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

We consider the Keller–Segel system of degenerate type View the MathML source(KS)m with m>1m>1 below. We prove the uniqueness of weak solutions of View the MathML source(KS)m with the regularity of View the MathML source∂tu∈Lloc1(R×(0,T)). In addition, we shall show that every weak solution of View the MathML source(KS)m has the property that ∂tu∂tu belongs to View the MathML sourceLlocp(R×(0,T)) for all View the MathML sourcep∈[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 View the MathML source(KS)m, i.e., there is a uniform boundedness from below of View the MathML source∂x2um−1.