Boundedness in a quasilinear fully parabolic Keller-Segel system with logistic source

In this paper, we consider the quasilinear chemotaxis system (⋆){ut=∇⋅(D(u)∇u)−∇⋅(S(u)∇v)+f(u),x∈Ω,t>0,vt=Δv−v+u,x∈Ω,t>0 in a bounded domain Ω⊂Rn(n≥2) under zero-flux boundary conditions, where the nonlinearities D,S∈C2([0,∞)) are supposed to generalize the prototypes D(u)=CD(u+1)m−1andS(u)=CSu(u+1)q−1 with CD,CS>0 and m,q∈R, and f∈C1([0,∞)) satisfies f(u)≤r−buγ with r≥0,b>0 and γ>1. It is shown that if q<{max{m+2n−1,m+γ2−n−1n}for1<γ≤n+2n,max{m+2γn+2−1,m2+γ(n+4)2(n+2)−1}forn+2n<γ