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

Abstract

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;Lr(Rn)) 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 Ln/2(Rn). 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);Ln/2(Rn)) enables us to obtain the only weak solution.