Capacitary representations of positive solutions of semilinear parabolic equations

Abstract

We give a global bilateral estimate on the maximal solution ${\bar{u}}_F$ of $ \prt_tu-Δu+u^q=0$ in $\BBR^N\times (0,\infty)$, $q>1$, $N\ge 1$, which vanishes at $t=0$ on the complement of a closed subset $F\subset \BBR^N$. This estimate is expressed by a Wiener test involving the Bessel capacity $C_{2/q,q^{\prime }}$. We deduce from this estimate that ${\bar{u}}_F$ is $\sigma$-moderate in Dynkin's sense.

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• Maximal solutions of equation u = uq in arbitrary domains