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 = \Delta u - \nabla \cdot (u \nabla v), \\ v_t = \Delta v - v + u, \end{ cases} \end{ align} considered in a quarter disc $\Omega = \{ \,(x_1, x_2) \in \mathbb R : x_1 > 0, x_2 > 0, x_1^2 + x_2^2 < R^2\,\}$, $R > 0$, in the following sense: For all reasonably smooth nonnegative initial data $u_0, v_0$ with $\int_\Omega u_0 < 2\pi$, 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 $\int_\Omega u_0 = m$ 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.
Categories: math.AP