Capacitary estimates of solutions of semilinear parabolic equations

We prove that any positive solution of in with initial trace (F, 0), where F is a closed subset of can be represented, up to two universal multiplicative constants, by a series involving the Bessel capacity . 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 , by using the "density" of F expressed in terms of the -Bessel capacity.