Capacitary representation of positive solutions of semilinear parabolic equations

We give a global bilateral estimate on the maximal solution ${\overline{u}}_F$ of ${?}_{t}u?\mathrm{\Delta }u+u^q=0$ in ${\mathbb{R}}^N\times (0,\infty )$, $q>1$, $N?1$, which vanishes at $t=0$ on the complement of a closed subset $F?{\mathbb{R}}^N$. This estimate is expressed by a Wiener test involving the Bessel capacity $C_{2/q,q^{\prime }}$. We deduce from this estimate that ${\overline{u}}_F$ is σ-moderate in Dynkin's sense. To cite this article: M. Marcus, L. Véron, C. R. Acad. Sci. Paris, Ser. I 342 (2006).