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

In this paper, we study the following chemotaxis system with singular sensitivity and logistic sourceu 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 + 4and r , χ > 0 satisfyingχ 2 < min2 r + r 2k , 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 tor μ1 / ( k − 1 ) ,r μ1 / ( k − 1 )as t → ∞ if μ is suitably large.