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

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 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), then the blow-up set of type I blow-up solutions is the union of an (n-1) -rectifiable set and a set of Hausdorff dimension at most n-3.