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.

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.

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