Corners and collapse: Some simple observations concerning critical masses and boundary blow-up in the fully parabolic Keller-Segel system

Our main result shows that the mass $2\pi$ is critical for the minimal Keller-Segel system \begin{align}\label{prob:abstract}\tag{$\star$} \begin{cases} u_t = Δu - \nabla \cdot (u \nabla v), \ v_t = Δv - v + u, \end{cases} \end{align} considered in a quarter disc $\mathrm{\Omega }=\{(x_1,x_2)\in \mathbb{R}:x_1>0,x_2>0,x_1^2+x_2^2<R^2\}$, $R>0$, in the following sense: For all reasonably smooth nonnegative initial data $u_0,v_0$ with ${\int }_{\mathrm{\Omega }}{u}_0<2\pi$, there exists a global classical solution to the Neumann initial boundary value problem associated to \eqref{prob:abstract}, while for all $m>2\pi$ there exist nonnegative initial data $u_0,v_0$ with ${\int }_{\mathrm{\Omega }}{u}_0=m$ so that the corresponding classical solution of this problem blows up in finite time. At the same time, this gives an example of boundary blow-up in \eqref{prob:abstract}. Up to now, precise values of critical masses had been observed in spaces of radially symmetric functions or for parabolic-elliptic simplifications of \eqref{prob:abstract} only.