Local solvability and dilation-critical singularities of supercritical fractional heat equations

TL;DR

This paper investigates the local solvability of supercritical fractional heat equations, introducing the concept of dilation-critical singularities (DCS) and providing conditions and formulas for their existence.

Contribution

It defines dilation-critical singularities for supercritical fractional heat equations and establishes their existence and precise characterization.

Findings

Dilation-critical singularities (DCS) always exist for a broad class of supercritical nonlinearities.

Necessary and sufficient conditions for local-in-time solvability are established.

Exact formulas for DCS are derived.

Abstract

We consider the Cauchy problem for fractional semilinear heat equations with supercritical nonlinearities and establish both necessary conditions and sufficient conditions for local-in-time solvability. We introduce the notion of a dilation-critical singularity (DCS) of the initial data and show that such singularities always exist for a large class of supercritical nonlinearities. Moreover, we provide exact formulae for such singularities.