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 C ([0, T] ; L^{frac(N (q - m), 2)} (R^N)) 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 u ∈ L^∞ (0, T ; L^{frac(N (q - m), 2)} (R^N)) and u^{frac(N (q - m), 2)} ∈ C_w ([0, T] ; L¹ (R^N)).