We prove that any positive solution of \partial_t u-\Delta u+u^q=0 (q>1) in R^N \times (0,\infty) with singular initial trace (F,0), where F is a closed subset of R^N can be represented, up to two universal multiplicative constants, by a series involving the Bessel capacity C_{2/q,q'}. As a consequence we prove that there exists a unique positive solution of the equation with such an initial trace. We also characterize the blow-up set of u(x,t) when t\downarrow 0 , by using the "density" of F expressed in terms of the C_{2/q,q'}-capacity.