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

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.