Abstract
We establish the uniqueness criterion for a solution to the operator $ \frac{\partial^{2p}}{\partial t^{2p}}-A(x,D) $ with the Dirichlet time-dependent boundary conditions and general boundary conditions in the space variables. The order of the differentiation operator $ \frac{\partial^{2p}}{\partial t^{2p}} $ is assumed even. Note that $ A(x,D) $ in the space variables is an arbitrary elliptic operator with some rather general boundary operators $ B_{j} $ obeying the conventional Agmon conditions. The Agmon conditions ensure the existence of a complete orthonormal system of eigenfunctions (in $ L_{2}(\Omega) $ ) provided that $ \Omega $ is a bounded domain with sufficiently smooth boundary.