Finite time blow-up in a parabolic–elliptic Keller–Segel system with flux dependent chemotactic coefficient

Abstract

A parabolic–elliptic Keller–Segel system ut=Δu−χ∇⋅(uf(|∇v|)∇v),0=Δv−M+u,with homogeneous Neumann boundary condition is considered in a radially symmetric domain Ω=BR(0)⊂RN(N≥3), where f(ξ)=(ξp−2(1+ξp)q−pp),ξ≥0,p≥2,1<q≤p<∞,and BR(0) is a N-dimensional ball of radius R>0. We assert that under a condition on the initial data, radial weak solutions blow-up in finite time when NN−1<q<2.