An initial-boundary-value problem for hyperbolic differential-operator equations on a finite interval. (Q873209)

An abstract interpretation of initial-boundary-value problems for hyperbolic type equations in Hilbert spaces is given. Under a series of assumptions imposed on operators appearing in the equations and the boundary conditions, the well-posedness of the Cauchy problem is proved. An expansion of the solutions by means of eigenvectors is shown in special cases. A generalization of the Fourier method is given as an application.