Large time behavior of solutions to a chemotaxis system with singular sensitivity and logistic source

Abstract

In this paper, we study the following chemotaxis system with singular sensitivity and logistic source trueright85.0pt{ut=Δu−χ∇·0trueuv∇v+ru−μuk,x∈Ω,t>0,vt=Δv−v+u,x∈Ω,t>0,in a smooth bounded convex domain Ω⊂double-struckRn (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 χ2<min2()r+r2k,4kfalse(k−1false)false(k−2false) (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 ()()rμ1/(k−1),()rμ1/(k−1) as t→∞ if μ is suitably large.