Eventual smoothness and stabilization of global weak solutions in parabolic–elliptic chemotaxis systems with logarithmic sensitivity. (October 2019)

• Record Type: Journal Article
• Authors: Ahn, Jaewook; Kang, Kyungkeun; Lee, Jihoon
• Is Part Of: Nonlinear analysis. Volume 49(2019)
• Page Start: 312
• Page End: 330
• Publication Date: 2019-10
• Subjects: Chemotaxis -- Eventual regularity -- Large time behavior -- Logarithmic sensitivity; Nonlinear functional analysis -- Periodicals; Analyse fonctionnelle non linéaire -- Périodiques; Nonlinear functional analysis; Periodicals; 515.7248
• DOI: 10.1016/j.nonrwa.2019.03.012
• Languages: English
• ISSNs: 1468-1218

Abstract

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.