^N , N≥2 under homogeneous Neumann boundary conditions. We show the existence of at least one global weak solution for χ<χ _N , N≥3, where χ _N ≔[Formula presented]. Moreover, under further assumptions of Ω and η, we prove that the constructed solution becomes smooth and stabilizes to a constant steady state after some waiting time if N=3,4. The stabilization of a global bounded solution, and the non-existence of non-constant steady states are also discussed in general dimensions.