Local Energy Decay for the Damped Wave Equation

Jean-Marc Bouclet (1) , Julien Royer (1)

1 IMT - Institut de Mathématiques de Toulouse UMR5219

We prove local energy decay for the damped wave equation on R^d. The problem which we consider is given by a long range metric perturbation of the Euclidean Laplacian with a short range absorption index. Under a geometric control assumption on the dissipation we obtain an almost optimal polynomial decay for the energy in suitable weighted spaces. The proof relies on uniform estimates for the corresponding "resolvent'', both for low and high frequencies. These estimates are given by an improved dissipative version of Mourre's commutators method.

Domaines: Physique mathématique [math-ph]; Equations aux dérivées partielles [math.AP]; Théorie spectrale [math.SP]

Jean-Marc Bouclet, Julien Royer. Local Energy Decay for the Damped Wave Equation. Journal of Functional Analysis, 2014, 266 (7), pp.Pages 4538-4615. ⟨10.1016/j.jfa.2014.01.028⟩. ⟨hal-00918736⟩