Stability vs. instability of singular steady states in the parabolic-elliptic Keller-Segel system on ${\mathbb{R}}^n$

The Cauchy problem in ${\mathbb{R}}^n$ is considered for \begin{eqnarray*} \left{ \begin{array}{l} u_t = Δu - \nabla \cdot (u\nabla v),\ 0 = Δv + u. \end{array} \right. \end{eqnarray*} For each $n\ge 10$, a statement on stability and attractiveness of the singular steady state given by $$u_{\star }(x):=\frac{2(n-2)}{\mathrm{\mid }x{\mathrm{\mid }}^2},x\in {\mathbb{R}}^n\setminus \{0\},$$ is derived within classes of nonnegative radial solutions emanating from initial data less concentrated than $u_{\star }$. In particular, for any such $n$ it is shown that infinite-time blow-up occurs for all radial initial data which are less concentrated than $u_{\star }$ and satisfy $$u_0(x)\ge \frac{2(n-2)}{\mathrm{\mid }x{\mathrm{\mid }}^2}-\frac{C}{\mathrm{\mid }x{\mathrm{\mid }}^{2+\theta }}forallx\in {\mathbb{R}}^n\setminus B_1(0)$$ with some $C>0$ and some $\theta >\frac{n-2+\sqrt{(n-2)(n-10)}}{2}$. This is complemented by a result which, in the case when $3\le n\le 9$, asserts instability of $u_{\star }$ as well as the existence of a bounded absorbing set for all radial trajectories initially less concentrated than $u_{\star }$. In particular, previous knowledge on stability properties of $u_{\star }$, as having been gained for $n\ge 11$ in [24], is thereby extended to any dimension $n\ge 3$.