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In this paper, we investigate the viscoelastic wave equation with variable coefficients and a delay, which is subject to nonlinear and nonlocal boundary dissipation. The existence of strong and weak solutions is obtained by means of Faedo-Galerkin approximation and denseness argument. By introducing an equivalent energy functional and using the Riemannian geometry method, we show the general decay rate of energy when kernel function satisfies some sufficient small conditions that we provide the range accurately. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the stabilization of Timoshenko systems with or without Kelvin–Voigt damping, in the presence of second sound. We have drawn conclusions that provide new insights into the effect of local Kelvin–Voigt damping. Our primary conclusion is that the exponential stability of the system achieved through local thermal damping will be destroyed by the local Kelvin–Voigt damping with a discontinuous coefficient at the interface. In particular, when local dissipation solely originates from the second sound, the system exhibits exponential stability. However, when Kelvin–Voigt damping is introduced into the local thermoelastic component, the system, surprisingly, loses its exponential stability. In this case, the corresponding solution decays polynomially with the rate t - 2 . [ABSTRACT FROM AUTHOR]

In this paper, we consider a semilinear wave equation with variable coefficients in a smooth domain, subject to acoustic boundary conditions and dissipative boundary memory feedback, where a general Borel measure is involved. We show that the decay rates of the energy associated to the semilinear systems are given implicitly as solutions of a first order nonlinear, dissipative ODE, which recovers and extends some of the results from the literature. To prove the main result we use energy multiplier methods, geometric analysis combined with Lasiecka and Tataru arguments. [ABSTRACT FROM AUTHOR]

Uniform stabilization of a structural acoustic model describing an acoustic chamber with flexible curved wall is addressed. The coupled nonlinear system consists of a variable-coefficient wave equation and a shallow shell model which is used to model the flexible curved wall (interface). The coupling between the wave and the shell takes place on the interface. Derivation of stability estimates for the variable-coefficient coupled system with the shallow shell depends on the Riemannian geometry method and the multiplier technique. The uniform energy decay rates of the overall interactive model are achieved by introducing nonlinear boundary feedbacks applied to the wave equation and the shell model. [ABSTRACT FROM AUTHOR]

We investigate the variable-coefficient wave equation with nonlinear acoustic boundary conditions and logarithmic nonlinearity source term. In addition, the damping is nonlinear, and plays a role only in part of the boundary. Using the semigroup theory, we state the existence and uniqueness of the local solutions. Accordingly, we prove that under some assumptions on the initial data, the local solution can be extended to be global. Particularly, it has been shown that the decay rates of the system are given implicitly as solutions to a first-order ODE by applying the Riemannian geometry method. [ABSTRACT FROM AUTHOR]

In this paper, we consider stabilization of transmission of the wave equation with dynamical boundary conditions that include a delay term acts on the boundary. By introducing suitable energy function, we prove under some assumptions that the closed‐loop system is exponentially stable. In addition, we also consider the situation where the above system is controlled by distributed feedback near the transmission boundary and can also obtain exponential stability estimates with fewer assumptions. [ABSTRACT FROM AUTHOR]

This article is devoted to the study of the exact controllability for a system of weakly coupled one dimensional degenerate wave equations. An internal locally control acts on only one equation. We show that, when the coupling parameter which depends on the degree of degradation is sufficiently small, the observation of the velocity of the first component of the solution on a left neighborhood of x = 1 allow us to get back a weakened energy of initial data of the second component of the solution for a sufficiently large time T . Using the Hilbert Uniqueness Method, we then establish an indirect exact controllability result. [ABSTRACT FROM AUTHOR]

In this paper, we study one‐dimensional linear degenerate wave equations with a distributed controller. We establish observability inequalities for degenerate wave equation by multiplier method. We also deduce the exact controllability for degenerate wave equation by Hilbert uniqueness method when the control acts on the nondegenerate boundary. Moreover, an explicit expression for the controllability time is given. [ABSTRACT FROM AUTHOR]

In this paper, we consider exact control problem for a coupled system of plate with Gurtin‐Pipkin equation. Using duality arguments, the problem is reduced to the obtention of suitable observability estimates for the dual system. Firstly, we obtain the observability inequality of the dual system by means of multiplier method. Then, we prove that the system is exactly controllable based on the Hilbert Unique Method. [ABSTRACT FROM AUTHOR]

We consider a Timoshenko system with partially Kelvin–Voigt damping, and with Cattaneo/Fourier type heat conduction. When the Timoshenko system with only heat conduction or only partially Kelvin–Voigt damping, some authors showed the non-exponential stability. In this paper, we add the Kelvin–Voigt damping in the thermoelastic Timoshenko system, and prove that the system is also not exponentially stable in cases of Fourier and Cattaneo type heat conduction, whether the wave speeds are equal or not. Additionally, we prove the semigroup decays with the optimal rate t − 1 2 . [ABSTRACT FROM AUTHOR]

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]

In this paper, we consider global nonexistence of a solution for coupled quasilinear system with damping and source under Dirichlet boundary condition. We obtain a global nonexistence result of solution by using the perturbed energy method, where the initial energy is assumed to be positive. Copyright © 2017 John Wiley & Sons, Ltd. [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]

The wave equation of variable coefficients in an exterior domain with a localized half-linear dissipation is considered. The energy decay estimate for the system is established by Riemannian geometry method. [ABSTRACT FROM AUTHOR]

A variable-coefficient wave equation with acoustic boundary conditions and a time-varying delay in the boundary feedback is considered. Applying the Riemannian geometry method, we show that the decay rates of the nonlinear system with a time-varying delay are described by solutions to a first order ODE. [ABSTRACT FROM AUTHOR]

Two types of open-loop systems of an Euler-Bernoulli equation with variable coefficient and Dirichlet boundary control and collocated observation are considered. The uncontrolled boundary is either hinged or clamped. It is shown, with the help of multiplier method on Riemannian manifold, that in both cases, systems are well-posed in the sense of D. Salamon and regular in the sense of G. Weiss. In addition, the feedthrough operators are found to be zero. The result implies that the exact controllability of open-loop is equivalent to the exponential stability of closed-loop under a proportional output feedback for these systems. Copyright © 2013 John Wiley & Sons, Ltd. [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]

In this paper, we study one-dimensional linear degenerate wave equations in domains with moving boundary. We deduce the exact controllability for degenerate wave equation by Hilbert Uniqueness Method when the control acts on the moving boundary. Moreover, an explicit expression for the controllability time is given. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we establish some observability inequalities from both boundary and distributed controls for the transmission of shallow shell with a middle surface of any shape. In particular, if only the boundary controls are allowed, we obtain the condition for the observability inequality on Young''s modulus and Poisson''s coefficient of the material which is similar as those in the case of wave and plate equations. [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]

We consider the stabilization of the shallow shell by boundary feedbacks where the model has a middle surface of any shape.First,we put the shallow shell model in a suitable semigroup scheme.The existence,the uniqueness,and the properties of solutions to the shallow shell are then treated by the semigroup approach and the regularity of elliptic boundary value problems.Finally, we establish the uniform energy decay rate for the shallow shell under some checkable geometric conditions on the middle surface. [ABSTRACT FROM AUTHOR]

This paper studies the decay and the asymptotic behavior of solutions to the 3D incompressible Navier–Stokes equations with nonlinear damping α | u | β − 1 u (α > 0 , β > 1). We will prove the L 2 decay of weak solutions for β > 1 with any α > 0. Moreover, we concern the asymptotic stability of the solutions to the system for β > 1 with any α > 0. [ABSTRACT FROM AUTHOR]

Internal exact controllability for the nonlinear wave equation in one space dimension y t t − y x x + g ( y ) y t = h is studied, where g ( ⋅ ) is a nonnegative function. For the case, lim sup | s | → ∞ g ( s ) ln | s | < γ , we obtain the global exact controllability for the equation with Dirichlet boundary condition. The proof is based on the combination of fixed-point arguments and explicit observability estimates for the linearized wave equation with a potential that depends on both x and t . For the case, g ( s ) = | s | , we only get a local exact controllability by means of Banach fixed-point theorem. [ABSTRACT FROM AUTHOR]

In this paper, we consider performance output tracking for a wave equation with a general boundary disturbance. The control and the disturbance are unmatched. Different from the existing results, the hidden regularity, instead of the high gain or variable structure, is utilized in the adaptive servomechanism design. As a result, we are able to cope with more complicated and general disturbances. Moreover, the performance output can track the reference signal exponentially, and at the same time, all the states of the subsystems involved are uniformly bounded. More specially, the overall closed-loop system is exponentially stable when the disturbance and reference are disconnected to the system. The numerical simulations are presented to illustrate the effect of the proposed scheme. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we consider a nonlinear viscoelastic wave equation with nonlinear boundary damping and source terms. Under some appropriate assumptions on the relaxation function and with certain initial data, the global existence of solutions and a general decay for the energy have been established. [Copyright &y& Elsevier]

Abstract: A wave equation with variable coefficients in principal part and memory conditions on the boundary is considered. The Riemannian geometry method is applied to prove the exponential decay of the energy provided the relaxation function also decays exponentially. [Copyright &y& Elsevier]

Abstract: We consider the system of nonlinear wave equations with initial and Dirichlet boundary conditions. Under some suitable assumptions on the functions and , the initial data and the parameters in the equations, the theorems of global existence and nonexistence are proved. [Copyright &y& Elsevier]