Identifying secondary critical mass levels in two- and higher-dimensional Keller-Segel systems

Abstract

The parabolic-elliptic Keller-Segel system \[ \left\{ \begin{array}{l} u_t = Δu - \nabla \cdot (u\nabla v), \\[1mm] 0 = Δv - μ+ u, \qquad μ:=\frac{1}{|Ω|} \int_Ωu, \end{array} \right. \qquad \qquad (\star) \] is considered under homogeneous Neumann boundary conditions in the ball $Ω=B_R(0)\subset {\mathbb R}^n$, where $n\ge 2$ and $R>0$. The main results identify a mass level $m_\star(n,R)>0$ as critical in the sense that $\bullet$ whenever $0 m_\star(n,R)$, then any solution of ($\star$) emanating from nonnegative and nonconstant continuous radial initial data $u_0$ with $\int_{B_R} u_0=m$ and $\frac{1}{|B_r|} \int_{B_r} u_0 \ge \frac{m}{|B_R|}$ for all $r\in (0,R)$ blows up in finite time. The value of $m_\star(n,R)$ can be determined in an essentially explicit manner in dependence on principal zeroes of appropriate Bessel functions of the first kind; in particular, $m_\star(2,R)\approx 14.68197 \cdot π$ for all $R>0$. Accordingly, as a supplement to known mass-dependent blow-up dichotomies these findings reveal a genuinely secondary critical mass phenomenon for ($\star$) especially also in its two-dimensional version in which no precedent of this form seems known.