Uniqueness and regularity of weak solutions for the 1-DD degenerate Keller–Segel systems

We consider the Keller–Segel system of degenerate type (KS)m with m>1m>1 below. We prove the uniqueness of weak solutions of (KS)m with the regularity of ∂tu∈Lloc1(R×(0,T)). In addition, we shall show that every weak solution of (KS)m has the property that ∂tu∂tu belongs to Llocp(R×(0,T)) for all p∈[1,m+1m). This implies that the weak solution uu actually becomes a strong solution. These results are obtained as applications of the Aronson–Bénilan type estimate to (KS)m, i.e.,   there is a uniform boundedness from below of ∂x2um−1.