Problems with integral conditions with respect to time variable for linear evolution equations and systems of equations. Metric approach to the problem of small denominators

Abstract

We give a survey of the papers concerning the research of the well-posedness, in a multidimensional layer, of the problems with general nonlocal conditions with respect to the time variable which contain integral terms in the form of the arbitrary order moments of the unknown function. The problems are considered for hyperbolic and parabolic equations and systems of equations, for mixed type equations, and for equations unsolved with respect to the higher time derivative in the class of functions that are almost periodic with respect to the spatial variables. Generally, these problems are conditionally well-posed, and their solvability is related to the problem of small denominators, which is solved by using the metric approach.