Point Dynamics in a Singular Limit of the Keller--Segel Model 2: Formation of the Concentration Regions

This paper continues the analysis started in the first part of this article (cf. [J. J. L. Velázquez, SIAM J. Appl. Math., 64 (2004), pp. 1198--1223]). It was seen there, using the method of matched asymptotics, that a regularized version of the Keller--Segel system admits, for a suitable asymptotic limit, solutions with some regions of high concentrations for the cell density. This paper considers the relation between the phenomenon of blow-up for the limit problem and the dynamics of the concentration regions described in [J. J. L. Velázquez, SIAM J. Appl. Math., 64 (2004), pp. 1198--1223]. In particular, this paper analyzes the precise way in which the regularization introduced in the Keller--Segel system stops the aggregation process and yields the formation of concentration regions.