Stabilization and Spectrum in Operator Differential Equations

We investigate the Cauchy problem for a second order non-stationary linear operator differential equation in a Hilbert space. We consider the case of an unbounded self-adjoint positive operator with a special regard to the Laplace operator with Dirichlet boundary conditions. The corresponding problem is a mixed problem for a wave equation. Introducing the energy class solution we prove its representation by the Bochner-Stiltjes integral. We establish the connection between spectral properties of the Laplace operator and stabilization for large time values of the solutions to the mixed problem of the wave equation. We investigate the asymptotic behavior in time of the local energy function for the various types of spectrum. For bounded domains where the spectrum of the Laplace operator is purely discrete we any solution with a local energy tends to zero in time is identically zero. For arbitrary domains in the case of operator with non-empty point spectrum we prove that there are smooth and finite initial functions for which the local energy function does not decay. In the cases of continuous and absolutely continuous spectrum of the Laplace operator we prove the mean decay and the decay of the local energy function respectively.

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