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We consider damped $s$-fractional Klein--Gordon equations on $\mathbb{R}^d$,where $s$ denotes the order of the fractional Laplacian. In the one-dimensionalcase $d = 1$, Green (2020) established that the exponential decay for $s \geq2$ and the polynomial decay of order $s/(4-2s)$ hold if and only if the dampingcoefficient function satisfies the so-called geometric control condition. Inthis note, we show that the $o(1)$ energy decay is also equivalent to theseconditions in the case $d=1$. Furthermore, we extend this result to thehigher-dimensional case: the logarithmic decay, the $o(1)$ decay, and thethickness of the damping coefficient are equivalent for $s \geq 2$. Inaddition, we also prove that the exponential decay holds for $0 < s < 2$ if andonly if the damping coefficient function has a positive lower bound, so inparticular, we cannot expect the exponential decay under the geometric controlcondition.

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