Decay rates for Kelvin-Voigt damped wave equations II: The geometric control condition

Abstract

We study in this article decay rates for Kelvin-Voigt damped wave equations under a geometric control condition. When the damping coefficient is sufficiently smooth ( C 1 C^1 vanishing nicely, see the following equation: | ∇ a | ⩽ C a 1 2 |\nabla a|\leqslant Ca^{\frac {1}{2}} ) we show that exponential decay follows from geometric control conditions (see Nicolas Burq and Hans Christianson [Comm. Math. Phys. 336 (2015), pp. 101–130] and Louis Tebou [C. R. Math. Acad. Sci. Paris 350 (2012), pp. 603–608] for similar results under stronger assumptions on the damping function).