On the energy decay rate of the fractional wave equation on ℝ with relatively dense damping

Green, Walton

doi:10.1090/proc/15100

On the energy decay rate of the fractional wave equation on $\mathbb {R}$ with relatively dense damping
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Proc. Amer. Math. Soc. 148 (2020), 4745-4753 Request permission

Abstract:

We establish upper bounds for the decay rate of the energy of the damped fractional wave equation when the averages of the damping coefficient on all intervals of a fixed length are bounded below. If the power of the fractional Laplacian, $s$, is between 0 and 2, the decay is polynomial. For $s \ge 2$, the decay is exponential. Our assumption is also necessary for energy decay. Second, we prove that exponential decay cannot hold for $s<2$ if the damping vanishes at all.

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