Equivalence between the energy decay of fractional damped Klein-Gordon equations and geometric conditions for damping coefficients

We consider damped $s$-fractional Klein–Gordon equations on ${\mathbb{R}}^d$, where $s$ denotes the order of the fractional Laplacian. In the one-dimensional case $d=1$, Green (2020) established that the exponential decay for $s\ge 2$ and the polynomial decay of order $s\mathrm{/}(4-2s)$ hold if and only if the damping coefficient function satisfies the so-called geometric control condition. In this note, we show that the $o(1)$ energy decay is also equivalent to these conditions in the case $d=1$. Furthermore, we extend this result to the higher-dimensional case: the logarithmic decay, the $o(1)$ decay, and the thickness of the damping coefficient are equivalent for $s\ge 2$. In addition, we also prove that the exponential decay holds for $0<s<2$ if and only if the damping coefficient function has a positive lower bound, so in particular, we cannot expect the exponential decay under the geometric control condition.