发表机构
Applied and Computational Mathematics, Caltech; Department of Mathematics, Brown University（加州理工学院应用与计算数学系; 布朗大学数学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该论文在高维（d≥11）Keller-Segel系统中构造了II型爆破解，通过调制分析和能量估计证明了在余维数(l-1)类初值下的稳定性，并揭示了稳态浓度速率的量子化层级。
AI 中文摘要
我们研究在维度$d\geq11$的$\mathbb{R}^d$上抛物-椭圆型Keller-Segel系统的有限时间爆破问题，此时该问题为质量超临界。对于每个整数$l\geq2$，我们构造光滑径向对称解，其径向质量变量在量子化尺度上集中归一化稳态$Q$。每个爆破机制均可由在其经典存在时间内具有非负人口密度的解实现。更精确地，在爆破时间$T$附近，\\[ u(t,r)=\frac{1}{\lambda^2(t)}\left[Q\left(\frac{r}{\lambda(t)}\right) +\epsilon\left(t,\frac{r}{\lambda(t)}\right)\right], \qquad \lambda(t)=c(T-t)^{\frac{l}{\gamma(d)}}(1+o(1)), \\] 其中$c>0$且$\gamma(d)=\frac12\bigl(d-2-\sqrt{(d-2)(d-10)}\bigr)$。余项$\epsilon$在局部$L^\infty$范数和一系列高阶齐次Sobolev范数下收敛到零。由于$2l>\gamma(d)$，集中尺度严格小于抛物尺度$\sqrt{T-t}$，因此产生的爆破为II型。第$l$个机制恰好有$l-1$个不稳定的径向调制方向，并且在余维数为$(l-1)$的适当正则径向初值类中是稳定的。证明结合了由$Q$的代数尾部驱动的广义核展开、调制分析、强制加权高阶能量估计以及有限维拓扑论证。这为高维Keller-Segel流产生了稳态浓度速率的量子化层级。
英文摘要
We study finite-time blowup for the parabolic--elliptic Keller--Segel system on $\mathbb{R}^d$ in dimensions $d\geq11$, where the problem is mass supercritical. For every integer $l\geq2$, 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 $γ(d)=\frac12\bigl(d-2-\sqrt{(d-2)(d-10)}\bigr)$. The remainder $ε$ converges to zero in local $L^\infty$ norms and in a range of high-order homogeneous Sobolev norms. Since $2l>γ(d)$, 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.
