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] ; Lfrac(N (q - m), 2) (RN)) 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 ; Lfrac(N (q - m), 2) (RN)) and ufrac(N (q - m), 2) ∈ Cw ([0, T] ; L1 (RN)).
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