Formal power series solutions of the heat equation in one spatial variable

Abstract

In this article we investigate formal power series solutions of the heat equation in one spatial variable. In previous work of Lutz, Miyake, and Schäfke, resp. of W. Balser, solutions of a Cauchy problem have been shown to be k-summable in a direction d if, and only if, the initial condition satisfies a certain condition. Here, we investigate the initial value problem for the spatial variable, finding new results especially for the case when the initial values are Gevrey functions of order larger than one, so that the corresponding power series solution diverges.