Analytic and summable solutions of inhomogeneous moment partial differential equations

TL;DR

This paper investigates conditions under which formal solutions to inhomogeneous linear moment partial differential equations are convergent, analytically continuable, and summable, focusing on solutions in one or two variables with complex coefficients.

Contribution

It provides new criteria linking the properties of inhomogeneities to the summability and analytic continuation of solutions for a broad class of moment PDEs.

Findings

Established sufficient conditions for convergence of solutions.

Derived criteria for analytic continuation of solutions.

Analyzed summability in one and two variables.

Abstract

We study the Cauchy problem for a general inhomogeneous linear moment partial differential equation of two complex variables with constant coefficients, where the inhomogeneity is given by the formal power series. We state sufficient conditions for the convergence, analytic continuation and summability of formal power series solutions in terms of properties of the inhomogeneity. We consider both the summability in one variable t (with coefficients belonging to some Banach space of Gevrey series with respect to the second variable z) and the summability in two variables (t,z).