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

TL;DR

This paper explores the relationship between energy decay rates of fractional damped Klein-Gordon equations and geometric conditions on damping coefficients, extending known results to higher dimensions and different fractional orders.

Contribution

It establishes equivalences between energy decay types and geometric conditions for damping coefficients across various fractional orders and dimensions.

Findings

Energy decay is equivalent to geometric control conditions in 1D.

Logarithmic and $o(1)$ decay are equivalent to damping thickness in higher dimensions.

Exponential decay occurs only if damping has a positive lower bound for $s<2$.

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.