Dirichlet-Neumann problem for the partial differential equations with deviation over the space argument

Abstract

Dirichlet-Neumann problem for the typeless high order partial differential equation with deviating over the space argument is studied in the domain, which is the Cartesian product of the segment $(0,T)$ and the unit circle $\Omega=\mathbb R/(2\pi \mathbb Z)$. Dirichlet-Neumann problem for hyperbolic equations and their systems in case with absent argument deviation $h$ has been studied by the authors before. Correct solvability conditions have been established for these problems for almost all (with respect to Lebesgue measure) numbers $T>0$ and for almost all (with respect to Lebesgue measure) vectors, constructed by coefficients of the equation. In this paper, the solvability conditions of the problem for $h\ne0$ are described and the influence of the deviation $h$ on the solvability of the problem is studied. The solution of the problem is constructed in the form of the series with respect to the systems of orthogonal functions. Metric estimations (of exponential type) are proved for small denominators appearing during construction of the problem solution. These estimations guarantee the correctness of the problem in Sobolev spaces for almost all (with respect to Lebesgue measure) values $ T> 0 $ and for almost all (with respect to Lebesgue measure) values $ h \in [0,2\pi) $. The obtained results are based on the fact that the corresponding characteristic determinant permits factorization in the form of the product of hyperbolic functions with integer parameters.

Keywords

• dirichlet-neumann problem
• partial differential equation
• deviation over the space argument
• small denominators
• metric approach