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

The parabolic-elliptic Keller-Segel system $$\{\begin{array}{l}{u}_t=\mathrm{\Delta }u-\mathrm{\nabla }\cdot (u\mathrm{\nabla }v),\\ 0=\mathrm{\Delta }v-\mu +u,\mu :=\frac{1}{\mathrm{\mid }\mathrm{\Omega }\mathrm{\mid }}{\int }_{\mathrm{\Omega }}u,\end{array}(\star )$$ is considered under homogeneous Neumann boundary conditions in the ball $\mathrm{\Omega }=B_R(0)\subset {\mathbb{R}}^n$, 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 $0<m<m_{\star }(n,R)$, the constant equilibrium of ($\star$) given by $u\equiv v\equiv \frac{m}{\mathrm{\mid }{B}_R\mathrm{\mid }}$ is locally asymptotically stable with respect to the norm in $L^{\mathrm{\infty }}(B_R)$, but that $\bullet$ if $m>m_{\star }(n,R)$, then any solution of ($\star$) emanating from nonnegative and nonconstant continuous radial initial data $u_0$ with ${\int }_{B_R}{u}_0=m$ and $\frac{1}{\mathrm{\mid }{B}_r\mathrm{\mid }}{\int }_{B_r}{u}_0\ge \frac{m}{\mathrm{\mid }{B}_R\mathrm{\mid }}$ 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, $m_{\star }(2,R)\approx 14.68197\cdot \pi$ 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.