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

In Rn (n≥ 3), we first define a notion of weak solutions to the Keller-Segel system of parabolic-elliptic type in the scaling invariant class Ls(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.