A new result for global existence and boundedness of solutions to a parabolic-parabolic Keller-Segel system with logistic source

We consider the following fully parabolic Keller-Segel system with logistic source(KS){ut=Δu−χ∇⋅(u∇v)+au−μu2,x∈Ω,t>0,vt=Δv−v+u,x∈Ω,t>0, over a bounded domain Ω⊂RN(N≥1), with smooth boundary ∂Ω, the parameters a∈R,μ>0,χ>0. It is proved that if μ>0, then (KS) admits a global weak solution, while if μ>(N−2)+NχCN2+11N2+1, then (KS) possesses a global classical solution which is bounded, where CN2+11N2+1 is a positive constant which is corresponding to the maximal Sobolev regularity. Apart from this, we also show that if a=0 and μ>(N−2)+NχCN2+11N2+1, then both u(⋅,t) and v(⋅,t) decay to zero with respect to the norm in L∞(Ω) as t→∞.