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

Abstract

A quasi-monotonicity formula for the solution to a semilinear parabolic equation $u_t=\mathrm{\Delta }u+V(x)|u{|}^{p-1}u$, $p>(N+2)/(N-2)$ in $\Omega \times (0,T)$ 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 $Q\subset \Omega \times (0,T)$ there exists a close subset $Q^{\prime }\subset Q$ such that $u$ is continuous in $Q^{\prime }$ and the $(N-\frac{4}{p-1})$-dimensional parabolic Hausdorff measure ${\mathcal{H}}^{(N-\frac{4}{p-1})}(Q\setminus Q^{\prime })$ of $Q\setminus Q^{\prime }$ is finite.