An Analytic Cauchy Problem in the Class of Functions with an Integral Metric in Spatial and Temporal Variables

Abstract

The article considers the complex Cauchy problem for general systems of linear differential equations in Lebesgue's Banach spaces with the weight of analytic functions with integral metrics. The functions from the solution spaces can admit power law-type singularities in integral sense when approaching the cone lateral boundary. The necessary and sufficient conditions under which the stated Cauchy problem is a locally well- posed one in the specified scale of functional spaces are derived. It turns out that these conditions for the orders of differential operators in case of extended singularities of solutions over the cone lateral boundary are identical with the Leray-Volevich conditions. In case of one equation containing the Kovalevskaya inequalities, these conditions are less restrictive than they are in the case of extended singularities of the solution over the cylinder lateral surface. Thus, the inequalities either exactly describe the structure of systems of linear differential equations for which the Cauchy problem is locally well-posed in the specified class, or they describe the scale of functional spaces in which the specified Cauchy problem is locally a well-posed one. In the course of proving the main theorem, new lemmas about the integral properties of analytic functions have been obtained, which may be of interest for the theory of functions of complex variable. These lemmas show an estimate for the norm of derivative from an analytic function through the norm of the function itself; they also show an estimate of the norm of integral with a parameter from the analytic function through the norm of the initial function, as well as other properties.