We consider a mixed problem for wave equations with a localized damping in the n-dimensional half space . We first propose a new type of Hardy inequality in the whole space by employing the Fourier transform. Then, this inequality combined with a multiplier method developed by [Sci. Math. Japon.55 (2002), 33–42] will be effectively applied to the half space problems of the damped wave equations with a variable coefficient to catch a fast energy decay property. The Fourier transform is partially used to obtain the -boundedness of the solution. The method introduced in this paper will compensate lack of an effective Hardy type inequality in the n-dimensional half space.
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