General decay of a transmission problem for kirchhoff type wave equations with boundary memory condition

In this paper, we investigate the influence of boundary dissipation on the decay property of solutions for a transmission problem of Kirchhoff type wave equation with boundary memory condition. By introducing suitable energy and Lyapunov functionals, we establish a general decay estimate for the energy, which depends on the behavior of relaxation function.     

一类具有动力边界条件的进化方程解的衰减性

本文研究了一类具有动力边界条件的方程解的衰减性.利用能量扰动法,得到了解的衰减性与外力f(x,t)之间的关系,即它们具有相同的指数衰减性和代数衰减性.     

Observabilité, contrôlabilité et stabilisation frontière du système d’élasticité linéaire

We study homogeneous but not necessarily isotropic linear elastodynamic systems. Applying the Hilbert Uniqueness Method (HUM), we establish their boundary observability and their exact boundary controllability. Then we construct boundary feedbacks leading to arbitrarily large energy decay rates. Finally, under some geometrical conditions, we prove that a “natural” boundary feedback leads also to exponentiel energy decay.     

Existence and decay rates for a coupled Balakrishnan-Taylor viscoelastic system with dynamic boundary conditions

In this paper, we are concerned with a coupled viscoelastic wave system with Balakrishnan-Taylor dampings, dynamic boundary conditions, source terms, and past histories. Under suitable assumptions on relaxation functions and source terms, we prove the global existence of solutions by potential well theory and we establish a more general decay result of energy, in which the exponential decay and polynomial decay are only special cases, by introducing suitable energy and perturbed Lyapunov functionals.     

Strong stability and uniform decay of solutions to a wave equation with semilinear porous acoustic boundary conditions

We consider a wave equation with semilinear porous acoustic boundary conditions. This is a coupled system of second and first order in time partial differential equations, with possibly semilinear boundary conditions on the interface. The results obtained are (i) strong stability for the linear model, (ii) exponential decay rates for the energy of the linear model, and (iii) local exponential decay rates for the energy of the semilinear model. This work builds on a previous result showing generation of a well-posed dynamical system. The main tools used in the proofs are (i) the Stability Theorem of Arendt-Batty, (ii) energy methods used in the study of a wave equation with boundary damping, and (iii) an abstract result of I. Lasiecka applicable to hyperbolic-like systems with nonlinearly perturbed boundary conditions.     

Stability of a transmission problem for Kirchhoff‐type wave equations with memory on the boundary

In this paper, we investigate the influence of boundary dissipation on the decay property of solutions for a transmission problem of Kirchhoff‐type wave equations with a memory condition on one part of the boundary. Without the condition u₀ = 0 on Γ₀, we establish a general decay of energy depending on the behavior of relaxation function by introducing suitable energy and Lyapunov functionals. This result allows a wider class of relaxation functions. Copyright © 2016 John Wiley & Sons, Ltd.     

Energy Decay of Solutions to a Nondegenerate Wave Equation with a Fractional Boundary Control

In this paper, we study the energy decay rate for a one-dimensional nondegenerate wave equation under a fractional control applied at the boundary. We proved the polynomial decay result with an estimation of the decay rates. Our result is established using the frequency-domain method and Borichev-Tomilov theorem.     

Existence and general decay for nondissipative hyperbolic differential inclusions with acoustic/memory boundary conditions

In this paper, we prove the existence and general energy decay rate of global solution to the mixed problem for nondissipative multi‐valued hyperbolic differential inclusions with memory boundary conditions on a portion of the boundary and acoustic boundary conditions on the rest of it. For the existence of solutions, we prove the global existence of weak solution by using Galerkin's method and compactness arguments. For the energy decay rates, we first consider the general nonlinear case of h satisfying a smallness condition, and prove the general energy decay rate by using perturbed modified energy method. Then, we consider the linear case of h: and prove the general decay estimates of equivalent energy.     

Nonlinear stabilization of abstract systems via a linear observability inequality and application to vibrating PDE's

This Note is concerned with the links between nonlinear stabilization of hyperbolic systems and linear observability for the unforced corresponding linear system, for locally distributed and boundary feedbacks as well. We show that if the linear system is observable through a locally distributed (resp. boundary) observation, then any dissipative nonlinear feedback locally distributed (resp. active only on a part of the boundary) stabilize the system and we give a general energy decay formula. Our results generalize previous results by Haraux (1989) and Ammari and Tucsnak (2001) for linear feedbacks. We show by this way that for the locally distributed case, one can combine the optimal geometric conditions of Bardos et al. (1992) and the method of Alabau-Boussouira (2005) to deduce energy decay rates for nonlinear damped systems.     

Stoxes方程初边值问题的Phragmen－Lindelof二择性原理

本文对非定常的Ｓｔｏｋｅｓ方程的初边值问题证明了Ｐｈｒａｇｍｅｎ－Ｌｉｎｄｅｌｏｆ二择性原理，即证明Ｓｔｏｋｅｓ流函数的能量，随着与带状区域有限端距离的增加必定或者按指数率增长或者按指数率衰减．对能量衰减情况建立了Ｓｔｏｋｅｓ流速度的最大模的点点估计．并提出求全能量上界的方法．     

On the decay of solutions to the 2D Neumann exterior problem for the wave equation

We consider the exterior problem in the plane for the wave equation with a Neumann boundary condition and study the asymptotic behavior of the solution for large times. For possible application we are interested to show a decay estimate which does not involve weighted norms of the initial data. In the paper we prove such an estimate, by a combination of the estimate of the local energy decay and decay estimates for the free space solution.     

Existence and general decay for nondissipative distributed systems with boundary frictional and memory dampings and acoustic boundary conditions

In this paper, we study the existence and general energy decay rate of global solutions for nondissipative distributed systems

$$u''-\triangle u+h(\nabla u)=0$$

with boundary frictional and memory dampings and acoustic boundary conditions. For the existence of solutions, we prove the global existence of weak solution by using Faedo–Galerkin’s method and compactness arguments. For the energy decay rate, we first consider the general nonlinear case of h satisfying a smallness condition and prove the general energy decay rate by using perturbed modified energy method. Then, we consider the linear case of h: \({h(\nabla u)=-\nabla\phi\cdot\nabla u}\) and prove the general decay estimates of equivalent energy.

     

Energy decay estimates and infinite blow‐up phenomena for a strongly damped semilinear wave equation with logarithmic nonlinear source

This paper deals with the energy decay estimates and infinite blow‐up phenomena for a strongly damped semilinear wave equation with logarithmic nonlinear source term under null Dirichlet boundary condition. By constructing a new family of potential wells, together with logarithmic Sobolev inequality and perturbation energy technique, we establish sufficient conditions to guarantee the solution exists globally or occurs infinite blow‐up and derive the polynomial or exponential energy decay estimates under some appropriate conditions.     

Boundary stabilization for a coupled system

We investigate in this work the global existence of weak solutions for a nonlinear coupled system with mixed type boundary conditions. More precisely, Dirichlet and feedback boundary conditions. Further, we also prove the exponential decay of the energy associated with these solutions.     

Stabilization of the uniform Timoshenko beam by one locally distributed feedback

In this article, we study the energy decay rate for an elastic Timoshenko system. This system consists of two coupled wave equations. Only the equation about the rotation angle is damped by one locally distributed feedback at the neighbourhood of the boundary. The equation for the transverse displacement of the beam is only indirectly damped through the coupling. First, we establish an exponential energy decay rate in the case of the same speed of propagation. Next, when the wave speeds are different, a polynomial-type decay rate is obtained. These results are proved by verifying the frequency domain conditions.     

On uniform decay of coupled wave equation of Kirchhoff type subject to memory condition on the boundary

In this paper we prove the existence of solution and uniform decay of energy to the mixed problem for coupled wave equation of Kirchhoff-type subject to memory condition on the boundary.     

Indirect stability of the wave equation with a dynamic boundary control

In this paper, we consider a damped wave equation with a dynamic boundary control. First, combining a general criteria of Arendt and Batty with Holmgren's theorem we show the strong stability of our system. Next, we show that our system is not uniformly stable in general, since it is the case for the unit disk. Hence, we look for a polynomial decay rate for smooth initial data for our system by applying a frequency domain approach. In a first step, by giving some sufficient conditions on the boundary of our domain and by using the exponential decay of the wave equation with a standard damping, we prove a polynomial decay in of the energy. In a second step, under appropriated conditions on the boundary, called the multiplier control conditions, we establish a polynomial decay in of the energy. Later, we show in a particular case that such a polynomial decay is available even if the previous conditions are not satisfied. For this aim, we consider our system on the unit square of the plane. Using a method based on a Fourier analysis and a specific analysis of the obtained 1‐d problems combining Ingham's inequality and an interpolation method, we establish a polynomial decay in of the energy for sufficiently smooth initial data. Finally, in the case of the unit disk, using the real part of the asymptotic expansion of eigenvalues of the damped system, we prove that the obtained decay is optimal in the domain of the operator.     

The global solvability and asymptotic behavior of a transmission problem for Kirchhoff‐type wave equations with memory source on the boundary

A transmission problem for Kirchhoff‐type wave equations with memory source term on one part of the boundary feedback is considered. By using the Faedo‐Galerkin approximation technique, the method of Lyapunov functional and the energy perturbation technique, we establish well‐posedness of global solution and derive a general decay estimate of the energy.     

Decay of Solutions to Damped Korteweg–de Vries Type Equation

In the present paper we establish results concerning the decay of the energy related to the damped Korteweg–de Vries equation posed on infinite domains. We prove the exponential decay rates of the energy when a initial value problem and a localized dissipative mechanism are in place. If this mechanism is effective in the whole line, we get a similar result in H ^k -level, k∈ℕ. In addition, the decay of the energy regarding a initial boundary value problem posed on the right half-line, is obtained considering convenient a smallness condition on the initial data but a more general dissipative effect.     

Global existence,asymptotic stability and blow-up of solutions for the generalized Boussinesq equation with nonlinear boundary condition

In this paper, we consider initial boundary value problem of the generalized Boussinesq equation with nonlinear interior source and boundary absorptive terms. We establish firstly the local existence of solutions by standard Galerkin method. Then we prove both the global existence of the solution and a general decay of the energy functions under some restrictions on the initial data. We also prove a blow-up result for solutions with positive and negative initial energy respectively.