Remarks on an elliptic problem arising in weighted energy estimates for wave equations with space-dependent damping term in an exterior domain

TL;DR

This paper extends weighted energy estimates and diffusion phenomena for wave equations with space-dependent damping in exterior domains, removing the restriction of radial symmetry in the damping coefficient.

Contribution

It modifies the elliptic problem approach to handle non-radially symmetric damping coefficients, broadening the applicability of previous methods.

Findings

Established weighted energy estimates for non-radial damping

Demonstrated diffusion phenomena in more general settings

Extended the elliptic problem framework to asymmetric coefficients

Abstract

This paper is concerned with weighted energy estimates and diffusion phenomena for the initial-boundary problem of the wave equation with space-dependent damping term in an exterior domain. In this analysis, an elliptic problem was introduced by Todorova and Yordanov. This attempt was quite useful when the coefficient of the damping term is radially symmetric. In this paper, by modifying their elliptic problem, we establish weighted energy estimates and diffusion phenomena even when the coefficient of the damping term is not radially symmetric.