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An initial-boundary-value problem for hyperbolic differential-operator equations on a finite interval. (English) Zbl 1164.34458
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.

34G10 Linear differential equations in abstract spaces
35L15 Initial value problems for second-order hyperbolic equations
35L20 Initial-boundary value problems for second-order hyperbolic equations