arXiv 2212.024812022-12-25math.APmath.CA

含空间阻尼的分数阶Klein-Gordon方程的测不准原理和能量衰减估计

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

Soichiro Suzuki

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

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摘要：

考虑了$\mathbb{R}^d$上具有空间依赖阻尼的$s$ -分数阶Klein-Gordon方程.最近的研究表明，所谓的几何控制条件（GCC）密切相关的半群估计的方程。特别地，在$d = 1$的情况下，对于任何$s > 0$，已知关于GCC的指数稳定性的充分必要条件。另一方面，在$d \geq 2$和$s \geq 2$的情况下，Green-Jaye-Mitkovski（2022）证明了“$1$ -GCC”对于指数稳定性是足够的，但也证明了如果$s$足够大，则不需要。本文证明了指数稳定性与傅立叶分析中的一种测不准原理的等价性。作为等价性的结果，我们证明了在$s \geq 4$的情况下，$1$ -GCC对于指数稳定性是不必要的。此外，我们还建立了关于$s$的外推结果。特别地，我们可以从某些$s > 2$的指数稳定性获得非分数情况$s = 2$的多项式稳定性。

英文摘要：

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 $s > 2$.

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