Initial trace of solutions of semilinear heat equation with absorption

TL;DR

This paper investigates the initial trace problem for positive solutions of semilinear heat equations with strong absorption, establishing measure-theoretic characterizations and analyzing specific subcritical cases.

Contribution

It provides a measure-theoretic description of initial traces for solutions and analyzes the subcritical case of the semilinear heat equation with absorption.

Findings

Initial trace is an outer regular Borel measure.

Characterization of initial trace in the subcritical case.

Analysis of solutions satisfying specific growth conditions.

Abstract

We study the initial trace problem for positive solutions of semilinear heat equations with strong absorption. We show that in general this initial trace is an outer regular Borel measure. We emphasize in particular the case where $u$ satisfies (E) $\partial_t u-\Delta u+t^\alpha |u|^{q-1}u=0$, with $q>1$ and $\alpha>-1$ and prove that in the subcritical case $1