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 $x = \infty$.
Ryo Ikehata. Takeshi Komatsu. "Fast energy decay for wave equations with variable damping coefficients in the $1$-D half line." Differential Integral Equations 29 (5/6) 421 - 440, May/June 2016. https://doi.org/10.57262/die/1457536885