A problem on longitudinal vibration in a short bar with dynamical boundary conditions

In this paper, we consider an initial-boundary problem with dynamical nonlocal boundary condition for a pseudohyperbolic fourth-order equation in a rectangular. Dynamical nonlocal boundary condition represents a relation between values of a required solution, its derivatives with respect of spacial variables, second-order derivatives with respect of time-variables and an integral term. 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. The main result lies in justification of solvability of this problem. Existence and uniqueness of a generalized solution are proved. The proof is based on the a priori estimates obtained in this paper, Galerkin's procedure and the properties of the Sobolev spaces.