Capacitary estimates of solutions of semilinear parabolic equations

We prove that any positive solution of $ \prt_tu-Δu+u^q=0$ ($q>1$) in $\BBR^N\ti(0,\infty)$ with initial trace $(F,0)$, where $F$ is a closed subset of $\BBR^N$ can be estimated from above and below and 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.

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