论文标题
不确定性原理和能量衰减估计分数Klein-gordon方程与空间依赖性阻尼
The uncertainty principle and energy decay estimates of the fractional Klein-Gordon equation with space-dependent damping
论文作者
论文摘要
我们考虑$ s元素klein-gordon方程，并在$ \ mathbb {r}^d $上进行空间依赖性阻尼。最近的研究表明，所谓的几何控制条件（GCC）与方程式的半群估计密切相关。特别是，在$ d = 1 $的情况下，就GCC而言，指数稳定性的必要条件是任何$ S> 0 $。另一方面，在情况下，$ d \ geq 2 $和$ s \ geq 2 $，绿色jaye-mitkovski（2022）证明，“ $ 1 $ -GCC”足以满足指数稳定性，但也认为如果$ S $足够大，则不必要。在本文中，我们证明了指数稳定性与傅立叶分析中一种不确定性原理之间的等效性。由于等效性，我们表明$ 1 $ -GCC对于$ s \ geq 4 $的指数稳定性不是必需的。此外，我们还建立了针对$ S $的外推结果。特别是，我们可以从某些$ s> 2 $的指数稳定性中获得非骨折的情况下的多项式稳定性$ s = 2 $。
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 $d \geq 2$ and $s \geq 2$, 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 $s \geq 4$. 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 $s > 2$.
