The logistic chemotaxis system with singular sensitivity and signal absorption in dimension two

Abstract

In this paper we study the non-flux chemotaxis system: ut=Δu−χ∇⋅(uv∇v)+bu−μu2, vt=Δv−uv, in a bounded and smooth domain Ω⊂R2, with b∈R and χ,μ>0. The existing literature, on the premise of relatively weak cross-diffusion with 0<χ<1, has established the global existence of classical solutions for arbitrary μ>0. In the present paper, we extend to prove that for any χ≥1, there exists μ∗(χ)>0 such that whenever μ>μ∗(χ), any reasonably smooth initial data emanates a unique global classical solution. Moreover, it is shown that there exist α(Ω)>0 and β(Ω)>0 such that when b>0, if μ>α(Ω)b+β(Ω) for 0<χ<1, or μ>max{μ∗(χ),α(Ω)b+β(Ω)} for χ≥1, then any classical solution must be globally bounded with the convergence that (u(⋅,t),|∇v(⋅,t)|v(⋅,t),v(⋅,t))→(bμ,0,0) in [L∞(Ω)]3 as t→∞. Instead, when b≤0, all classical solutions remain uniformly bounded in time as long as μ>0 for 0<χ<1, or μ>μ∗(χ) for χ≥1, and it also holds that (u(⋅,t),|∇v(⋅,t)|v(⋅,t),v(⋅,t))→(0,0,λ) in [L∞(Ω)]3 as t→∞ with 0≤λ<1|Ω|∫Ωv(⋅,0).