Boundedness in a quasilinear fully parabolic Keller–Segel system with logistic source

In this paper, we consider the quasilinear chemotaxis system (⋆){u_t=∇⋅(D(u)∇u)−∇⋅(S(u)∇v)+f(u),x∈Ω,t>0,v_t=Δv−v+u,x∈Ω,t>0 in a bounded domain Ω⊂Rⁿ(n≥2) under zero-flux boundary conditions, where the nonlinearities D,S∈C²([0,∞)) are supposed to generalize the prototypes D(u)=C_D(u+1)^m−1andS(u)=C_Su(u+1)^q−1 with C_D,C_S>0 and m,q∈R, and f∈C¹([0,∞)) satisfies f(u)≤r−bu^γ with r≥0,b>0 and γ>1. It is shown that if [Formula presented], then (⋆) has a unique globally bounded classical solution.