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

Abstract

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. IntroductionThe problem of summability of formal solutions of linear PDEs was mainly studied under the assumption that the Cauchy data are convergent, see Balser The case of more general initial data was investigated only for the complex heat equation (see Balser [1,4]). In [1] Balser considered the case of entire initial data with an appropriate growth condition and he gave some preliminary results for divergent initial data, too. Next, these results were extended in [4], where a characterisation of summable formal power series solutions of the complex heat equation in terms of properties of divergent Cauchy data was given.The aim of our paper is a generalisation of Balser's results [1, 4] to homogeneous linear partial differential equations with constant coefficients.Namely, we consider the initial value problem for a general linear partial differential equation with constant coefficients in two complex variables (t, z) P (∂ t , ∂ z ) u = 0, ∂ j t u(0, z) = ϕ j (z) (j = 0, . . . , n − 1),2010 Mathematics Subject Classification. 35C10, 35C15, 35E15, 40G10.

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 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.…”

“…As in the previous papers [10,11] we use the moment pseudodi¤erential operators l i ðq m 2 ; z Þ to factorise the operator P as Pðq m 1 ; t ; q m 2 ; z Þ ¼ P 0 ðq m 2 ; z Þðq m 1 ; t À l 1 ðq m 2 ; z ÞÞ n 1 . .…”