Optimality of energy decay rates for fractional wave equations with Hölder damping

Construct, on the circle T = R/2πZ, sequences of real numbers λ_n → ∞ and L2-normalized functions u_n such that for the damping χ_β(x) = (cos x)^{2β} 1_{(-π/2, π/2)}(x) with β ∈ [0,1], the residual of the stationary operator P(λ_n) = |D|^α − i λ_n χ_β − λ_n^2 satisfies ||P(λ_n) u_n||_{L^2} ≤ C λ_n^{2(1+ν#/α)}, where ν# = min(−1, 2β + α/2 − 2). This establishes the optimality of the polynomial energy decay rate E(u,t) ≤ C⟨t⟩^{−γ#} proved in the main theorem for the fractional wave equation (∂_t^2 + χ ∂_t + |D|^α)u = 0 under the geometric control condition.

Background

The paper proves explicit polynomial energy decay rates for solutions to the damped fractional wave equation (∂_t^2 + χ ∂_t + |D|^α)u = 0 on a closed Riemannian manifold when the geometric control condition holds and the damping satisfies √χ ∈ C^{0,β}. The decay exponent depends on the Hölder regularity β of the damping.

The authors explicitly raise the question of optimality and conjecture that their decay rates are optimal. Using results of Anantharaman–Léautaud, they reduce the optimality question on the circle T = R/2πZ to constructing quasimodes with a specific residual bound for a model damping χβ(x) = (cos x)^{2β} 1{(-π/2, π/2)}(x). Achieving this construction would show that the resolvent estimate obtained in the paper is sharp and thus that the associated polynomial energy decay cannot be improved.