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

Abstract

The present paper is concerned with the parabolic–parabolic Keller–Segel system

with degenerate critical diffusion $q=q_{\star }:=(N-2)/N$ in space dimension $N\ge 3$, the underlying domain $\Omega$ being either $\mathrm{\Omega }={\mathbb{R}}^N$ or the open ball $\mathrm{\Omega }=B_R(0)$ of ${\mathbb{R}}^N$ 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 $0=\mathrm{\Delta }v-\alpha v+u$. Assuming that $N=3,4$ and $\alpha >0$, we prove that radially symmetric solutions with negative initial energy blow up in finite time in $\mathrm{\Omega }={\mathbb{R}}^N$ and in $\mathrm{\Omega }=B_R(0)$ under mixed Neumann–Dirichlet boundary conditions. Moreover, if $\mathrm{\Omega }=B_R(0)$ and Neumann boundary conditions are imposed on both $u$ and $v$, we show the existence of a positive constant $C$ depending only on $N$, $\Omega$, and the mass of $u_0$ 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.