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This paper is devoted to giving definitions of Besov spaces on an arbitrary open set of R n via the spectral theorem for the Schrödinger operator with the Dirichlet boundary condition. The crucial point is to introduce some test function spaces on Ω. The fundamental properties of Besov spaces are also shown, such as embedding relations and duality, etc. Furthermore, the isomorphism relations are established among the Besov spaces in which regularity of functions is measured by the Dirichlet Laplacian and the Schrödinger operators.

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We shall find asymptotic profiles for strictly hyperbolic equations with time-dependent coefficients which are of Lipschitz class and have some stability condition. More precisely, it will be shown that there exists a solution which is not asymptotically free provided that the coefficient tends slowly to some constant.