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Initial-boundary value problem for systems of equations describing wave motion in a fluid layer on an elastic base. (English. Russian original) Zbl 0781.76010
Math. Notes 52, No. 1, 748-750 (1992); translation from Mat. Zametki 52, No. 1, 150-153 (1992).
The title initial-boundary value problem is $$\nabla^ 2\psi= 0$$, $$-H< z< 0$$, $$\nabla^ 2 U+ (c^ 2- 1)\nabla\text{ div } U= \mu^ 2 U_{tt}$$, $$z<- H$$, $$\psi_{tt}+ \psi_ z= 0$$, $$z= 0$$, where $$\psi$$ is the potential of the moving liquid $$(-H\leq z\leq 0)$$ and $$U$$ is the displacement vector in the elastic half-space $$z\leq -H$$. A sketch of the proof of the following statement is given: the system in question can be reduced to the form $$\partial^ 2 W/\partial t^ 2={\mathcal L}W$$, where $$\mathcal L$$ is a certain essentially self-adjoint operator generated by an explicitly written linear system of equations. The spectrum of $$\widehat{\mathcal L}$$ consists of discrete and continuous parts, and the corresponding eigenfunctions form a complete orthonormal system with respect to a certain scalar product.
##### MSC:
 76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction 74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.) 35Q35 PDEs in connection with fluid mechanics
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##### References:
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