In this paper, we consider two internal stabilization problems for the multi-dimensional wave equation with a boundary time-delay. We prove that the first problem is well-posed in an appropriate functional space. Subsequently, we numerically study the exponential stability in a two-dimensional case under Geometric Control Condition (GCC) derived by Lebeau. In addition, we provide a numerical investigation of the second wave system, which corresponds to the two-dimensional variant of the system studied by Datko et al.
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