Equivalence between the energy decay of fractional damped Klein-Gordon equations and geometric conditions for damping coefficients
Kotaro Inami, Soichiro Suzuki
TL;DR
This work characterizes how geometric properties of the damping coefficient $\gamma$ govern energy decay for damped $s$-fractional Klein--Gordon equations on $\mathbb{R}^d$. It couples semigroup theory with an uncertainty-principle-based resolvent framework to connect spectral conditions (absence of spectrum on the imaginary axis and resolvent bounds) to decay rates, identifying GCC in $d=1$ and thickness in $d\ge 2$ as the sharp geometric criteria. The authors prove that exponential decay for $s\ge 2$ is equivalent to GCC (or thickness) and that for $0<s<2$ exponential decay requires $\inf \gamma>0$, while logarithmic decay in higher dimensions corresponds to thick damping. A key technical ingredient is a Kovrijkine-type resolvent estimate on thick sets, which, via Burq--Joly theory, yields precise decay rates and clarifies how geometry dictates energy dissipation in fractional damping regimes.
Abstract
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.