On solvability of the initial-boundary value problems for a nonlocal hyperbolic equation with periodic boundary conditions

Abstract

In this paper, the solvability of initial-boundary value problems for a nonlocal analogue of a hyperbolic equation in a cylindrical domain is studied. The elliptic part of the considered equation involves a nonlocal Laplace operator, which is introduced using involution-type mappings. Two types of boundary conditions are considered. These conditions are specified as a relationship between the values of the unknown function at points in one half of the lateral part of the cylinder and the values at points in the other part of the cylinder boundary. The boundary conditions specified in this form generalize known periodic and antiperiodic boundary conditions for circular domains. The unknown function is represented in the form u(x) = v(x) + w(x), where v(x) is the even part of the function and w(x) is the odd part of the function with respect to the mapping. Using the properties of these functions, we obtain auxiliary initial-boundary value problems with classical hyperbolic equations. In this case, the boundary conditions of these problems are specified in the form of the Dirichlet and Neumann conditions. Further, using the known assertions for the auxiliary problems, theorems on the existence and uniqueness of the solution to the main problems are proved. The solutions to the problems are constructed as a series in systems of eigenfunctions of the Dirichlet and Neumann problems for the classical Laplace operator.