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 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 N ( q − m ) 2 ( R N ) ) and u N ( q − m ) 2 ∈ C w ( [ 0 , T ] ; L 1 ( R N ) ) .