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

TL;DR

This paper investigates the long-term behavior of solutions to the wave equation with space-dependent damping, demonstrating that solutions can be asymptotically expanded using solutions of related parabolic equations, with precise remainder estimates.

Contribution

It introduces a novel asymptotic expansion method for damped wave equations with space-dependent damping, utilizing a decomposition approach and weighted energy estimates.

Findings

Solutions exhibit asymptotic expansion in terms of parabolic solutions

Effective damping leads to specific large-time behavior

Weighted energy methods provide precise remainder estimates

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 method with suitable supersolutions of the corresponding parabolic problem.