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

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 $s\ge 2$ 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 $s\ge 2$. 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.

arXiv comment9 pages. We corrected several mistakes in the previous version

References in corpus (2)

• Semi-uniform stability of operator semigroups and energy decay of damped waves
• The uncertainty principle and energy decay estimates of the fractional Klein-Gordon equation with space-dependent damping