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

We consider a quasilinear parabolic-parabolic Keller-Segel system involving a source term of logistic type,(0.1){ut=∇⋅(ϕ(u)∇u)−∇⋅(ψ(u)∇v)+g(u),(x,t)∈Ω×(0,T),vt=Δv−v+u,(x,t)∈Ω×(0,T), with nonnegative initial data under Neumann boundary condition in a smooth bounded domain Ω⊂Rn, n⩾1. Here, ϕ and ψ are supposed to be smooth positive functions satisfying c1sp⩽ϕ and c1sq⩽ψ(s)⩽c2sq when s⩾s0 with some s0>1, and we assume that g is smooth on [0,∞) fulfilling g(0)⩾0 and g(s)⩽as−μs2 for all s>0 with constants a⩾0 and μ>0. Within this framework, it is proved that whenever q<1, for any sufficiently smooth initial data there exists a unique classical solution which is global in time and bounded. Our result is independent of p.