Stabilization of linear waves with inhomogeneous Neumann boundary conditions

TL;DR

This paper investigates the stabilization of linear wave equations with inhomogeneous Neumann boundary conditions, establishing global existence and decay results, and providing numerical insights into energy behavior under various boundary data scenarios.

Contribution

It introduces new analysis techniques for wave equations with inhomogeneous Neumann boundary conditions, proving global existence and uniform stabilization, which were not fully understood before.

Findings

Proved global existence of solutions.

Established uniform decay rates for solutions.

Numerical simulations illustrate energy behavior with improper boundary data.

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 stabilization of solutions with decay rates characterized 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.