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

Abstract

We consider damped s s -fractional Klein–Gordon equations on R d \mathbb {R}^d , where s s denotes the order of the fractional Laplacian. In the one-dimensional case d = 1 d = 1 , Green (2020) established that the exponential decay for s ≥ 2 s \geq 2 and the polynomial decay of order s / ( 4 − 2 s ) 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 ) o(1) energy decay is also equivalent to these conditions in the case d = 1 d=1 . Furthermore, we extend this result to the higher-dimensional case: the logarithmic decay, the o ( 1 ) o(1) decay, and the thickness of the damping coefficient are equivalent for s ≥ 2 s \geq 2 . In addition, we also prove that the exponential decay holds for 0 > s > 2 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.