Remarks on an elliptic problem arising in weighted energy estimates for   wave equations with space-dependent damping term in an exterior domain
  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. 
Motohiro Sobajima and Yuta Wakasugi
math.AP