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

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.