Existence and uniqueness theorem on weak solutions to the parabolic-elliptic Keller-Segel system

In R-n (n >= 3), we first define a notion of weak solutions to the Keller-Segel system of parabolic-elliptic type in the scaling invariant class L-s(0, T; L-r (R-n)) for 2/s + n/r = 2 with n/2 < r < n. Any condition on derivatives of solutions is not required at all. The local existence theorem of weak solutions is established for every initial data in L-n/2(R-n). We prove also their uniqueness. As for the marginal case when r = n/2, we show that if n >= 4, then the class C([0, T); L-n/2(R-n)) enables us to obtain the only weak solution. (c) 2012 Elsevier Inc. All rights reserved.