Large time behavior to a chemotaxis–consumption model with singular sensitivity and logistic source

This paper studies the chemotaxis–consumption model with singular sensitivity and logistic sourceu t = △ u − χ ∇ ·f ( u ) v ∇ v + κ u − μ u 2x ∈ Ω , t > 0 ,v t = ϵ △ v − g ( u ) v , x ∈ Ω , t > 0 ,under homogenous Neumann boundary condition in a smooth bounded domain Ω ⊂ ℝ n ( n ≥ 2 ) , with positive parameters χ ,  ϵ ,  μ , and f ( u ) ≃  u α , g ( u ) ≃  u β for α ∈ ℝ and β  > 0 . It is shown that the system possesses a global classical solution when α  < 1 and 0 < β < 4 n + 2 . Furthermore, the asymptotic behavior of solutions is determined for 0 <  α  < 1 and 0 <  β  <  α in a two‐dimensional setting, provided a sufficiently large μ .