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This paper is concerned with the uniform energy decay rates of a structural acoustic model which describes the interactions between the acoustic medium and the elastic structural in an acoustic chamber where one “wall” is flexible and curved. The coupled system considered consists of a variable-coefficient wave equation and a variable-coefficient plate equation defined on the Riemannian manifold with the coupling on the interface between the acoustic medium and the flexible wall. Both the components of the dynamics are subject to nonlinear boundary damping. Furthermore, a nonlinear delay term acting in the boundary feedbacks of the structure component is considered. The Riemannian geometry method is applied to derive the energy estimates of the nonlinear coupled system with variable coefficients. The uniform energy decay rates of the nonlinear variable-coefficient structural acoustic system with delay effects on the Riemannian manifold are quantified by a solution to a constructed nonlinear ODE. [ABSTRACT FROM AUTHOR]

This paper deals with a Cauchy problem for the elasticity system with a structural damping. The decay rates of the total energy of the system are presented by applying the Fourier transform and multiplier in the Fourier space. Furthermore, almost optimal decay estimates of the total energy are obtained based on the Haraux–Komornik inequality. [ABSTRACT FROM AUTHOR]

This paper is concerned with the existence and energy decay estimates of solutions to the variable-coefficient Euler-Bernoulli plate with a delay in localized nonlinear internal feedback. The existence of solutions to the nonlinear system is obtained by using Galerkin method combined with some energy estimates. Furthermore, applying Riemannian geometry method, we establish uniform decay rates of the plate equation with variable coefficients and a nonlinear delay term. [ABSTRACT FROM AUTHOR]

Abstract: A system of Naghdi''s shell equation with Dirichlet boundary control and collocated observation is considered. Results on the associated nonhomogeneous boundary value problem are presented. Based on these results, it is shown that the system is well-posed in the sense of D. Salamon and regular in the sense of G. Weiss. The expression of the corresponding feedthrough operator is explicitly found by means of Riemannian geometric method and partial Fourier transform. These properties make this partial differential control system parallel in many ways finite-dimensional ones in the general framework of well-posed and regular infinite-dimensional systems. [Copyright &y& Elsevier]

Abstract: We present the uniqueness in the Cauchy problem for the Koiter shell where the model is described by the Riemannian geometry. The Carleman estimates are established by the Bochner technique which yields the uniqueness for the Koiter shell. [Copyright &y& Elsevier]

Abstract: A thermoelastic plate with variable coefficients in the absence of dissipative boundary mechanisms is considered. Using -semigroup theory and some techniques in partial differential equations, it is shown that the corresponding -semigroup is analytic and hence the exponential stability of the system is concluded. [Copyright &y& Elsevier]

Abstract: We consider the stabilization of the transmission problem of Naghdi''s model by boundary feedbacks where the model has a middle surface of any shape. The exponential decay rate for the problem is established under some checkable geometric conditions on the middle surface. [Copyright &y& Elsevier]