How unstable is spatial homogeneity in Keller-Segel systems? A new critical mass phenomenon in two- and higher-dimensional parabolic-elliptic cases

Abstract

The parabolic-elliptic Keller-Segel system $$\begin{aligned} \left\{ \begin{array}{ll} u_t = \Delta u - \nabla \cdot (u\nabla v),&{} \\ 0 = \Delta v - \mu + u, &{}\quad \mu :=\frac{1}{|\Omega |} \int _\Omega u, \end{array} \right. \qquad \qquad (\star ) \end{aligned}$$ is considered under homogeneous Neumann boundary conditions in the ball $$\Omega =B_R(0)\subset {\mathbb {R}}^n$$ . The main objective is to reveal that in the context of radially symmetric solutions, this problem exhibits an apparently novel type of critical mass phenomenon: It is shown, namely, that for any choice of $$n\ge 2$$ and $$R>0$$ there exists a positive number $$m_c=m_c(n,R)$$ with the following properties: In consequence, precisely at mass levels above $$m_c$$ the constant steady states of ( $$\star $$ ) possess the extreme instability property of repelling arbitrary concentration-increasing perturbations in such a drastic sense that corresponding trajectories collapse in finite time.