This article considers the following higher-dimensional quasilinear parabolic-parabolic-ODE chemotaxis system with generalized Logistic source and homogeneous Neumann boundary conditions
{ut=∇⋅(D(u)∇u)−∇⋅(S(u)∇v)+f(u),x∈Ω,t>0vt=Δv+w−v,x∈Ω,t>0,wt=u−w,x∈Ω,t>0,
in a bounded domain
Ω⊂Rn(n≥2) with smooth boundary
∂Ω, where the diffusion coefficient
D(u) and the chemotactic sensitivity function
S(u) are supposed to satisfy
D(u)≥M1(u+1)−α and
S(u)≤M2(u+1)β, respectively, where
M1,M2>0 and
α,β∈R. Moreover, the logistic source
f(u) is supposed to satisfy
f(u)≤a−μuγ with
μ>0,
γ≥1, and
a≥0. As
α+2β<γ−1+2γn, we show that the solution of the above chemotaxis system with sufficiently smooth nonnegative initial data is uniformly bounded.
