Let $q\geq 1+\frac{2}{N}$. We prove that any positive solution of (E) $\prt_t u-\xD u+u^q=0$ in $\mathbb{R}^N\times(0,\infty)$ admits an initial trace which is a nonnegative Borel measure, outer regular with respect to the fine topology associated to the Bessel capacity $C_{\frac{2}{q},q'}$ in $\BBR^N$ ($q'=q/q-1)$) and absolutely continuous with respect to this capacity. If $\nu$ is a nonnegative Borel measure in $\BBR^N$ with the above properties we construct a positive solution $u$ of (E) with initial trace $\gn$ and we prove that this solution is the unique $\gs$-moderate solution of (E) with such an initial trace. Finally we prove that every positive solution of (E) is $\gs$-moderate.