Sharp Decay Rates for the Weakly Coupled Hyperbolic System with One Internal Damping

Abstract

This paper analyzes the longtime behavior of a system of two hyperbolic equations that are coupled by a localized zero order term with a coupling function $c(\cdot)$, with either Dirichlet boundary conditions or Neumann boundary conditions. Only one hyperbolic equation is supposed to be damped with a damping function $d(\cdot)$. Under the assumption of supp $c(\cdot)\cap$ supp $d(\cdot)\ne \emptyset$, it is shown that sufficiently smooth solutions of the system decay logarithmically at infinity without any geometric conditions on the effective damping domain. The proof of these decay results relies on the interpolation inequalities for the coupled elliptic system and makes use of the estimate on the resolvent operator for the coupled hyperbolic system. The main tool to derive the desired interpolation inequalities is the global Carleman estimate.