Existence and asymptotic behaviour for the parabolic-parabolic Keller-Segel system with singular data

Abstract

This work considers the Keller-Segel system of parabolic-parabolic type in R(n) for n >= 2. We prove existence results in a new framework and with initial data in N(r,lambda,infinity)(-beta) x (B) over dot(infinity,infinity)(0). This initial data class is larger than the previous ones, e.g., Kozono-Sugiyama (2008 Indiana Univ. Math. J. 57 1467-500) and Biler (1998 Adv. Math. Sci. Appl. 8 715-43), and covers physical cases of initial aggregation at points (Diracs) and on filaments. Self-similar solutions are obtained for initial data with the correct homogeneity and a certain value of parameter gamma. We also show an asymptotic behaviour result, which provides a basin of attraction around each self-similar solution.