Large time behavior of solutions to a chemotaxis system with singular sensitivity and logistic source. Issue 7 (23rd April 2021)

• Record Type: Journal Article
• Authors: Li, Jiaqin; Li, Zhongping
• Is Part Of: Mathematische Nachrichten. Volume 294:Issue 7(2021)
• Page Start: 1374
• Page End: 1383
• Publication Date: 2021-04-23
• Subjects: chemotaxis -- large time behavior -- logistic source -- singular sensitivity; Mathematics -- Periodicals; 510.5
• DOI: 10.1002/mana.201900302
• Languages: English
• ISSNs: 0025-584X

Abstract

Abstract: In this paper, we study the following chemotaxis system with singular sensitivity and logistic source u t = Δ u − χ ∇ · u v ∇ v + r u − μ u k, x ∈ Ω, t > 0, v t = Δ v − v + u, x ∈ Ω, t > 0, in a smooth bounded convex domain Ω ⊂ R n ( n ≥ 2 ) with the non‐flux boundary, where χ, r, μ > 0, k > 2 . The boundedness of solutions has been proved in the case that n ≥ 2, k > 3 ( n + 2 ) n + 4 and r, χ > 0 satisfying χ 2 < min 2 r + r 2 k, 4 k ( k − 1 ) ( k − 2 ) (Zhao and Zheng, 2019). This paper mainly aims to give the large time behavior of bounded solutions and prove that the global classical solution will exponentially converge to r μ 1 / ( k − 1 ), r μ 1 / ( k − 1 ) as t → ∞ if μ is suitably large.