The uncertainty principle and energy decay estimates of the fractional Klein-Gordon equation with space-dependent damping

Abstract

We consider the $s$-fractional Klein-Gordon equation with space-dependent damping on ${\mathbb{R}}^d$. Recent studies reveal that the so-called geometric control conditions (GCC) are closely related to semigroup estimates of the equation. Particularly, in the case $d=1$, a necessary and sufficient condition for the exponential stability in terms of GCC is known for any $s>0$. On the other hand, in the case $d\ge 2$ and $s\ge 2$, Green-Jaye-Mitkovski (2022) proved that an `$1$-GCC' is sufficient for the exponential stability, but also conjectured that it is not necessary if $s$ is sufficiently large. In this paper, we prove the equivalence between the exponential stability and a kind of the uncertainty principle in Fourier analysis. As a consequence of the equivalence, we show that the $1$-GCC is not necessary for the exponential stability in the case $s\ge 4$. Furthermore, we also establish an extrapolation result with respect to $s$. In particular, we can obtain the polynomial stability for the non-fractional case $s=2$ from the exponential stability for some $s>2$.

26 pages. Added an explicit estimate of $\omega$ in Theorem 1.2 and simplified arguments in Section 3.2. Also fixed some typos

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