On the energy decay rates for the 1D damped fractional Klein–Gordon equation

Abstract

We consider the fractional Klein-Gordon equation in one spatial dimension, subjected to a damping coefficient, which is non-trivial and periodic, or more generally strictly positive on a periodic set. We show that the energy of the solution decays at the polynomial ratefor 0 < < 2 and at some exponential rate when ≥ 2. Our approach is based on the asymptotic theory of 0 semigroups in which one can relate the decay rate of the energy in terms of the resolvent growth of the semigroup generator. The main technical result is a new observability estimate for the fractional Laplacian, which may be of independent interest. K E Y W O R D S damped wave equation, fractional derivative, geometric control condition M S C ( 2 0 1 0 ) 35B35, 35B40, 35G30

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

“…Thus, the interest in this problem is in interpolating between these cases. The fractional model (1) was introduced recently by Malhi and Stanislavova in [9]. Our results are inspired by their paper, but we are even able to recover what is known in the classical case of s = 2 from a new perspective.…”

“…Proof of Theorem 2. Now, to prove the threshold value (Theorem 2), we use the fact that exponential decay yields (8) with c independent of λ, from which (9) follows. Suppose that s < 2.…”

“…The above result does not say anything about the optimality of the rates. However, we can answer the question posed in [9] concerning the value of the threshold between exponential and polynomial decay. In the final section, we show that exponential decay neccesitates that s be greater than 2 (as long as γ is not bounded away from zero), thus establishing s = 2 as the threshold.…”

“…The initial investigation into the fractional case in [9] required the restrictive assumption that the set {x ∈ R : γ(x) ≥ ε} contains a periodic set. In this case, it was proved that the rate of decay is polynomial if s < 2 and exponential if s ≥ 2.…”

Uncertainty Principles Associated to Sets Satisfying the Geometric Control Condition

“…Suppose that E satisfies the k-GCC. Then for any δ > 0 and β > 0, there is a constant C > 0 such that for every R > 0, We consider the fractional damped wave equation recently introduced by Malhi and Stanislavova in [22].…”