Diffusion phenomena for the wave equation with space-dependent damping term growing at infinity

TL;DR

This paper investigates the long-term behavior of solutions to a wave equation with space-dependent damping that increases at infinity, showing they resemble solutions to a heat equation over time.

Contribution

It introduces a novel analysis of wave equations with unbounded damping, connecting their asymptotics to heat equation solutions using semigroup and weighted energy estimates.

Findings

Solutions asymptotically behave like heat equation solutions

Weighted energy estimates are effective for unbounded damping

Elliptic problem analysis aids in constructing weight functions

Abstract

In this paper, we study the asymptotic behavior of solutions to the wave equation with damping depending on the space variable and growing at the spatial infinity. We prove that the solution is approximated by that of the corresponding heat equation as time tends to infinity. The proof is based on semigroup estimates for the corresponding heat equation and weighted energy estimates for the damped wave equation. To construct a suitable weight function for the energy estimates, we study a certain elliptic problem.