Asymptotic expansion of solutions to the wave equation with space-dependent damping

Article type: Research Article

Authors: Sobajima, Motohiro^a | Wakasugi, Yuta^{b; *}

Affiliations: [a] Department of Mathematics, Faculty of Science and Technology, Tokyo University of Science, 2641 Yamazaki, Noda-shi, Chiba, 278-8510, Japan | [b] Laboratory of Mathematics, Graduate School of Advanced Science and Engineering, Hiroshima University, Higashi-Hiroshima, 739-8527, Japan

Correspondence: [*] Corresponding author. E-mail: [email protected].

Abstract: We study the large time behavior of solutions to the wave equation with space-dependent damping in an exterior domain. We show that if the damping is effective, then the solution is asymptotically expanded in terms of solutions of corresponding parabolic equations. The main idea to obtain the asymptotic expansion is the decomposition of the solution of the damped wave equation into the solution of the corresponding parabolic problem and the time derivative of the solution of the damped wave equation with certain inhomogeneous term and initial data. The estimate of the remainder term is an application of weighted energy methods with suitable supersolutions of the corresponding parabolic problem.

Keywords: Wave equation, space-dependent damping, asymptotic expansion

DOI: 10.3233/ASY-231834

Journal: Asymptotic Analysis, vol. 134, no. 1-2, pp. 241-279, 2023