Gevrey Order of Formal Power Series Solutions of Inhomogeneous Partial Differential Equations with Constant Coefficients

Abstract

In an earlier paper, the first author showed that certain normalized formal solutions of homogeneous linear partial differential equations with constant coefficients are multisummable, with a multisummability type that can be determined from a Newton polygon associated with the PDE. In this article, some of the results obtained there are extended in several directions: First of all, arbitrary formal solutions of inhomogeous PDE are considered, and it is shown that, in some sense, they can be computed completely explicitly. Secondly, the Gevrey order of these formal solutions is determined. Finally, formal power series are discussed that, in general, do not satisfy a PDE with constant coefficients, but instead may be considered as solutions of singularly perturbed ODE, or integro-differential equations of a certain form. Introduction In [3, 6], the first author introduced and studied normalized formal solutions of a Cauchy problem for general homogeneous linear partial differential equations in two variables having constant coefficients. Multisummability of these formal power series was then investigated in [4]. In detail, it has been shown that, under the assumption that the initial condition used is holomorphic near the origin, one can determine a multisummability type corresponding naturally to the PDE under consideration. The normalized formal solution then is multisummable in a given multidirection, provided that the initial condition can be continued into finitely many (small) sectors, and in every such sector is at most of a certain exponential growth that, in general, depends upon the sector. The multisummability type, the location of the sectors, and the corresponding ∗Institut fur Angewandte Analysis, Universitat Ulm, D–89069 Ulm, Germany †Department of Mathematics, Graduate School of Science, Hiroshima University, 1-3-1 Kagamiyama, Higashi-Hiroshima 739-8526 Japan