Existence and uniqueness theorem on mild solutions to the Keller-Segel system in the scaling invariant space

We prove the existence and uniqueness of solutions (u,v) to the Keller-Segel system of parabolic-parabolic type in Rn for n≥. 3 in the scaling invariant class u∈Lq(0,T;Lr(Rn)), v∈Lr~(0,T;Hβ,r~(Rn)), where 2/. q+. n/. r= 2, 2/q~+n/r~=2β provided the initial data (u0,v0) is chosen as u0∈Ln/2(Rn), v0∈H2α,n/2α(Rn) for n/2(n+. 2) < α ≤ 1/2. In particular, our uniqueness result holds for all n≥. 2 even though we impose an assumption only on u such as Lq(0,T;Lr(Rn)) for 2/. q+. n/. r= 2 with n/2 < r. As for the marginal case when r= n/2, we show that if n≥. 3 and if u∈C([0,T);Ln/2(Rn)), ∇;v∈C([0,T);Ln(Rn)), then (u,v) is the only solution.