Fast energy decay for wave equations with a localized damping in the n-D half space

Article type: Research Article

Authors: Ikehata, Ryo

Affiliations: Department of Mathematics, Graduate School of Education, Hiroshima University, Higashi-Hiroshima 739-8524, Japan. E-mail: ikehatar@hiroshima-u.ac.jp

Abstract: We consider a mixed problem for wave equations with a localized damping in the n-dimensional half space R+n. 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 L2-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.

Keywords: Localized damping, wave equation, half space, weighted initial data, multiplier method, Fourier analysis, total energy decay

DOI: 10.3233/ASY-171420

Journal: Asymptotic Analysis, vol. 103, no. 1-2, pp. 77-94, 2017