Summary: We consider a simplified version of a fluid-structure PDE model – in fact, a heat-structure interaction PDE-model. It is intended to be a first step toward a more realistic fluid-structure PDE model which has been of longstanding interest within the mathematical and biological sciences. The simplified model replaces the linear dynamic Stokes equation with a linear \(n\)-dimensional heat equation (heat-structure interaction). The entire dynamics manifests both hyperbolic and parabolic features. Our main result is as follows: Given smooth initial data – i.e., data in the domain of the associated semigroup generator – the corresponding solutions decay at the rate \(o(t^{-\frac{1}{2}}) \). The basis of our proof is the recently derived resolvent criterion in [A. Borichev and Y. Tomilov, Math. Ann. 347, No. 2, 455–478 (2010; Zbl 1185.47044)]. In order to apply it, however, suitable PDE-estimates need to be established for each component by also making critical use of the interface conditions.