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 u 0 in L n /2 w (ℝ n ), the weak L n /2 ‐space on ℝ n . If ‖ u 0 ‖   L   n /2  w(ℝ   n )is sufficiently small, then our solution exists globally in time. Our motivation to construct solutions in L n /2 w (ℝ n ) stems from obtaining a self‐similar solution which does not belong to any usual L p (ℝ n ). Furthermore, the characterization of local existence of solutions gives us an explicit blow‐up rate of ‖ u ( t )‖   L   p (ℝ   n )for n /2 < p < ∞ as t → T max , where T max denotes the maximal existence time (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)