Two-point boundary value problem for a partial differential equation in spaces of periodic functions

Abstract

We investigate the two-point in time boundary value problem for the partial differential equations of the second-order with one spatial variable and constant coefficients. The problem is considered in in the spaces of functions which Fourier coefficients are characterized by exponential behavior on the Cartesian product of the time interval and spatial domain $\mathbb{R}/2\pi\mathbb{Z}$. The correct solvability of the problem is established, the formulas for solutions are presented, the kernel is described and the smoothness of the solution is established in the spaces of functions that are periodic in one spatial variable. We have established the conditions which are close to the necessary conditions of solvability of the problem in scale of spaces of functions with exponentially increasing (or decreasing) Fourier coefficients.We also found the asymptotic estimates demonstrating the absence of the problem of small denominators, which arises of many spatial variables and makes the boundary value problem incorrect. We have established sufficient conditions of the finite-dimensionality of the kernel of the problem and found upper bounds for its dimension. The results are obtained under the condition of minimum smoothness on the right-hand sides of two-point conditions, which is close to the necessary condition.