Optimal singularities of initial data of a fractional semilinear heat equation in open sets

TL;DR

This paper investigates the precise conditions under which a fractional semilinear heat equation has solutions, focusing on the optimal singularities of initial data in various open set geometries.

Contribution

It establishes necessary and sufficient conditions for solvability, identifying the optimal singularity strength of initial data in open sets with boundary considerations.

Findings

Characterization of optimal initial data singularities for solvability

Differentiation of conditions between interior and boundary points

Conditions applicable to possibly unbounded and disconnected sets

Abstract

We consider necessary conditions and sufficient conditions on the solvability of the Cauchy--Dirichlet problem for a fractional semilinear heat equation in open sets (possibly unbounded and disconnected) with a smooth boundary. Our conditions enable us to identify the optimal strength of the admissible singularity of initial data for the local-in-time solvability and they differ in the interior of the set and on the boundary of the set.