Year: 2025
Type: article
Abstract: We study some problems on self-similar solutions to the Fujita equation when p > ( n + 2 ) / ( n − 2 ) p>(n+2)/(n-2) , especially, the characterization of constant solutions by its 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 nonzero self-similar solutions, the constant solutions have 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 ) p>(n+2)/(n-2) , then the blow-up set of type I blow-up solutions is the union of an ( n − 1 ) (n-1) -rectifiable set and a set of Hausdorff dimension at most n − 3 n-3 .
Authors: Kelei Wang, Juncheng Wei, Ke Wu
Institutions: Wuhan University, Chinese University of Hong Kong
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4407702957