Logarithmic Decay of Hyperbolic Equations with Arbitrary Small Boundary Damping

Abstract

This paper addresses an analysis on the longtime behavior of the hyperbolic equations with a partially boundary damping, under sharp regularity assumptions on the coefficients appeared in the equation. Based on a global Carleman estimate, we establish an estimate on the underlying resolvent operator of the equation, via which we show the logarithmic decay rate for solutions of the hyperbolic equations without any geometric assumption on the subboundary in which the damping is effective.