Boundedness of solutions to a quasilinear parabolic–elliptic Keller–Segel system with logistic source

We study global solutions of a class of chemotaxis systems generalizing the prototype{ut=∇⋅((u+1)m−1∇u)−χ∇⋅(u(u+1)q−1∇v)+au−bur,x∈Ω,t>0,0=Δv−v+u,x∈Ω,t>0,in a bounded domain Ω⊂RN(N≥1)Ω⊂RN(N≥1) with smooth boundary, with parameters m≥1,r>1,a≥0,b,q,χ>0m≥1,r>1,a≥0,b,q,χ>0. It is shown that when q+1b0:=N[r−m]−2(r−m)N+2(r−2)χ  if  q+1=r,then for any sufficiently smooth initial data there exists a classical solution which is global in time and bounded. The results improve the results of Wang et al. (2014) [37] and Cao and Zheng (2014) [6].