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)), , where 2/q+n/r=2, provided the initial data (u0,v0) is chosen as u0∈Ln/2(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