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SUMMARYIn this paper we consider the question of stabilization of a fluid-structure model that describes the interaction between a 3-D incompressible fluid and a 2-D plate, the interface, which coincides with a flat flexible part of the surface of the vessel containing the fluid. The mathematical model comprises the Stokes equations and the equations for the longitudinal deflections of the plate with inclusion of the shear stress, which the fluid exerts on the plate. We show that the energy associated with the model decays strongly when the interface is equipped with a locally supported dissipative mechanism. Our main tool is an abstract resolvent criterion due to Tomilov.

This paper is concerned with a model which describes the interaction of sound and elastic waves in a structural acoustic chamber in which one "wall" is flexible and flat. The model is new in the sense that the composite dynamics of the three-dimensional structure is described by the linearized equations for a gas defined on the interior of the chamber and the Reissner-Mindlin plate equations on the two-dimensional flat wall of the chamber, while, if a two-dimensional acoustic chamber is considered, the Timoshenko beam equations describe the deflections of the one-dimensional "wall." With a view to achieving uniform stabilization of the structure linear feedback boundary damping is incorporated in the model, viz. in the wave equation for the gas and in the system of equations for the vibrations of the elastic medium. We present the uniform stability result for the case of a twodimensional chamber and outline the method for the three-dimensional model which shows strong resemblance with the system of dynamic plane elasticity.

In obtaining in Theorem 4.1 of the original paper, the strong asymptotic decay of the semigroup exp(−tC −1 A) associated with P r (P), the argumentation following (4.5) on p. 1060 is incorrect. Also, with regard to the polynomial decay of the semigroup associated with the model, the conclusion (4.15) on p. 1061 from (4.14) requires substantiation. Whereas the strong asymptotic stability of the original model P r (P) can be achieved by rectification of the argumentation, it turns out, with regard to the polynomial stability, that the conclusion (4.15) can only be accomplished when [ψ, φ], the vector of shear angles, is divergence-free. This requires the analysis of a modification of P r (P) in which the constitutive equations include the equation ∇ · [ψ, φ] = 0 in the domain Ω. For the semigroup associated with the modified model, P r (P), we are able to establish a decay rate of (1 t) 1 2 , which is a faster decay rate than the rate (1 t) 1 4 achieved for P r (P) in the original paper. 1