This paper analyzes the long time behavior of a linearized model for fluid-structure interaction. The space domain consists of two parts in which the evolution is governed by the heat equation and the wave equation respectively, with transmission conditions at the interface. Based on the construction of ray-like solutions by means of Geometric Optics expansions and a careful analysis of the transfer of the energy at the interface, we show the lack of uniform decay of solutions in general domains. Also, we prove the polynomial decay result for smooth solutions under a suitable Geometric Control Condition. This condition requires that all rays propagating in the wave domain reach the interface in a uniform time after, possibly, bouncing in the exterior boundary. © 2004 Elsevier SAS. All rights reserved.
Rauch, J., Zhang, X., & Zuazua, E. (2005). Polynomial decay for a hyperbolic-parabolic coupled system. Journal Des Mathematiques Pures et Appliquees, 84(4), 407–470. https://doi.org/10.1016/j.matpur.2004.09.006