(untitled) (Journal of Partial Differential Equations, 2023)
In this paper, we consider the quasilinear chemotaxis system of parabolic-elliptic type cases u_t= abla·(D(u) abla u)- abla·(f(u) abla v), & xΩ, t>0, 0=Δ v-μ(t)+g(u), & xΩ, t>0 casesunder homogeneous Neumann boundary conditions in a smooth bounded domain ΩR^n, n≥1. The nonlinear diffusivity D(ξ) and chemosensitivity f(ξ) as well as nonlinear signal production g(ξ) are supposed to extend the prototypes D(ξ)=C_0(1+ξ)^-m, f(ξ)=K(1+ξ)^k, g(ξ)=L(1+ξ)^l, C_0>0,ξ≥ 0,K,k,L,l>0,mR. We proved that if m+k+l>1+2n, then there exists nonnegative radially symmetric initial data u_0 such that the corresponding solutions blow up in finite time. However, the system admits a global bounded classical solution for arbitrary initial datum when m+k+l