Stabilisation of linear waves with inhomogeneous Neumann boundary conditions

Abstract

We study linear damped and viscoelastic wave equations evolving on a bounded domain. For both models, we assume that waves are subject to an inhomogeneous Neumann boundary condition on a portion of the domain's boundary. The analysis of these models presents additional interesting features and challenges compared to their homogeneous counterparts. In the present context, energy depends on the boundary trace of velocity. It is not clear in advance how this quantity should be controlled based on the given data, due to regularity issues. However, we establish global existence and also prove uniform stabilisation of solutions with decay rates characterised by the Neumann input. We supplement these results with numerical simulations in which the data do not necessarily satisfy the given assumptions for decay. These simulations provide, at a numerical level, insights into how energy could possibly change in the presence of, for example, improper data.