论文标题
抑制klein-gordon方程的能量衰减与阻尼系数的几何条件之间的等效性
Equivalence between the energy decay of fractional damped Klein-Gordon equations and geometric conditions for damping coefficients
论文作者
论文摘要
我们认为$ \ mathbb {r}^d $上的damped $ s-s $ frestional klein- gordon方程，其中$ s $表示分数laplacian的顺序。在一维情况下，$ d = 1 $，绿色（2020年）确定，$ s \ geq 2 $的指数衰减以及订单$ s/（4-2s）$ hold的多项式衰减，并且仅当减速系数功能满足所谓的几何控制条件时。在本说明中，我们表明$ o（1）$能量衰减也等同于这些情况，在$ d = 1 $的情况下。此外，我们将此结果扩展到更高维的情况：对数衰减，$ o（1）$衰减和阻尼系数的厚度等效于$ s \ geq 2 $。此外，我们还证明，指数衰减的价格为$ 0 <s <2 $，并且仅当减速系数函数具有正下限，因此，特别是，我们不能指望在几何控制条件下指数衰减。
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.
