Uncertainty Principles Associated to Sets Satisfying the Geometric Control Condition

Semi-uniform stability of operator semigroups and energy decay of damped waves

“…The classical damped wave equation discussed here is, of course, just one (prototypical) example of how abstract semigroup results may be applied to concrete PDEs. We conclude by mentioning a small selection of other interesting applications of semigroup methods to differential equations, along with sample references: damped wave equations on unbounded domains/manifolds [4,39,51,83,85,104,117]; local energy decay for damped wave equations [31,130]; Klein-Gordon and Kelvin-Vogt type equations [7,36,151]; energy decay for non-linear damped wave equations [24,81,82,119]; vectorial damped wave equations [86]; damped wave equations with unbounded and/or indefinite dampings [1,12,25,61,62]; viscoelastic boundary dampings [140]; wave equations with periodic (or even general non-stationary) dampings [83,96]; fractional damped wave equations [65,66]; damped wave equations on manifolds with rough metrics [143]. Even though the subject of damped wave equations is already vast, we hope and expect that the stream of substantial advances in this area, whether obtained by abstract techniques or otherwise, will continue for many years to come.…”

Exponential decay for damped Klein–Gordon equations on asymptotically cylindrical and conic manifolds

“…In [27] the results of [8,31] were extended to highly oscillatory periodic dampings. See also, inexhaustively listed here, [9,14,15,22,23] for recent development on Euclidean spaces.…”