On diffusion phenomena for the linear wave equation with space-dependent damping

TL;DR

This paper demonstrates that solutions to the linear wave equation with space-dependent damping asymptotically behave like solutions to the heat equation, establishing a diffusion phenomenon in the $L^2$ sense.

Contribution

It proves the diffusion phenomenon for the wave equation with space-dependent damping, linking wave dynamics to heat equation behavior.

Findings

Solution asymptotically approaches heat equation profile

Diffusion phenomenon established in $L^2$-sense

Provides rigorous proof for space-dependent damping case

Abstract

In this paper, we prove the diffusion phenomenon for the linear wave equation with space-dependent damping. We prove that the asymptotic profile of the solution is given by a solution of the corresponding heat equation in the $L^2$-sense.