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

This paper deals with the higher dimension quasilinear parabolic-parabolic Keller-Segel system involving a source term of variable logistic type ut=∇⋅(ϕ(u)∇u)−∇⋅(ψ(u)∇υ)+g(x,u), −Δυ=−υ+u, in (x,t)∈Ω×(0,T), subject to nonnegative initial data and homogeneous Neumann boundary condition, where Ω is a smooth and bounded domain in RN (N≥1) and g∈C1(Ω‾×[0,∞)) function satisfies g(x,s)≤ηsk(x)−μsm(x),s>0 with g(x,0)≥0, x∈Ω and η≥0,μ>0 are constants, k,m are measurable functions with 1<k−:=ess infx∈Ωk(x)≤k(x)≤k+:=esssup k(x)x∈Ω<+∞, 2≤m−:=ess infx∈Ωm(x)≤m(x)≤m+:=esssup m(x)x∈Ω<+∞. Positive functions ψ,ϕ∈C2([0,∞)) satisfy νsq≤ψ≤χsq and ϕ≥σsp with p,q∈R, ν,χ,σ>0 when s≥0. It is proved that if q=m−−1 (critical case) and [Formula presented] then there exists a classical solution which is global in time and bounded. Moreover, if q∈(k+−1,m−−1), we obtain that there is a classical solution of the above system uniformly in time bounded without any restrictions on m− and μ.