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

We consider a quasilinear parabolic–parabolic Keller–Segel system with a logistic type source ut=∇⋅(ϕ(u)∇u)−∇⋅(ψ(u)∇v)+g(u)ut=∇⋅(ϕ(u)∇u)−∇⋅(ψ(u)∇v)+g(u), vt=Δv−v+uvt=Δv−v+u in a smooth bounded domain Ω⊂RnΩ⊂Rn, n≥1n≥1, subject to nonnegative initial data and homogeneous Neumann boundary conditions, where ϕ,ψϕ,ψ and gg are smooth positive functions satisfying c1sp≤ϕ(s)c1sp≤ϕ(s) and c1sq≤ψ(s)≤c2sqc1sq≤ψ(s)≤c2sq for p,q∈Rp,q∈R and s≥s0>1s≥s0>1, g(s)≤as−μskg(s)≤as−μsk for s>0s>0, with constants a≥0a≥0, μ,c1,c2>0μ,c1,c2>0, and the extended logistic exponent k>1k>1 instead of the ordinary k=2k=2. It is proved that if qμ0μ>μ0 for some μ0>0μ0>0, then there exists a classical solution which is global in time and bounded. This shows the exact way of the logistic exponent k>1k>1 effecting the behavior of solutions.