Summability of Formal Power Series Solutions of Partial Differential Equations with Constant Coefficients

We study Gevrey properties and summability of power series in two variables that are formal solutions of a Cauchy problem for general linear partial differential equations with constant coefficents. Doing so, we extend earlier results in two articles of , resp. , for the complex heat equation, as well as in a paper of , who have investigated the same questions for a certain class of linear PDE with constant coefficients subject to some restrictive assumptions. Moreover, we also present an example of a PDE, where the formal solution of the Cauchy problem is not -summable for whatever value of , but instead is multi-summable with two levels under corresponding conditions upon the Cauchy data. That this can occur has not been observed up to now