Summability of formal solutions of linear partial differential equations with divergent initial data

TL;DR

This paper investigates the summability of formal solutions to linear PDEs with divergent initial data, establishing conditions for summability in one and two variables within the framework of moment-PDEs.

Contribution

It provides necessary and sufficient conditions for summability of formal solutions based on properties of divergent initial data, extending the theory to moment-PDEs.

Findings

Conditions for summability in one variable t.

Conditions for summability in two variables t and z.

Application to moment-PDEs framework.

Abstract

We study the Cauchy problem for a general homogeneous linear partial differential equation in two complex variables with constant coefficients and with divergent initial data. We state necessary and sufficient conditions for the summability of formal power series solutions in terms of properties of divergent Cauchy data. 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). The results are presented in the general framework of moment-PDEs.