Problem on vibration of a bar with nonlinear second-order boundary damping

In this paper, we study the initial-boundary problem with nonlinear dynamical boundary condition for the pseudohyperbolic equation. This problem represents a mathematical model of longitudinal vibration in a thick short bar with dynamic nonlinear second-order boundary damping. The existence and uniqueness of a generalized solution are proved. The proof is based on a priori estimates and Galerkin procedure. This approach allows to construct approximation in the suitable for practical application form.