On an initial-boundary value problem arising in the dynamics

A linearized problem of small motions of a system of layers of incompressible ideal stratified fluids with a free surface, covered with the elastic ice that is modeled by the elastic plate, is considered. With using the method of the orthogonal projecting the boundary conditions on the moving surface, the original initial-boundary value problem is reduced to the equivalent Cauchy problem for an ordinary differential equation of the second order in some Hilbert space. We find the conditions under which there exists a time-strong solution to the initial-boundary value problem describing the evolution of the original hydrodynamics system.