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

Soichiro Suzuki
Dec 2022
摘要
We consider the $s$-fractional Klein-Gordon equation with space-dependentdamping on $\mathbb{R}^d$. Recent studies reveal that the so-called geometriccontrol conditions (GCC) are closely related to semigroup estimates of theequation. Particularly, in the case $d = 1$, a necessary and sufficientcondition 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 theexponential stability, but also conjectured that it is not necessary if $s$ issufficiently large. In this paper, we prove the equivalence between theexponential stability and a kind of the uncertainty principle in Fourieranalysis. As a consequence of the equivalence, we show that the 1-GCC is notnecessary for the exponential stability in the case $s \geq 4$.
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