Global existence and asymptotic behavior to a chemotaxis-consumption system with singular sensitivity and logistic source

We consider a chemotaxis-consumption system with singular sensitivity and logistic source: ut=Δu−∇⋅(uϕ(v)∇v)+ru−μuk, vt=Δv−uv in a smooth bounded domain Ω⊂Rn(n≥1), where r,μ>0, k>1, and ϕ(s)∈C1(0,∞) satisfying ϕ(s)→∞ as s→0. It is proved that there exists a global classical solution if k>1 for n=1 or k>1+n2 for n≥2. The asymptotic behavior of solutions is determined as well for ϕ(v)=1v, n=2 that if k>2, there exists μ∗>0 such that (u,v,|∇v|v)→((rμ)1k−1,0,0) as t→∞ provided μ>μ∗.