Note on the Exponential Stability of the Viscoelastic Wave Equation

Abstract: In this article, we study the viscoelastic wave equation with a localized history memory term, and without frictional damping. Under the geometric control condition on the memory region, we prove exponential decay of the viscoelastic energy using the Gearhart-Pr{ü}ss-Huang Theorem. We prove in high frequency that a loss of derivative is compensated by a decay in frequency.