Fast energy decay for wave equations with variable damping coefficients in the 1-D half line

TL;DR

This paper establishes rapid energy decay rates for one-dimensional wave equations with spatially variable damping that vanishes at the boundary and is effective at infinity, providing insights into energy dissipation in such systems.

Contribution

It introduces new decay estimates for wave equations with localized variable damping in the half line, especially where damping vanishes near the boundary and is effective at infinity.

Findings

Derived fast decay estimates for energy in the half line

Demonstrated critical damping effectiveness at infinity

Analyzed the impact of variable damping vanishing at boundary

Abstract

We derive fast decay estimates of the total energy for wave equations with localized variable damping coefficients, which are dealt with in the one dimensional half line $(0,\infty)$. The variable damping coefficient vanishes near the boundary $x = 0$, and is effective critically near spatial infinity $x = \infty$.