Classification of positive solutions of heat equation with supercritical absorption

TL;DR

This paper characterizes the initial traces of positive solutions to a supercritical heat equation, showing they correspond to certain measures, and establishes existence, uniqueness, and regularity results for these solutions.

Contribution

It introduces a comprehensive framework for the initial trace of positive solutions to the supercritical heat equation, including measure characterization and solution construction.

Findings

Initial trace is a nonnegative Borel measure with specific regularity.

Constructs solutions for given initial measures with these properties.

Proves all positive solutions are $ ext{gs}$-moderate and unique under these conditions.

Abstract

Let $q\geq 1+\frac{2}{N}$. We prove that any positive solution of (E) $\prt_t u-\xD u+u^q=0$ in $\mathbb{R}^N\times(0,\infty)$ admits an initial trace which is a nonnegative Borel measure, outer regular with respect to the fine topology associated to the Bessel capacity $C_{\frac{2}{q},q'}$ in $\BBR^N$ ($q'=q/q-1)$) and absolutely continuous with respect to this capacity. If $\nu$ is a nonnegative Borel measure in $\BBR^N$ with the above properties we construct a positive solution $u$ of (E) with initial trace $\gn$ and we prove that this solution is the unique $\gs$-moderate solution of (E) with such an initial trace. Finally we prove that every positive solution of (E) is $\gs$-moderate.