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

### Gao-Feng Zheng

Department of Mathematics, Huazhong Normal University, Wuhan 430079, PR China

## Abstract

A quasi-monotonicity formula for the solution to a semilinear parabolic equation $u_{t}=Δu+V(x)∣u∣_{p−1}u$, $p>(N+2)/(N−2)$ 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 $(N−p−14 )$-dimensional parabolic Hausdorff measure $H_{(N−p−14)}(Q∖Q_{′})$ of $Q∖Q_{′}$ is finite.

## Cite this article

Gao-Feng Zheng, A quasi-monotonicity formula and partial regularity for borderline solutions to a parabolic equation. Ann. Inst. H. Poincaré Anal. Non Linéaire 27 (2010), no. 6, pp. 1333–1360

DOI 10.1016/J.ANIHPC.2010.07.001