Finite time blowup for the parabolic-parabolic Keller-Segel system with critical diffusion

The present paper is concerned with the parabolic-parabolic Keller-Segel system∂tu=div(∇uq+1−u∇v),t>0,x∈Ω,∂tv=Δv−αv+u,t>0,x∈Ω,(u,v)(0)=(u0,v0)≥0,x∈Ω, with degenerate critical diffusion q=q⋆:=(N−2)/N in space dimension N≥3, the underlying domain Ω being either Ω=RN or the open ball Ω=BR(0) of RN with suitable boundary conditions. It has remained open whether there exist solutions blowing up in finite time, the existence of such solutions being known for the parabolic-elliptic reduction with the second equation replaced by 0=Δv−αv+u. Assuming that N=3,4 and α>0, we prove that radially symmetric solutions with negative initial energy blow up in finite time in Ω=RN and in Ω=BR(0) under mixed Neumann-Dirichlet boundary conditions. Moreover, if Ω=BR(0) and Neumann boundary conditions are imposed on both u and v, we show the existence of a positive constant C depending only on N, Ω, and the mass of u0 such that radially symmetric solutions blow up in finite time if the initial energy does not exceed −C. The criterion for finite time blowup is satisfied by a large class of initial data.