Stabilization of hyperbolic equations with mixed boundary conditions

Abstract

This paper is devoted to study decay properties of solutions to hyperbolic equations in a bounded domain with two types of dissipative mechanisms, i.e. either with a small boundary or an internal damping. Both of the equations are equipped with the mixed boundary conditions. When the Geometric Control Condition on the dissipative region is not satisfied, we show that sufficiently smooth solutions to the equations decay logarithmically, under sharp regularity assumptions on the coefficients, the damping and the boundary of the domain involved in the equations. Our decay results rely on an analysis of the size of resolvent operators for hyperbolic equations on the imaginary axis. To derive this kind of resolvent estimates, we employ global Carleman estimates for elliptic equations with mixed boundary conditions.