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

We study global solutions of a class of chemotaxis systems generalizing the prototype{ut=∇⋅(ϕ(u)∇u)−χ∇⋅(ψ(u)∇v)+au−bur,x∈Ω,t>0,vt=Δv−v+u,x∈Ω,t>0 in a bounded domain Ω⊂RN(N≥1) with smooth boundary ∂Ω, ϕ(u)=(u+1)−αϕ(u)=(u+1)−α, ψ(u)=u(u+1)β−1ψ(u)=u(u+1)β−1, and the parameters r>1r>1, a≥0a≥0, b,χ>0b,χ>0, and α,β∈Rα,β∈R. It is proved that if 0<α+β