A quasi-monotonicity formula and partial regularity for borderline solutions to a parabolic equation

A quasi-monotonicity formula for the solution to a semilinear parabolic equation ut=Δu+V(x)|u|p−1u, p>(N+2)/(N−2) in Ω×(0,T) with 0-Dirichlet boundary condition is obtained. As an application, it is shown that for some suitable global weak solution u and any compact set Q⊂Ω×(0,T) there exists a close subset Q′⊂Q such that u is continuous in Q′ and the -dimensional parabolic Hausdorff measure of Q∖Q′ is finite.