Finite-time blow-up in a higher-dimensional quasilinear parabolic-elliptic chemotaxis system with space dependent logistic source

Abstract

This paper deals with a quasilinear parabolic-elliptic chemotaxis system with space dependent logistic source{ut=∇⋅(φ(u)∇u)−χ∇⋅(ψ(u)∇v)+κ(|x|)u−μ(|x|)uθ,0=Δv−m(t)|Ω|+u,m(t):=∫Ωu(⋅,t) under homogeneous Neumann boundary conditions in a ball Ω=BR(0)⊂Rn, n≥2, where R>0 for θ≥1, χ>0, sufficiently smooth functions κ,μ:[0,R]→[0,∞) and the nonlinear diffusivity φ(u) and chemosensitivity ψ(u) are supposed to extend the prototypesφ(u)=(u+1)−p,ψ(u)=uq,p≥0,q∈R. We proved that whenever μ′,−κ′>0 and μ(s)≤μ1sα for all s∈[0,R] and some μ1>0, α≥n(θ−1), as well as 0≤p<1,1≤q<2−p, then for all ∫Ωu0>4(n2−n)n1−p−qa(n)qχ, there exist nonnegative radially symmetric initial data such that the corresponding solutions blow up in finite time, where a(n) is the volume of the unit ball in Rn.