Diffusion phenomena for the wave equation with space-dependent damping in an exterior domain

TL;DR

This paper studies how solutions to a damped wave equation in an exterior domain behave over time, showing they resemble heat equation solutions when damping is effective, with proven decay rates and optimality.

Contribution

It establishes the asymptotic equivalence between damped wave solutions and heat solutions in exterior domains with space-dependent damping, including decay rate optimality.

Findings

Solutions approximate heat equation solutions asymptotically

Decay rates are proven to be optimal

Effective damping leads to heat-like behavior

Abstract

In this paper, we consider the asymptotic behavior of solutions to the wave equation with space-dependent damping in an exterior domain. We prove that when the damping is effective, the solution is approximated by that of the corresponding heat equation as time tends to infinity. Our proof is based on semigroup estimates for the corresponding heat equation and weighted energy estimates for the damped wave equation. The optimality of the decay late for solutions is also established.