Critical mass in a quasilinear parabolic-elliptic Keller-Segel model

Abstract

The parabolic-elliptic Keller-Segel system{ut=∇⋅(ϕ(u)∇u)−∇⋅(ψ(u)∇v),x∈Ω,t>0,0=Δv−v+u,x∈Ω,t>0,(⋆) is considered under homogeneous Neumann boundary conditions in a bounded domain Ω⊂Rn (n≥3) with smooth boundary. Here ϕ and ψ are positive functions which satisfy ψ(s)/ϕ(s) grows like s2n as s→∞. We show that there exist m⁎,m⁎>0 such that if the total mass ∫Ωu0<m⁎, (⋆) admits a global bounded solution; if Ω is radially symmetric and ∫Ωu0>m⁎, we can find initial data u0(x)=u0(|x|) such that the corresponding solution must be unbounded.