Logarithmic decay of the energy for an hyperbolic-parabolic coupled system

Laboratoire LMV, Université de Versailles Saint-Quentin-en-Yvelines, 45 Avenue des États-Unis, Bâtiment Fermat, 78035 Versailles, France. ines.fathallah@math.uvsq.fr

Received: 24 December 2008
Revised: 14 November 2009
Revised: 31 March 2010

Abstract

This paper is devoted to the study of a coupled system which consists of a wave equation and a heat equation coupled through a transmission condition along a steady interface. This system is a linearized model for fluid-structure interaction introduced by Rauch, Zhang and Zuazua for a simple transmission condition and by Zhang and Zuazua for a natural transmission condition. Using an abstract theorem of Burq and a new Carleman estimate proved near the interface, we complete the results obtained by Zhang and Zuazua and by Duyckaerts. We prove, without a Geometric Control Condition, a logarithmic decay of the energy.