On the Energy Decay Rate of the Fractional Wave Equation on $\mathbb{R}$ with Relatively Dense Damping

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. Second, we show that our assumption on the damping is necessary for the energy to decay exponentially.