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

Abstract 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 ≔ 4 + N 2 4 + N 2 − 4 . 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.