Asymptotic behavior of solutions to two-dimensional chemotaxis system with logistic source and singular sensitivity

In this paper we study the asymptotic behavior of solutions to the parabolic-elliptic chemotaxis system with singular sensitivity and logistic source: ut=Δu−χ∇⋅(uv∇v)+ru−μu2, 0=Δv−v+u, subject to homogeneous Neumann boundary conditions in a bounded domain Ω⊂R2 with smooth boundary, where χ,μ>0 and r∈R. It is proved that for any nonnegative initial date u0∈C(Ω¯) such that ∫Ωu0−1<16μη|Ω|2/χ2 with the constant η relying on Ω, the solution (u(⋅,t),v(⋅,t)) converges asymptotically to the constant equilibrium (r/μ,r/μ) in the L∞-norm as t→∞ if r>2(χ+1−1)2+χ2/(16η|Ω|).