F-stability, entropy and energy gap for supercritical Fujita equation

TL;DR

This paper introduces F-stability, entropy, and energy gap concepts for supercritical Fujita equations, revealing that constant solutions minimize entropy among bounded self-similar solutions and establishing geometric properties of blow-up sets.

Contribution

It defines new stability and entropy notions for supercritical Fujita equations and proves entropy gaps and geometric structure results for blow-up solutions.

Findings

Constant solutions have the lowest entropy among bounded self-similar solutions.

There exists a gap in entropy between constant and non-constant solutions.

The blow-up set of type I solutions is composed of a rectifiable set and a lower-dimensional set.

Abstract

We study some problems on self similar solutions to the Fujita equation when $p>(n+2)/(n-2)$, especially, the characterization of constant solutions by the energy. Motivated by recent advances in mean curvature flows, we introduce the notion of $F-$functional, $F$-stability and entropy for solutions of supercritical Fujita equation. Using these tools, we prove that among bounded positive self similar solutions, the constant solution has the lowest entropy. Furthermore, there is also a gap between the entropy of constant and non-constant solutions. As an application of these results, we prove that if $p>(n+2)/(n-2)$, then the blow up set of type I blow up solutions is the union of a $(n-1)-$ rectifiable set and a set of Hausdorff dimension at most $n-3$.