Strong solutions to the Keller-Segel system with the weak L ⁿ₂ 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 L ⁿ_2w (Rⁿ), the weak L ⁿ₂ -space on Rn. If ||u₀||L ⁿ₂ w (Rⁿ) is sufficiently small, then our solution exists globally in time. Our motivation to construct solutions in L ⁿ_2w (Rn) stems from obtaining a self-similar solution which does not belong to any usual L^p(Rⁿ). Furthermore, the characterization of local existence of solutions gives us an explicit blow-up rate of ||u(t)||L^p(Rⁿ) for n 2 < p≤∞as t → Tmax, where Tmax denotes the maximal existence time.