Strong solutions to the Keller-Segel system with the weak L n /2 initial data and its application to the blow-up rate

We shall show an exact time interval for the existence of local strong solutions to the Keller-Segel system with the initial data u0 in Ln /2w (ℝn), the weak Ln /2-space on ℝn. If ‖u0‖ is sufficiently small, then our solution exists globally in time. Our motivation to construct solutions in Ln /2w (ℝn) stems from obtaining a self-similar solution which does not belong to any usual Lp(ℝn). Furthermore, the characterization of local existence of solutions gives us an explicit blow-up rate of ‖u (t)‖ for n /2 < p < ∞ as t → Tmax, where Tmax denotes the maximal existence time (© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)