Capacitary estimates of solutions of semilinear parabolic equations

TL;DR

This paper develops a new representation formula for positive solutions of semilinear heat equations with power absorption, linking initial conditions to Wiener tests and Bessel capacities.

Contribution

It introduces an almost-everywhere representation formula involving Wiener tests and Bessel capacities for solutions with specific initial traces.

Findings

Representation formula involving Wiener test and Bessel capacities

Initial trace characterized by indicatrix function of a compact set

Provides criteria for solution behavior based on capacity estimates

Abstract

We prove an almoste representation formula for positive solutions of semilinear heat equations with power-type absorption the initial trace of which is the indicatrix function of a compact set. The representations involves a Wiener test via Bessel capacities.