Point Dynamics in a Singular Limit of the Keller--Segel Model 1: Motion of the Concentration Regions

Abstract

The purpose of this paper is to study a singular perturbation limit of a Keller--Segel system that generates blow-up in finite time. The main question that is addressed is the description of the evolution of the solutions of this problem beyond the blow-up time for the limit problem if a suitable parameter $\varepsilon>0$ approaches zero. This problem is studied using matched asymptotic expansions. The resulting limit solution can be described beyond the blow-up time by means of the motion of a set of points whose dynamics is coupled with a parabolic-elliptic system of equations.