Long-time behavior of a coupled heat-wave system arising in fluid-structure interaction

This paper is devoted to the analysis of the long-time behavior of a coupled wave-heat system in which a wave and a heat equation evolve in two bounded domains, with natural transmission conditions at a common interface. These conditions couple, in particular, the heat unknown with the velocity of the wave solution. This model may be viewed as a simplified version of linearized models which arise in fluid-structure interaction. First, we show the strong asymptotic stability of solutions to this system. Then, based on the construction of ray-like solutions by means of geometric optics expansions and a careful analysis of the transfer of energy at the interface, we show the lack of uniform decay in general domains. Further, we obtain a polynomial decay result for smooth solutions of the system under a suitable geometric assumption which guarantees that the heat domain envelopes the wave domain. Finally, in the absence of geometric conditions we show a logarithmic decay result for the same system but with simplified transmission conditions at the interface. We also analyze the difficulty there is to extend this result to the more natural transmission conditions.