  In this work, the dynamics of a multilayered structure-fluid interaction
(FSI) PDE system is considered. Here, the coupling of 3D Stokes and 3D elastic
dynamics is realized via an additional 2D elastic equation on the boundary
interface. Such modeling PDE systems appear in the mathematical modeling of
eukaryotic cells and vascular blood flow in mammalian arteries. We analyze the
long time behavior of solutions to such FSI coupled system in the sense of
strong stability.
  Our proof is based on an analysis of the spectrum of the associated semigroup
generator $\mathbf{A}$ which in particular entails the elimination of all three
parts of the spectrum of $\mathbf{A}$ from the imaginary axis. In order to
avoid steady states in our stability analysis, we firstly show that zero is an
eigenvalue for the operator $\mathbf{A}$, and we provide a characterization of
the (one dimensional) zero eigenspace $Null(\mathbf{A}).$ In turn, we address
the issue of asymptotic decay of the solution to the zero state for any initial
data taken from the orthogonal complement of the zero eigenspace
$Null(\mathbf{A})^{\bot}.$
