Regularity and asymptotic behaviour for a damped plate-membrane transmission problem

TL;DR

This paper investigates the stability and regularity of solutions in a coupled damped plate and wave equation system, demonstrating exponential stability in fully damped cases and polynomial stability otherwise, with improved regularity results.

Contribution

It establishes stability conditions for a coupled plate-wave system and proves higher regularity of solutions in the weak formulation.

Findings

Exponential stability in damped-damped systems

Polynomial stability in damped-undamped systems

Solutions have higher Sobolev regularity than initially defined

Abstract

We consider a transmission problem where a structurally damped plate equation is coupled with a damped or undamped wave equation by transmission conditions. We show that exponential stability holds in the damped-damped situation and polynomial stability (but no exponential stability) holds in the damped-undamped case. Additionally, we show that the solutions first defined by the weak formulation, in fact have higher Sobolev space regularity.