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

TL;DR

This paper investigates how the energy of a damped fractional wave equation on the real line decays over time, establishing conditions under which decay is polynomial or exponential based on the fractional Laplacian's order.

Contribution

It provides new upper bounds for energy decay rates and demonstrates the necessity of certain damping conditions for exponential decay in fractional wave equations.

Findings

Energy decay is polynomial for 0 < s < 2.

Energy decay is exponential for s ≥ 2.

Damping conditions are necessary for exponential decay.

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