Energy decay for a locally undamped wave equation

Abstract

We study the decay rate for the energy of solutions of a damped wave equation in a situation where the Geometric Control Condition is violated. We assume that the set of undamped trajectories is a flat torus of positive codimension and that the metric is locally flat around this set. We further assume that the damping function enjoys locally a prescribed homogeneity near the undamped set in traversal directions. We prove a sharp decay estimate at a polynomial rate that depends on the homogeneity of the damping function. Our method relies on a refined microlocal analysis linked to a second microlocalization procedure to cut the phase space into tiny regions respecting the uncertainty principle but way too small to enter a standard semiclassical analysis localization. Using a multiplier method, we obtain the energy estimates in each region and we then patch the microlocal estimates together.

Concentration of Laplace Eigenfunctions and Stabilization of Weakly Damped Wave Equation

“…Remark 1.5. -Notice that in Theorem 1.4 the decay rate is worst than the rates exhibited by Leautaud-Lerner [18] in the particular case when the submanifold Σ is a torus (and the metric of M is flat near Σ). We shall exhibit below examples showing that the rate (1.6) is optimal in general.…”

“…Here we want to explore the question when some trajectories are trapped and exhibit decay rates (assuming more regularity on the initial data). This latter question was previously studied in a general setting in [19] and on tori in [11,21,1] (see also [12,13]) and more recently by Leautaud-Lerner [18]. According to the works by Borichev-Tomilov [3], stabilization results for the wave equation are equivalent to resolvent estimates.…”

Rational invariant tori and band edge spectra for non-selfadjoint operators

“…While in the considerations of Section 7, we are concerned with operators on the one-dimensional torus, it will be convenient to analyze the case of R first. See also [15].…”

Optimal decay rate for the wave equation on a square with constant damping on a strip

“…Unfortunately they could not characterize the exact decay rate in terms of properties of a. A breakthrough into this direction was achieved in [6] in a slightly different situation (there S degenerates to a line).…”