Abstract
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.
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Communicated by A. V. Balakrishnan
This research was supported in part by the Air Force Office of Scientific Research under Grant AFOSR-78-3350 (1st author) and Grant AFOSR-77-3338 (2nd author).
This research was performed while the author was visiting the Department of System Science, University of California, Los Angeles.
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Lasiecka, I., Triggiani, R. A cosine operator approach to modelingL 2(0,T; L 2 (Γ))—Boundary input hyperbolic equations. Appl Math Optim 7, 35–93 (1981). https://doi.org/10.1007/BF01442108
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DOI: https://doi.org/10.1007/BF01442108