Stabilization of the weakly coupled wave-plate system with one internal damping

TL;DR

This paper investigates the stabilization of a coupled wave-plate system with only one damped component, demonstrating logarithmic decay of solutions without geometric constraints, using advanced Carleman estimates and resolvent analysis.

Contribution

It establishes logarithmic decay for a coupled wave-plate system with minimal damping assumptions, employing novel interpolation inequalities derived from Carleman estimates.

Findings

Solutions decay logarithmically at infinity

No geometric conditions needed on damping domain

Uses Carleman estimates for interpolation inequalities

Abstract

This paper is addressed to a stabilization problem of a system coupled by a wave and a Euler-Bernoulli plate equation. Only one equation is supposed to be damped. Under some assumption about the damping and the coupling terms, it is shown that sufficiently smooth solutions of the system decay logarithmically at infinity without any geometric conditions on the effective damping domain. The proofs of these decay results rely on the interpolation inequalities for the coupled elliptic-parabolic systems and make use of the estimate of the resolvent operator for the coupled system. The main tools to derive the desired interpolation inequalities are global Carleman estimates.