article

Title: The Keller-Segel system of parabolic-parabolic type with initial data in weak $L^{n/2}(\mathbb{R}^n)$ and its application to self-similar solutions

Authors: Hideo Kozono and Yoshie Sugiyama

Issue: Volume 57 (2008), Issue 4, 1467-1500

Abstract:

$We shall show the existence of a \emph{global} strong solution to the semilinear Keller-Segel system in $\mathbb{R}^{n}$, $n \ge 3$ of \emph{parabolic-parabolic type} with small initial data $u_{0} \in L_{w}^{n/2}(\mathbb{R}^{n})$ and $v_{0} \in \mathrm{BMO}$. Our method is based on the perturbation of linearization together with the $L^{p} - L^{q}$-estimates of the heat semigroup and the fractional powers of the Laplace operator. As a by-product of our method, we shall construct a self-similar solution and prove the smoothing effect. Furthermore, the stability problem on our strong solutions will be also discussed.$