Conjugation problem with initial-nonlocal conditions for factorized higher order equations

Abstract

In this paper, we consider a problem in the cylindrical domain $(-\alpha,\beta)\times(\mathbb{R}/2\pi\mathbb{Z})$ separated by the hyperplane $\{0\}\times(\mathbb{R}/2\pi\mathbb{Z})$ into nonoverlapping cylindrical subdomains. In particular, this problem can be interpreted as the problem of finding the solution of two factorized PDEs defined in these subdomains respectively, which satisfies the conjugation conditions on the hyperplane as well as the initial-nonlocal conditions on the bottom and top surfaces of the domain.

By the method of separation of variables, the solution can be represented formally as Fourier series, but there is a question about the convergence of the given series in Sobolev spaces of periodic functions with respect to the spatial variables. This convergence is related to the problem of small denominators and may be unstable with respect to small variations of the coefficients of the problem or the parameters of the domain.

We establish the estimates for the small denominators ensuring the convergence of the solutions, from which we obtain the sufficient conditions for the solvability of the problem in Sobolev spaces. The obtained results show that the solvability of the problem depends on the coefficients of the differential equations.