On the Stability of Type II Blowup for the Keller-Segel System in High Dimensions

Abstract

We study finite-time blowup for the parabolic--elliptic Keller--Segel system on ${\mathbb{R}}^d$ in dimensions $d\ge 11$, where the problem is mass supercritical. For every integer $l\ge 2$, we construct smooth radially symmetric solutions whose radial mass variable concentrates the normalized stationary state $Q$ at a quantized scale. Each blowup regime can be realized by solutions with nonnegative population density throughout their classical lifespan. More precisely, near the blowup time $T$, \[ u(t,r)=\frac{1}{λ^2(t)}\left[Q\left(\frac{r}{λ(t)}\right) +ε\left(t,\frac{r}{λ(t)}\right)\right], \qquad λ(t)=c(T-t)^{\frac{l}{γ(d)}}(1+o(1)), \] where $c>0$ and $\gamma (d)=\frac{1}{2}(d-2-\sqrt{(d-2)(d-10)})$. The remainder $\epsilon$ converges to zero in local $L^{\infty }$ norms and in a range of high-order homogeneous Sobolev norms. Since $2l>\gamma (d)$, the concentration scale is strictly smaller than the parabolic scale $\sqrt{T-t}$, and the resulting blowup is of type II. The $l$-th regime has exactly $l-1$ unstable radial modulation directions and is stable within a codimension-$(l-1)$ class of suitably regular radial initial data. The proof combines a generalized-kernel expansion driven by the algebraic tail of $Q$, modulation analysis, coercive weighted high-order energy estimates, and a finite-dimensional topological argument. This yields a quantized hierarchy of stationary-state concentration rates for the high-dimensional Keller--Segel flow.