Unbounded mass radial solutions for the Keller-Segel equation in the disk

TL;DR

This paper constructs radial solutions to the Keller-Segel equation in a disk that blow up at the center and concentrate on the boundary as a parameter approaches zero, revealing solutions with unbounded mass.

Contribution

It demonstrates the existence of unbounded mass solutions for small parameters, showing new blow-up and concentration phenomena in the Keller-Segel model.

Findings

Solutions blow up at the origin as λ→0

Solutions concentrate on the boundary as λ→0

Mass of solutions becomes unbounded as λ→0

Abstract

We consider the boundary value problem $$ \left\{ \begin{array}{rcll} -\Delta u+ u -\lambda e^u&=&0,\ u>0 & \mathrm{in}\ B_1(0)\\ \partial_\nu u&=&0&\mathrm{on}\ \partial B_1(0), \end{array}\right. $$ whose solutions correspond to steady states of the Keller--Segel system for chemotaxis. Here $B_1(0)$ is the unit disk, $\nu$ the outer normal to $\partial B_1(0)$, and $\lambda>0$ is a parameter. We show that, provided $\lambda$ is sufficiently small, there exists a family of radial solutions $u_\lambda$ to this system which blow up at the origin and concentrate on $\partial B_1(0)$, as $\lambda\to 0$. These solutions satisfy $$ \lim_{\lambda\to 0} \frac{u_\lambda(0)}{|\ln\lambda|}=0\quad \mbox{and}\quad 0<\lim_{\lambda\to 0} \frac{1}{|\ln\lambda|}\int_{B_1(0)}\lambda e^{u_\lambda(x)}dx<\infty, $$ having in particular unbounded mass, as $\lambda\to 0$.