Analytic Solutions of Moment Partial Differential Equations with Constant Coefficients

Abstract

Abstract. We consider the Cauchy problem for linear moment partial di¤erential equations with constant coe‰cients in two complex variables. We construct an integral representation of the solution of this problem and study its analyticity. As a result we derive a characterisation of multisummable formal solutions of the Cauchy problem.

Analytic and Summable Solutions of Inhomogeneous Moment Partial Differential Equations

“…The theory of moment partial di¤erential equations was developed by the author [10,11]. In [10] formal, analytic and summable solutions of homogeneous moment partial di¤erential equations were studied.…”

“…In [10] formal, analytic and summable solutions of homogeneous moment partial di¤erential equations were studied. In the same paper the author also calculated normalised formal solutions in the inhomogeneous case and he determined their Gevrey orders.…”

“…In the same paper the author also calculated normalised formal solutions in the inhomogeneous case and he determined their Gevrey orders. Next, in [11] the author extended the results of [10] to homogeneous equations with divergent Cauchy data. Moreover, Lastra, Malek and Sanz [7] generalised the results of [10] about homogeneous moment equations to the case when moment functions are determined by given strongly regular moment sequences.…”

“…The present paper is a natural generalisation of [10,11] to the inhomogeneous case with the inhomogeneity given by the formal power series. More precisely, we consider the initial value problem for a general inhomogeneous linear moment partial di¤erential equation of two complex variables ðt; zÞ with constant coe‰cients Pðq m 1 ; t ; q m 2 ; z Þû u ¼f f ðt; zÞ; q j m 1 ; tû uð0; zÞ ¼ 0 for j ¼ 0; .…”

“…; n À 1; ð1Þ where Pðl; zÞ is a polynomial of two variables of degree n with respect to l and the inhomogeneityf f ðt; zÞ is a formal power series in both variables of Gevrey order ðs s 1 ;s s 2 Þ A R 2 , and m 1 , m 2 are moment functions of orders s 1 , s 2 respectively. Similarly to [11], we consider the wider class of moment functions than in [6] or [10], which creates the group with respect to multiplication. This extension allows us to study some integro-di¤erential operators as moment di¤erential operators.…”