A problem with nonlocal condition for one-dimensional hyperbolic equation

In this paper, we study the problem with a dynamic nonlocal condition for the one-dimensional hyperbolic equation, which occurs in the study of rod vibrations. This problem may be used as a mathematical model of longitudinal vibration in a thick short bar and illustrates a nonlocal approach to such processes. Conditions have been obtained for input data, providing unambiguous resolution of the task, proof of the existence and singularity of the problem in the space of Sobolev. The proof is based on the a priori estimates obtained in this paper, Galerkin’s procedure and the properties of the Sobolev spaces.