Necessary conditions for the solvability of fractional semilinear heat equations in the very weak framework

TL;DR

This paper establishes necessary initial value conditions for solving fractional semilinear heat equations within the very weak solution framework, extending previous results from integral solutions and introducing a new proof method.

Contribution

It introduces a novel proof technique to derive necessary conditions for very weak solutions, broadening the applicability beyond integral solutions.

Findings

Necessary conditions for solvability in very weak framework

New proof method applicable to fractional semilinear heat equations

Conditions are more general than previous integral solution results

Abstract

In this paper we obtain necessary conditions on the initial value for the solvability of the Cauchy problem for semilinear heat equations. These necessary conditions were already obtained in the framework of integral solutions, but not in that of very weak ones. We establish a new proof method, which can derive the desired conditions in the framework of very weak solutions. In particular, since any integral solution is a very weak solution, our conditions are more general.