an interaction of a homogeneous viscous incompressible fluid which occupies a domain ${{\cal O}}$ bounded by the (solid) walls of the container $S$ and a horizontal (flat) boundary ${{\Omega}}$ on which a thin (nonlinear) elastic shell is placed. The motion of the fluid is described by linearized 3D Navier–Stokes equations. To describe deformations of the shell we use the full von Karman shallow shell model which accounts for both transversal and in-plane displacements. For details concerning the shell model chosen we refer to [@Vor1957; @lagnese; @lag-2] and also to the papers [@las-SIAM98; @las-AA98; @las-CPDE99; @puel-AMO96; @Sed91b; @Sed_1997] and the references therein.

This fluid-structure interaction model assumes that large deflections of the shell produce small effect on the fluid. This corresponds to the case when the fluid fills the container which is large in comparison with the size of the plate.

We note that the mathematical studies of the problem of fluid–structure interaction in the case of viscous fluids and elastic plates/bodies have a long history. We refer to [@CDEG05; @Chu_2010; @ChuRyz2011; @Ggob-jmfm08; @Ggob-aa09; @Ggob-mmas09; @Kop98] and the references therein for the case of plates/membranes, to [@CS06] in the case of moving elastic bodies, and to [@avalos-amo07; @aval-tri07; @aval-tri09; @BGLT07; @BGLT08; 