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This paper investigates the regularity properties of the solution of a second-order hyperbolic equation defined over a bounded domain Ω with boundary Γ, under the action of a boundary forcing term inL 2(0,T; L 2(Γ)). Both Dirichlet and Neumann nonhomogeneous cases are considered. A functional analytic model based on cosine operator functions is presented, which provides an input-solution formula to be interpreted in appropriate topologies. With the help of this model, it is shown, for example, that the solution of the nonhomogeneous Dirichlet problem is inL 2(0,T; L 2(Ω)), when Ω is either a parallelepiped or a sphere, while the solution of the nonhomogeneous Neumann problem is inL 2(0,T; H 3/4-e(Ω)) when Ω is a parallelepiped and inL 2(0,T; H 2/3(Ω) when Ω is a sphere. The Dirichlet case for general domains is studied by means of pseudodifferential operator techniques.
Applied Mathematics and Optimization – Springer Journals
Published: Mar 23, 2005
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