L^p-tauberian theorems and L^p-rates for energy decay

Charles Batty (1) , Alexander Borichev (2) , Yuri Tomilov (3)

1 Centre for Mathematical Biology, Mathematical Institute
2 I2M - Institut de Mathématiques de Marseille
3 MIMUW - Faculty of Mathematics, Informatics, and Mechanics [Warsaw]

We prove $L^p$-analogues of the classical tauberian theorem of Ingham and Karamata, and its variations giving rates of decay. These results are applied to derive $L^p$-decay of operator families arising in the study of the decay of energy for damped wave equations and local energy for wave equations in exterior domains. By constructing some examples of critical behaviour we show that the $L^p$-rates of decay obtained in this way are best possible under our assumptions.

Mots clés: Tauberian theorem; Energy decay; Damped wave equation; Laplace transforms

Domaines: Analyse fonctionnelle [math.FA]; Equations aux dérivées partielles [math.AP]; Variables complexes [math.CV]; Systèmes dynamiques [math.DS]

Charles Batty, Alexander Borichev, Yuri Tomilov. L^p-tauberian theorems and L^p-rates for energy decay. Journal of Functional Analysis, 2016, 270 (3), pp.1153-1201. ⟨10.1016/j.jfa.2015.12.003⟩. ⟨hal-01258057⟩