ε-Regularity theorem and its application to the blow-up solutions of Keller–Segel systems in higher dimensions

Let us consider (KS)m below for all N⩾2 and general exponents m and q. In particular, the 2-D semi-linear case such as N=2, m=1 and q=2 is included. We establish an ε-regularity theorem for weak solutions. As an application, we give an extension criterion in which coincides with a scaling invariant class of weak solutions associated with (KS)m. In addition, the Hausdorff dimension of its singular set is zero if and .