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

Abstract

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|).