We consider the wave equation in a bounded region with a smooth boundary with distributed delay on the boundary or into the domain. In both cases, under suitable assumptions, we prove the exponential stability of the solution. These results are obtained by introducing suitable energies and by proving some observability inequalities. For an internal distributed delay, we further show some instability results.
Serge Nicaise. Cristina Pignotti. "Stabilization of the wave equation with boundary or internal distributed delay." Differential Integral Equations 21 (9-10) 935 - 958, 2008. https://doi.org/10.57262/die/1356038593