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

Zheng, Gao-Feng

Annales de l'I.H.P. Analyse non linéaire, Tome 27 (2010) no. 6, pp. 1333-1360

A quasi-monotonicity formula for the solution to a semilinear parabolic equation $u_t=\Delta u+V\left(x\right){\left|u\right|}^{p-1}u$, $p>(N+2)/(N-2)$ 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 $Q^{\text{'}}\subset Q$ such that u is continuous in $Q^{\text{'}}$ and the $(N-\frac{4}{p-1})$-dimensional parabolic Hausdorff measure ${\mathscr{H}}^{(N-\frac{4}{p-1})}(Q\setminus Q^{\text{'}})$ of $Q\setminus Q^{\text{'}}$ is finite.

Keywords: Quasi-monotonicity formula, Partial regularity, Borderline solutions, Semilinear parabolic equations, Potential

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