In this talk, we will consider certain fluid-structure PDE models of longstanding interest in the mathematical and biological sciences. Within the structure of such fluid-structure dynamics, their respective PDE components will generally come into contact via a boundary interface. Consequently, the fluid-structure PDE entire will manifest both hyperbolic and parabolic features. For these PDE models, we will discuss spectral, higher regularity, and stability properties. In particular, we will discuss the problem of attaining rational rates of decay for smooth solutions.