부산대학교 도서관

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 \geq 2 and the polynomial decay of order s/(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 \geq 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. [ABSTRACT FROM AUTHOR]