Formation and Behavior of Dirac Singularities in the Parabolic-Elliptic Keller-Segel System in Dimensions $n\geq 3$

TL;DR

This paper investigates the formation and evolution of Dirac singularities in solutions to the Keller-Segel system in dimensions three and higher, revealing continuous mass concentration behavior unlike the two-dimensional case.

Contribution

It extends the analysis of blow-up solutions to measure-valued solutions in higher dimensions, demonstrating continuous Dirac mass formation and asymptotic mass absorption at the origin.

Findings

Existence of measure-valued solutions with a Dirac mass component.

Dirac mass formation occurs continuously, not instantaneously.

All mass asymptotically concentrates at the origin.

Abstract

We consider nonnegative radially symmetric solutions of the parabolic-elliptic Keller-Segel system \begin{align*} \left\lbrace \begin{array}{r@{}l@{\quad}l} &u_t=\Delta u-\nabla \cdot \big(u\nabla v\big),\\ &0=\Delta v -\mu + u , \\ \end{array}\right. \end{align*} where $\mu$ is the spatial average of $u$, under homogeneous Neumann boundary conditions in a ball in $\mathbb R^n$ for $n\geq 3$. In two dimensions, it is well established that solutions blowing up in finite time converge to a Dirac profile in the vague topology. In contrast, for $n\geq 3$, blow-up solutions with finite existence time do not appear to exhibit such concentration behavior. By generalizing to measure-valued solutions corresponding to accumulated densities of $u$, we extend the analysis beyond the blow-up time. Within this framework, we establish the existence of a minimal solution \[ u(t)=\theta(t)\delta_0 + \rho(\cdot,t) dx, \qquad t \geq 0, \] where $\rho$ is integrable and $\theta$ is increasing and right-continuous. We further construct a class of initial data for which $\theta(t_0)>0$ for some $t_0>0$, thereby establishing the formation of a Dirac mass at the origin. Unlike in the case $n=2$, the singular mass does not jump to a positive level instantaneously; instead, $\theta$ becomes positive continuously. Moreover, $\theta$ is strictly increasing on $[t_0,\infty)$, and the entire mass is asymptotically absorbed at the origin.