Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system

TL;DR

This paper demonstrates that in higher dimensions, the fully parabolic Keller-Segel system can exhibit finite-time blow-up for a large set of initial conditions, highlighting the system's potential for singularity formation.

Contribution

It proves finite-time blow-up for radially symmetric solutions in higher dimensions and shows the set of initial data causing blow-up is dense in relevant function spaces.

Findings

Finite-time blow-up occurs for certain initial data in dimensions n ≥ 3.

The set of initial data leading to blow-up is dense in specific function spaces.

Explicit blow-up criteria are provided for the Keller-Segel system.

Abstract

We study the Neumann initial-boundary value problem for the fully parabolic Keller-Segel system u_t=\Delta u - \nabla \cdot (u\nabla v), \qquad x\in\Omega, \ t>0, [1mm] v_t=\Delta v-v+u, \qquad x\in\Omega, \ t>0, where $\Omega$ is a ball in ${\mathbb{R}}^n$ with $n\ge 3$. It is proved that for any prescribed $m>0$ there exist radially symmetric positive initial data $(u_0,v_0) \in C^0(\bar\Omega) \times W^{1,\infty}(\Omega)$ with $\int_\Omega u_0=m$ such that the corresponding solution blows up in finite time. Moreover, by providing an essentially explicit blow-up criterion it is shown that within the space of all radial functions, the set of such blow-up enforcing initial data indeed is large in an appropriate sense; in particular, this set is dense with respect to the topology of $L^p(\Omega) \times W^{1,2}(\Omega)$ for any $p \in (1,\frac{2n}{n+2})$.