Energy decay in a wave guide with dissipation at infinity

¹ Laboratoire de Mathématique: Modélisation Déterministe et Aléatoire (LAMMDA), École Supérieure des Sciences et de la Technologie de Hammam Sousse, Université de Sousse, Rue Lamine Abassi, 4011 H. Sousse, Tunisia
mallougm70@gmail.com
² Institut de Mathématiques de Toulouse, Université Toulouse 3, 118 route de Narbonne, 31062 Toulouse Cedex 9, France

^a Corresponding author: julien.royer@math.univ-toulouse.fr

Received: 15 June 2016
Revised: 1 June 2017
Accepted: 12 August 2017

Abstract

We prove local and global energy decay for the wave equation in a wave guide with damping at infinity. More precisely, the absorption index is assumed to converge slowly to a positive constant, and we obtain the diffusive phenomenon typical for the contribution of low frequencies when the damping is effective at infinity. On the other hand, the usual Geometric Control Condition is not necessarily satisfied so we may have a loss of regularity for the contribution of high frequencies. Since our results are new even in the Euclidean space, we also state a similar result in this case.